2b266655.2be1a5de.js

(window.webpackJsonp=window.webpackJsonp||[]).push([[14],{113:function(a,e,t){"use strict";t.r(e),t.d(e,"frontMatter",(function(){return b})),t.d(e,"metadata",(function(){return p})),t.d(e,"rightToc",(function(){return c})),t.d(e,"default",(function(){return O}));var s=t(2),m=t(6),n=(t(0),t(146)),b={id:"math",title:"The Math Behind PMM",sidebar_label:"The Math Behind PMM"},p={id:"math",isDocsHomePage:!1,title:"The Math Behind PMM",description:"Core PMM",source:"@site/docs/math.md",permalink:"/docs/docs/math",editUrl:"https://github.com/DODOEX/docs/edit/master/docs/math.md",sidebar_label:"The Math Behind PMM",sidebar:"someSidebar",previous:{title:"Core Concepts",permalink:"/docs/docs/coreConcept"},next:{title:"Smart Contract Framework",permalink:"/docs/docs/framework"}},c=[{value:"The Price Curve Integral",id:"the-price-curve-integral",children:[]},{value:"Solving the quadratic equation for trading",id:"solving-the-quadratic-equation-for-trading",children:[]},{value:"Solving the quadratic equation for regression targets",id:"solving-the-quadratic-equation-for-regression-targets",children:[]},{value:"Trades",id:"trades",children:[]},{value:"Deposit",id:"deposit",children:[]},{value:"Withdrawal",id:"withdrawal",children:[]}],r={rightToc:c};function O(a){var e=a.components,t=Object(m.a)(a,["components"]);return Object(n.b)("wrapper",Object(s.a)({},r,t,{components:e,mdxType:"MDXLayout"}),Object(n.b)("h1",{id:"core-pmm"},"Core PMM"),Object(n.b)("p",null,"The core of PMM is essentially ",Object(n.b)("strong",{parentName:"p"},Object(n.b)("em",{parentName:"strong"},"calculating one integral and solving two quadratic equations")),". The smart contract implementation can be found ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"https://github.com/DODOEX/dodo-smart-contract/blob/master/contracts/lib/DODOMath.sol"}),"here"),"."),Object(n.b)("h2",{id:"the-price-curve-integral"},"The Price Curve Integral"),Object(n.b)("p",null,"For traders, the most important thing is the average transaction price. The average transaction price is the integral of the marginal price ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"P"),Object(n.b)("mrow",{parentName:"msub"},Object(n.b)("mi",{parentName:"mrow"},"m"),Object(n.b)("mi",{parentName:"mrow"},"a"),Object(n.b)("mi",{parentName:"mrow"},"r"),Object(n.b)("mi",{parentName:"mrow"},"g"),Object(n.b)("mi",{parentName:"mrow"},"i"),Object(n.b)("mi",{parentName:"mrow"},"n")))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"P_{margin}")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.969438em",verticalAlign:"-0.286108em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.13889em"}}),"P"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.311664em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.13889em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"m"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"a"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.02778em"}}),"r"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.03588em"}}),"g"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"i"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"n"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.286108em"}}),Object(n.b)("span",{parentName:"span"})))))))))),". Let's take the base token shortage scenario as an example."),Object(n.b)("p",null,Object(n.b)("img",Object(s.a)({parentName:"p"},{src:"https://dodoex.github.io/docs/img/dodo_integrate.jpeg",alt:null}))),Object(n.b)("p",null,Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"Q"),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("msubsup",{parentName:"mrow"},Object(n.b)("mo",{parentName:"msubsup"},"\u222b"),Object(n.b)("msub",{parentName:"msubsup"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("msub",{parentName:"msubsup"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"2"))),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"P"),Object(n.b)("mrow",{parentName:"msub"},Object(n.b)("mi",{parentName:"mrow"},"m"),Object(n.b)("mi",{parentName:"mrow"},"a"),Object(n.b)("mi",{parentName:"mrow"},"r"),Object(n.b)("mi",{parentName:"mrow"},"g"),Object(n.b)("mi",{parentName:"mrow"},"i"),Object(n.b)("mi",{parentName:"mrow"},"n"))),Object(n.b)("mi",{parentName:"mrow"},"d"),Object(n.b)("mi",{parentName:"mrow"},"B")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\Delta Q =\\int^{B_2}_{B_1}P_{margin}dB")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.4921509999999998em",verticalAlign:"-0.45592em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mop"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mop op-symbol small-op",style:{marginRight:"0.19445em",position:"relative",top:"-0.0005599999999999772em"}}),"\u222b"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.036231em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.34418em",marginLeft:"-0.19445em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"}))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.2579em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"})))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.45592em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.16666666666666666em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.13889em"}}),"P"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.311664em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.13889em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"m"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"a"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.02778em"}}),"r"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.03588em"}}),"g"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"i"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"n"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.286108em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"d"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B")))))),Object(n.b)("p",null,Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("msubsup",{parentName:"mrow"},Object(n.b)("mo",{parentName:"msubsup"},"\u222b"),Object(n.b)("msub",{parentName:"msubsup"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("msub",{parentName:"msubsup"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"2"))),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("mn",{parentName:"mrow"},"1"),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),")"),Object(n.b)("mi",{parentName:"mrow"},"i"),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mi",{parentName:"mrow"},"i"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"0")),Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"/"),Object(n.b)("mi",{parentName:"mrow"},"B"),Object(n.b)("msup",{parentName:"mrow"},Object(n.b)("mo",Object(s.a)({parentName:"msup"},{stretchy:"false"}),")"),Object(n.b)("mn",{parentName:"msup"},"2")),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mi",{parentName:"mrow"},"d"),Object(n.b)("mi",{parentName:"mrow"},"B")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"= \\int^{B_2}_{B_1}(1-k)i+i(B_0/B)^2kdB")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.36687em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.4921509999999998em",verticalAlign:"-0.45592em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mop"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mop op-symbol small-op",style:{marginRight:"0.19445em",position:"relative",top:"-0.0005599999999999772em"}}),"\u222b"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.036231em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.34418em",marginLeft:"-0.19445em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"}))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.2579em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"})))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.45592em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"i"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.064108em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"i"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"/"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8141079999999999em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.063em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"d"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B")))))),Object(n.b)("p",null,Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mi",{parentName:"mrow"},"i"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"2")),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),")"),Object(n.b)("mo",{parentName:"mrow"},"\u2217"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("mn",{parentName:"mrow"},"1"),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("msubsup",{parentName:"mfrac"},Object(n.b)("mi",{parentName:"msubsup"},"B"),Object(n.b)("mn",{parentName:"msubsup"},"0"),Object(n.b)("mn",{parentName:"msubsup"},"2")),Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"2")))),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),")")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"= i(B_2-B_1)*(1-k+k\\frac{B_0^2}{B_1B_2})")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.36687em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"i"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2217"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.77777em",verticalAlign:"-0.08333em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.5792199999999998em",verticalAlign:"-0.44509999999999994em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.1341199999999998em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.655em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"}))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.5102em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8913142857142857em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.214em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.931em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.286em"}}),Object(n.b)("span",{parentName:"span"})))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.44509999999999994em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")")))))),Object(n.b)("p",null,"This tells the trader how much they should pay if they buy ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"2"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"B_1-B_2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"}))))))))))," amount of base tokens."),Object(n.b)("p",null,"Rearranging the equation above, the average transaction price is thus:\n",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",{parentName:"mrow"},"P"),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"Q")),Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"2")),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")))),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mi",{parentName:"mrow"},"i"),Object(n.b)("mo",{parentName:"mrow"},"\u2217"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("mn",{parentName:"mrow"},"1"),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("msubsup",{parentName:"mfrac"},Object(n.b)("mi",{parentName:"msubsup"},"B"),Object(n.b)("mn",{parentName:"msubsup"},"0"),Object(n.b)("mn",{parentName:"msubsup"},"2")),Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"2")))),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),")")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"P=\\frac{\\Delta Q}{B_2-B_1}=i*(1-k+k\\frac{B_0^2}{B_1B_2})")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.68333em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.13889em"}}),"P"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.369539em",verticalAlign:"-0.44509999999999994em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.924439em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.655em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin mtight"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"}))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.446108em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"Q"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.44509999999999994em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.65952em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"i"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2217"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.77777em",verticalAlign:"-0.08333em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.5792199999999998em",verticalAlign:"-0.44509999999999994em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.1341199999999998em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.655em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"}))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.5102em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8913142857142857em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.214em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.931em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.286em"}}),Object(n.b)("span",{parentName:"span"})))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.44509999999999994em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")")))))),Object(n.b)("p",null,"We found that the average transaction price is only dependent on the state of the system before and after the transaction, so the price calculation methods for both buying and selling are the same: integrating ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"P"),Object(n.b)("mrow",{parentName:"msub"},Object(n.b)("mi",{parentName:"mrow"},"m"),Object(n.b)("mi",{parentName:"mrow"},"a"),Object(n.b)("mi",{parentName:"mrow"},"r"),Object(n.b)("mi",{parentName:"mrow"},"g"),Object(n.b)("mi",{parentName:"mrow"},"i"),Object(n.b)("mi",{parentName:"mrow"},"n")))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"P_{margin}")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.969438em",verticalAlign:"-0.286108em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.13889em"}}),"P"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.311664em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.13889em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"m"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"a"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.02778em"}}),"r"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.03588em"}}),"g"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"i"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"n"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.286108em"}}),Object(n.b)("span",{parentName:"span"})))))))))),"."),Object(n.b)("h2",{id:"solving-the-quadratic-equation-for-trading"},"Solving the quadratic equation for trading"),Object(n.b)("p",null,"Without the loss of generality, the integral becomes the following when there is a shorage of quote tokens:"),Object(n.b)("p",null,Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"B"),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mn",{parentName:"mfrac"},"1"),Object(n.b)("mi",{parentName:"mfrac"},"i")),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"2")),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),")"),Object(n.b)("mo",{parentName:"mrow"},"\u2217"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("mn",{parentName:"mrow"},"1"),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("msubsup",{parentName:"mfrac"},Object(n.b)("mi",{parentName:"msubsup"},"Q"),Object(n.b)("mn",{parentName:"msubsup"},"0"),Object(n.b)("mn",{parentName:"msubsup"},"2")),Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"2")))),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),")")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\Delta B = \\frac{1}{i}(Q_2-Q_1)*(1-k+k\\frac{Q_0^2}{Q_1Q_2})")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.68333em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.190108em",verticalAlign:"-0.345em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.845108em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.6550000000000002em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"i")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.394em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.345em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2217"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.77777em",verticalAlign:"-0.08333em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.6152279999999997em",verticalAlign:"-0.481108em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.1341199999999998em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.6550000000000002em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"0em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"0em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"}))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.5102em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8913142857142857em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.214em",marginLeft:"0em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.931em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.286em"}}),Object(n.b)("span",{parentName:"span"})))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.481108em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")")))))),Object(n.b)("p",null,"Let's derive how to calculate the price when there is a shortage of quote tokens and only the number of base tokens you want to buy or sell (i.e. ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"B")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\Delta B")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.68333em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"))))),") is given."),Object(n.b)("p",null,"Now that ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"B"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{separator:"true"}),","),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"0")),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{separator:"true"}),","),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\Delta B, Q_0, Q_1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mpunct"}),","),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.16666666666666666em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mpunct"}),","),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.16666666666666666em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"}))))))))))," are given, we need to calculate ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"2"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"Q_2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))))))),", which is found by solving a quadratic equation. Transforming the equation into standard form:"),Object(n.b)("p",null,Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("mn",{parentName:"mrow"},"1"),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),")"),Object(n.b)("msubsup",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msubsup"},"Q"),Object(n.b)("mn",{parentName:"msubsup"},"2"),Object(n.b)("mn",{parentName:"msubsup"},"2")),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("msubsup",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msubsup"},"Q"),Object(n.b)("mn",{parentName:"msubsup"},"0"),Object(n.b)("mn",{parentName:"msubsup"},"2"))),Object(n.b)("msub",{parentName:"mfrac"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1"))),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"i"),Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"B"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),")"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"2")),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("msubsup",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msubsup"},"Q"),Object(n.b)("mn",{parentName:"msubsup"},"0"),Object(n.b)("mn",{parentName:"msubsup"},"2")),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mn",{parentName:"mrow"},"0")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"(1-k)Q_2^2+(\\frac{kQ_0^2}{Q_1}-Q_1+kQ_1-i\\Delta B)Q_2-kQ_0^2=0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.064108em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8141079999999999em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.4518920000000004em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.063em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.24810799999999997em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.6152279999999997em",verticalAlign:"-0.481108em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.1341199999999998em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.6550000000000002em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"0em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"}))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.5102em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8913142857142857em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.214em",marginLeft:"0em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.931em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.286em"}}),Object(n.b)("span",{parentName:"span"})))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.481108em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8888799999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"i"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.0622159999999998em",verticalAlign:"-0.24810799999999997em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8141079999999999em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.4518920000000004em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.063em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.24810799999999997em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"0")))))),Object(n.b)("p",null,Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",{parentName:"mrow"},"l"),Object(n.b)("mi",{parentName:"mrow"},"e"),Object(n.b)("mi",{parentName:"mrow"},"t"),Object(n.b)("mtext",{parentName:"mrow"},"\xa0"),Object(n.b)("mi",{parentName:"mrow"},"a"),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mn",{parentName:"mrow"},"1"),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{separator:"true"}),","),Object(n.b)("mtext",{parentName:"mrow"},"\xa0"),Object(n.b)("mi",{parentName:"mrow"},"b"),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("msubsup",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msubsup"},"Q"),Object(n.b)("mn",{parentName:"msubsup"},"0"),Object(n.b)("mn",{parentName:"msubsup"},"2"))),Object(n.b)("msub",{parentName:"mfrac"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1"))),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"i"),Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"B"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{separator:"true"}),","),Object(n.b)("mtext",{parentName:"mrow"},"\xa0"),Object(n.b)("mi",{parentName:"mrow"},"c"),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("msubsup",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msubsup"},"Q"),Object(n.b)("mn",{parentName:"msubsup"},"0"),Object(n.b)("mn",{parentName:"msubsup"},"2"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"let \\ a=1-k, \\ b=\\frac{kQ_0^2}{Q_1}-Q_1+kQ_1-i\\Delta B, \\ c=-kQ_0^2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.69444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.01968em"}}),"l"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"e"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"t"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace"}),"\xa0"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"a"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.72777em",verticalAlign:"-0.08333em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8888799999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mpunct"}),","),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.16666666666666666em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace"}),"\xa0"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"b"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.6152279999999997em",verticalAlign:"-0.481108em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.1341199999999998em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.6550000000000002em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"0em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"}))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.5102em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8913142857142857em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.214em",marginLeft:"0em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.931em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.286em"}}),Object(n.b)("span",{parentName:"span"})))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.481108em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8888799999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"i"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mpunct"}),","),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.16666666666666666em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace"}),"\xa0"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"c"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.0622159999999998em",verticalAlign:"-0.24810799999999997em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8141079999999999em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.4518920000000004em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.063em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.24810799999999997em"}}),Object(n.b)("span",{parentName:"span"}))))))))))),Object(n.b)("p",null,"Because ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"2")),Object(n.b)("mo",{parentName:"mrow"},">"),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mn",{parentName:"mrow"},"0")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"Q_2>=0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),">")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.36687em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"0"))))),", we discard the negative root, and so"),Object(n.b)("p",null,Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"2")),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"b"),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("msqrt",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"msqrt"},Object(n.b)("msup",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msup"},"b"),Object(n.b)("mn",{parentName:"msup"},"2")),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mn",{parentName:"mrow"},"4"),Object(n.b)("mi",{parentName:"mrow"},"a"),Object(n.b)("mi",{parentName:"mrow"},"c")))),Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("mn",{parentName:"mrow"},"2"),Object(n.b)("mi",{parentName:"mrow"},"a")))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"Q_2=\\frac{-b+\\sqrt{b^2-4ac}}{2a}")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.384482em",verticalAlign:"-0.345em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.039482em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.6550000000000002em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"a")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.3939999999999997em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"b"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin mtight"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord sqrt mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.9221171428571429em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"svg-align",style:{top:"-3em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight",style:{paddingLeft:"0.833em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"b"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.7463142857142857em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.786em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin mtight"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"4"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"a"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"c"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.882117142857143em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"hide-tail mtight",style:{minWidth:"0.853em",height:"1.08em"}}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{d:"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.11788285714285718em"}}),Object(n.b)("span",{parentName:"span"}))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.345em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"})))))))),Object(n.b)("p",null,"It can be proven that:"),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},"When ",Object(n.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"B"),Object(n.b)("mo",{parentName:"mrow"},">"),Object(n.b)("mn",{parentName:"mrow"},"0")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\Delta B>0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.72243em",verticalAlign:"-0.0391em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),">"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"0"))))),", ",Object(n.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"2")),Object(n.b)("mo",{parentName:"mrow"},">"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"Q_2>Q_1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),">"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))))))),"; trader buy base token, and should pay ",Object(n.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"2")),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"Q_2-Q_1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"}))))))))))),Object(n.b)("li",{parentName:"ul"},"When ",Object(n.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"B"),Object(n.b)("mo",{parentName:"mrow"},"<"),Object(n.b)("mn",{parentName:"mrow"},"0")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\Delta B<0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.72243em",verticalAlign:"-0.0391em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"<"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"0"))))),", ",Object(n.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"2")),Object(n.b)("mo",{parentName:"mrow"},"<"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"Q_2<Q_1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"<"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))))))),"; trader sell base token, and will receive ",Object(n.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"2"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"Q_1-Q_2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"}))))))))))),Object(n.b)("li",{parentName:"ul"},"When ",Object(n.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"B"),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mn",{parentName:"mrow"},"0")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\Delta B=0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.68333em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"0"))))),", ",Object(n.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"2")),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"Q_2=Q_1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))))))),".")),Object(n.b)("h2",{id:"solving-the-quadratic-equation-for-regression-targets"},"Solving the quadratic equation for regression targets"),Object(n.b)("p",null,"When the system is not in the equilibrium state, changes to the oracle price will bring profit or loss. For example, assume that shortage of base tokens is the current state, and the oracle price goes up. It is clear that the excess quote tokens cannot buy enough base tokens to return the base token balance to the base token regression target. Thus, LPs who deposited base tokens will suffer a loss. Conversely, if the oracle price drops, the excess quote tokens can buy more base tokens, causing the base token balance to exceed the base token regression target, and LPs who deposited base tokens will make a profit."),Object(n.b)("p",null,"In summary, the regression target is influenced by the oracle price. To calculate the regression target at a certain oracle price, we make the following derivation:"),Object(n.b)("p",null,"Given ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"Q"),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mi",{parentName:"mrow"},"i"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"2")),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),")"),Object(n.b)("mo",{parentName:"mrow"},"\u2217"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("mn",{parentName:"mrow"},"1"),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("msubsup",{parentName:"mfrac"},Object(n.b)("mi",{parentName:"msubsup"},"B"),Object(n.b)("mn",{parentName:"msubsup"},"0"),Object(n.b)("mn",{parentName:"msubsup"},"2")),Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"2")))),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),")")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\Delta Q = i(B_2-B_1)*(1-k+k\\frac{B_0^2}{B_1B_2})")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"i"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2217"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.77777em",verticalAlign:"-0.08333em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.5792199999999998em",verticalAlign:"-0.44509999999999994em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.1341199999999998em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.655em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"}))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.5102em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8913142857142857em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.214em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.931em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.286em"}}),Object(n.b)("span",{parentName:"span"})))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.44509999999999994em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")")))))),Object(n.b)("p",null,"Since we are doing regression, ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"2")),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"0"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"B_2=B_0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))))))),". Rearraging the equation with respect to ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"0"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"B_0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"}))))))))))," gives"),Object(n.b)("p",null,Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mi",{parentName:"mfrac"},"k"),Object(n.b)("msub",{parentName:"mfrac"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1"))),Object(n.b)("msubsup",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msubsup"},"B"),Object(n.b)("mn",{parentName:"msubsup"},"0"),Object(n.b)("mn",{parentName:"msubsup"},"2")),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("mn",{parentName:"mrow"},"1"),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mn",{parentName:"mrow"},"2"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),")"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"0")),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"["),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"("),Object(n.b)("mn",{parentName:"mrow"},"1"),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),")"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"Q")),Object(n.b)("mi",{parentName:"mfrac"},"i")),Object(n.b)("mo",Object(s.a)({parentName:"mrow"},{stretchy:"false"}),"]"),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mn",{parentName:"mrow"},"0")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\frac{k}{B_1}B_0^2+(1-2k)B_0-[(1-k)B_1+\\frac{\\Delta Q}{i}] = 0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.325208em",verticalAlign:"-0.44509999999999994em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8801079999999999em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.655em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.31731428571428577em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.357em",marginLeft:"-0.05017em",marginRight:"0.07142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.5em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.143em"}}),Object(n.b)("span",{parentName:"span"}))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.394em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.03148em"}}),"k"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.44509999999999994em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8141079999999999em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.4518920000000004em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.063em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.24810799999999997em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"2"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"["),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen"}),"("),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),")"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.269439em",verticalAlign:"-0.345em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.924439em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.6550000000000002em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"i")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.446108em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"Q"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.345em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose"}),"]"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"0")))))),Object(n.b)("p",null,"The negative root does not make sense and is discarded, so ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"0"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"B_0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"}))))))))))," is:"),Object(n.b)("p",null,Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"0")),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("msqrt",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"msqrt"},Object(n.b)("mn",{parentName:"mrow"},"1"),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("mn",{parentName:"mrow"},"4"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"Q")),Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mi",{parentName:"mrow"},"i"))))),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mn",{parentName:"mrow"},"1")),Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("mn",{parentName:"mrow"},"2"),Object(n.b)("mi",{parentName:"mrow"},"k")))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"B_0=B_1+B_1\\frac{\\sqrt{1+\\frac{4k\\Delta Q}{B_1 i}}-1}{2k}")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.8746em",verticalAlign:"-0.345em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.5296em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.6550000000000002em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.03148em"}}),"k")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.7386274999999998em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord sqrt mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.1299607142857147em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"svg-align",style:{top:"-3.7142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3.7142857142857144em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight",style:{paddingLeft:"1.4285714285714286em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin mtight"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter sizing reset-size3 size6"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.9584142857142857em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.656em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.3448em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.3447999999999998em",marginLeft:"-0.05017em",marginRight:"0.1em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.64444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.29964em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"i")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.2255000000000003em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line mtight",style:{borderBottomWidth:"0.049em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.4623857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"4"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"Q"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.5580285714285714em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter sizing reset-size3 size6"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.1019607142857146em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3.7142857142857144em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"hide-tail mtight",style:{minWidth:"1.02em",height:"1.8285714285714287em"}}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{width:"400em",height:"1.8285714285714287em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{d:"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.6123249999999999em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin mtight"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.345em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"})))))))),Object(n.b)("p",null,"In this case, ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"Q"),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("mi",{parentName:"mrow"},"Q"),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"0"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\Delta Q=Q-Q_0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))))))),". It can be proven that, when ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"Q"),Object(n.b)("mo",{parentName:"mrow"},"\u2265"),Object(n.b)("mn",{parentName:"mrow"},"0")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\Delta Q \\ge 0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"\u2265"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.64444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"0"))))),", ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"0")),Object(n.b)("mo",{parentName:"mrow"},"\u2265"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"B_0\\ge B_1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"\u2265"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))))))),". "),Object(n.b)("p",null,"This fact is extremely important, because it ensures that the base token balance and the quote token balance will never be greater than the regression target simultaneously, or less than the regression target simultaneously. This means that PMM will only switch between the three states discussed in the Core Concepts section."),Object(n.b)("p",null,"Similarly, the formula for quote token regression target ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"0"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"Q_0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"}))))))))))," is"),Object(n.b)("p",null,Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"0")),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",{parentName:"mrow"},"\u2217"),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("msqrt",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"msqrt"},Object(n.b)("mn",{parentName:"mrow"},"1"),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("mn",{parentName:"mrow"},"4"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"B"),Object(n.b)("mi",{parentName:"mrow"},"i")),Object(n.b)("msub",{parentName:"mfrac"},Object(n.b)("mi",{parentName:"msub"},"Q"),Object(n.b)("mn",{parentName:"msub"},"1"))))),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mn",{parentName:"mrow"},"1")),Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("mn",{parentName:"mrow"},"2"),Object(n.b)("mi",{parentName:"mrow"},"k")))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"Q_0=Q_1+Q_1*\\frac{\\sqrt{1+\\frac{4k\\Delta B i}{Q_1}}-1}{2k}")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"0em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2217"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.8746000000000003em",verticalAlign:"-0.345em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.5296000000000003em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.6550000000000002em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.03148em"}}),"k")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.7660625000000003em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord sqrt mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.0907678571428572em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"svg-align",style:{top:"-3.7142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3.7142857142857144em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight",style:{paddingLeft:"1.4285714285714286em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin mtight"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter sizing reset-size3 size6"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8800285714285714em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.656em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"Q"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.3448em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.3447999999999998em",marginLeft:"0em",marginRight:"0.1em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.64444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.29964em"}}),Object(n.b)("span",{parentName:"span"}))))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.2255000000000003em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line mtight",style:{borderBottomWidth:"0.049em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.384em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"4"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"i"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.5580285714285714em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter sizing reset-size3 size6"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.062767857142857em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3.7142857142857144em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"hide-tail mtight",style:{minWidth:"1.02em",height:"1.8285714285714287em"}}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{width:"400em",height:"1.8285714285714287em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{d:"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.6515178571428574em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin mtight"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.345em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"})))))))),Object(n.b)("h1",{id:"peripheral"},"Peripheral"),Object(n.b)("p",null,"This section will deal with the math pertaining to the peripheral functioning of PMM."),Object(n.b)("h2",{id:"trades"},"Trades"),Object(n.b)("p",null,"As mentioned above, the regression target depends on the oracle price, and the price curve in turn depend on the regression target. So in every trade, we should calculate the regression target well in advance to make the price curve fixed."),Object(n.b)("p",null,"In addition, since the price curve given by PMM is segmented, if a transaction involves different states (for example, when a trader sells an astronomical amount of base tokens during a base token shortage and forces the state into a quote token shortage), the price needs to be calculated in segments as well."),Object(n.b)("p",null,"Please be advised that this calculation requires a high degree of accuracy. The smart contract provides six trading functions for the three possible states. You can find the most important logic of cross-state trading ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"https://github.com/DODOEX/dodo-smart-contract/blob/master/contracts/impl/Trader.sol"}),"here"),"."),Object(n.b)("h2",{id:"deposit"},"Deposit"),Object(n.b)("p",null,"Depositing and withdrawing base token when there is a shortage of base tokens, or quote tokens when there is a shortage of quote token, will change the price curve. This requires us to process the deposit and withdrawal with caution and care in order to keep the capital pool sustainable and fair."),Object(n.b)("p",null,"We will analyze what happens when an LP makes a deposit when there is a shortage of base tokens."),Object(n.b)("p",null,"According to the calculation formula of ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"0"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"B_0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"}))))))))))," derived above\n",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"0")),Object(n.b)("mo",{parentName:"mrow"},"="),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mo",{parentName:"mrow"},"\u2217"),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("msqrt",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"msqrt"},Object(n.b)("mn",{parentName:"mrow"},"1"),Object(n.b)("mo",{parentName:"mrow"},"+"),Object(n.b)("mfrac",{parentName:"mrow"},Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("mn",{parentName:"mrow"},"4"),Object(n.b)("mi",{parentName:"mrow"},"k"),Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"Q")),Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1")),Object(n.b)("mi",{parentName:"mrow"},"i"))))),Object(n.b)("mo",{parentName:"mrow"},"\u2212"),Object(n.b)("mn",{parentName:"mrow"},"1")),Object(n.b)("mrow",{parentName:"mfrac"},Object(n.b)("mn",{parentName:"mrow"},"2"),Object(n.b)("mi",{parentName:"mrow"},"k")))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"B_0=B_1+B_1*\\frac{\\sqrt{1+\\frac{4k\\Delta Q}{B_1 i}}-1}{2k}")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mrel"}),"="),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2777777777777778em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin"}),"\u2217"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mspace",style:{marginRight:"0.2222222222222222em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.8746em",verticalAlign:"-0.345em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.5296em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.6550000000000002em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"2"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.03148em"}}),"k")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.23em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line",style:{borderBottomWidth:"0.04em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.7386274999999998em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord sqrt mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"1.1299607142857147em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"svg-align",style:{top:"-3.7142857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3.7142857142857144em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight",style:{paddingLeft:"1.4285714285714286em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin mtight"}),"+"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mopen nulldelimiter sizing reset-size3 size6"})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mfrac"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.9584142857142857em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.656em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.3448em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.3447999999999998em",marginLeft:"-0.05017em",marginRight:"0.1em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.64444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.29964em"}}),Object(n.b)("span",{parentName:"span"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"i")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.2255000000000003em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"frac-line mtight",style:{borderBottomWidth:"0.049em"}}))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.4623857142857144em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size3 size1 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"4"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight",style:{marginRight:"0.03148em"}}),"k"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"Q"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.5580285714285714em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter sizing reset-size3 size6"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.1019607142857146em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3.7142857142857144em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"hide-tail mtight",style:{minWidth:"1.02em",height:"1.8285714285714287em"}}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{width:"400em",height:"1.8285714285714287em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{d:"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z"})))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.6123249999999999em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mbin mtight"}),"\u2212"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1"))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.345em"}}),Object(n.b)("span",{parentName:"span"}))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mclose nulldelimiter"})))))))),Object(n.b)("p",null,"After an LP deposit ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",{parentName:"mrow"},"b")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"b")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.69444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"b")))))," base tokens, ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"B_1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"}))))))))))," increases by ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",{parentName:"mrow"},"b")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"b")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.69444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"b"))))),", and ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"0"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"B_0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"}))))))))))," increases more than than ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",{parentName:"mrow"},"b")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"b")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.69444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"b"))))),"'s magnitude. It means that this deposit helps all LPs who provided base token make a profit. The reason why is that the deposit makes the price curve smoother, and the same amount of ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"\u0394"),Object(n.b)("mi",{parentName:"mrow"},"Q")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\Delta Q")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8777699999999999em",verticalAlign:"-0.19444em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u0394"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"Q")))))," can now buy more base tokens."),Object(n.b)("p",null,"In this case, as soon as the LP makes a deposit, the LP makes a profit. This is referred to as the deposit reward. The essential source of this reward is the slippage paid by the trader who made the system deviate from equilibrium state."),Object(n.b)("div",{className:"admonition admonition-note alert alert--secondary"},Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M6.3 5.69a.942.942 0 0 1-.28-.7c0-.28.09-.52.28-.7.19-.18.42-.28.7-.28.28 0 .52.09.7.28.18.19.28.42.28.7 0 .28-.09.52-.28.7a1 1 0 0 1-.7.3c-.28 0-.52-.11-.7-.3zM8 7.99c-.02-.25-.11-.48-.31-.69-.2-.19-.42-.3-.69-.31H6c-.27.02-.48.13-.69.31-.2.2-.3.44-.31.69h1v3c.02.27.11.5.31.69.2.2.42.31.69.31h1c.27 0 .48-.11.69-.31.2-.19.3-.42.31-.69H8V7.98v.01zM7 2.3c-3.14 0-5.7 2.54-5.7 5.68 0 3.14 2.56 5.7 5.7 5.7s5.7-2.55 5.7-5.7c0-3.15-2.56-5.69-5.7-5.69v.01zM7 .98c3.86 0 7 3.14 7 7s-3.14 7-7 7-7-3.12-7-7 3.14-7 7-7z"})))),"note")),Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(n.b)("p",{parentName:"div"},"It is important to note that deposit rewards are not risk-free arbitrage trading opportunities. "))),Object(n.b)("h2",{id:"withdrawal"},"Withdrawal"),Object(n.b)("p",null,"Similarly, after an LP withdraws ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",{parentName:"mrow"},"b")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"b")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.69444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"b")))))," base tokens, ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"1"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"B_1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"1")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"}))))))))))," decreases by ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",{parentName:"mrow"},"b")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"b")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.69444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"b"))))),", and ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msub",{parentName:"mrow"},Object(n.b)("mi",{parentName:"msub"},"B"),Object(n.b)("mn",{parentName:"msub"},"0"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"B_0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.83333em",verticalAlign:"-0.15em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05017em"}}),"B"),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.30110799999999993em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.5500000000000003em",marginLeft:"-0.05017em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.15em"}}),Object(n.b)("span",{parentName:"span"}))))))))))," decreases by more than ",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("mi",{parentName:"mrow"},"b")),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"b")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.69444em",verticalAlign:"0em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"b"))))),"'s magnitude. This withdrawal causes all LPs who owes Base Tokens to suffer losses. This is because this withdrawal makes the price curve more steep, and the excess quote tokens have less purchasing power in terms of base tokens."),Object(n.b)("p",null,"The PMM algorithm stipulates that a withdrawal fee is required to withdraw tokens in this case. The magnitude of the fee is equal to the aggregate loss of all LPs caused by the withdrawal. This fee will be directly distributed to all LPs that have not yet withdrawn."),Object(n.b)("p",null,"Factoring in the deposit reward from the previous section, if an LP makes a withdrawal immediately after depositing, the withdrawal fee will be greater than the deposit reward, thus eliminating any possibility of risk-free arbitrage trading."),Object(n.b)("p",null,"It is worth noting that both deposit reward and withdrawal fee are only significant when the system deviates very far from the equilibrium state and the deposit/withdrawal amount is large. Traders thus often overlook the existence of this gain/loss. Of course, traders are also welcome to extract value from the system by exploiting this if they so wish. In order to do that, they can first deposit to earn deposit rewards when the system deviates from the equilibrium, and then withdraw once the system returns to the equilibrium to avoid the withdrawal fee."))}O.isMDXComponent=!0},146:function(a,e,t){"use strict";t.d(e,"a",(function(){return j})),t.d(e,"b",(function(){return i}));var s=t(0),m=t.n(s);function n(a,e,t){return e in a?Object.defineProperty(a,e,{value:t,enumerable:!0,configurable:!0,writable:!0}):a[e]=t,a}function b(a,e){var t=Object.keys(a);if(Object.getOwnPropertySymbols){var s=Object.getOwnPropertySymbols(a);e&&(s=s.filter((function(e){return Object.getOwnPropertyDescriptor(a,e).enumerable}))),t.push.apply(t,s)}return t}function p(a){for(var e=1;e<arguments.length;e++){var t=null!=arguments[e]?arguments[e]:{};e%2?b(Object(t),!0).forEach((function(e){n(a,e,t[e])})):Object.getOwnPropertyDescriptors?Object.defineProperties(a,Object.getOwnPropertyDescriptors(t)):b(Object(t)).forEach((function(e){Object.defineProperty(a,e,Object.getOwnPropertyDescriptor(t,e))}))}return a}function c(a,e){if(null==a)return{};var t,s,m=function(a,e){if(null==a)return{};var t,s,m={},n=Object.keys(a);for(s=0;s<n.length;s++)t=n[s],e.indexOf(t)>=0||(m[t]=a[t]);return m}(a,e);if(Object.getOwnPropertySymbols){var n=Object.getOwnPropertySymbols(a);for(s=0;s<n.length;s++)t=n[s],e.indexOf(t)>=0||Object.prototype.propertyIsEnumerable.call(a,t)&&(m[t]=a[t])}return m}var r=m.a.createContext({}),O=function(a){var e=m.a.useContext(r),t=e;return a&&(t="function"==typeof a?a(e):p(p({},e),a)),t},j=function(a){var e=O(a.components);return m.a.createElement(r.Provider,{value:e},a.children)},N={inlineCode:"code",wrapper:function(a){var e=a.children;return m.a.createElement(m.a.Fragment,{},e)}},l=m.a.forwardRef((function(a,e){var t=a.components,s=a.mdxType,n=a.originalType,b=a.parentName,r=c(a,["components","mdxType","originalType","parentName"]),j=O(t),l=s,i=j["".concat(b,".").concat(l)]||j[l]||N[l]||n;return t?m.a.createElement(i,p(p({ref:e},r),{},{components:t})):m.a.createElement(i,p({ref:e},r))}));function i(a,e){var t=arguments,s=e&&e.mdxType;if("string"==typeof a||s){var n=t.length,b=new Array(n);b[0]=l;var p={};for(var c in e)hasOwnProperty.call(e,c)&&(p[c]=e[c]);p.originalType=a,p.mdxType="string"==typeof a?a:s,b[1]=p;for(var r=2;r<n;r++)b[r]=t[r];return m.a.createElement.apply(null,b)}return m.a.createElement.apply(null,t)}l.displayName="MDXCreateElement"}}]);