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Biliard_Torus _ A simulation has been developed to show the results obtained in the paper “Trajectory tracking in rectangular billiards by unfolding the billiard table” of the University of Rome Tor Verga.html
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Biliard_Torus _ A simulation has been developed to show the results obtained in the paper “Trajectory tracking in rectangular billiards by unfolding the billiard table” of the University of Rome Tor Verga.html
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<title>Biliard_Torus | A simulation has been developed to show the results obtained in the paper “Trajectory tracking in rectangular billiards by unfolding the billiard table” of the University of Rome Tor Vergata. In particular, the motion of a ball inside a billiard table with elastic walls can be mapped onto the surface of a torus.</title>
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<h1><a href="https://gabro29.github.io/Biliard_Torus/">Biliard_Torus</a></h1>
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<h1 id="unfolding-the-billiard">Unfolding the Billiard</h1>
<p><strong>Introduzione</strong></p>
<p>È stata sviluppata una simulazione volta a mostrare i risultati ottenuti nel paper <a href="https://www.sciencedirect.com/science/article/pii/S240589632032317XRL"><em>“Trajectory tracking in rectangular billiards by unfolding the billiard table”</em></a>. In particolare, il moto di una pallina all’interno di un biliardo, con pareti elastiche, può essere mappato sulla superficie di un toroide.</p>
<p><strong>Tavolo del Biliardo</strong></p>
<p>Innanzitutto, mediante l’uso di Matplot Lib, una libreria di Python, vengono tracciate delle linee di riferimento che individuano il perimetro di un quadrilatero di lati <em>b</em> e <em>h</em>. Tale oggetto rappresenta il “campo da gioco” della simulazione ossia il <em>biliardo</em>. Successivamente, si definiscono le coordinate di un punto detto <em>spawn point</em>, il quale giace all’interno del <em>biliardo</em> definito precedentemente, che costituisce il punto di partenza della simulazione, di fatto una pallina.</p>
<p>Il moto della pallina è costruito analiticamente: si fissa una direzione arbitraria e si traccia una retta che ne indica la traiettoria. È chiaro che, durante il moto, la pallina urterà le pareti del <em>biliardo</em>. Per rendere più semplice il nostro lavoro è stato implementato l’effetto pac-man sulle pareti. Per l’equivalenza dei due fenomeni di veda l’immagine seguente.</p>
<p><img src="https://gabro29.github.io/Biliard_Torus/image1.png" alt=""></p>
<p>La pallina sbuca fuori dalla parte opposta, senza variare la sua direzione.</p>
<p><strong>Costruzione delle traiettorie</strong></p>
<p>A partire dalla prima traiettoria vengono tracciate le successive sulla base della seguente logica:</p>
<p><img src="https://gabro29.github.io/Biliard_Torus/image2.PNG" alt=""></p>
<p>Tale logica implementa l’effetto pac-man: il punto d’intersezione con la parete del biliardo viene traslato dalla parte opposta per disegnare la nuova traiettoria. La pendenza non viene cambiata mentre, tramite la formula inversa della retta si ricava l’intercetta.</p>
<p>È importante notare che nel nostro algoritmo l’angolo d’incidenza traiettoria-parete, rispetto alla normale della parete stessa, non può essere pari a $\frac{\pi}{2}$. Il motivo è che la pendenza di una retta è pari alla tangente dell’angolo che essa forma con l’asse delle ascisse</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-1-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>m</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mi>g</mi></mrow><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2" class="mjx-mrow"><span id="MJXc-Node-3" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span><span id="MJXc-Node-4" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-5" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-6" class="mjx-mrow"><span id="MJXc-Node-7" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-8" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.003em;">g</span></span></span></span><span id="MJXc-Node-9" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-10" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-11" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>m</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mi>g</mi></mrow><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-1">m = {tg}(\alpha)</script>
<p>Ricordando il dominio della funzione tangente risulta chiaro che
l’angolo di incidenza non può essere mai pari a <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-2-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>&#x03C0;</mi><mn>2</mn></mfrac></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-12" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-13" class="mjx-mrow"><span id="MJXc-Node-14" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.547em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.773em; top: -1.084em;"><span id="MJXc-Node-15" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.773em; bottom: -0.6em;"><span id="MJXc-Node-16" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.547em;"></span></span><span class="mjx-vsize" style="height: 1.191em; vertical-align: -0.425em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-2">\frac{\pi}{2}</script> .
Inoltre, si precisa che: nel caso in cui la pallina urti perfettamente
uno degli spigoli del <em>biliardo</em>, le traiettorie sarebbero tutte
sovrapposte e quindi di poco d’interesse.</p>
<p><strong>Traiettorie Dense</strong></p>
<p>Innanzitutto si osservi l’animazione riportata di seguito:</p>
<p><img src="https://gabro29.github.io/Biliard_Torus/image3.gif" alt=""></p>
<p>Si nota che a partire da un segmento, unendo le due estremità, si giunge
ad una circonferenza. Si passa poi ad un Toroide facendo ruotare la
circonferenza attorno ad un asse e preservandone la traccia durante il
suo moto (colorata in arcobaleno).</p>
<p>Tale animazione è intesa come strumento esemplificativo per comprendere
che le considerazioni relative al caso unidimensionale possono essere
estese al caso tridimensionale.</p>
<p>Si consideri dunque la funzione:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-3-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow></msub><mo>:</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">&#x2192;</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mtext>&#xA0;</mtext><mi>c</mi><mi>o</mi><mi>n</mi><mtext>&#xA0;</mtext><mi>&#x03B8;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">R</mi></mrow><mo>.</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-17" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-18" class="mjx-mrow"><span id="MJXc-Node-19" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-20" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-21" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-22" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-23" class="mjx-texatom" style=""><span id="MJXc-Node-24" class="mjx-mrow"><span id="MJXc-Node-25" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span><span id="MJXc-Node-26" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">:</span></span><span id="MJXc-Node-27" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-28" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-29" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-30" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-31" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-32" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.224em; padding-bottom: 0.335em;">→</span></span><span id="MJXc-Node-33" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-34" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-35" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-36" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-37" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-38" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-39" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">c</span></span><span id="MJXc-Node-40" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">o</span></span><span id="MJXc-Node-41" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-42" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-43" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-44" class="mjx-texatom"><span id="MJXc-Node-45" class="mjx-mrow"><span id="MJXc-Node-46" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-47" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.279em;">R</span></span></span></span><span id="MJXc-Node-48" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext> </mtext><mtext> </mtext><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow></msub><mo>:</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mtext> </mtext><mi>c</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mi>θ</mi><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-3">\ \ T_{\theta}:\lbrack 0,1) \rightarrow \lbrack 0,1)\ con\ \theta \mathbb{\in R}.</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display"><span id="MathJax-Element-4-Frame" class="mjx-full-width mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mspace linebreak="newline" /><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mi>x</mi><mo stretchy="false">&#x21A6;</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; min-width: 6.141em; position: relative;"><span id="MJXc-Node-49" class="mjx-math" aria-hidden="true" style="width: 100%;"><span id="MJXc-Node-50" class="mjx-mrow" style="width: 100%;"><span class="mjx-stack" style="width: 100%; vertical-align: 0px;"><span class="mjx-block" style="text-align: center;"><span class="mjx-box"><span id="MJXc-Node-51" class="mjx-mspace" style="width: 0px; height: 0px;"></span><span id="MJXc-Node-52" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-53" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-54" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-55" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-56" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.224em; padding-bottom: 0.335em;">↦</span></span><span id="MJXc-Node-57" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-58" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-59" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-60" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-61" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mspace linebreak="newline"></mspace><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>x</mi><mo stretchy="false">↦</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>θ</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-4">\\ \ \ \ x \mapsto \{x + \theta\}</script>
<p>Dove <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-5-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-62" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-63" class="mjx-mrow"><span id="MJXc-Node-64" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-65" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-66" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-67" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-68" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>θ</mi><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-5">\{x + \theta\}</script> indica la parte frazionaria. Per semplicità si considera un segmento di lunghezza unitaria e si fissa un punto di partenza <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-6-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&#x2208;</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-69" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-70" class="mjx-mrow"><span id="MJXc-Node-71" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-72" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-73" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-74" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-75" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-76" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-77" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-6">x \in \lbrack 0,1)</script>
Definiamo:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-7-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-78" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-79" class="mjx-mrow"><span id="MJXc-Node-80" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-81" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-85" class="mjx-texatom" style=""><span id="MJXc-Node-86" class="mjx-mrow"><span id="MJXc-Node-87" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-82" class="mjx-texatom" style=""><span id="MJXc-Node-83" class="mjx-mrow"><span id="MJXc-Node-84" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-88" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-89" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-90" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-91" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-92" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-7">T_{\theta}^{0}(x) = x</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-8-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-93" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-94" class="mjx-mrow"><span id="MJXc-Node-95" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-96" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-100" class="mjx-texatom" style=""><span id="MJXc-Node-101" class="mjx-mrow"><span id="MJXc-Node-102" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-97" class="mjx-texatom" style=""><span id="MJXc-Node-98" class="mjx-mrow"><span id="MJXc-Node-99" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-103" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-104" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-105" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-106" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-107" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-108" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-109" class="mjx-texatom" style=""><span id="MJXc-Node-110" class="mjx-mrow"><span id="MJXc-Node-111" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span><span id="MJXc-Node-112" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-113" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-114" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-8">T_{\theta}^{1}(x) = T_{\theta}(x)</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-9-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-115" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-116" class="mjx-mrow"><span id="MJXc-Node-117" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-118" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-122" class="mjx-texatom" style=""><span id="MJXc-Node-123" class="mjx-mrow"><span id="MJXc-Node-124" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-119" class="mjx-texatom" style=""><span id="MJXc-Node-120" class="mjx-mrow"><span id="MJXc-Node-121" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-125" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-126" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-127" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-128" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-129" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-130" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-131" class="mjx-texatom" style=""><span id="MJXc-Node-132" class="mjx-mrow"><span id="MJXc-Node-133" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span><span id="MJXc-Node-134" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-135" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-136" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-137" class="mjx-texatom" style=""><span id="MJXc-Node-138" class="mjx-mrow"><span id="MJXc-Node-139" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span><span id="MJXc-Node-140" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-141" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-142" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-143" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-9">T_{\theta}^{2}(x) = T_{\theta}( T_{\theta}(x) )</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-10-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo>=</mo><munder><mrow class="MJX-TeXAtom-OP MJX-fixedlimits"><munder><mrow><mo stretchy="false">(</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow></msub><mo>&#x2218;</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow></msub><mo>&#x2218;</mo><mo>&#x22EF;</mo><mo>&#x2218;</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo>&#x23DF;</mo></munder></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mtext>&#xA0;volte</mtext></mrow></munder><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-144" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-145" class="mjx-mrow"><span id="MJXc-Node-146" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-147" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-151" class="mjx-texatom" style=""><span id="MJXc-Node-152" class="mjx-mrow"><span id="MJXc-Node-153" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-148" class="mjx-texatom" style=""><span id="MJXc-Node-149" class="mjx-mrow"><span id="MJXc-Node-150" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-154" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-155" class="mjx-munderover MJXc-space3"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-op"><span id="MJXc-Node-156" class="mjx-texatom"><span id="MJXc-Node-157" class="mjx-mrow"><span id="MJXc-Node-158" class="mjx-munderover"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-op"><span id="MJXc-Node-159" class="mjx-mrow"><span id="MJXc-Node-160" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-161" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-162" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-163" class="mjx-texatom" style=""><span id="MJXc-Node-164" class="mjx-mrow"><span id="MJXc-Node-165" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span><span id="MJXc-Node-166" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∘</span></span><span id="MJXc-Node-167" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-168" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-169" class="mjx-texatom" style=""><span id="MJXc-Node-170" class="mjx-mrow"><span id="MJXc-Node-171" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span><span id="MJXc-Node-172" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∘</span></span><span id="MJXc-Node-173" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.335em;">⋯</span></span><span id="MJXc-Node-174" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∘</span></span><span id="MJXc-Node-175" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-176" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-177" class="mjx-texatom" style=""><span id="MJXc-Node-178" class="mjx-mrow"><span id="MJXc-Node-179" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span><span id="MJXc-Node-180" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="padding-top: 0.12em;"><span id="MJXc-Node-181" class="mjx-mo"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.108em; padding-bottom: 0.488em; margin: 0px 0.01em 0px 0.024em;"></span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.108em; padding-bottom: 0.488em; margin: 0px 0.085em 0px -0.065em; letter-spacing: -0.081em;"></span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: 0px; padding-bottom: 0.488em; margin-right: 0.01em; padding-top: 0.108em; margin-bottom: 0px;"></span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.108em; padding-bottom: 0.488em; margin: 0px 0.085em 0px -0.065em; letter-spacing: -0.081em;"></span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.108em; padding-bottom: 0.488em; margin-right: 0.024em; margin-bottom: 0px; margin-top: 0px;"></span></span></span></span></span></span></span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 3.905em;"><span id="MJXc-Node-182" class="mjx-texatom" style=""><span id="MJXc-Node-183" class="mjx-mrow"><span id="MJXc-Node-184" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-185" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;"> volte</span></span></span></span></span></span></span></span><span id="MJXc-Node-186" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-187" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-188" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo>=</mo><munder><mrow class="MJX-TeXAtom-OP MJX-fixedlimits"><munder><mrow><mo stretchy="false">(</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow></msub><mo>∘</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow></msub><mo>∘</mo><mo>⋯</mo><mo>∘</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mtext> volte</mtext></mrow></munder><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-10">T_{\theta}^{n} = \underbrace{(T_{\theta} \circ T_{\theta} \circ \cdots \circ T_{\theta})}_{n \text{ volte}}(x)</script>
<p><strong><em>Lemma 1</em></strong></p>
<p>Si ha <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-11-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-189" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-190" class="mjx-mrow"><span id="MJXc-Node-191" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-192" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-196" class="mjx-texatom" style=""><span id="MJXc-Node-197" class="mjx-mrow"><span id="MJXc-Node-198" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-193" class="mjx-texatom" style=""><span id="MJXc-Node-194" class="mjx-mrow"><span id="MJXc-Node-195" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-199" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-200" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-201" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-202" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-203" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-204" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-205" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-206" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-207" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-208" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-11">T_{\theta}^{n}(x) = \{ x + n\theta \}</script> per ogni <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-12-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">N</mi></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-209" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-210" class="mjx-mrow"><span id="MJXc-Node-211" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-212" class="mjx-texatom"><span id="MJXc-Node-213" class="mjx-mrow"><span id="MJXc-Node-214" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-215" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.335em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-12">n\mathbb{\in N}</script>. Si dimostra per induzione.</p>
<p>Per <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-13-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>0</mn></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-216" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-217" class="mjx-mrow"><span id="MJXc-Node-218" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-219" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-220" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-13">n = 0</script>, si ha per definizione che:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-14-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-221" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-222" class="mjx-mrow"><span id="MJXc-Node-223" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-224" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-228" class="mjx-texatom" style=""><span id="MJXc-Node-229" class="mjx-mrow"><span id="MJXc-Node-230" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-225" class="mjx-texatom" style=""><span id="MJXc-Node-226" class="mjx-mrow"><span id="MJXc-Node-227" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-231" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-232" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-233" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-234" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-235" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-236" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-237" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-238" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-239" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-240" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-241" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>θ</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-14">T_{\theta}^{0}(x) = x = \{x + \theta\}</script>
<p>Assumiamo sia:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-15-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-242" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-243" class="mjx-mrow"><span id="MJXc-Node-244" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-245" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-249" class="mjx-texatom" style=""><span id="MJXc-Node-250" class="mjx-mrow"><span id="MJXc-Node-251" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-246" class="mjx-texatom" style=""><span id="MJXc-Node-247" class="mjx-mrow"><span id="MJXc-Node-248" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-252" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-253" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-254" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-255" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-256" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-257" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-258" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-259" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-260" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-261" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-15">T_{\theta}^{n}(x) = \{ x + n\theta \}</script>
<p>Si applica <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-16-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-262" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-263" class="mjx-mrow"><span id="MJXc-Node-264" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-265" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-266" class="mjx-texatom" style=""><span id="MJXc-Node-267" class="mjx-mrow"><span id="MJXc-Node-268" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-16">T_{\theta}</script> ambo i membri:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-17-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow></msub><mo>&#x2218;</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow></msub><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-269" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-270" class="mjx-mrow"><span id="MJXc-Node-271" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-272" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-273" class="mjx-texatom" style=""><span id="MJXc-Node-274" class="mjx-mrow"><span id="MJXc-Node-275" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span><span id="MJXc-Node-276" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∘</span></span><span id="MJXc-Node-277" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-278" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-282" class="mjx-texatom" style=""><span id="MJXc-Node-283" class="mjx-mrow"><span id="MJXc-Node-284" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-279" class="mjx-texatom" style=""><span id="MJXc-Node-280" class="mjx-mrow"><span id="MJXc-Node-281" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-285" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-286" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-287" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-288" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-289" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-290" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-291" class="mjx-texatom" style=""><span id="MJXc-Node-292" class="mjx-mrow"><span id="MJXc-Node-293" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span><span id="MJXc-Node-294" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-295" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-296" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-297" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-298" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-299" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-300" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-301" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow></msub><mo>∘</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow></msub><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">}</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-17">T_{\theta} \circ T_{\theta}^{n}(x) = T_{\theta}( \{ x + n\theta \} )</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-18-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo><mo>+</mo><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-302" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-303" class="mjx-mrow"><span id="MJXc-Node-304" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-305" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-309" class="mjx-texatom" style=""><span id="MJXc-Node-310" class="mjx-mrow"><span id="MJXc-Node-311" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-312" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-313" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-306" class="mjx-texatom" style=""><span id="MJXc-Node-307" class="mjx-mrow"><span id="MJXc-Node-308" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-314" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-315" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-316" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-317" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-318" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-319" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-320" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-321" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-322" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-323" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-324" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-325" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-326" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-327" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">}</mo><mo>+</mo><mi>θ</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-18">T_{\theta}^{n + 1}(x) = \{ \{ x + n\theta \} + \theta \}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-19-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mo fence="false" stretchy="false">{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo><mo>+</mo><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-328" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-329" class="mjx-mrow"><span id="MJXc-Node-330" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-331" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-332" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-333" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-334" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-335" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-336" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-337" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-338" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-339" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-340" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-341" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-342" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-343" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-344" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span><span id="MJXc-Node-345" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-346" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-347" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mo fence="false" stretchy="false">{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo stretchy="false">)</mo><mo>−</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">]</mo><mo>+</mo><mi>θ</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-19">= \{ (x + n\theta) - \lbrack x + n\theta\rbrack + \theta \}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-20-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>&#x03B8;</mi><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-348" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-349" class="mjx-mrow"><span id="MJXc-Node-350" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-351" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-352" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-353" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-354" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-355" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-356" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-357" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-358" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-359" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-360" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-361" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-362" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-363" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-364" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-365" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-366" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span><span id="MJXc-Node-367" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>θ</mi><mo>−</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">]</mo><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-20">= \{ x + (n + 1)\theta - \lbrack x + n\theta\rbrack \}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-21-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>&#x03B8;</mi><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>&#x03B8;</mi><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo><mo fence="false" stretchy="false">]</mo><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-368" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-369" class="mjx-mrow"><span id="MJXc-Node-370" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-371" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-372" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-373" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-374" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-375" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-376" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-377" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-378" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-379" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-380" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-381" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-382" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; 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padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-390" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-391" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-392" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-393" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-394" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-395" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-396" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-397" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-398" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-399" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-400" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-401" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-402" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-403" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span><span id="MJXc-Node-404" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span><span id="MJXc-Node-405" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>θ</mi><mo>−</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">]</mo><mo>−</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>θ</mi><mo>−</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">]</mo><mo fence="false" stretchy="false">]</mo><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-21">= \{ x + (n + 1)\theta - \lbrack x + n\theta\rbrack - \lbrack x + (n + 1)\theta - \lbrack x + n\theta\rbrack \rbrack \}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-22-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>&#x03B8;</mi><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo><mo>+</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-406" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-407" class="mjx-mrow"><span id="MJXc-Node-408" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-409" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-410" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-411" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-412" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-413" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-414" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-415" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-416" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-417" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-418" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-419" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-420" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-421" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-422" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-423" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span><span id="MJXc-Node-424" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-425" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-426" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-427" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-428" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-429" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-430" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-431" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-432" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-433" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-434" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span><span id="MJXc-Node-435" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-436" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-437" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-438" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-439" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-440" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-441" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>θ</mi><mo>−</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">]</mo><mo>−</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>θ</mi><mo fence="false" stretchy="false">]</mo><mo>+</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">]</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-22">= x + (n + 1)\theta - \lbrack x + n\theta\rbrack - \lbrack x + (n + 1)\theta \rbrack + \lbrack x + n\theta\rbrack</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-23-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-442" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-443" class="mjx-mrow"><span id="MJXc-Node-444" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-445" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-446" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-447" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-448" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-449" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-450" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-451" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-452" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-453" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-454" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>θ</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-23">= \{ x + (n + 1)\theta \}</script>
<blockquote>
<p>◻</p>
</blockquote>
<p>Si definisce poil’immagine della successione <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-24-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-455" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-456" class="mjx-mrow"><span id="MJXc-Node-457" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-458" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-459" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-463" class="mjx-texatom" style=""><span id="MJXc-Node-464" class="mjx-mrow"><span id="MJXc-Node-465" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-460" class="mjx-texatom" style=""><span id="MJXc-Node-461" class="mjx-mrow"><span id="MJXc-Node-462" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-466" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-467" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-468" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-469" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-470" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-471" class="mjx-texatom" style=""><span id="MJXc-Node-472" class="mjx-mrow"><span id="MJXc-Node-473" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-24">\{ T_{\theta}^{n}(x) \}_{n}</script> , cioè l’insieme <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-25-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C4;</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">N</mi></mrow><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-474" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-475" class="mjx-mrow"><span id="MJXc-Node-476" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.08em;">τ</span></span><span id="MJXc-Node-477" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-478" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-479" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-480" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-481" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-482" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-483" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-487" class="mjx-texatom" style=""><span id="MJXc-Node-488" class="mjx-mrow"><span id="MJXc-Node-489" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-484" class="mjx-texatom" style=""><span id="MJXc-Node-485" class="mjx-mrow"><span id="MJXc-Node-486" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-490" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-491" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-492" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-493" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">:</span></span><span id="MJXc-Node-494" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-495" class="mjx-texatom"><span id="MJXc-Node-496" class="mjx-mrow"><span id="MJXc-Node-497" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-498" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.335em;">N</span></span></span></span><span id="MJXc-Node-499" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-25">\tau(x) = \{ T_{\theta}^{n}(x):n\mathbb{\in N} \}</script>.
Vale che <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-26-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C4;</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x2286;</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-500" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-501" class="mjx-mrow"><span id="MJXc-Node-502" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.08em;">τ</span></span><span id="MJXc-Node-503" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-504" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-505" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-506" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.335em; padding-bottom: 0.501em;">⊆</span></span><span id="MJXc-Node-507" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-508" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-509" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-510" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-511" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊆</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-26">\tau(x) \subseteq \lbrack 0,1)</script>.</p>
<p><strong><em>Teorema 1</em></strong></p>
<p>La successione <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-27-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-512" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-513" class="mjx-mrow"><span id="MJXc-Node-514" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-515" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-516" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-520" class="mjx-texatom" style=""><span id="MJXc-Node-521" class="mjx-mrow"><span id="MJXc-Node-522" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-517" class="mjx-texatom" style=""><span id="MJXc-Node-518" class="mjx-mrow"><span id="MJXc-Node-519" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-523" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-524" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-525" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-526" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-527" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-528" class="mjx-texatom" style=""><span id="MJXc-Node-529" class="mjx-mrow"><span id="MJXc-Node-530" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-27">\{ T_{\theta}^{n}(x) \}_{n}</script> è periodica <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-28-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">&#x21D4;</mo><mi>&#x03B8;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">Q</mi></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-531" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-532" class="mjx-mrow"><span id="MJXc-Node-533" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.224em; padding-bottom: 0.39em;">⇔</span></span><span id="MJXc-Node-534" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-535" class="mjx-texatom"><span id="MJXc-Node-536" class="mjx-mrow"><span id="MJXc-Node-537" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-538" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.501em;">Q</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">⇔</mo><mi>θ</mi><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-28">\Leftrightarrow \theta \mathbb{\in Q}</script>.</p>
<p>DIM:</p>
<p>Se la successione è periodica, allora esiste un <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-29-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">N</mi></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-539" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-540" class="mjx-mrow"><span id="MJXc-Node-541" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-542" class="mjx-texatom"><span id="MJXc-Node-543" class="mjx-mrow"><span id="MJXc-Node-544" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-545" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.335em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-29">N\mathbb{\in N}</script> tale che:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-30-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>=</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>N</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mi>N</mi><mi>&#x03B8;</mi><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>N</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-546" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-547" class="mjx-mrow"><span id="MJXc-Node-548" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-549" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-553" class="mjx-texatom" style=""><span id="MJXc-Node-554" class="mjx-mrow"><span id="MJXc-Node-555" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-550" class="mjx-texatom" style=""><span id="MJXc-Node-551" class="mjx-mrow"><span id="MJXc-Node-552" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-556" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-557" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-558" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-559" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-560" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-561" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-562" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-563" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-567" class="mjx-texatom" style=""><span id="MJXc-Node-568" class="mjx-mrow"><span id="MJXc-Node-569" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-564" class="mjx-texatom" style=""><span id="MJXc-Node-565" class="mjx-mrow"><span id="MJXc-Node-566" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-570" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-571" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-572" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-573" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-574" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-575" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-576" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-577" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-578" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-579" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-580" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-581" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-582" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-583" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-584" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>=</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>N</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mi>N</mi><mi>θ</mi><mo>−</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>N</mi><mi>θ</mi><mo fence="false" stretchy="false">]</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-30">T_{\theta}^{0}(x) = x = T_{\theta}^{N}(x) = x + N\theta - \lbrack x + N\theta\rbrack</script>
<p>Da cui:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-31-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>&#x03B8;</mi><mo>=</mo><mfrac><mrow><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>N</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo></mrow><mi>N</mi></mfrac></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-585" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-586" class="mjx-mrow"><span id="MJXc-Node-587" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-588" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-589" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.907em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.907em; top: -1.533em;"><span id="MJXc-Node-590" class="mjx-mrow"><span id="MJXc-Node-591" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-592" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-593" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-594" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-595" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-596" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span></span></span><span class="mjx-denominator" style="width: 3.907em; bottom: -0.716em;"><span id="MJXc-Node-597" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 3.907em;"></span></span><span class="mjx-vsize" style="height: 2.249em; vertical-align: -0.716em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>θ</mi><mo>=</mo><mfrac><mrow><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>N</mi><mi>θ</mi><mo fence="false" stretchy="false">]</mo></mrow><mi>N</mi></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-31">\theta = \frac{\lbrack x + N\theta\rbrack}{N}</script>
<p>Essendo <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-32-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-598" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-599" class="mjx-mrow"><span id="MJXc-Node-600" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-32">\theta</script> dato dal rapporto di due numeri interi, allora è un numero razionale. Nell’altra direzione, se si suppone <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-33-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">Q</mi></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-601" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-602" class="mjx-mrow"><span id="MJXc-Node-603" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-604" class="mjx-texatom"><span id="MJXc-Node-605" class="mjx-mrow"><span id="MJXc-Node-606" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-607" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.501em;">Q</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-33">\theta\mathbb{\in Q}</script>, allora</p>
<p><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-34-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi><mo>=</mo><mfrac><mi>N</mi><mi>M</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">Q</mi></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-608" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-609" class="mjx-mrow"><span id="MJXc-Node-610" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-611" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-612" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.885em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.251em; top: -1.325em;"><span id="MJXc-Node-613" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.251em; bottom: -0.617em;"><span id="MJXc-Node-614" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.081em;">M</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.885em;"></span></span><span class="mjx-vsize" style="height: 1.373em; vertical-align: -0.437em;"></span></span><span id="MJXc-Node-615" class="mjx-texatom"><span id="MJXc-Node-616" class="mjx-mrow"><span id="MJXc-Node-617" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-618" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.501em;">Q</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi>N</mi><mi>M</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-34">\theta = \frac{N}{M}\mathbb{\in Q}</script> con $N,M\in\mathbb{Z}, e\, M\neq 0$</p>
<p>Per <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-35-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi><mo>=</mo><mn>0</mn></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-619" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-620" class="mjx-mrow"><span id="MJXc-Node-621" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-622" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-623" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-35">\theta = 0</script>, si ha il caso base della definizione ed il teorema è
chiaramente verificato. Per gli altri casi basta trovare anche un solo esempio in cui si nota la periodicità. Al passo <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-36-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-624" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-625" class="mjx-mrow"><span id="MJXc-Node-626" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.081em;">M</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-36">M</script> si avrà:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-37-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>M</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>M</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>M</mi><mfrac><mi>N</mi><mi>M</mi></mfrac><mo fence="false" stretchy="false">}</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>N</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><mi>x</mi><mo>=</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-627" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-628" class="mjx-mrow"><span id="MJXc-Node-629" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-630" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-634" class="mjx-texatom" style=""><span id="MJXc-Node-635" class="mjx-mrow"><span id="MJXc-Node-636" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.081em;">M</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-631" class="mjx-texatom" style=""><span id="MJXc-Node-632" class="mjx-mrow"><span id="MJXc-Node-633" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-637" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-638" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-639" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-640" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-641" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-642" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-643" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-644" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-645" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-646" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-647" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-648" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-649" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-650" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-651" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-652" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.251em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.251em; top: -1.385em;"><span id="MJXc-Node-653" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span><span class="mjx-denominator" style="width: 1.251em; bottom: -0.716em;"><span id="MJXc-Node-654" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.081em;">M</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 1.251em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.716em;"></span></span><span id="MJXc-Node-655" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-656" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-657" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-658" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-659" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-660" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-661" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-662" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-663" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-664" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-665" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-666" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-667" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-668" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-669" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-670" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-674" class="mjx-texatom" style=""><span id="MJXc-Node-675" class="mjx-mrow"><span id="MJXc-Node-676" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-671" class="mjx-texatom" style=""><span id="MJXc-Node-672" class="mjx-mrow"><span id="MJXc-Node-673" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-677" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-678" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-679" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-680" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>M</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>M</mi><mi>θ</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>M</mi><mfrac><mi>N</mi><mi>M</mi></mfrac><mo fence="false" stretchy="false">}</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>N</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><mi>x</mi><mo>=</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-37">T_{\theta}^{M}(x) = \{ x + M\theta \} = \{ x + M\frac{N}{M} \} = \{ x + N \} = \{ x \} = x = T_{\theta}^{0}(x).</script>
<blockquote>
<p>◻</p>
</blockquote>
<p><strong><em>Lemma 2</em></strong></p>
<p>Su <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-38-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-681" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-682" class="mjx-mrow"><span id="MJXc-Node-683" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-684" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-685" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-686" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-687" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-38">\lbrack 0,1)</script> si definisce la distanza tra due punti
<span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-39-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mi>y</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-688" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-689" class="mjx-mrow"><span id="MJXc-Node-690" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-691" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-692" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mi>y</mi></math></span></span><script type="math/tex" id="MathJax-Element-39">x,y</script>, la quantità:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-40-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>m</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mo>&#x2212;</mo><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mo>&#x2212;</mo><mi>y</mi><mo>+</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mo>&#x2212;</mo><mi>y</mi><mo>&#x2212;</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo fence="false" stretchy="false">}</mo><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-693" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-694" class="mjx-mrow"><span id="MJXc-Node-695" class="mjx-texatom"><span id="MJXc-Node-696" class="mjx-mrow"><span id="MJXc-Node-697" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">d</span></span></span></span><span id="MJXc-Node-698" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-699" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-700" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-701" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-702" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-703" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-704" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span><span id="MJXc-Node-705" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em;">i</span></span><span id="MJXc-Node-706" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-707" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-708" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-709" class="mjx-texatom"><span id="MJXc-Node-710" class="mjx-mrow"><span id="MJXc-Node-711" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">|</span></span></span></span><span id="MJXc-Node-712" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-713" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-714" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-715" class="mjx-texatom"><span id="MJXc-Node-716" class="mjx-mrow"><span id="MJXc-Node-717" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">|</span></span></span></span><span id="MJXc-Node-718" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-719" class="mjx-texatom MJXc-space1"><span id="MJXc-Node-720" class="mjx-mrow"><span id="MJXc-Node-721" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">|</span></span></span></span><span id="MJXc-Node-722" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-723" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-724" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-725" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-726" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-727" class="mjx-texatom"><span id="MJXc-Node-728" class="mjx-mrow"><span id="MJXc-Node-729" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">|</span></span></span></span><span id="MJXc-Node-730" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-731" class="mjx-texatom MJXc-space1"><span id="MJXc-Node-732" class="mjx-mrow"><span id="MJXc-Node-733" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">|</span></span></span></span><span id="MJXc-Node-734" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-735" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-736" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-737" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-738" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-739" class="mjx-texatom"><span id="MJXc-Node-740" class="mjx-mrow"><span id="MJXc-Node-741" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">|</span></span></span></span><span id="MJXc-Node-742" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-743" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>m</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mo>−</mo><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mo>−</mo><mi>y</mi><mo>−</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo fence="false" stretchy="false">}</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-40">\mathbb{d}(x,y) = min(\{ |x - y|,|x - y + 1|,|x - y - 1| \})</script>
<p>Possiamo mostrare che:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-41-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-744" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-745" class="mjx-mrow"><span id="MJXc-Node-746" class="mjx-texatom"><span id="MJXc-Node-747" class="mjx-mrow"><span id="MJXc-Node-748" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">d</span></span></span></span><span id="MJXc-Node-749" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-750" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-751" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-752" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-753" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-754" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-755" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-756" class="mjx-mrow"><span id="MJXc-Node-757" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">d</span></span></span></span><span id="MJXc-Node-758" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-759" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-760" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-764" class="mjx-texatom" style=""><span id="MJXc-Node-765" class="mjx-mrow"><span id="MJXc-Node-766" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-761" class="mjx-texatom" style=""><span id="MJXc-Node-762" class="mjx-mrow"><span id="MJXc-Node-763" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-767" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-768" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-769" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-770" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-771" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-772" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-776" class="mjx-texatom" style=""><span id="MJXc-Node-777" class="mjx-mrow"><span id="MJXc-Node-778" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-773" class="mjx-texatom" style=""><span id="MJXc-Node-774" class="mjx-mrow"><span id="MJXc-Node-775" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-779" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-780" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-781" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-782" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-41">\mathbb{d}(x,y) = \mathbb{d}( T_{\theta}^{n}(x), T_{\theta}^{n}(y) )</script>
<p><img src="https://gabro29.github.io/Biliard_Torus/image4.gif" alt=""></p>
<p>Quando a partire dal segmento si giunge alla circonferenza, si hanno due modi per collegare due punti: si formano due archi. La distanza è data quindi dall’arco di lunghezza minima. La facoltà di scegliere il percorso minore è contemplata quando si considera l’offset di <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-42-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x00B1;</mo><mn>1</mn></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-783" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-784" class="mjx-mrow"><span id="MJXc-Node-785" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">±</span></span><span id="MJXc-Node-786" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-42">\pm 1</script> a seconda dei casi.</p>
<p><strong><em>Teorema 2</em></strong></p>
<p>Le seguenti tre proposizioni sono equivalneti:
(i) La successione <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-43-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-787" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-788" class="mjx-mrow"><span id="MJXc-Node-789" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-790" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-791" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-795" class="mjx-texatom" style=""><span id="MJXc-Node-796" class="mjx-mrow"><span id="MJXc-Node-797" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-792" class="mjx-texatom" style=""><span id="MJXc-Node-793" class="mjx-mrow"><span id="MJXc-Node-794" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-798" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-799" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-800" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-801" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-802" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-803" class="mjx-texatom" style=""><span id="MJXc-Node-804" class="mjx-mrow"><span id="MJXc-Node-805" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-43">\{ T_{\theta}^{n}(x) \}_{n}</script> non è periodica
(ii) <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-44-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">R</mi><mo>&#x2216;</mo><mi mathvariant="double-struck">Q</mi></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-806" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-807" class="mjx-mrow"><span id="MJXc-Node-808" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-809" class="mjx-texatom"><span id="MJXc-Node-810" class="mjx-mrow"><span id="MJXc-Node-811" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-812" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.279em;">R</span></span><span id="MJXc-Node-813" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∖</span></span><span id="MJXc-Node-814" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.501em;">Q</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>∖</mo><mi mathvariant="double-struck">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-44">\theta\mathbb{\in R \smallsetminus Q}</script>
(iii) L’immagine <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-45-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C4;</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>$</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#xE8;</mo></mrow><mi>d</mi><mi>e</mi><mi>n</mi><mi>s</mi><mi>a</mi><mi>i</mi><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>$</mo></mrow><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-815" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-816" class="mjx-mrow"><span id="MJXc-Node-817" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.08em;">τ</span></span><span id="MJXc-Node-818" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-819" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-820" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-821" class="mjx-texatom"><span id="MJXc-Node-822" class="mjx-mrow"><span id="MJXc-Node-823" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.39em;">$</span></span></span></span><span id="MJXc-Node-824" class="mjx-texatom"><span id="MJXc-Node-825" class="mjx-mrow"><span id="MJXc-Node-826" class="mjx-mo"><span class="mjx-char" style="padding-top: 0.556em; padding-bottom: 0.224em;"><span class="mjx-charbox MJXc-TeX-unknown-R" style="padding-bottom: 0.277em; width: 0.487em;">è</span></span></span></span></span><span id="MJXc-Node-827" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-828" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">e</span></span><span id="MJXc-Node-829" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-830" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">s</span></span><span id="MJXc-Node-831" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">a</span></span><span id="MJXc-Node-832" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em;">i</span></span><span id="MJXc-Node-833" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-834" class="mjx-texatom"><span id="MJXc-Node-835" class="mjx-mrow"><span id="MJXc-Node-836" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.39em;">$</span></span></span></span><span id="MJXc-Node-837" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-838" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-839" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-840" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-841" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>$</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>è</mo></mrow><mi>d</mi><mi>e</mi><mi>n</mi><mi>s</mi><mi>a</mi><mi>i</mi><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>$</mo></mrow><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-45">\tau(x)$ è densa in $\lbrack 0,1)</script>.</p>
<p>DIM:
L’equivalenza di (i) e (ii) viene dal lemma. Per mostrare che (iii) implica (i) e (ii) basta notare che se <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-46-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-842" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-843" class="mjx-mrow"><span id="MJXc-Node-844" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-845" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-846" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-850" class="mjx-texatom" style=""><span id="MJXc-Node-851" class="mjx-mrow"><span id="MJXc-Node-852" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-847" class="mjx-texatom" style=""><span id="MJXc-Node-848" class="mjx-mrow"><span id="MJXc-Node-849" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-853" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-854" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-855" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-856" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-857" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-858" class="mjx-texatom" style=""><span id="MJXc-Node-859" class="mjx-mrow"><span id="MJXc-Node-860" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-46">\{ T_{\theta}^{n}(x) \}_{n}</script> è periodica, allora <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-47-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C4;</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-861" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-862" class="mjx-mrow"><span id="MJXc-Node-863" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.08em;">τ</span></span><span id="MJXc-Node-864" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-865" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-866" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-47">\tau(x)</script> è finito e non può essere denso. Rimane da dimostrare che (ii) (oppure (i)) implica (iii). Facciamo la dimostrazione in due passi.</p>
<ol>
<li>Si verifica innanzitutto che gli elementi di <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-48-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-867" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-868" class="mjx-mrow"><span id="MJXc-Node-869" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-870" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-874" class="mjx-texatom" style=""><span id="MJXc-Node-875" class="mjx-mrow"><span id="MJXc-Node-876" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-871" class="mjx-texatom" style=""><span id="MJXc-Node-872" class="mjx-mrow"><span id="MJXc-Node-873" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-877" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-878" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-879" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-48">T_{\theta}^{n}(x)</script> sono tutti diversi.</li>
</ol>
<p>Se così non fosse sarebbe:</p>
<p><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-49-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>m</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-880" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-881" class="mjx-mrow"><span id="MJXc-Node-882" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-883" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-887" class="mjx-texatom" style=""><span id="MJXc-Node-888" class="mjx-mrow"><span id="MJXc-Node-889" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-884" class="mjx-texatom" style=""><span id="MJXc-Node-885" class="mjx-mrow"><span id="MJXc-Node-886" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-890" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-891" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-892" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-893" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-894" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-895" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-899" class="mjx-texatom" style=""><span id="MJXc-Node-900" class="mjx-mrow"><span id="MJXc-Node-901" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-896" class="mjx-texatom" style=""><span id="MJXc-Node-897" class="mjx-mrow"><span id="MJXc-Node-898" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-902" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-903" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-904" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>m</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-49">T_{\theta}^{n}(x) = T_{\theta}^{m}(x)</script>per qualche $n,m\mathbb{\in N}$</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-50-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>m</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-905" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-906" class="mjx-mrow"><span id="MJXc-Node-907" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-908" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-909" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-910" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-911" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-912" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-913" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-914" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-915" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-916" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-917" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span><span id="MJXc-Node-918" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-919" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>x</mi><mo>+</mo><mi>m</mi><mi>θ</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-50">\{ x + n\theta \} = \{ x + m\theta \}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-51-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo><mo>=</mo><mi>x</mi><mo>+</mo><mi>m</mi><mi>&#x03B8;</mi><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>m</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-920" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-921" class="mjx-mrow"><span id="MJXc-Node-922" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-923" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-924" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-925" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-926" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-927" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-928" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-929" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-930" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-931" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-932" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span><span id="MJXc-Node-933" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-934" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-935" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-936" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span><span id="MJXc-Node-937" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-938" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-939" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-940" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-941" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-942" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span><span id="MJXc-Node-943" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-944" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo>−</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">]</mo><mo>=</mo><mi>x</mi><mo>+</mo><mi>m</mi><mi>θ</mi><mo>−</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>m</mi><mi>θ</mi><mo fence="false" stretchy="false">]</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-51">x + n\theta - \lbrack x + n\theta\rbrack = x + m\theta - \lbrack x + m\theta\rbrack</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-52-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>&#x03B8;</mi><mo>=</mo><mfrac><mrow><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>m</mi><mi>&#x03B8;</mi><mo fence="false" stretchy="false">]</mo></mrow><mrow><mi>n</mi><mo>&#x2212;</mo><mi>m</mi></mrow></mfrac></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-945" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-946" class="mjx-mrow"><span id="MJXc-Node-947" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-948" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-949" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 8.539em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 8.539em; top: -1.533em;"><span id="MJXc-Node-950" class="mjx-mrow"><span id="MJXc-Node-951" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-952" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-953" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-954" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-955" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-956" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span><span id="MJXc-Node-957" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-958" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-959" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-960" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-961" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span><span id="MJXc-Node-962" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-963" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span></span></span><span class="mjx-denominator" style="width: 8.539em; bottom: -0.793em;"><span id="MJXc-Node-964" class="mjx-mrow"><span id="MJXc-Node-965" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-966" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-967" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 8.539em;"></span></span><span class="mjx-vsize" style="height: 2.326em; vertical-align: -0.793em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>θ</mi><mo>=</mo><mfrac><mrow><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>n</mi><mi>θ</mi><mo fence="false" stretchy="false">]</mo><mo>−</mo><mo fence="false" stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>m</mi><mi>θ</mi><mo fence="false" stretchy="false">]</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>m</mi></mrow></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-52">\theta = \frac{\lbrack x + n\theta\rbrack - \lbrack x + m\theta\rbrack}{n - m}</script>
<p>In questo caso <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-53-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-968" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-969" class="mjx-mrow"><span id="MJXc-Node-970" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-53">\theta</script> è dato dal rapporto di due numeri interi per cui <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-54-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">Q</mi></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-971" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-972" class="mjx-mrow"><span id="MJXc-Node-973" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-974" class="mjx-texatom"><span id="MJXc-Node-975" class="mjx-mrow"><span id="MJXc-Node-976" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-977" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.501em;">Q</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-54">\theta\mathbb{\in Q}</script>. Ma si era detto che <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-55-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi><mo>&#x2208;</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi><mo>&#x2216;</mo><mi mathvariant="double-struck">Q</mi></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-978" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-979" class="mjx-mrow"><span id="MJXc-Node-980" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-981" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-982" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-983" class="mjx-mrow"><span id="MJXc-Node-984" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.279em;">R</span></span><span id="MJXc-Node-985" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∖</span></span><span id="MJXc-Node-986" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.501em;">Q</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi><mo>∖</mo><mi mathvariant="double-struck">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-55">\theta \in \mathbb{R \smallsetminus Q}</script>. Quindi gli elementi della successione sono tutti diversi <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-56-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">R</mi><mo>&#x2216;</mo><mi mathvariant="double-struck">Q</mi></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-987" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-988" class="mjx-mrow"><span id="MJXc-Node-989" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-990" class="mjx-texatom"><span id="MJXc-Node-991" class="mjx-mrow"><span id="MJXc-Node-992" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-993" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.279em;">R</span></span><span id="MJXc-Node-994" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∖</span></span><span id="MJXc-Node-995" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.501em;">Q</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>∖</mo><mi mathvariant="double-struck">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-56">\theta\mathbb{\in R \smallsetminus Q}</script>.</p>
<ol>
<li>Si verifica adesso che l’immagine <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-57-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C4;</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-996" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-997" class="mjx-mrow"><span id="MJXc-Node-998" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.08em;">τ</span></span><span id="MJXc-Node-999" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1000" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1001" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-57">\tau(x)</script> è densa in <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-58-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1002" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1003" class="mjx-mrow"><span id="MJXc-Node-1004" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-1005" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-1006" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1007" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-1008" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-58">\lbrack 0,1)</script>.</li>
</ol>
<p>Si divida il segmento unitario, in cui è definita la successione, in $N$ intervallini di lunghezza <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-59-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>N</mi></mfrac></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1009" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1010" class="mjx-mrow"><span id="MJXc-Node-1011" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.769em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.088em; top: -1.308em;"><span id="MJXc-Node-1012" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.088em; bottom: -0.617em;"><span id="MJXc-Node-1013" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.769em;"></span></span><span class="mjx-vsize" style="height: 1.361em; vertical-align: -0.437em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>N</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-59">\frac{1}{N}</script>. Se si considera il fatto che la
successione parte dal passo <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-60-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1014" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1015" class="mjx-mrow"><span id="MJXc-Node-1016" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1017" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1021" class="mjx-texatom" style=""><span id="MJXc-Node-1022" class="mjx-mrow"><span id="MJXc-Node-1023" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1018" class="mjx-texatom" style=""><span id="MJXc-Node-1019" class="mjx-mrow"><span id="MJXc-Node-1020" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1024" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1025" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1026" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-60">T_{\theta}^{0}(x)</script> ed arriva al paso
<span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-61-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>N</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1027" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1028" class="mjx-mrow"><span id="MJXc-Node-1029" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1030" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1034" class="mjx-texatom" style=""><span id="MJXc-Node-1035" class="mjx-mrow"><span id="MJXc-Node-1036" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1031" class="mjx-texatom" style=""><span id="MJXc-Node-1032" class="mjx-mrow"><span id="MJXc-Node-1033" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1037" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1038" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1039" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>N</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-61">T_{\theta}^{N}(x)</script>, allora saranno generati <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-62-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>+</mo><mn>1</mn></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1040" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1041" class="mjx-mrow"><span id="MJXc-Node-1042" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-1043" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-1044" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-62">N + 1</script> numeri. Affinché questi stiano all’interno della precedente suddivisione, deve essere che almeno un intervallino ne contenga almeno due. Siano questi:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-63-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em" /><mtext>e</mtext><mspace width="1em" /><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em" /><mtext>con</mtext><mspace width="1em" /><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>&#x2208;</mo><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><mi>N</mi><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1045" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1046" class="mjx-mrow"><span id="MJXc-Node-1047" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1048" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1052" class="mjx-texatom" style=""><span id="MJXc-Node-1053" class="mjx-mrow"><span id="MJXc-Node-1054" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1055" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1056" class="mjx-texatom" style=""><span id="MJXc-Node-1057" class="mjx-mrow"><span id="MJXc-Node-1058" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1049" class="mjx-texatom" style=""><span id="MJXc-Node-1050" class="mjx-mrow"><span id="MJXc-Node-1051" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1059" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1060" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1061" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1062" class="mjx-mspace" style="width: 1em; height: 0px;"></span><span id="MJXc-Node-1063" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">e</span></span><span id="MJXc-Node-1064" class="mjx-mspace" style="width: 1em; height: 0px;"></span><span id="MJXc-Node-1065" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1066" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1070" class="mjx-texatom" style=""><span id="MJXc-Node-1071" class="mjx-mrow"><span id="MJXc-Node-1072" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1073" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1074" class="mjx-texatom" style=""><span id="MJXc-Node-1075" class="mjx-mrow"><span id="MJXc-Node-1076" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1067" class="mjx-texatom" style=""><span id="MJXc-Node-1068" class="mjx-mrow"><span id="MJXc-Node-1069" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1077" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1078" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1079" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1080" class="mjx-mspace" style="width: 1em; height: 0px;"></span><span id="MJXc-Node-1081" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">con</span></span><span id="MJXc-Node-1082" class="mjx-mspace" style="width: 1em; height: 0px;"></span><span id="MJXc-Node-1083" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1084" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1085" class="mjx-texatom" style=""><span id="MJXc-Node-1086" class="mjx-mrow"><span id="MJXc-Node-1087" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1088" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1089" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-1090" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1091" class="mjx-texatom" style=""><span id="MJXc-Node-1092" class="mjx-mrow"><span id="MJXc-Node-1093" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1094" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-1095" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-1096" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-1097" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1098" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-1099" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1100" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span><span id="MJXc-Node-1101" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1102" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">…</span></span><span id="MJXc-Node-1103" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1104" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-1105" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mtext>e</mtext><mspace width="1em"></mspace><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mtext>con</mtext><mspace width="1em"></mspace><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>∈</mo><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-63">T_{\theta}^{n_{0}}(x) \quad \text{e} \quad T_{\theta}^{m_{0}}(x) \quad \text{con} \quad n_{0}, m_{0} \in \{0, 1, 2, \ldots, N\}</script>
<p>Vale quindi che:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-64-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext>&#xA0;</mtext><mo>&#x2212;</mo><mtext>&#xA0;</mtext><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>&lt;</mo><mfrac><mn>1</mn><mi>N</mi></mfrac></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1106" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1107" class="mjx-mrow"><span id="MJXc-Node-1108" class="mjx-texatom"><span id="MJXc-Node-1109" class="mjx-mrow"><span id="MJXc-Node-1110" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">|</span></span></span></span><span id="MJXc-Node-1111" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1112" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1116" class="mjx-texatom" style=""><span id="MJXc-Node-1117" class="mjx-mrow"><span id="MJXc-Node-1118" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1119" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1120" class="mjx-texatom" style=""><span id="MJXc-Node-1121" class="mjx-mrow"><span id="MJXc-Node-1122" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1113" class="mjx-texatom" style=""><span id="MJXc-Node-1114" class="mjx-mrow"><span id="MJXc-Node-1115" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1123" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1124" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1125" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1126" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1127" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-1128" class="mjx-mtext MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1129" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1130" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1134" class="mjx-texatom" style=""><span id="MJXc-Node-1135" class="mjx-mrow"><span id="MJXc-Node-1136" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1137" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1138" class="mjx-texatom" style=""><span id="MJXc-Node-1139" class="mjx-mrow"><span id="MJXc-Node-1140" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1131" class="mjx-texatom" style=""><span id="MJXc-Node-1132" class="mjx-mrow"><span id="MJXc-Node-1133" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1141" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1142" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1143" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1144" class="mjx-texatom"><span id="MJXc-Node-1145" class="mjx-mrow"><span id="MJXc-Node-1146" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">|</span></span></span></span><span id="MJXc-Node-1147" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;"><</span></span><span id="MJXc-Node-1148" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.088em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.088em; top: -1.368em;"><span id="MJXc-Node-1149" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span><span class="mjx-denominator" style="width: 1.088em; bottom: -0.716em;"><span id="MJXc-Node-1150" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 1.088em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.716em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mo>−</mo><mtext> </mtext><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo><</mo><mfrac><mn>1</mn><mi>N</mi></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-64">| T_{\theta}^{n_{0}}(x)\ - \ T_{\theta}^{m_{0}}(x) | < \frac{1}{N}</script>
<p>e quindi</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-65-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><mtext>&#xA0;</mtext><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mtext>&#xA0;</mtext><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>&lt;</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>.</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1151" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1152" class="mjx-mrow"><span id="MJXc-Node-1153" class="mjx-texatom"><span id="MJXc-Node-1154" class="mjx-mrow"><span id="MJXc-Node-1155" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">d</span></span></span></span><span id="MJXc-Node-1156" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1157" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1158" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1159" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1163" class="mjx-texatom" style=""><span id="MJXc-Node-1164" class="mjx-mrow"><span id="MJXc-Node-1165" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1166" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1167" class="mjx-texatom" style=""><span id="MJXc-Node-1168" class="mjx-mrow"><span id="MJXc-Node-1169" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1160" class="mjx-texatom" style=""><span id="MJXc-Node-1161" class="mjx-mrow"><span id="MJXc-Node-1162" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1170" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1171" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1172" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1173" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1174" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1175" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1176" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1180" class="mjx-texatom" style=""><span id="MJXc-Node-1181" class="mjx-mrow"><span id="MJXc-Node-1182" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1183" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1184" class="mjx-texatom" style=""><span id="MJXc-Node-1185" class="mjx-mrow"><span id="MJXc-Node-1186" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1177" class="mjx-texatom" style=""><span id="MJXc-Node-1178" class="mjx-mrow"><span id="MJXc-Node-1179" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1187" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1188" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1189" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1190" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1191" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;"><</span></span><span id="MJXc-Node-1192" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.088em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.088em; top: -1.368em;"><span id="MJXc-Node-1193" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span><span class="mjx-denominator" style="width: 1.088em; bottom: -0.716em;"><span id="MJXc-Node-1194" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 1.088em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.716em;"></span></span><span id="MJXc-Node-1195" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><mtext> </mtext><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo><</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-65">\mathbb{d}( \ T_{\theta}^{m_{0}}(x),\ T_{\theta}^{n_{0}}(x) ) < \frac{1}{N}.</script>
<p>Si applica <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-66-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mrow class="MJX-TeXAtom-ORD"><mo>$</mo></mrow><mo stretchy="false">(</mo><mi>c</mi><mi>o</mi><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>$</mo></mrow><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>&gt;</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1196" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1197" class="mjx-mrow"><span id="MJXc-Node-1198" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1199" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1203" class="mjx-texatom" style=""><span id="MJXc-Node-1204" class="mjx-mrow"><span id="MJXc-Node-1205" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-1206" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1207" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1208" class="mjx-texatom" style=""><span id="MJXc-Node-1209" class="mjx-mrow"><span id="MJXc-Node-1210" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1200" class="mjx-texatom" style=""><span id="MJXc-Node-1201" class="mjx-mrow"><span id="MJXc-Node-1202" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1211" class="mjx-texatom"><span id="MJXc-Node-1212" class="mjx-mrow"><span id="MJXc-Node-1213" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.39em;">$</span></span></span></span><span id="MJXc-Node-1214" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1215" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">c</span></span><span id="MJXc-Node-1216" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">o</span></span><span id="MJXc-Node-1217" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-1218" class="mjx-texatom"><span id="MJXc-Node-1219" class="mjx-mrow"><span id="MJXc-Node-1220" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.39em;">$</span></span></span></span><span id="MJXc-Node-1221" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1222" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1223" class="mjx-texatom" style=""><span id="MJXc-Node-1224" class="mjx-mrow"><span id="MJXc-Node-1225" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1226" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">></span></span><span id="MJXc-Node-1227" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-1228" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1229" class="mjx-texatom" style=""><span id="MJXc-Node-1230" class="mjx-mrow"><span id="MJXc-Node-1231" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mrow class="MJX-TeXAtom-ORD"><mo>$</mo></mrow><mo stretchy="false">(</mo><mi>c</mi><mi>o</mi><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>$</mo></mrow><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>></mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-66">T_{\theta}^{- m_{0}}$ (con $n_{0} > m_{0}</script>) e ricordando il Lemma 2 si ottiene:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-67-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><msub><mi>m</mi><mn>0</mn></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x2218;</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>m</mi><mn>0</mn></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><msub><mi>m</mi><mn>0</mn></msub></mrow></msubsup><mo>&#x2218;</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mn>0</mn></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>&lt;</mo><mfrac><mn>1</mn><mi>N</mi></mfrac></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1232" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1233" class="mjx-mrow"><span id="MJXc-Node-1234" class="mjx-texatom"><span id="MJXc-Node-1235" class="mjx-mrow"><span id="MJXc-Node-1236" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">d</span></span></span></span><span id="MJXc-Node-1237" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1238" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1239" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1243" class="mjx-texatom" style=""><span id="MJXc-Node-1244" class="mjx-mrow"><span id="MJXc-Node-1245" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-1246" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1247" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1248" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1240" class="mjx-texatom" style=""><span id="MJXc-Node-1241" class="mjx-mrow"><span id="MJXc-Node-1242" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1249" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1250" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1251" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1252" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∘</span></span><span id="MJXc-Node-1253" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1254" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1258" class="mjx-texatom" style=""><span id="MJXc-Node-1259" class="mjx-mrow"><span id="MJXc-Node-1260" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1261" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1262" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1255" class="mjx-texatom" style=""><span id="MJXc-Node-1256" class="mjx-mrow"><span id="MJXc-Node-1257" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1263" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1264" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1265" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1266" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1267" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1268" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1272" class="mjx-texatom" style=""><span id="MJXc-Node-1273" class="mjx-mrow"><span id="MJXc-Node-1274" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-1275" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1276" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1277" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1269" class="mjx-texatom" style=""><span id="MJXc-Node-1270" class="mjx-mrow"><span id="MJXc-Node-1271" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1278" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∘</span></span><span id="MJXc-Node-1279" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1280" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1284" class="mjx-texatom" style=""><span id="MJXc-Node-1285" class="mjx-mrow"><span id="MJXc-Node-1286" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1287" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1288" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1281" class="mjx-texatom" style=""><span id="MJXc-Node-1282" class="mjx-mrow"><span id="MJXc-Node-1283" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1289" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1290" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1291" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1292" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1293" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;"><</span></span><span id="MJXc-Node-1294" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.088em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.088em; top: -1.368em;"><span id="MJXc-Node-1295" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span><span class="mjx-denominator" style="width: 1.088em; bottom: -0.716em;"><span id="MJXc-Node-1296" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 1.088em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.716em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><msub><mi>m</mi><mn>0</mn></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∘</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>m</mi><mn>0</mn></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><msub><mi>m</mi><mn>0</mn></msub></mrow></msubsup><mo>∘</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mn>0</mn></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo><</mo><mfrac><mn>1</mn><mi>N</mi></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-67">\mathbb{d}(T_{\theta}^{-m_0}(x) \circ T_{\theta}^{m_0}(x), T_{\theta}^{-m_0} \circ T_{\theta}^{n_0}(x)) < \frac{1}{N}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-68-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>&#x2212;</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>&lt;</mo><mfrac><mn>1</mn><mi>N</mi></mfrac></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1297" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1298" class="mjx-mrow"><span id="MJXc-Node-1299" class="mjx-texatom"><span id="MJXc-Node-1300" class="mjx-mrow"><span id="MJXc-Node-1301" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">d</span></span></span></span><span id="MJXc-Node-1302" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1303" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1304" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1305" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1306" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1310" class="mjx-texatom" style=""><span id="MJXc-Node-1311" class="mjx-mrow"><span id="MJXc-Node-1312" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1313" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1314" class="mjx-texatom" style=""><span id="MJXc-Node-1315" class="mjx-mrow"><span id="MJXc-Node-1316" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1317" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-1318" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1319" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1320" class="mjx-texatom" style=""><span id="MJXc-Node-1321" class="mjx-mrow"><span id="MJXc-Node-1322" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1307" class="mjx-texatom" style=""><span id="MJXc-Node-1308" class="mjx-mrow"><span id="MJXc-Node-1309" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1323" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1324" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1325" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1326" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1327" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;"><</span></span><span id="MJXc-Node-1328" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.088em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.088em; top: -1.368em;"><span id="MJXc-Node-1329" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span><span class="mjx-denominator" style="width: 1.088em; bottom: -0.716em;"><span id="MJXc-Node-1330" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 1.088em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.716em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>−</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo><</mo><mfrac><mn>1</mn><mi>N</mi></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-68">\mathbb{d}( x,T_{\theta}^{n_{0} - m_{0}}(x) ) < \frac{1}{N}</script>
<p>Si consideri ora la successione:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-69-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi></mrow></msub><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>j</mi></mrow></msub><mo>=</mo><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>&#x2212;</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1331" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1332" class="mjx-mrow"><span id="MJXc-Node-1333" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-1334" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1335" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1336" class="mjx-texatom" style=""><span id="MJXc-Node-1337" class="mjx-mrow"><span id="MJXc-Node-1338" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">j</span></span></span></span></span></span><span id="MJXc-Node-1339" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1340" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1341" class="mjx-texatom" style=""><span id="MJXc-Node-1342" class="mjx-mrow"><span id="MJXc-Node-1343" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">j</span></span></span></span></span></span><span id="MJXc-Node-1344" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1345" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-1346" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1347" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1351" class="mjx-texatom" style=""><span id="MJXc-Node-1352" class="mjx-mrow"><span id="MJXc-Node-1353" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">j</span></span><span id="MJXc-Node-1354" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1355" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1356" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1357" class="mjx-texatom" style=""><span id="MJXc-Node-1358" class="mjx-mrow"><span id="MJXc-Node-1359" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1360" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-1361" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1362" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1363" class="mjx-texatom" style=""><span id="MJXc-Node-1364" class="mjx-mrow"><span id="MJXc-Node-1365" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1366" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1348" class="mjx-texatom" style=""><span id="MJXc-Node-1349" class="mjx-mrow"><span id="MJXc-Node-1350" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1367" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1368" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1369" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1370" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi></mrow></msub><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>j</mi></mrow></msub><mo>=</mo><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>−</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-69">\{x_{j}\}_{j} = \{T_{\theta}^{j(n_{0} - m_{0})}(x)\}</script>
<p>Si nota che questa è una sotto-successione di <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-70-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow class="MJX-TeXAtom-ORD"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub><mo>.</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1371" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1372" class="mjx-mrow"><span id="MJXc-Node-1373" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-1374" class="mjx-texatom"><span id="MJXc-Node-1375" class="mjx-mrow"><span id="MJXc-Node-1376" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1377" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1381" class="mjx-texatom" style=""><span id="MJXc-Node-1382" class="mjx-mrow"><span id="MJXc-Node-1383" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1378" class="mjx-texatom" style=""><span id="MJXc-Node-1379" class="mjx-mrow"><span id="MJXc-Node-1380" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span></span></span><span id="MJXc-Node-1384" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1385" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1386" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1387" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1388" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1389" class="mjx-texatom" style=""><span id="MJXc-Node-1390" class="mjx-mrow"><span id="MJXc-Node-1391" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span></span><span id="MJXc-Node-1392" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow class="MJX-TeXAtom-ORD"><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-70">\{ {T_{\theta}^{n}}(x) \}_{n}.</script></p>
<p>Si applichi ora il <em>Lemma 2</em> tra due punti della sotto-successione:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-71-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>&#x2212;</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>&#x2212;</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1393" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1394" class="mjx-mrow"><span id="MJXc-Node-1395" class="mjx-texatom"><span id="MJXc-Node-1396" class="mjx-mrow"><span id="MJXc-Node-1397" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">d</span></span></span></span><span id="MJXc-Node-1398" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1399" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1400" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1404" class="mjx-texatom" style=""><span id="MJXc-Node-1405" class="mjx-mrow"><span id="MJXc-Node-1406" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">j</span></span><span id="MJXc-Node-1407" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1408" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1409" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1410" class="mjx-texatom" style=""><span id="MJXc-Node-1411" class="mjx-mrow"><span id="MJXc-Node-1412" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1413" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-1414" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1415" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1416" class="mjx-texatom" style=""><span id="MJXc-Node-1417" class="mjx-mrow"><span id="MJXc-Node-1418" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1419" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1401" class="mjx-texatom" style=""><span id="MJXc-Node-1402" class="mjx-mrow"><span id="MJXc-Node-1403" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1420" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1421" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1422" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1423" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1424" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1425" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1429" class="mjx-texatom" style=""><span id="MJXc-Node-1430" class="mjx-mrow"><span id="MJXc-Node-1431" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1432" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">j</span></span><span id="MJXc-Node-1433" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-1434" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-1435" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1436" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1437" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1438" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1439" class="mjx-texatom" style=""><span id="MJXc-Node-1440" class="mjx-mrow"><span id="MJXc-Node-1441" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1442" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-1443" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1444" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1445" class="mjx-texatom" style=""><span id="MJXc-Node-1446" class="mjx-mrow"><span id="MJXc-Node-1447" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1448" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1426" class="mjx-texatom" style=""><span id="MJXc-Node-1427" class="mjx-mrow"><span id="MJXc-Node-1428" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1449" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1450" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1451" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1452" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>−</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>−</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-71">\mathbb{d}( T_{\theta}^{j( n_{0} - m_{0} )}(x),T_{\theta}^{(j + 1)( n_{0} - m_{0} )}(x) )</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-72-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>&#x2212;</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>&#x2212;</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>&#x2212;</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1453" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1454" class="mjx-mrow"><span id="MJXc-Node-1455" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1456" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-1457" class="mjx-mrow"><span id="MJXc-Node-1458" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">d</span></span></span></span><span id="MJXc-Node-1459" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1460" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1461" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1465" class="mjx-texatom" style=""><span id="MJXc-Node-1466" class="mjx-mrow"><span id="MJXc-Node-1467" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">j</span></span><span id="MJXc-Node-1468" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1469" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1470" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1471" class="mjx-texatom" style=""><span id="MJXc-Node-1472" class="mjx-mrow"><span id="MJXc-Node-1473" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1474" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-1475" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1476" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1477" class="mjx-texatom" style=""><span id="MJXc-Node-1478" class="mjx-mrow"><span id="MJXc-Node-1479" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1480" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1462" class="mjx-texatom" style=""><span id="MJXc-Node-1463" class="mjx-mrow"><span id="MJXc-Node-1464" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1481" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1482" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1483" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1484" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1485" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1486" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1490" class="mjx-texatom" style=""><span id="MJXc-Node-1491" class="mjx-mrow"><span id="MJXc-Node-1492" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">j</span></span><span id="MJXc-Node-1493" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1494" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1495" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1496" class="mjx-texatom" style=""><span id="MJXc-Node-1497" class="mjx-mrow"><span id="MJXc-Node-1498" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1499" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-1500" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1501" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1502" class="mjx-texatom" style=""><span id="MJXc-Node-1503" class="mjx-mrow"><span id="MJXc-Node-1504" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1505" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1487" class="mjx-texatom" style=""><span id="MJXc-Node-1488" class="mjx-mrow"><span id="MJXc-Node-1489" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1506" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1507" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1508" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1512" class="mjx-texatom" style=""><span id="MJXc-Node-1513" class="mjx-mrow"><span id="MJXc-Node-1514" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1515" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1516" class="mjx-texatom" style=""><span id="MJXc-Node-1517" class="mjx-mrow"><span id="MJXc-Node-1518" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1519" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-1520" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1521" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1522" class="mjx-texatom" style=""><span id="MJXc-Node-1523" class="mjx-mrow"><span id="MJXc-Node-1524" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1509" class="mjx-texatom" style=""><span id="MJXc-Node-1510" class="mjx-mrow"><span id="MJXc-Node-1511" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1525" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1526" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1527" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1528" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1529" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>−</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>−</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>−</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-72">= \mathbb{d}( T_{\theta}^{j( n_{0} - m_{0} )}(x),T_{\theta}^{j( n_{0} - m_{0} )}( T_{\theta}^{n_{0} - m_{0}}(x) ) )</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-73-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B8;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>&#x2212;</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>&lt;</mo><mfrac><mn>1</mn><mi>N</mi></mfrac></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1530" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1531" class="mjx-mrow"><span id="MJXc-Node-1532" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1533" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-1534" class="mjx-mrow"><span id="MJXc-Node-1535" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">d</span></span></span></span><span id="MJXc-Node-1536" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1537" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1538" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1539" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-1540" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.343em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-1544" class="mjx-texatom" style=""><span id="MJXc-Node-1545" class="mjx-mrow"><span id="MJXc-Node-1546" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1547" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1548" class="mjx-texatom" style=""><span id="MJXc-Node-1549" class="mjx-mrow"><span id="MJXc-Node-1550" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1551" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-1552" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1553" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1554" class="mjx-texatom" style=""><span id="MJXc-Node-1555" class="mjx-mrow"><span id="MJXc-Node-1556" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1541" class="mjx-texatom" style=""><span id="MJXc-Node-1542" class="mjx-mrow"><span id="MJXc-Node-1543" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span></span></span></span><span id="MJXc-Node-1557" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1558" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1559" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1560" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1561" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;"><</span></span><span id="MJXc-Node-1562" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.088em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.088em; top: -1.368em;"><span id="MJXc-Node-1563" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span><span class="mjx-denominator" style="width: 1.088em; bottom: -0.716em;"><span id="MJXc-Node-1564" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 1.088em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.716em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">d</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mi>θ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>−</mo><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo><</mo><mfrac><mn>1</mn><mi>N</mi></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-73">= \mathbb{d}( x,T_{\theta}^{n_{0} - m_{0}}(x) ) < \frac{1}{N}</script>
<p>Quindi si è ottenuta una sotto-successione che ha un passo più breve di <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-74-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>N</mi></mfrac></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1565" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1566" class="mjx-mrow"><span id="MJXc-Node-1567" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.769em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.088em; top: -1.308em;"><span id="MJXc-Node-1568" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.088em; bottom: -0.617em;"><span id="MJXc-Node-1569" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.769em;"></span></span><span class="mjx-vsize" style="height: 1.361em; vertical-align: -0.437em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>N</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-74">\frac{1}{N}</script>. Poiché è possibile scegliere <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-75-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1570" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1571" class="mjx-mrow"><span id="MJXc-Node-1572" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-75">N</script> grande quanto si vuole, allora anche il passo può essere piccolo quanto si vuole.</p>
<p>Concludiamo che <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-76-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C4;</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1573" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1574" class="mjx-mrow"><span id="MJXc-Node-1575" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.08em;">τ</span></span><span id="MJXc-Node-1576" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1577" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1578" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-76">\tau(x)</script> è densa in $$\lbrack 0,1)$.</p>
<p><strong>Mappatura sul Toroide</strong></p>
<p>Un Toroide può essere definito mediante le seguenti equazioni
parametriche:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-77-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mi>x</mi><mo stretchy="false">(</mo><mi>&#x03C9;</mi><mo>,</mo><mi>&#x03C6;</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>R</mi><mo>+</mo><mi>r</mi><mi>cos</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>&#x03C9;</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>cos</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>&#x03C6;</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>y</mi><mo stretchy="false">(</mo><mi>&#x03C9;</mi><mo>,</mo><mi>&#x03C6;</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>R</mi><mo>+</mo><mi>r</mi><mi>cos</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>&#x03C9;</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>sin</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>&#x03C6;</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>z</mi><mo stretchy="false">(</mo><mi>&#x03C9;</mi><mo>,</mo><mi>&#x03C6;</mi><mo stretchy="false">)</mo><mo>=</mo><mi>r</mi><mi>sin</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>&#x03C9;</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow><mspace width="1em" /><mtext>con&#xA0;</mtext><mi>&#x03C9;</mi><mo>,</mo><mi>&#x03C6;</mi><mo>&#x2208;</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>&#x03C0;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1579" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1580" class="mjx-mrow"><span id="MJXc-Node-1581" class="mjx-mrow"><span id="MJXc-Node-1582" class="mjx-mo" style="vertical-align: -0.639em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.666em; padding-bottom: 0.279em; margin-bottom: -0.081em;">⎧</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.943em; padding-bottom: 0.943em;">⎨</span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: -0.299em; padding-bottom: 1.164em;">⎩</span></span></span><span id="MJXc-Node-1583" class="mjx-mtable" style="vertical-align: -1.563em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-1584" class="mjx-mtr" style="height: 1.175em;"><span id="MJXc-Node-1585" class="mjx-mtd" style="padding: 0px; text-align: left; width: 13.456em;"><span id="MJXc-Node-1586" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-1587" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1588" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1589" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">ω</span></span><span id="MJXc-Node-1590" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1591" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em;">φ</span></span><span id="MJXc-Node-1592" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1593" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1594" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1595" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em;">R</span></span><span id="MJXc-Node-1596" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-1597" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">r</span></span><span id="MJXc-Node-1598" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">cos</span></span><span id="MJXc-Node-1599" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-1600" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1601" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">ω</span></span><span id="MJXc-Node-1602" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1603" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1604" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">cos</span></span><span id="MJXc-Node-1605" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-1606" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1607" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em;">φ</span></span><span id="MJXc-Node-1608" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-1609" class="mjx-mtr" style="height: 1.275em;"><span id="MJXc-Node-1610" class="mjx-mtd" style="padding: 0.1em 0px 0px; text-align: left;"><span id="MJXc-Node-1611" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-1612" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-1613" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1614" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">ω</span></span><span id="MJXc-Node-1615" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1616" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em;">φ</span></span><span id="MJXc-Node-1617" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1618" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1619" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1620" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em;">R</span></span><span id="MJXc-Node-1621" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-1622" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">r</span></span><span id="MJXc-Node-1623" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">cos</span></span><span id="MJXc-Node-1624" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-1625" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1626" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">ω</span></span><span id="MJXc-Node-1627" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1628" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1629" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">sin</span></span><span id="MJXc-Node-1630" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-1631" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1632" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em;">φ</span></span><span id="MJXc-Node-1633" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-1634" class="mjx-mtr" style="height: 1.175em;"><span id="MJXc-Node-1635" class="mjx-mtd" style="padding: 0.1em 0px 0px; text-align: left;"><span id="MJXc-Node-1636" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-1637" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">z</span></span><span id="MJXc-Node-1638" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1639" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">ω</span></span><span id="MJXc-Node-1640" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1641" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em;">φ</span></span><span id="MJXc-Node-1642" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1643" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1644" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">r</span></span><span id="MJXc-Node-1645" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">sin</span></span><span id="MJXc-Node-1646" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-1647" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1648" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">ω</span></span><span id="MJXc-Node-1649" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-1650" class="mjx-mo" style="width: 0.12em;"></span></span><span id="MJXc-Node-1651" class="mjx-mspace" style="width: 1em; height: 0px;"></span><span id="MJXc-Node-1652" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">con </span></span><span id="MJXc-Node-1653" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">ω</span></span><span id="MJXc-Node-1654" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1655" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em;">φ</span></span><span id="MJXc-Node-1656" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-1657" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-1658" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-1659" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1660" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span><span id="MJXc-Node-1661" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-1662" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mi>x</mi><mo stretchy="false">(</mo><mi>ω</mi><mo>,</mo><mi>φ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>R</mi><mo>+</mo><mi>r</mi><mi>cos</mi><mo></mo><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>cos</mi><mo></mo><mo stretchy="false">(</mo><mi>φ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>y</mi><mo stretchy="false">(</mo><mi>ω</mi><mo>,</mo><mi>φ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>R</mi><mo>+</mo><mi>r</mi><mi>cos</mi><mo></mo><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>sin</mi><mo></mo><mo stretchy="false">(</mo><mi>φ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>z</mi><mo stretchy="false">(</mo><mi>ω</mi><mo>,</mo><mi>φ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>r</mi><mi>sin</mi><mo></mo><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow><mspace width="1em"></mspace><mtext>con </mtext><mi>ω</mi><mo>,</mo><mi>φ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-77">\begin{cases}
x(\omega, \varphi) = (R + r \cos(\omega)) \cos(\varphi) \\
y(\omega, \varphi) = (R + r \cos(\omega)) \sin(\varphi) \\
z(\omega, \varphi) = r \sin(\omega)
\end{cases}
\quad \text{con } \omega, \varphi \in [0, 2\pi)</script>
<p>Dove <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-78-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C9;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1663" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1664" class="mjx-mrow"><span id="MJXc-Node-1665" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">ω</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math></span></span><script type="math/tex" id="MathJax-Element-78">\omega</script>, <em>φ</em> sono gli angoli di rotazione, <em>r</em> è il raggio del tubo del Toroide, <em>R</em> è la distanza dal centro del tubo al centro di rotazione del Toroide.</p>
<p>Il motivo per il quale si giunge al Toroide è dato dalla seguente
animazione esemplificativa:</p>
<p><img src="https://gabro29.github.io/Biliard_Torus/image5.gif" alt=""></p>
<p>È possibile ottenere un Toroide a partire da un rettangolo se si incollano tra di loro il bordo superiore e inferiore e poi quello destro e sinistro</p>
<p>Ricordando che la traiettoria della pallina è costruita mediante rette,
e che queste vengono disegnate operativamente unendo dei punti tra di
loro, è possibile sfruttare tali punti per mappare una traiettoria
curvilinea sul Toroide. Passando quindi da un grafico in 2D ad uno
tridimensionale mediante le seguenti proporzioni:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-79-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mi>&#x03B8;</mi><mo>:</mo><mn>2</mn><mi>&#x03C0;</mi><mo>=</mo><mi>x</mi><mo>:</mo><mi>b</mi></mtd></mtr><mtr><mtd><mi>&#x03C6;</mi><mo>:</mo><mn>2</mn><mi>&#x03C0;</mi><mo>=</mo><mi>y</mi><mo>:</mo><mi>h</mi></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1666" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1667" class="mjx-mrow"><span id="MJXc-Node-1668" class="mjx-mrow"><span id="MJXc-Node-1669" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.22em; padding-bottom: 1.22em;">{</span></span><span id="MJXc-Node-1670" class="mjx-mtable" style="vertical-align: -0.872em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-1671" class="mjx-mtr" style="height: 1.1em;"><span id="MJXc-Node-1672" class="mjx-mtd" style="padding: 0px; text-align: left; width: 5.8em;"><span id="MJXc-Node-1673" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-1674" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-1675" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">:</span></span><span id="MJXc-Node-1676" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span><span id="MJXc-Node-1677" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-1678" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1679" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-1680" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">:</span></span><span id="MJXc-Node-1681" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em;">b</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-1682" class="mjx-mtr" style="height: 1.143em;"><span id="MJXc-Node-1683" class="mjx-mtd" style="padding: 0.1em 0px 0px; text-align: left;"><span id="MJXc-Node-1684" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-1685" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em;">φ</span></span><span id="MJXc-Node-1686" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">:</span></span><span id="MJXc-Node-1687" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span><span id="MJXc-Node-1688" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-1689" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1690" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-1691" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">:</span></span><span id="MJXc-Node-1692" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em;">h</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-1693" class="mjx-mo" style="width: 0.12em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mi>θ</mi><mo>:</mo><mn>2</mn><mi>π</mi><mo>=</mo><mi>x</mi><mo>:</mo><mi>b</mi></mtd></mtr><mtr><mtd><mi>φ</mi><mo>:</mo><mn>2</mn><mi>π</mi><mo>=</mo><mi>y</mi><mo>:</mo><mi>h</mi></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-79">\begin{cases}
\theta : 2\pi = x : b \\
\varphi : 2\pi = y : h \\
\end{cases}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-80-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mi>&#x03B8;</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>&#x03C0;</mi><mo>&#x22C5;</mo><mi>x</mi></mrow><mi>b</mi></mfrac></mtd></mtr><mtr><mtd><mi>&#x03C6;</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>&#x03C0;</mi><mo>&#x22C5;</mo><mi>y</mi></mrow><mi>h</mi></mfrac></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1694" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1695" class="mjx-mrow"><span id="MJXc-Node-1696" class="mjx-mrow"><span id="MJXc-Node-1697" class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.551em; padding-bottom: 1.551em;">{</span></span><span id="MJXc-Node-1698" class="mjx-mtable" style="vertical-align: -1.303em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-1699" class="mjx-mtr" style="height: 1.484em;"><span id="MJXc-Node-1700" class="mjx-mtd" style="padding: 0px; text-align: left; width: 3.675em;"><span id="MJXc-Node-1701" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-1702" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-1703" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1704" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.501em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.123em; top: -1.319em;"><span id="MJXc-Node-1705" class="mjx-mrow" style=""><span id="MJXc-Node-1706" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span><span id="MJXc-Node-1707" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-1708" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.335em;">⋅</span></span><span id="MJXc-Node-1709" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.123em; bottom: -0.639em;"><span id="MJXc-Node-1710" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em;">b</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 1.501em;"></span></span><span class="mjx-vsize" style="height: 1.384em; vertical-align: -0.452em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-1711" class="mjx-mtr" style="height: 1.622em;"><span id="MJXc-Node-1712" class="mjx-mtd" style="padding: 0.1em 0px 0px; text-align: left;"><span id="MJXc-Node-1713" class="mjx-mrow"><span id="MJXc-Node-1714" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em;">φ</span></span><span id="MJXc-Node-1715" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1716" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.447em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.047em; top: -1.513em;"><span id="MJXc-Node-1717" class="mjx-mrow" style=""><span id="MJXc-Node-1718" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span><span id="MJXc-Node-1719" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-1720" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.335em;">⋅</span></span><span id="MJXc-Node-1721" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.047em; bottom: -0.639em;"><span id="MJXc-Node-1722" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em;">h</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 1.447em;"></span></span><span class="mjx-vsize" style="height: 1.522em; vertical-align: -0.452em;"></span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-1723" class="mjx-mo" style="width: 0.12em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi><mo>⋅</mo><mi>x</mi></mrow><mi>b</mi></mfrac></mtd></mtr><mtr><mtd><mi>φ</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi><mo>⋅</mo><mi>y</mi></mrow><mi>h</mi></mfrac></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-80">\begin{cases}
\theta = \frac{2\pi \cdot x}{b} \\
\varphi = \frac{2\pi \cdot y}{h}
\end{cases}</script>
<p>Dove <em>b</em> e <em>h</em> sono rispettivamente la base e l’altezza del rettangolo. Infine sfruttando le coordinate parametriche di prima si costruisce il Toroide.</p>
<p><strong>Traiettorie dense nel toroide</strong></p>
<p>Si consideri un punto <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-81-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mo>&#x2208;</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1724" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1725" class="mjx-mrow"><span id="MJXc-Node-1726" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1727" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1728" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1729" class="mjx-texatom" style=""><span id="MJXc-Node-1730" class="mjx-mrow"><span id="MJXc-Node-1731" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1732" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1733" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-1734" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1735" class="mjx-texatom" style=""><span id="MJXc-Node-1736" class="mjx-mrow"><span id="MJXc-Node-1737" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1738" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1739" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-1740" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-1741" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-1742" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1743" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-1744" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1745" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1746" class="mjx-texatom" style=""><span id="MJXc-Node-1747" class="mjx-mrow"><span id="MJXc-Node-1748" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mo>∈</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-81">( x_{0},y_{0} ) \in \lbrack 0,1)^{2}</script> e due numeri <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-82-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B1;</mi><mo>,</mo><mi>&#x03B2;</mi><mo>&#x2208;</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow><mo class="MJX-variant">&#x2216;</mo><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1749" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1750" class="mjx-mrow"><span id="MJXc-Node-1751" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-1752" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1753" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-1754" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-1755" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-1756" class="mjx-mrow"><span id="MJXc-Node-1757" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.279em;">R</span></span></span></span><span id="MJXc-Node-1758" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∖</span></span><span id="MJXc-Node-1759" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-1760" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-1761" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow><mo class="MJX-variant">∖</mo><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-82">\alpha, \beta \in \mathbb{R} \setminus \{0\}</script>.</p>
<p>Quindi si definisce la mappa:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-83-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>&#x03D5;</mi><mrow class="MJX-TeXAtom-ORD"><mo mathvariant="double-struck">:</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">&#x2192;</mo></mrow><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1762" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1763" class="mjx-mrow"><span id="MJXc-Node-1764" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span><span id="MJXc-Node-1765" class="mjx-texatom"><span id="MJXc-Node-1766" class="mjx-mrow"><span id="MJXc-Node-1767" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">:</span></span><span id="MJXc-Node-1768" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.279em;">R</span></span><span id="MJXc-Node-1769" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.224em; padding-bottom: 0.335em;">→</span></span></span></span><span id="MJXc-Node-1770" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-1771" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-1772" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1773" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-1774" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1775" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1776" class="mjx-texatom" style=""><span id="MJXc-Node-1777" class="mjx-mrow"><span id="MJXc-Node-1778" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>ϕ</mi><mrow class="MJX-TeXAtom-ORD"><mo mathvariant="double-struck">:</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">→</mo></mrow><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-83">\phi\mathbb{:R \rightarrow}\lbrack 0,1)^{2}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-84-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mi>t</mi><mo stretchy="false">&#x21A6;</mo><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B1;</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo><mo>,</mo><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B2;</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1779" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1780" class="mjx-mrow"><span id="MJXc-Node-1781" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1782" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1783" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1784" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1785" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1786" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1787" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1788" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1789" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1790" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1791" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1792" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1793" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1794" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1795" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1796" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1797" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1798" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1799" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1800" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1801" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1802" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1803" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1804" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1805" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1806" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1807" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1808" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1809" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1810" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1811" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1812" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1813" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1814" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1815" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1816" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1817" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.224em; padding-bottom: 0.335em;">↦</span></span><span id="MJXc-Node-1818" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1819" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-1820" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1821" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1822" class="mjx-texatom" style=""><span id="MJXc-Node-1823" class="mjx-mrow"><span id="MJXc-Node-1824" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1825" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-1826" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-1827" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1828" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-1829" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1830" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-1831" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-1832" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1833" class="mjx-texatom" style=""><span id="MJXc-Node-1834" class="mjx-mrow"><span id="MJXc-Node-1835" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1836" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-1837" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-1838" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1839" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-1840" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>t</mi><mo stretchy="false">↦</mo><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>α</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo><mo>,</mo><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>β</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-84">\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t \mapsto ( \{ x_{0} + \alpha t \},\{ y_{0} + \beta t \} )</script>
<p>Tale mappa rappresenta un moto di natura continua sul Toroide, con velocità:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-85-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mover><mi>v</mi><mo>&#x2192;</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo>,</mo><mi>&#x03B2;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1841" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1842" class="mjx-mrow"><span id="MJXc-Node-1843" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="padding-bottom: 0.06em;"><span id="MJXc-Node-1845" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.224em; padding-bottom: 0.335em;">→</span></span></span><span class="mjx-op" style="padding-left: 0.23em;"><span id="MJXc-Node-1844" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">v</span></span></span></span></span><span id="MJXc-Node-1846" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1847" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1848" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-1849" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1850" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-1851" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mover><mi>v</mi><mo>→</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-85">\overrightarrow{v} = (\alpha,\beta)</script>
<p>Per semplicità si scrive:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-86-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow class="MJX-TeXAtom-ORD"><mi>&#x03D5;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B1;</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1852" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1853" class="mjx-mrow"><span id="MJXc-Node-1854" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1855" class="mjx-texatom"><span id="MJXc-Node-1856" class="mjx-mrow"><span id="MJXc-Node-1857" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.375em; padding-right: 0.071em;"><span id="MJXc-Node-1858" class="mjx-texatom" style=""><span id="MJXc-Node-1859" class="mjx-mrow"><span id="MJXc-Node-1860" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span><span id="MJXc-Node-1861" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1862" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1863" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1864" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1865" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-1866" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1867" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1868" class="mjx-texatom" style=""><span id="MJXc-Node-1869" class="mjx-mrow"><span id="MJXc-Node-1870" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1871" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-1872" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-1873" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1874" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow class="MJX-TeXAtom-ORD"><mi>ϕ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>α</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-86">{\phi}_{1}(t) = \{ x_{0} + \alpha t \}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-87-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow class="MJX-TeXAtom-ORD"><mi>&#x03D5;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B2;</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1875" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1876" class="mjx-mrow"><span id="MJXc-Node-1877" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1878" class="mjx-texatom"><span id="MJXc-Node-1879" class="mjx-mrow"><span id="MJXc-Node-1880" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.375em; padding-right: 0.071em;"><span id="MJXc-Node-1881" class="mjx-texatom" style=""><span id="MJXc-Node-1882" class="mjx-mrow"><span id="MJXc-Node-1883" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span></span></span></span><span id="MJXc-Node-1884" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1885" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1886" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1887" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1888" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-1889" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-1890" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1891" class="mjx-texatom" style=""><span id="MJXc-Node-1892" class="mjx-mrow"><span id="MJXc-Node-1893" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1894" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-1895" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-1896" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1897" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow class="MJX-TeXAtom-ORD"><mi>ϕ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>β</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-87">{\phi}_{2}(t) = \{ y_{0} + \beta t \}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-88-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><mi>&#x03D5;</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>&#x03D5;</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>&#x03D5;</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-1898" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1899" class="mjx-mrow"><span id="MJXc-Node-1900" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1901" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1902" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1903" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1904" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1905" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1906" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span><span id="MJXc-Node-1907" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span><span id="MJXc-Node-1908" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1909" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1910" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1911" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1912" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1913" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1914" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1915" class="mjx-texatom" style=""><span id="MJXc-Node-1916" class="mjx-mrow"><span id="MJXc-Node-1917" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span><span id="MJXc-Node-1918" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1919" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1920" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1921" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1922" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-1923" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1924" class="mjx-texatom" style=""><span id="MJXc-Node-1925" class="mjx-mrow"><span id="MJXc-Node-1926" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span></span></span></span><span id="MJXc-Node-1927" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1928" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1929" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1930" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>ϕ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>ϕ</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>ϕ</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-88">\ \ \ \ \ \ \ \phi(t) = ( \phi_{1}(t),\phi_{2}(t) )</script>
<p><strong><em>Teorema 3</em></strong></p>
<p><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-89-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03D5;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1931" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1932" class="mjx-mrow"><span id="MJXc-Node-1933" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></span></span><script type="math/tex" id="MathJax-Element-89">\phi</script> è periodica <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-90-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">&#x21D4;</mo><mfrac><mi>&#x03B1;</mi><mi>&#x03B2;</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">Q</mi></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1934" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1935" class="mjx-mrow"><span id="MJXc-Node-1936" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.224em; padding-bottom: 0.39em;">⇔</span></span><span id="MJXc-Node-1937" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.594em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.84em; top: -1.095em;"><span id="MJXc-Node-1938" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.84em; bottom: -0.833em;"><span id="MJXc-Node-1939" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.594em;"></span></span><span class="mjx-vsize" style="height: 1.363em; vertical-align: -0.589em;"></span></span><span id="MJXc-Node-1940" class="mjx-texatom"><span id="MJXc-Node-1941" class="mjx-mrow"><span id="MJXc-Node-1942" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-1943" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.501em;">Q</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">⇔</mo><mfrac><mi>α</mi><mi>β</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-90">\Leftrightarrow \frac{\alpha}{\beta}\mathbb{\in Q}</script>.</p>
<p>DIM:
Supponiamo <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-91-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03D5;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1944" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1945" class="mjx-mrow"><span id="MJXc-Node-1946" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></span></span><script type="math/tex" id="MathJax-Element-91">\phi</script> sia periodica con periodo <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-92-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>&gt;</mo><mn>0</mn></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1947" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1948" class="mjx-mrow"><span id="MJXc-Node-1949" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1950" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">></span></span><span id="MJXc-Node-1951" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-92">t>0</script>, allora:</p>
<p>Se <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-93-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03D5;</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1952" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1953" class="mjx-mrow"><span id="MJXc-Node-1954" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span><span id="MJXc-Node-1955" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1956" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1957" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1958" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1959" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1960" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1961" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1962" class="mjx-texatom" style=""><span id="MJXc-Node-1963" class="mjx-mrow"><span id="MJXc-Node-1964" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1965" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-1966" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-1967" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1968" class="mjx-texatom" style=""><span id="MJXc-Node-1969" class="mjx-mrow"><span id="MJXc-Node-1970" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-1971" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-93">\phi(t) = ( x_{0},y_{0} )</script>, cioè <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-94-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi>&#x03D5;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1972" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1973" class="mjx-mrow"><span id="MJXc-Node-1974" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1975" class="mjx-texatom"><span id="MJXc-Node-1976" class="mjx-mrow"><span id="MJXc-Node-1977" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.375em; padding-right: 0.071em;"><span id="MJXc-Node-1978" class="mjx-texatom" style=""><span id="MJXc-Node-1979" class="mjx-mrow"><span id="MJXc-Node-1980" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span><span id="MJXc-Node-1981" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-1982" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-1983" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-1984" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-1985" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-1986" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1987" class="mjx-texatom" style=""><span id="MJXc-Node-1988" class="mjx-mrow"><span id="MJXc-Node-1989" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi>ϕ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-94">{\phi}_{1}(t) = x_{0}</script> e <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-95-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi>&#x03D5;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-1990" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1991" class="mjx-mrow"><span id="MJXc-Node-1992" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1993" class="mjx-texatom"><span id="MJXc-Node-1994" class="mjx-mrow"><span id="MJXc-Node-1995" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.375em; padding-right: 0.071em;"><span id="MJXc-Node-1996" class="mjx-texatom" style=""><span id="MJXc-Node-1997" class="mjx-mrow"><span id="MJXc-Node-1998" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span></span></span></span><span id="MJXc-Node-1999" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2000" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2001" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2002" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2003" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2004" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2005" class="mjx-texatom" style=""><span id="MJXc-Node-2006" class="mjx-mrow"><span id="MJXc-Node-2007" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi>ϕ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-95">{ \phi}_{2}(t) = y_{0}</script>.</p>
<p>Dalla prima si ha che:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-96-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B1;</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2008" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2009" class="mjx-mrow"><span id="MJXc-Node-2010" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-2011" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2012" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2013" class="mjx-texatom" style=""><span id="MJXc-Node-2014" class="mjx-mrow"><span id="MJXc-Node-2015" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2016" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2017" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-2018" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2019" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-2020" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2021" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-2022" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2023" class="mjx-texatom" style=""><span id="MJXc-Node-2024" class="mjx-mrow"><span id="MJXc-Node-2025" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>α</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-96">\{ x_{0} + \alpha t \} = x_{0}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-97-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B1;</mi><mi>t</mi><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B1;</mi><mi>t</mi><mo fence="false" stretchy="false">]</mo><mo>=</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2026" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2027" class="mjx-mrow"><span id="MJXc-Node-2028" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2029" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2030" class="mjx-texatom" style=""><span id="MJXc-Node-2031" class="mjx-mrow"><span id="MJXc-Node-2032" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2033" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2034" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-2035" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2036" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-2037" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-2038" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2039" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2040" class="mjx-texatom" style=""><span id="MJXc-Node-2041" class="mjx-mrow"><span id="MJXc-Node-2042" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2043" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2044" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-2045" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2046" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span><span id="MJXc-Node-2047" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2048" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-2049" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2050" class="mjx-texatom" style=""><span id="MJXc-Node-2051" class="mjx-mrow"><span id="MJXc-Node-2052" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>α</mi><mi>t</mi><mo>−</mo><mo fence="false" stretchy="false">[</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>α</mi><mi>t</mi><mo fence="false" stretchy="false">]</mo><mo>=</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-97">x_{0} + \alpha t - \lbrack x_{0} + \alpha t \rbrack = x_{0}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-98-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>t</mi><mo>=</mo><mfrac><mrow><mo fence="false" stretchy="false">[</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B1;</mi><mi>t</mi><mo fence="false" stretchy="false">]</mo></mrow><mi>&#x03B1;</mi></mfrac></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2053" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2054" class="mjx-mrow"><span id="MJXc-Node-2055" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2056" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2057" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.94em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.94em; top: -1.533em;"><span id="MJXc-Node-2058" class="mjx-mrow"><span id="MJXc-Node-2059" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-2060" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2061" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2062" class="mjx-texatom" style=""><span id="MJXc-Node-2063" class="mjx-mrow"><span id="MJXc-Node-2064" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2065" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2066" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-2067" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2068" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span></span></span><span class="mjx-denominator" style="width: 3.94em; bottom: -0.722em;"><span id="MJXc-Node-2069" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 3.94em;"></span></span><span class="mjx-vsize" style="height: 2.255em; vertical-align: -0.722em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>t</mi><mo>=</mo><mfrac><mrow><mo fence="false" stretchy="false">[</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>α</mi><mi>t</mi><mo fence="false" stretchy="false">]</mo></mrow><mi>α</mi></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-98">t = \frac{\lbrack x_{0} + \alpha t \rbrack}{\alpha}</script>
<p>Dalla seconda si ha che:
<span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-99-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B2;</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2070" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2071" class="mjx-mrow"><span id="MJXc-Node-2072" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-2073" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2074" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2075" class="mjx-texatom" style=""><span id="MJXc-Node-2076" class="mjx-mrow"><span id="MJXc-Node-2077" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2078" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2079" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-2080" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2081" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-2082" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2083" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2084" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2085" class="mjx-texatom" style=""><span id="MJXc-Node-2086" class="mjx-mrow"><span id="MJXc-Node-2087" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>β</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-99">\{ y_{0} + \beta t \} = y_{0}</script></p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-100-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B2;</mi><mi>t</mi><mo>&#x2212;</mo><mo fence="false" stretchy="false">[</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B2;</mi><mi>t</mi><mo fence="false" stretchy="false">]</mo><mo>=</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2088" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2089" class="mjx-mrow"><span id="MJXc-Node-2090" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2091" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2092" class="mjx-texatom" style=""><span id="MJXc-Node-2093" class="mjx-mrow"><span id="MJXc-Node-2094" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2095" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2096" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-2097" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2098" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">−</span></span><span id="MJXc-Node-2099" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-2100" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2101" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2102" class="mjx-texatom" style=""><span id="MJXc-Node-2103" class="mjx-mrow"><span id="MJXc-Node-2104" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2105" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2106" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-2107" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2108" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span><span id="MJXc-Node-2109" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2110" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2111" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2112" class="mjx-texatom" style=""><span id="MJXc-Node-2113" class="mjx-mrow"><span id="MJXc-Node-2114" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>β</mi><mi>t</mi><mo>−</mo><mo fence="false" stretchy="false">[</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>β</mi><mi>t</mi><mo fence="false" stretchy="false">]</mo><mo>=</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-100">y_{0} + \beta t - \lbrack y_{0} + \beta t \rbrack = y_{0}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-101-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>t</mi><mo>=</mo><mfrac><mrow><mo fence="false" stretchy="false">[</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B2;</mi><mi>t</mi><mo fence="false" stretchy="false">]</mo></mrow><mi>&#x03B2;</mi></mfrac></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2115" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2116" class="mjx-mrow"><span id="MJXc-Node-2117" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2118" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2119" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.791em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.791em; top: -1.533em;"><span id="MJXc-Node-2120" class="mjx-mrow"><span id="MJXc-Node-2121" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-2122" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2123" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2124" class="mjx-texatom" style=""><span id="MJXc-Node-2125" class="mjx-mrow"><span id="MJXc-Node-2126" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2127" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2128" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-2129" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2130" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">]</span></span></span></span><span class="mjx-denominator" style="width: 3.791em; bottom: -0.932em;"><span id="MJXc-Node-2131" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 3.791em;"></span></span><span class="mjx-vsize" style="height: 2.465em; vertical-align: -0.932em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>t</mi><mo>=</mo><mfrac><mrow><mo fence="false" stretchy="false">[</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>β</mi><mi>t</mi><mo fence="false" stretchy="false">]</mo></mrow><mi>β</mi></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-101">t = \frac{\lbrack y_{0} + \beta t \rbrack}{\beta}</script>
<p>Essendo che la <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-102-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2132" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2133" class="mjx-mrow"><span id="MJXc-Node-2134" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></span></span><script type="math/tex" id="MathJax-Element-102">t</script> è la stessa in entrambi i casi, si possono eguagliare le equazioni:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-103-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>t</mi><mo>=</mo><mfrac><mi>n</mi><mi>&#x03B1;</mi></mfrac><mo>=</mo><mfrac><mi>m</mi><mi>&#x03B2;</mi></mfrac></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2135" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2136" class="mjx-mrow"><span id="MJXc-Node-2137" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2138" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2139" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.84em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.84em; top: -1.144em;"><span id="MJXc-Node-2140" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-denominator" style="width: 0.84em; bottom: -0.722em;"><span id="MJXc-Node-2141" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.84em;"></span></span><span class="mjx-vsize" style="height: 1.865em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-2142" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2143" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.078em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.078em; top: -1.144em;"><span id="MJXc-Node-2144" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-denominator" style="width: 1.078em; bottom: -0.932em;"><span id="MJXc-Node-2145" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 1.078em;"></span></span><span class="mjx-vsize" style="height: 2.075em; vertical-align: -0.932em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>t</mi><mo>=</mo><mfrac><mi>n</mi><mi>α</mi></mfrac><mo>=</mo><mfrac><mi>m</mi><mi>β</mi></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-103">t = \frac{n}{\alpha} = \frac{m}{\beta}</script>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-104-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>&#x03B1;</mi><mi>&#x03B2;</mi></mfrac><mo>=</mo><mfrac><mi>n</mi><mi>m</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="double-struck">&#xA0;</mtext><mo>&#x2208;</mo><mi mathvariant="double-struck">Q</mi></mrow><mo>.</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2146" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2147" class="mjx-mrow"><span id="MJXc-Node-2148" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.84em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.84em; top: -1.144em;"><span id="MJXc-Node-2149" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-denominator" style="width: 0.84em; bottom: -0.932em;"><span id="MJXc-Node-2150" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.84em;"></span></span><span class="mjx-vsize" style="height: 2.075em; vertical-align: -0.932em;"></span></span><span id="MJXc-Node-2151" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2152" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.078em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.078em; top: -1.144em;"><span id="MJXc-Node-2153" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-denominator" style="width: 1.078em; bottom: -0.722em;"><span id="MJXc-Node-2154" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 1.078em;"></span></span><span class="mjx-vsize" style="height: 1.865em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-2155" class="mjx-texatom"><span id="MJXc-Node-2156" class="mjx-mrow"><span id="MJXc-Node-2157" class="mjx-mtext"><span class="mjx-char MJXc-TeX-ams-R" style="margin-top: -0.218em; padding-bottom: 0.279em;"> </span></span><span id="MJXc-Node-2158" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-2159" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.501em;">Q</span></span></span></span><span id="MJXc-Node-2160" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>α</mi><mi>β</mi></mfrac><mo>=</mo><mfrac><mi>n</mi><mi>m</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="double-struck"> </mtext><mo>∈</mo><mi mathvariant="double-struck">Q</mi></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-104">\frac{\alpha}{\beta} = \frac{n}{m}\mathbb{\ \in Q}.</script>
<p>D’altra parte, se <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-105-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>&#x03B1;</mi><mi>&#x03B2;</mi></mfrac><mo>=</mo><mfrac><mi>n</mi><mi>m</mi></mfrac></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2161" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2162" class="mjx-mrow"><span id="MJXc-Node-2163" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.594em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.84em; top: -1.095em;"><span id="MJXc-Node-2164" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.84em; bottom: -0.833em;"><span id="MJXc-Node-2165" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.594em;"></span></span><span class="mjx-vsize" style="height: 1.363em; vertical-align: -0.589em;"></span></span><span id="MJXc-Node-2166" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2167" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.762em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.078em; top: -1.095em;"><span id="MJXc-Node-2168" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.078em; bottom: -0.524em;"><span id="MJXc-Node-2169" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.762em;"></span></span><span class="mjx-vsize" style="height: 1.144em; vertical-align: -0.37em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>α</mi><mi>β</mi></mfrac><mo>=</mo><mfrac><mi>n</mi><mi>m</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-105">\frac{\alpha}{\beta} = \frac{n}{m}</script> con <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-106-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>,</mo><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">N</mi><mo mathvariant="double-struck">,</mo><mi mathvariant="double-struck">n</mi><mo>&#x2260;</mo><mn mathvariant="double-struck">0</mn></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2170" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2171" class="mjx-mrow"><span id="MJXc-Node-2172" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-2173" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2174" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span><span id="MJXc-Node-2175" class="mjx-texatom"><span id="MJXc-Node-2176" class="mjx-mrow"><span id="MJXc-Node-2177" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-2178" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.335em;">N</span></span><span id="MJXc-Node-2179" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2180" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">n</span></span><span id="MJXc-Node-2181" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.556em;">≠</span></span><span id="MJXc-Node-2182" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>,</mo><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo mathvariant="double-struck">,</mo><mi mathvariant="double-struck">n</mi><mo>≠</mo><mn mathvariant="double-struck">0</mn></mrow></math></span></span><script type="math/tex" id="MathJax-Element-106">n,m\mathbb{\in N, n \neq 0 }</script> si ha:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-107-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>&#x03D5;</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo><mo>,</mo><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mrow><mi>n</mi><mi>&#x03B2;</mi></mrow><mi>&#x03B1;</mi></mfrac><mo fence="false" stretchy="false">}</mo><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2183" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2184" class="mjx-mrow"><span id="MJXc-Node-2185" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span><span id="MJXc-Node-2186" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2187" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2188" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2189" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2190" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2191" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-2192" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2193" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2194" class="mjx-texatom" style=""><span id="MJXc-Node-2195" class="mjx-mrow"><span id="MJXc-Node-2196" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2197" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2198" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-2199" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-2200" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2201" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-2202" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2203" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2204" class="mjx-texatom" style=""><span id="MJXc-Node-2205" class="mjx-mrow"><span id="MJXc-Node-2206" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2207" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2208" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.373em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.373em; top: -1.432em;"><span id="MJXc-Node-2209" class="mjx-mrow"><span id="MJXc-Node-2210" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-2211" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span></span><span class="mjx-denominator" style="width: 1.373em; bottom: -0.722em;"><span id="MJXc-Node-2212" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 1.373em;"></span></span><span class="mjx-vsize" style="height: 2.154em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-2213" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-2214" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>ϕ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo><mo>,</mo><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mrow><mi>n</mi><mi>β</mi></mrow><mi>α</mi></mfrac><mo fence="false" stretchy="false">}</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-107">\phi(t) = ( \{ x_{0} + n \},\{ y_{0} + \frac{n\beta}{\alpha} \} )</script>
<p><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-108-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo><mo>,</mo><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>m</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2215" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2216" class="mjx-mrow"><span id="MJXc-Node-2217" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2218" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2219" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-2220" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2221" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2222" class="mjx-texatom" style=""><span id="MJXc-Node-2223" class="mjx-mrow"><span id="MJXc-Node-2224" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2225" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2226" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-2227" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-2228" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2229" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-2230" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2231" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2232" class="mjx-texatom" style=""><span id="MJXc-Node-2233" class="mjx-mrow"><span id="MJXc-Node-2234" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2235" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2236" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span><span id="MJXc-Node-2237" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-2238" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2239" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2240" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2241" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2242" class="mjx-texatom" style=""><span id="MJXc-Node-2243" class="mjx-mrow"><span id="MJXc-Node-2244" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2245" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2246" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2247" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2248" class="mjx-texatom" style=""><span id="MJXc-Node-2249" class="mjx-mrow"><span id="MJXc-Node-2250" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2251" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo><mo>,</mo><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>m</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-108">= ( \{ x_{0} + n \},\{ y_{0} + m\} = ( x_{0},y_{0} )</script>
per ogni <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-109-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>,</mo><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">Z</mi><mo>&#x2216;</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow><mo>.</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2252" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2253" class="mjx-mrow"><span id="MJXc-Node-2254" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-2255" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2256" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">m</span></span><span id="MJXc-Node-2257" class="mjx-texatom"><span id="MJXc-Node-2258" class="mjx-mrow"><span id="MJXc-Node-2259" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-2260" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.279em;">Z</span></span><span id="MJXc-Node-2261" class="mjx-mo"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∖</span></span></span></span><span id="MJXc-Node-2262" class="mjx-texatom"><span id="MJXc-Node-2263" class="mjx-mrow"><span id="MJXc-Node-2264" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span><span id="MJXc-Node-2265" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>,</mo><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">Z</mi><mo>∖</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-109">n,m\mathbb{\in Z \smallsetminus}{ 0 }.</script></p>
<p><strong><em>Teorema 4</em></strong></p>
<p>L’immagine di <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-110-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03D5;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2266" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2267" class="mjx-mrow"><span id="MJXc-Node-2268" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></span></span><script type="math/tex" id="MathJax-Element-110">\phi</script> riempie densamente <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-111-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo stretchy="false">&#x21D4;</mo><mfrac><mi>&#x03B1;</mi><mi>&#x03B2;</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">R</mi><mo>&#x2216;</mo><mi mathvariant="double-struck">Q</mi></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2269" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2270" class="mjx-mrow"><span id="MJXc-Node-2271" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-2272" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-2273" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2274" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-2275" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2276" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2277" class="mjx-texatom" style=""><span id="MJXc-Node-2278" class="mjx-mrow"><span id="MJXc-Node-2279" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span></span></span></span><span id="MJXc-Node-2280" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.224em; padding-bottom: 0.39em;">⇔</span></span><span id="MJXc-Node-2281" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.594em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.84em; top: -1.095em;"><span id="MJXc-Node-2282" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.84em; bottom: -0.833em;"><span id="MJXc-Node-2283" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.594em;"></span></span><span class="mjx-vsize" style="height: 1.363em; vertical-align: -0.589em;"></span></span><span id="MJXc-Node-2284" class="mjx-texatom"><span id="MJXc-Node-2285" class="mjx-mrow"><span id="MJXc-Node-2286" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-2287" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.279em;">R</span></span><span id="MJXc-Node-2288" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∖</span></span><span id="MJXc-Node-2289" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.501em;">Q</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo stretchy="false">⇔</mo><mfrac><mi>α</mi><mi>β</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>∖</mo><mi mathvariant="double-struck">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-111">\lbrack 0,1)^{2} \Leftrightarrow \frac{\alpha}{\beta}\mathbb{\in R \smallsetminus Q}</script>.</p>
<p>DIM:
Sia <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-112-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>&#x03B1;</mi><mi>&#x03B2;</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">R</mi><mo>&#x2216;</mo><mi mathvariant="double-struck">Q</mi></mrow><mo>.</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2290" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2291" class="mjx-mrow"><span id="MJXc-Node-2292" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.594em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.84em; top: -1.095em;"><span id="MJXc-Node-2293" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.84em; bottom: -0.833em;"><span id="MJXc-Node-2294" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.594em;"></span></span><span class="mjx-vsize" style="height: 1.363em; vertical-align: -0.589em;"></span></span><span id="MJXc-Node-2295" class="mjx-texatom"><span id="MJXc-Node-2296" class="mjx-mrow"><span id="MJXc-Node-2297" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-2298" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.279em;">R</span></span><span id="MJXc-Node-2299" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∖</span></span><span id="MJXc-Node-2300" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.501em;">Q</span></span></span></span><span id="MJXc-Node-2301" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>α</mi><mi>β</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>∖</mo><mi mathvariant="double-struck">Q</mi></mrow><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-112">\frac{\alpha}{\beta}\mathbb{\in R \smallsetminus Q}.</script>
Sia <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-113-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>&#x2208;</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2302" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2303" class="mjx-mrow"><span id="MJXc-Node-2304" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2305" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2306" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2307" class="mjx-texatom" style=""><span id="MJXc-Node-2308" class="mjx-mrow"><span id="MJXc-Node-2309" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span><span id="MJXc-Node-2310" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2311" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2312" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2313" class="mjx-texatom" style=""><span id="MJXc-Node-2314" class="mjx-mrow"><span id="MJXc-Node-2315" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span><span id="MJXc-Node-2316" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2317" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-2318" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-2319" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-2320" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2321" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-2322" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2323" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2324" class="mjx-texatom" style=""><span id="MJXc-Node-2325" class="mjx-mrow"><span id="MJXc-Node-2326" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>∈</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-113">( x_{1},y_{1} ) \in \lbrack 0,1)^{2}</script>, essendo <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-114-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B2;</mi><mo>&gt;</mo><mn>0</mn></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2327" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2328" class="mjx-mrow"><span id="MJXc-Node-2329" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-2330" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">></span></span><span id="MJXc-Node-2331" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>></mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-114">\beta > 0</script> si trova un tempo <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-115-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2332" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2333" class="mjx-mrow"><span id="MJXc-Node-2334" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2335" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2336" class="mjx-texatom" style=""><span id="MJXc-Node-2337" class="mjx-mrow"><span id="MJXc-Node-2338" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-115">t_{0}</script> tale che
<span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-116-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi>&#x03D5;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>.</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2339" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2340" class="mjx-mrow"><span id="MJXc-Node-2341" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2342" class="mjx-texatom"><span id="MJXc-Node-2343" class="mjx-mrow"><span id="MJXc-Node-2344" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.375em; padding-right: 0.071em;"><span id="MJXc-Node-2345" class="mjx-texatom" style=""><span id="MJXc-Node-2346" class="mjx-mrow"><span id="MJXc-Node-2347" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span></span></span></span><span id="MJXc-Node-2348" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2349" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2350" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2351" class="mjx-texatom" style=""><span id="MJXc-Node-2352" class="mjx-mrow"><span id="MJXc-Node-2353" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2354" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2355" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2356" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2357" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2358" class="mjx-texatom" style=""><span id="MJXc-Node-2359" class="mjx-mrow"><span id="MJXc-Node-2360" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span><span id="MJXc-Node-2361" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi>ϕ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-116">{\phi}_{2}( t_{0} ) = y_{1}.</script></p>
<p>Allora si avrà che</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-117-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>&#x03D5;</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mi>n</mi><mi>&#x03B2;</mi></mfrac><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B2;</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B2;</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo fence="false" stretchy="false">}</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2362" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2363" class="mjx-mrow"><span id="MJXc-Node-2364" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2365" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2366" class="mjx-texatom" style=""><span id="MJXc-Node-2367" class="mjx-mrow"><span id="MJXc-Node-2368" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span></span></span></span><span id="MJXc-Node-2369" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2370" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2371" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2372" class="mjx-texatom" style=""><span id="MJXc-Node-2373" class="mjx-mrow"><span id="MJXc-Node-2374" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2375" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2376" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.8em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.8em; top: -1.144em;"><span id="MJXc-Node-2377" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-denominator" style="width: 0.8em; bottom: -0.932em;"><span id="MJXc-Node-2378" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.8em;"></span></span><span class="mjx-vsize" style="height: 2.075em; vertical-align: -0.932em;"></span></span><span id="MJXc-Node-2379" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2380" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2381" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-2382" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2383" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2384" class="mjx-texatom" style=""><span id="MJXc-Node-2385" class="mjx-mrow"><span id="MJXc-Node-2386" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2387" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2388" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-2389" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2390" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2391" class="mjx-texatom" style=""><span id="MJXc-Node-2392" class="mjx-mrow"><span id="MJXc-Node-2393" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2394" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2395" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-2396" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-2397" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2398" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-2399" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2400" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2401" class="mjx-texatom" style=""><span id="MJXc-Node-2402" class="mjx-mrow"><span id="MJXc-Node-2403" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2404" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2405" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-2406" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2407" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2408" class="mjx-texatom" style=""><span id="MJXc-Node-2409" class="mjx-mrow"><span id="MJXc-Node-2410" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2411" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>ϕ</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mi>n</mi><mi>β</mi></mfrac><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>β</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>β</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-117">\phi_{2}( t_{0} + \frac{n}{\beta} ) = \{ y_{0} + \beta t_{0} + n \} = \{ y_{0} + \beta t_{0} \}</script>
<p><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-118-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><msub><mi>&#x03D5;</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2412" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2413" class="mjx-mrow"><span id="MJXc-Node-2414" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2415" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-2416" class="mjx-texatom"><span id="MJXc-Node-2417" class="mjx-mrow"><span id="MJXc-Node-2418" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2419" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2420" class="mjx-texatom" style=""><span id="MJXc-Node-2421" class="mjx-mrow"><span id="MJXc-Node-2422" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span></span></span></span><span id="MJXc-Node-2423" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2424" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2425" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2426" class="mjx-texatom" style=""><span id="MJXc-Node-2427" class="mjx-mrow"><span id="MJXc-Node-2428" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2429" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2430" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2431" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.439em; padding-right: 0.071em;"><span id="MJXc-Node-2432" class="mjx-texatom" style=""><span id="MJXc-Node-2433" class="mjx-mrow"><span id="MJXc-Node-2434" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><msub><mi>ϕ</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-118">= {\phi_{2}( t_{0} ) = y}_{1}</script> per ogni <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-119-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">Z</mi></mrow><mo>.</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2435" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2436" class="mjx-mrow"><span id="MJXc-Node-2437" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-2438" class="mjx-texatom"><span id="MJXc-Node-2439" class="mjx-mrow"><span id="MJXc-Node-2440" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-2441" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.279em;">Z</span></span></span></span><span id="MJXc-Node-2442" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">Z</mi></mrow><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-119">n\mathbb{\in Z}.</script></p>
<p>In altre parole, ad intervalli di tempo <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-120-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>&#x03B2;</mi></mfrac></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2443" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2444" class="mjx-mrow"><span id="MJXc-Node-2445" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.547em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.773em; top: -1.308em;"><span id="MJXc-Node-2446" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.773em; bottom: -0.833em;"><span id="MJXc-Node-2447" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.547em;"></span></span><span class="mjx-vsize" style="height: 1.514em; vertical-align: -0.589em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>β</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-120">\frac{1}{\beta}</script> la traiettoria della pallina interseca la retta <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-121-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2448" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2449" class="mjx-mrow"><span id="MJXc-Node-2450" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-2451" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2452" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2453" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2454" class="mjx-texatom" style=""><span id="MJXc-Node-2455" class="mjx-mrow"><span id="MJXc-Node-2456" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-121">y = y_{1}</script>.</p>
<p>Considerando la coordinata <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-122-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03D5;</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2457" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2458" class="mjx-mrow"><span id="MJXc-Node-2459" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2460" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2461" class="mjx-texatom" style=""><span id="MJXc-Node-2462" class="mjx-mrow"><span id="MJXc-Node-2463" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span><span id="MJXc-Node-2464" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2465" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2466" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2467" class="mjx-texatom" style=""><span id="MJXc-Node-2468" class="mjx-mrow"><span id="MJXc-Node-2469" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2470" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-122">\phi_{1}( t_{0} )</script> al tempo <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-123-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mi>&#x03B2;</mi></mfrac></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2471" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2472" class="mjx-mrow"><span id="MJXc-Node-2473" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.566em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.8em; top: -1.095em;"><span id="MJXc-Node-2474" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.8em; bottom: -0.833em;"><span id="MJXc-Node-2475" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.566em;"></span></span><span class="mjx-vsize" style="height: 1.363em; vertical-align: -0.589em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mi>β</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-123">\frac{n}{\beta}</script> , si ha che:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-124-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow class="MJX-TeXAtom-ORD"><mi>&#x03D5;</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mi>n</mi><mi>&#x03B2;</mi></mfrac><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B1;</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x22C5;</mo><mtext>&#xA0;</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>&#x03B2;</mi><mo accent="false">&#x00AF;</mo></mover></mrow><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B1;</mi></mrow></msubsup><mo fence="false" stretchy="false">}</mo><mo>=</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>&#x03B1;</mi><mi>&#x03B2;</mi></mfrac></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B1;</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mo>.</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2476" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2477" class="mjx-mrow"><span id="MJXc-Node-2478" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2479" class="mjx-texatom"><span id="MJXc-Node-2480" class="mjx-mrow"><span id="MJXc-Node-2481" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.375em; padding-right: 0.071em;"><span id="MJXc-Node-2482" class="mjx-texatom" style=""><span id="MJXc-Node-2483" class="mjx-mrow"><span id="MJXc-Node-2484" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span><span id="MJXc-Node-2485" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2486" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2487" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2488" class="mjx-texatom" style=""><span id="MJXc-Node-2489" class="mjx-mrow"><span id="MJXc-Node-2490" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2491" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2492" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.8em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.8em; top: -1.144em;"><span id="MJXc-Node-2493" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-denominator" style="width: 0.8em; bottom: -0.932em;"><span id="MJXc-Node-2494" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.8em;"></span></span><span class="mjx-vsize" style="height: 2.075em; vertical-align: -0.932em;"></span></span><span id="MJXc-Node-2495" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2496" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2497" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-2498" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2499" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2500" class="mjx-texatom" style=""><span id="MJXc-Node-2501" class="mjx-mrow"><span id="MJXc-Node-2502" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2503" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2504" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-2505" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2506" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2507" class="mjx-texatom" style=""><span id="MJXc-Node-2508" class="mjx-mrow"><span id="MJXc-Node-2509" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2510" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2511" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-2512" class="mjx-texatom"><span id="MJXc-Node-2513" class="mjx-mrow"><span id="MJXc-Node-2514" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-2515" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.335em;">⋅</span></span><span id="MJXc-Node-2516" class="mjx-mtext MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.274em; padding-bottom: 0.335em;"> </span></span></span></span></span><span class="mjx-stack" style="vertical-align: -0.603em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2522" class="mjx-texatom" style=""><span id="MJXc-Node-2523" class="mjx-mrow"><span id="MJXc-Node-2524" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-2517" class="mjx-texatom" style=""><span id="MJXc-Node-2518" class="mjx-mrow"><span id="MJXc-Node-2519" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em; padding-left: 0.042em;"><span id="MJXc-Node-2521" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.066em 0px -0.205em; letter-spacing: -0.312em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-2520" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-2525" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span><span id="MJXc-Node-2526" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2527" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-2528" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.388em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-2534" class="mjx-texatom" style=""><span id="MJXc-Node-2535" class="mjx-mrow"><span id="MJXc-Node-2536" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-2529" class="mjx-texatom" style=""><span id="MJXc-Node-2530" class="mjx-mrow"><span id="MJXc-Node-2531" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.594em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.84em; top: -1.123em;"><span id="MJXc-Node-2532" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.84em; bottom: -0.862em;"><span id="MJXc-Node-2533" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.1px solid; top: -0.292em; width: 0.594em;"></span></span><span class="mjx-vsize" style="height: 1.404em; vertical-align: -0.61em;"></span></span></span></span></span></span></span><span id="MJXc-Node-2537" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2538" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2539" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2540" class="mjx-texatom" style=""><span id="MJXc-Node-2541" class="mjx-mrow"><span id="MJXc-Node-2542" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2543" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2544" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-2545" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2546" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2547" class="mjx-texatom" style=""><span id="MJXc-Node-2548" class="mjx-mrow"><span id="MJXc-Node-2549" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2550" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2551" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow class="MJX-TeXAtom-ORD"><mi>ϕ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mi>n</mi><mi>β</mi></mfrac><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>α</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>⋅</mo><mtext> </mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>β</mi><mo accent="false">¯</mo></mover></mrow><mrow class="MJX-TeXAtom-ORD"><mi>α</mi></mrow></msubsup><mo fence="false" stretchy="false">}</mo><mo>=</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>α</mi><mi>β</mi></mfrac></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>α</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-124">{\phi}_{1}( t_{0} + \frac{n}{\beta} ) = \{ x_{0} + \alpha t_{0} + {n \cdot \ }_{\overline{\beta}}^{\alpha} \} = T_{\frac{\alpha}{\beta}}^{n}( x_{0} + \alpha t_{0} ).</script>
<p>Dal <em>Teorema 2</em> sappiamo che <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-125-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>&#x03B1;</mi><mi>&#x03B2;</mi></mfrac></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>&#x03B1;</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2552" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2553" class="mjx-mrow"><span id="MJXc-Node-2554" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">{</span></span><span id="MJXc-Node-2555" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.12em;"><span id="MJXc-Node-2556" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em; padding-right: 0.12em;">T</span></span></span><span class="mjx-stack" style="vertical-align: -0.388em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.291em; padding-right: 0.071em;"><span id="MJXc-Node-2562" class="mjx-texatom" style=""><span id="MJXc-Node-2563" class="mjx-mrow"><span id="MJXc-Node-2564" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-2557" class="mjx-texatom" style=""><span id="MJXc-Node-2558" class="mjx-mrow"><span id="MJXc-Node-2559" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.594em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.84em; top: -1.123em;"><span id="MJXc-Node-2560" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.84em; bottom: -0.862em;"><span id="MJXc-Node-2561" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.1px solid; top: -0.292em; width: 0.594em;"></span></span><span class="mjx-vsize" style="height: 1.404em; vertical-align: -0.61em;"></span></span></span></span></span></span></span><span id="MJXc-Node-2565" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2566" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2567" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2568" class="mjx-texatom" style=""><span id="MJXc-Node-2569" class="mjx-mrow"><span id="MJXc-Node-2570" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2571" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2572" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-2573" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2574" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2575" class="mjx-texatom" style=""><span id="MJXc-Node-2576" class="mjx-mrow"><span id="MJXc-Node-2577" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2578" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2579" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2580" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">}</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2581" class="mjx-texatom" style=""><span id="MJXc-Node-2582" class="mjx-mrow"><span id="MJXc-Node-2583" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msubsup><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>α</mi><mi>β</mi></mfrac></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mi>α</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-125">\{ T_{\frac{\alpha}{\beta}}^{n}( x_{0} + \alpha t_{0} ) \}_{n}</script> è densa in <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-126-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2584" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2585" class="mjx-mrow"><span id="MJXc-Node-2586" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-2587" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-2588" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2589" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-2590" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-126">\lbrack 0,1)</script> se <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-127-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>&#x03B1;</mi><mi>&#x03B2;</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mo>&#x2208;</mo><mi mathvariant="double-struck">R</mi><mo>&#x2216;</mo><mi mathvariant="double-struck">Q</mi></mrow></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2591" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2592" class="mjx-mrow"><span id="MJXc-Node-2593" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.594em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.84em; top: -1.095em;"><span id="MJXc-Node-2594" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.84em; bottom: -0.833em;"><span id="MJXc-Node-2595" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.594em;"></span></span><span class="mjx-vsize" style="height: 1.363em; vertical-align: -0.589em;"></span></span><span id="MJXc-Node-2596" class="mjx-texatom"><span id="MJXc-Node-2597" class="mjx-mrow"><span id="MJXc-Node-2598" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-2599" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.279em;">R</span></span><span id="MJXc-Node-2600" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">∖</span></span><span id="MJXc-Node-2601" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.501em;">Q</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>α</mi><mi>β</mi></mfrac><mrow class="MJX-TeXAtom-ORD"><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>∖</mo><mi mathvariant="double-struck">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-127">\frac{\alpha}{\beta}\mathbb{\in R \smallsetminus Q}</script>.</p>
<p>Se ne conclude che:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-128-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow><mo>&#x220B;</mo><mi>n</mi><mo stretchy="false">&#x21A6;</mo><mi>&#x03D5;</mi><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mi>n</mi><mi>&#x03B1;</mi></mfrac><mo stretchy="false">)</mo><mo>&#x2208;</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>.</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2602" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2603" class="mjx-mrow"><span id="MJXc-Node-2604" class="mjx-texatom"><span id="MJXc-Node-2605" class="mjx-mrow"><span id="MJXc-Node-2606" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.445em; padding-bottom: 0.335em;">N</span></span></span></span><span id="MJXc-Node-2607" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∋</span></span><span id="MJXc-Node-2608" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span><span id="MJXc-Node-2609" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.224em; padding-bottom: 0.335em;">↦</span></span><span id="MJXc-Node-2610" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span><span id="MJXc-Node-2611" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2612" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2613" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2614" class="mjx-texatom" style=""><span id="MJXc-Node-2615" class="mjx-mrow"><span id="MJXc-Node-2616" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span></span></span></span></span><span id="MJXc-Node-2617" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.445em;">+</span></span><span id="MJXc-Node-2618" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.84em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.84em; top: -1.144em;"><span id="MJXc-Node-2619" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">n</span></span></span><span class="mjx-denominator" style="width: 0.84em; bottom: -0.722em;"><span id="MJXc-Node-2620" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.84em;"></span></span><span class="mjx-vsize" style="height: 1.865em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-2621" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2622" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-2623" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-2624" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-2625" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2626" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-2627" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2628" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2629" class="mjx-texatom" style=""><span id="MJXc-Node-2630" class="mjx-mrow"><span id="MJXc-Node-2631" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">2</span></span></span></span></span></span><span id="MJXc-Node-2632" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow><mo>∋</mo><mi>n</mi><mo stretchy="false">↦</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mi>n</mi><mi>α</mi></mfrac><mo stretchy="false">)</mo><mo>∈</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-128">\mathbb{N} \ni n \mapsto \phi( t_{0} + \frac{n}{\alpha} ) \in \lbrack 0,1)^{2}.</script>
<p>Riempie densamente il segmento <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-129-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2633" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2634" class="mjx-mrow"><span id="MJXc-Node-2635" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2636" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-2637" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2638" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2639" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2640" class="mjx-texatom" style=""><span id="MJXc-Node-2641" class="mjx-mrow"><span id="MJXc-Node-2642" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span><span id="MJXc-Node-2643" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-129">( x,y_{1} )</script> con
<span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-130-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&#x2208;</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>.</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2644" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2645" class="mjx-mrow"><span id="MJXc-Node-2646" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-2647" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.39em;">∈</span></span><span id="MJXc-Node-2648" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">[</span></span><span id="MJXc-Node-2649" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.39em;">0</span></span><span id="MJXc-Node-2650" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2651" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span><span id="MJXc-Node-2652" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2653" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mo fence="false" stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-130">x \in \lbrack 0,1).</script></p>
<p>Ma allora <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-131-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">&#x21A6;</mo><mi>&#x03D5;</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2654" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2655" class="mjx-mrow"><span id="MJXc-Node-2656" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2657" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.224em; padding-bottom: 0.335em;">↦</span></span><span id="MJXc-Node-2658" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.501em;">ϕ</span></span><span id="MJXc-Node-2659" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2660" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.39em; padding-bottom: 0.279em;">t</span></span><span id="MJXc-Node-2661" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">↦</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-131">t \mapsto \phi(t)</script> si avvicina quanto si vuole anche a <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-132-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2662" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2663" class="mjx-mrow"><span id="MJXc-Node-2664" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2665" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2666" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2667" class="mjx-texatom" style=""><span id="MJXc-Node-2668" class="mjx-mrow"><span id="MJXc-Node-2669" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span><span id="MJXc-Node-2670" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2671" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2672" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2673" class="mjx-texatom" style=""><span id="MJXc-Node-2674" class="mjx-mrow"><span id="MJXc-Node-2675" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span><span id="MJXc-Node-2676" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-132">( x_{1},y_{1} )</script>.</p>
<p><strong>Relazione tra Biliardo e Intervallo</strong></p>
<p>Sulla base dei precedenti teoremi è possibile ora mettere in relazione <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-133-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>&#x03B1;</mi><mi>&#x03B2;</mi></mfrac></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2677" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2678" class="mjx-mrow"><span id="MJXc-Node-2679" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.594em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.84em; top: -1.095em;"><span id="MJXc-Node-2680" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.84em; bottom: -0.833em;"><span id="MJXc-Node-2681" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.594em;"></span></span><span class="mjx-vsize" style="height: 1.363em; vertical-align: -0.589em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>α</mi><mi>β</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-133">\frac{\alpha}{\beta}</script> del caso tridimensionale, con il passo <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-134-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2682" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2683" class="mjx-mrow"><span id="MJXc-Node-2684" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-134">\theta</script> del caso unidimensionale.</p>
<p>Si ricorda che la traiettoria della pallina viene definita impostando una determinata pendenza, e che se non ci fosse alcuna parete si avrebbe una retta. Le limitazioni del calcolatore trattano tale retta come un
grande segmento, in particolare tale segmento rappresenta l’ipotenusa di un triangolo rettangolo. Il passo <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-135-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2685" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2686" class="mjx-mrow"><span id="MJXc-Node-2687" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-135">\theta</script> del caso unidimensionale può
essere calcolato considerando la proiezione dell’ipotenusa di lunghezza <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-136-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2688" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2689" class="mjx-mrow"><span id="MJXc-Node-2690" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em;">l</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math></span></span><script type="math/tex" id="MathJax-Element-136">l</script> sull’asse delle ascisse.</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-137-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>&#x03B8;</mi><mo>=</mo><mi>l</mi><mi>cos</mi><mo>&#x2061;</mo><mi>&#x03B3;</mi></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2691" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2692" class="mjx-mrow"><span id="MJXc-Node-2693" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span><span id="MJXc-Node-2694" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2695" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.445em; padding-bottom: 0.279em;">l</span></span><span id="MJXc-Node-2696" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.169em; padding-bottom: 0.335em;">cos</span></span><span id="MJXc-Node-2697" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2698" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.025em;">γ</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>θ</mi><mo>=</mo><mi>l</mi><mi>cos</mi><mo></mo><mi>γ</mi></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-137">\theta = l\cos\gamma</script>
<p>Dove <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-138-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B3;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2699" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2700" class="mjx-mrow"><span id="MJXc-Node-2701" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.025em;">γ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math></span></span><script type="math/tex" id="MathJax-Element-138">\gamma</script> è l’angolo che la retta forma con l’asse delle ascisse.</p>
<p>Per quanto riguarda la pendenza, si ricorda che:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-139-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mover><mi>v</mi><mo>&#x2192;</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo>,</mo><mi>&#x03B2;</mi><mo stretchy="false">)</mo><mo>.</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2702" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2703" class="mjx-mrow"><span id="MJXc-Node-2704" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="padding-bottom: 0.06em;"><span id="MJXc-Node-2706" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.224em; padding-bottom: 0.335em;">→</span></span></span><span class="mjx-op" style="padding-left: 0.23em;"><span id="MJXc-Node-2705" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">v</span></span></span></span></span><span id="MJXc-Node-2707" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2708" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2709" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span><span id="MJXc-Node-2710" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.556em;">,</span></span><span id="MJXc-Node-2711" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-2712" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2713" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mover><mi>v</mi><mo>→</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-139">\overrightarrow{v} = (\alpha,\beta).</script>
<p>Per cui se si considera l’equazione della traiettoria si deduce che:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-140-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>f</mi><mrow class="MJX-TeXAtom-ORD"><msup><mi></mi><mo>&#x2032;</mo></msup></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>&#x03B1;</mi><mi>&#x03B2;</mi></mfrac><mo>.</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2714" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2715" class="mjx-mrow"><span id="MJXc-Node-2716" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.06em;"><span id="MJXc-Node-2717" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.501em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0.181em; padding-right: 0.071em;"><span id="MJXc-Node-2718" class="mjx-texatom" style=""><span id="MJXc-Node-2719" class="mjx-mrow"><span id="MJXc-Node-2720" class="mjx-msup"><span class="mjx-base"><span id="MJXc-Node-2721" class="mjx-mi"></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2722" class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.279em; padding-bottom: 0.335em;">′</span></span></span></span></span></span></span></span><span id="MJXc-Node-2723" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">(</span></span><span id="MJXc-Node-2724" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">x</span></span><span id="MJXc-Node-2725" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.611em;">)</span></span><span id="MJXc-Node-2726" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2727" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.84em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.84em; top: -1.144em;"><span id="MJXc-Node-2728" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-denominator" style="width: 0.84em; bottom: -0.932em;"><span id="MJXc-Node-2729" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.84em;"></span></span><span class="mjx-vsize" style="height: 2.075em; vertical-align: -0.932em;"></span></span><span id="MJXc-Node-2730" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>f</mi><mrow class="MJX-TeXAtom-ORD"><msup><mi></mi><mo>′</mo></msup></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>α</mi><mi>β</mi></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-140">f^{'}(x) = \frac{\alpha}{\beta}.</script>
<p>Un’ulteriore conferma di ciò si può ricavare dal <em>Teorema 4</em>: ad intervalli di tempo <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-141-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>&#x03B2;</mi></mfrac></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2731" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2732" class="mjx-mrow"><span id="MJXc-Node-2733" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.547em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.773em; top: -1.308em;"><span id="MJXc-Node-2734" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.773em; bottom: -0.833em;"><span id="MJXc-Node-2735" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.547em;"></span></span><span class="mjx-vsize" style="height: 1.514em; vertical-align: -0.589em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>β</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-141">\frac{1}{\beta}</script> la traiettoria della pallina interseca la retta <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-142-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2736" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2737" class="mjx-mrow"><span id="MJXc-Node-2738" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-2739" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2740" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.006em;"><span id="MJXc-Node-2741" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.501em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2742" class="mjx-texatom" style=""><span id="MJXc-Node-2743" class="mjx-mrow"><span id="MJXc-Node-2744" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-142">y = y_{1}</script>. Dunque si può ricavare <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-143-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B2;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2745" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2746" class="mjx-mrow"><span id="MJXc-Node-2747" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math></span></span><script type="math/tex" id="MathJax-Element-143">\beta</script> come:</p>
<span class="MathJax_Preview" style="color: inherit;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-144-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>&#x03B2;</mi><mo>=</mo><mfrac><mn>1</mn><mi>&#x03B8;</mi></mfrac><mo>.</mo></math>" role="presentation" style="font-size: 113%; text-align: center; position: relative;"><span id="MJXc-Node-2748" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2749" class="mjx-mrow"><span id="MJXc-Node-2750" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span><span id="MJXc-Node-2751" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2752" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-2753" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.39em; padding-bottom: 0.335em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.748em;"><span id="MJXc-Node-2754" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.115em; vertical-align: -0.748em;"></span></span><span id="MJXc-Node-2755" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.163em; padding-bottom: 0.335em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>β</mi><mo>=</mo><mfrac><mn>1</mn><mi>θ</mi></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-144">\beta = \frac{1}{\theta}.</script>
<p>Infine <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-145-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B1;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2756" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2757" class="mjx-mrow"><span id="MJXc-Node-2758" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math></span></span><script type="math/tex" id="MathJax-Element-145">\alpha</script> può essere ricavato in due modi differenti: considerando il reciproco della proiezione dell’ipotenusa sull’asse delle ordinate, oppure considerando che <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-146-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>&#x03B1;</mi><mi>&#x03B2;</mi></mfrac><mo>=</mo><mi>&#x03B8;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2759" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2760" class="mjx-mrow"><span id="MJXc-Node-2761" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.594em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.84em; top: -1.095em;"><span id="MJXc-Node-2762" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em;">α</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.84em; bottom: -0.833em;"><span id="MJXc-Node-2763" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.445em; padding-right: 0.007em;">β</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.285em; width: 0.594em;"></span></span><span class="mjx-vsize" style="height: 1.363em; vertical-align: -0.589em;"></span></span><span id="MJXc-Node-2764" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.058em; padding-bottom: 0.335em;">=</span></span><span id="MJXc-Node-2765" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.501em; padding-bottom: 0.279em;">θ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>α</mi><mi>β</mi></mfrac><mo>=</mo><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-146">\frac{\alpha}{\beta} = \theta</script>.</p>
<p><img src="https://gabro29.github.io/Biliard_Torus/image6.PNG" alt=""></p>
<p><strong>Simulazioni</strong></p>
<p>Di seguito si riporta una simulazione relativa al caso di moto periodico.</p>
<p><img src="https://gabro29.github.io/Biliard_Torus/image7.gif" alt=""></p>
<p>Il moto periodico produce una linea chiusa sul Toroide. Il punto di spawn della pallina è (0.2,0.4), mentre la pendenza è 0.2.</p>
<p>Si riportano inoltre i valori che caratterizzano tale simulazione.</p>
<table>
<thead>
<tr>
<th>Intersezione</th>
<th><strong>m</strong></th>
<th><strong>α</strong></th>
<th><strong>β</strong></th>
<th><strong>θ</strong></th>
</tr>
</thead>
<tbody>
<tr>
<td>(0.2, 0.4)</td>
<td>0.2</td>
<td>1</td>
<td>0.2</td>
<td>5.0</td>
</tr>
<tr>
<td>(0.2, 0.4)</td>
<td>0.2</td>
<td>1</td>
<td>0.2</td>
<td>5.0</td>
</tr>
<tr>
<td>(0.2, 0.4)</td>
<td>0.2</td>
<td>1</td>
<td>0.2</td>
<td>5.0</td>
</tr>
<tr>
<td>(0.2, 0.4)</td>
<td>0.2</td>
<td>1</td>
<td>0.2</td>
<td>5.0</td>
</tr>
<tr>
<td>(0.2, 0.4)</td>
<td>0.2</td>
<td>1</td>
<td>0.2</td>
<td>5.0</td>
</tr>
<tr>
<td>(0.2, 0.4)</td>
<td>0.2</td>
<td>1</td>
<td>0.2</td>
<td>5.0</td>
</tr>
</tbody>
</table>
<p>Di seguito si riporta una simulazione relativa al caso di moto non periodico.</p>
<p><img src="https://gabro29.github.io/Biliard_Torus/image8.gif" alt=""></p>
<p>Il moto non periodico riempie densamente il Toroide. Il punto di spawn della pallina è (0.2,0.4), mentre la pendenza è π.</p>
<p>Si riportano inoltre alcuni valori che caratterizzano tale simulazione.</p>
<table>
<thead>
<tr>
<th>Intersezione</th>
<th><strong>m</strong></th>
<th><strong>α</strong></th>
<th><strong>β</strong></th>
<th><strong>θ</strong></th>
</tr>
</thead>
<tbody>
<tr>
<td>(0.882, 0.4)</td>
<td><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-147-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2766" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2767" class="mjx-mrow"><span id="MJXc-Node-2768" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-147">\pi</script></td>
<td>1</td>
<td><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-148-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2769" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2770" class="mjx-mrow"><span id="MJXc-Node-2771" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-148">\pi</script></td>
<td>0.31830988618379086</td>
</tr>
<tr>
<td>(0.563, 0.4)</td>
<td><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-149-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2772" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2773" class="mjx-mrow"><span id="MJXc-Node-2774" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-149">\pi</script></td>
<td>1</td>
<td><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-150-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2775" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2776" class="mjx-mrow"><span id="MJXc-Node-2777" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-150">\pi</script></td>
<td>0.31830988618379086</td>
</tr>
<tr>
<td>(0.245, 0.4)</td>
<td><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-151-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2778" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2779" class="mjx-mrow"><span id="MJXc-Node-2780" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-151">\pi</script></td>
<td>1</td>
<td><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-152-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2781" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2782" class="mjx-mrow"><span id="MJXc-Node-2783" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-152">\pi</script></td>
<td>0.31830988618379086</td>
</tr>
<tr>
<td>(0.927, 0.4)</td>
<td><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-153-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2784" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2785" class="mjx-mrow"><span id="MJXc-Node-2786" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-153">\pi</script></td>
<td>1</td>
<td><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-154-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2787" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2788" class="mjx-mrow"><span id="MJXc-Node-2789" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-154">\pi</script></td>
<td>0.31830988618379086</td>
</tr>
<tr>
<td>(0.608, 0.4)</td>
<td><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-155-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2790" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2791" class="mjx-mrow"><span id="MJXc-Node-2792" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-155">\pi</script></td>
<td>1</td>
<td><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-156-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2793" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2794" class="mjx-mrow"><span id="MJXc-Node-2795" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-156">\pi</script></td>
<td>0.31830988618379086</td>
</tr>
<tr>
<td>(0.290, 0.4)</td>
<td><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-157-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2796" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2797" class="mjx-mrow"><span id="MJXc-Node-2798" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-157">\pi</script></td>
<td>1</td>
<td><span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-158-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" role="presentation" style="font-size: 113%; position: relative;"><span id="MJXc-Node-2799" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2800" class="mjx-mrow"><span id="MJXc-Node-2801" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.224em; padding-bottom: 0.279em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-158">\pi</script></td>
<td>0.31830988618379086</td>
</tr>
</tbody>
</table>
<p><strong>Riferimenti bibliografici</strong></p>
<table>
<thead>
<tr>
<th>Fonti Utilizzate</th>
</tr>
</thead>
<tbody>
<tr>
<td>Appunti presi durante le lezioni di Analisi I tenute dal Professore Matteo Dalla Riva</td>
</tr>
<tr>
<td>L. Menini, C. Possieri and A. Tornambè (2020). “Trajectory tracking in rectangular billiards”</td>
</tr>
</tbody>
</table>
</div>
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