https://www.binius.xyz/assets/arithmetic-pu4VQXkj.js

1import{u as n,j as s}from"./index-Bpb9-9J-.js";const m={title:"Arithmetic Shifts",description:"Handling the high sign bits"};function l(a){const e={a:"a",annotation:"annotation",div:"div",em:"em",h1:"h1",h2:"h2",h3:"h3",header:"header",li:"li",math:"math",mi:"mi",mn:"mn",mo:"mo",mover:"mover",mrow:"mrow",mstyle:"mstyle",msub:"msub",msup:"msup",mtable:"mtable",mtd:"mtd",mtext:"mtext",mtr:"mtr",munderover:"munderover",p:"p",path:"path",semantics:"semantics",span:"span",strong:"strong",svg:"svg",ul:"ul",...n(),...a.components};return s.jsxs(s.Fragment,{children:[s.jsxs(e.header,{children:[s.jsxs(e.h1,{id:"arithmetic-shifts",children:["Arithmetic Shifts",s.jsx(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#arithmetic-shifts",children:s.jsx(e.div,{"data-autolink-icon":!0})})]}),s.jsx(e.div,{role:"doc-subtitle",children:"Handling the high sign bits"})]}),`
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11`,s.jsxs(e.p,{children:["We first define the following auxiliary function, for each input ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"i"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mi,{children:"s"}),s.jsx(e.mo,{stretchy:"false",children:")"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"(i, s)"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),s.jsx(e.span,{className:"mopen",children:"("}),s.jsx(e.span,{className:"mord mathnormal",children:"i"}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"s"}),s.jsx(e.span,{className:"mclose",children:")"})]})})]})," ",s.jsx(e.em,{children:"in"})," the cube ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"script",children:"B"}),s.jsx(e.mn,{children:"6"})]}),s.jsx(e.mo,{children:"×"}),s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"script",children:"B"}),s.jsx(e.mn,{children:"6"})]})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\mathcal{B}_6 \\times \\mathcal{B}_6"})]})})}),s.jsxs(e.span,{className:"katex-html","aria-hidden":"true",children:[s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathcal",style:{marginRight:"0.03041em"},children:"B"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3011em"},children:s.jsxs(e.span,{style:{top:"-2.55em",marginLeft:"-0.0304em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord mtight",children:"6"})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.15em"},children:s.jsx(e.span,{})})})]})})]}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),s.jsx(e.span,{className:"mbin",children:"×"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathcal",style:{marginRight:"0.03041em"},children:"B"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3011em"}

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29`,s.jsxs(e.p,{children:["Clearly, ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsx(e.mi,{mathvariant:"sans-serif",children:"a"}),s.jsx(e.mo,{children:"="}),s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"sans-serif",children:"a"}),s.jsx(e.mn,{children:"6"})]})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\mathsf{a} = \\mathsf{a}_6"})]})})}),s.jsxs(e.span,{className:"katex-html","aria-hidden":"true",children:[s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.4444em"}}),s.jsx(e.span,{className:"mord mathsf",children:"a"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),s.jsx(e.span,{className:"mrel",children:"="}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.5944em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathsf",children:"a"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3011em"}

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31`,s.jsx(e.p,{children:"Once again, we get a recursive substructure:"}),`
32`,s.jsxs(e.ul,{children:[`
33`,s.jsxs(e.li,{children:[`
34`,s.jsx(e.strong,{children:"Base Case."}),`
35`,s.jsxs(e.ul,{children:[`
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37`]}),`
38`]}),`
39`,s.jsxs(e.li,{children:[`
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52,children:s.jsxs(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03148em"},children:"k"})})]})})})})})]})]})]})]})," overflows modulo ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsxs(e.msup,{children:[s.jsx(e.mn,{children:"2"}),s.jsx(e.mi,{children:"k"})]})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"2^k"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.8491em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord",children:"2"}),s.jsx(e.span,{className:"msupsub",children:s.jsx(e.span,{className:"vlist-t",children:s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.8491em"},children:s.jsxs(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03148em"},children:"k"})})]})})})})})]})]})})]}),". If both summands' most-significant bits are 1, then we definitely get an overflow; if neither are, then we definitely don't. If exactly one of the bits ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsxs(e.msub,{children:[s.jsx(e.mi,{children:"i"}),s.jsxs(e.mrow,{children:[s.jsx(e.mi,{children:"k"}),s.jsx(e.mo,{children:"−"}),s.jsx(e.mn,{children:"1"})]})]})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"i_{k - 1}"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.8679em",verticalAlign:"-0.2083em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathnormal",children:"i"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3361em"},children:s.jsxs(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsxs(e.span,{className:"mord mtight",children:[s.jsx(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03148em"},children:"k"}),s.jsx(e.span,{className:"mbin mtight",children:"−"}),s.jsx(e.span,{className:"mord mtight",children:"1"})]})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.2083em"},children:s.jsx(e.span,{})})})]})})]})]})})]})," and ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsxs(e.msub,{children:[s.jsx(e.mi,{children:"s"}),s.jsxs(e.mrow,{children:[s.jsx(e.mi,{children:"k"}),s.jsx(e.mo,{children:"−"}),s.jsx(e.mn,{children:"1"})]})]})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"s_{k - 1}"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.6389em",verticalAlign:"-0.2083em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathnormal",children:"s"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3361em"}

52,children:s.jsxs(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsxs(e.span,{className:"mord mtight",children:[s.jsx(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03148em"},children:"k"}),s.jsx(e.span,{className:"mbin mtight",children:"−"}),s.jsx(e.span,{className:"mord mtight",children:"1"})]})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.2083em"},children:s.jsx(e.span,{})})})]})})]})]})})]})," equals 1, then we get an overflow if and only if the less-significant substrings (excluding the most-significant bits), when added together, produce an overflow."]}),`
53`,s.jsxs(e.h3,{id:"the-arithmetizations",children:["The Arithmetizations",s.jsx(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#the-arithmetizations",children:s.jsx(e.div,{"data-autolink-icon":!0})})]}),`
54`,s.jsxs(e.p,{children:["Using this substructure, we once again get arithmetizations for the multilinear extensions ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsxs(e.msub,{children:[s.jsxs(e.mover,{accent:"true",children:[s.jsx(e.mi,{children:"a"}),s.jsx(e.mo,{stretchy:"true",children:"~"})]}),s.jsx(e.mi,{children:"k"})]}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"I"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mi,{children:"S"}),s.jsx(e.mo,{stretchy:"false",children:")"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\widetilde{a}_k(I, S)"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord accent",children:s.jsx(e.span,{className:"vlist-t",children:s.jsx(e.span,{className:"vlist-r",children:s.jsxs(e.span,{className:"vlist",style:{height:"0.6906em"},children:[s.jsxs(e.span,{style:{top:"-3em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"3em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"a"})]}),s.jsxs(e.span,{className:"svg-align",style:{top:"-3.4306em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"3em"}}),s.jsx(e.span,{style:{height:"0.26em"},children:s.jsx(e.svg,{xmlns:"http://www.w3.org/2000/svg",width:"100%",height:"0.26em",viewBox:"0 0 600 260",preserveAspectRatio:"none",children:s.jsx(e.path,{d:`M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
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59`,s.jsxs(e.ul,{children:[`
60`,s.jsxs(e.li,{children:[`
61`,s.jsx(e.strong,{children:"Base Case."}),`
62`,s.jsxs(e.ul,{children:[`
63`,s.jsxs(e.li,{children:[s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsxs(e.msub,{children:[s.jsxs(e.mover,{accent:"true",children:[s.jsx(e.mi,{mathvariant:"sans-serif",children:"a"}),s.jsx(e.mo,{stretchy:"true",children:"~"})]}),s.jsx(e.mn,{children:"0"})]}),s.jsx(e.mo,{children:"="}),s.jsx(e.mn,{children:"0"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\widetilde{\\mathsf{a}}_0 = 0"})]})})}),s.jsxs(e.span,{className:"katex-html","aria-hidden":"true",children:[s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.8544em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord accent",children:s.jsx(e.span,{className:"vlist-t",children:s.jsx(e.span,{className:"vlist-r",children:s.jsxs(e.span,{className:"vlist",style:{height:"0.7044em"},children:[s.jsxs(e.span,{style:{top:"-3em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"3em"}}),s.jsx(e.span,{className:"mord mathsf",children:"a"})]}),s.jsxs(e.span,{className:"svg-align",style:{top:"-3.4444em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"3em"}}),s.jsx(e.span,{style:{height:"0.26em"},children:s.jsx(e.svg,{xmlns:"http://www.w3.org/2000/svg",width:"100%",height:"0.26em",viewBox:"0 0 600 260",preserveAspectRatio:"none",children:s.jsx(e.path,{d:`M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
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66 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128
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68`]}),`
69`]}),`
70`,s.jsxs(e.li,{children:[`
71`,s.jsxs(e.p,{children:[s.jsx(e.strong,{children:"Inductive Case."})," For each ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsx(e.mi,{children:"k"}),s.jsx(e.mo,{children:"∈"}),s.jsx(e.mo,{stretchy:"false",children:"{"}),s.jsx(e.mn,{children:"1"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mo,{children:"…"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mn,{children:"6"}),s.jsx(e.mo,{stretchy:"false",children:"}"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"k \\in \\{1, \\ldots , 6\\}"})]})})}),s.jsxs(e.span,{className:"katex-html","aria-hidden":"true",children:[s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.7335em",verticalAlign:"-0.0391em"}}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.03148em"},children:"k"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),s.jsx(e.span,{className:"mrel",children:"∈"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),s.jsx(e.span,{className:"mopen",children:"{"}),s.jsx(e.span,{className:"mord",children:"1"}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"minner",children:"…"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord",children:"6"}),s.jsx(e.span,{className:"mclose",children:"}"})]})]})]})," and each ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"i"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mi,{children:"s"}),s.jsx(e.mo,{stretchy:"false",children:")"}),s.jsx(e.mo,{children:"∈"}),s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"script",children:"B"}),s.jsx(e.mi,{children:"k"})]}),s.jsx(e.mo,{children:"×"}),s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"script",children:"B"}),s.jsx(e.mi,{children:"k"})]})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"(i, s) \\in \\mathcal{B}_k \\times \\mathcal{B}_k"})]})})}),s.jsxs(e.span,{className:"katex-html","aria-hidden":"true",children:[s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),s.jsx(e.span,{className:"mopen",children:"("}),s.jsx(e.span,{className:"mord mathnormal",children:"i"}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"s"}),s.jsx(e.span,{className:"mclose",children:")"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),s.jsx(e.span,{className:"mrel",children:"∈"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathcal",style:{marginRight:"0.03041em"},children:"B"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3361em"}

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