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<div id="p1" class="ltx_para">
<table id="S0.Ex1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="m1" class="ltx_Math" alttext="a+b" display="block"><semantics id="m1a"><mrow id="m1.4" xref="m1.4.cmml"><mi id="m1.1" xref="m1.1.cmml">a</mi><mo id="m1.2" xref="m1.2.cmml">+</mo><mi id="m1.3" xref="m1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="m1b"><apply id="m1.4.cmml" xref="m1.4"><plus id="m1.2.cmml" xref="m1.2"/><ci id="m1.1.cmml" xref="m1.1">𝑎</ci><ci id="m1.3.cmml" xref="m1.3">𝑏</ci></apply></annotation-xml></semantics></math></td><td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr>
</table>
<table id="S0.Ex2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="m2" class="ltx_Math" alttext="F(a)" display="block"><semantics id="m2a"><mrow id="m2.3b" xref="m2.3.cmml"><mi id="m2.1" xref="m2.1.cmml">F</mi><mo id="m2.3c" xref="m2.3.cmml"></mo><mrow id="m2.3.1" xref="m2.3.cmml"><mo stretchy="false" id="m2.3" xref="m2.3.cmml">(</mo><mi id="m2.2" xref="m2.2.cmml">a</mi><mo stretchy="false" id="m2.3a" xref="m2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="m2b"><apply id="m2.3.cmml" xref="m2.3b"><ci id="m2.1.cmml" xref="m2.1">𝐹</ci><ci id="m2.2.cmml" xref="m2.2">𝑎</ci></apply></annotation-xml></semantics></math></td><td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr>
</table>
<table id="S0.Ex3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="m3" class="ltx_Math" alttext="\frac{1}{(1+a_{0})}" display="block"><semantics id="m3a"><mfrac id="m3.4" xref="m3.4.cmml"><mn id="m3.2" xref="m3.2.cmml">1</mn><mrow id="m3.3b" xref="m3.1.cmml"><mo stretchy="false" id="m3.3" xref="m3.1.cmml">(</mo><mrow id="m3.1" xref="m3.1.cmml"><mn id="m3.1.1" xref="m3.1.1.cmml">1</mn><mo id="m3.1.2" xref="m3.1.2.cmml">+</mo><msub id="m3.1.5" xref="m3.1.5.cmml"><mi id="m3.1.3" xref="m3.1.3.cmml">a</mi><mn id="m3.1.4" xref="m3.1.4.cmml">0</mn></msub></mrow><mo stretchy="false" id="m3.3a" xref="m3.1.cmml">)</mo></mrow></mfrac><annotation-xml encoding="MathML-Content" id="m3b"><apply id="m3.4.cmml" xref="m3.4"><divide id="m3.4.1.cmml" xref="m3.4"/><cn type="integer" id="m3.2.cmml" xref="m3.2">1</cn><apply id="m3.1.cmml" xref="m3.3b"><plus id="m3.1.2.cmml" xref="m3.1.2"/><cn type="integer" id="m3.1.1.cmml" xref="m3.1.1">1</cn><apply id="m3.1.5.cmml" xref="m3.1.5"><csymbol cd="ambiguous" id="m3.1.5.1.cmml" xref="m3.1.5">subscript</csymbol><ci id="m3.1.3.cmml" xref="m3.1.3">𝑎</ci><cn type="integer" id="m3.1.4.cmml" xref="m3.1.4">0</cn></apply></apply></apply></annotation-xml></semantics></math></td><td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr>
</table>
<table id="S0.Ex4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="m4" class="ltx_Math" alttext="\int_{a}^{b}F(x)dx" display="block"><semantics id="m4a"><mrow id="m4.13" xref="m4.13.cmml"><msubsup id="m4.6" xref="m4.6.cmml"><mo largeop="true" symmetric="true" id="m4.3" xref="m4.3.cmml">∫</mo><mi id="m4.4" xref="m4.4.cmml">a</mi><mi id="m4.5" xref="m4.5.cmml">b</mi></msubsup><mrow id="m4.12" xref="m4.12.cmml"><mrow id="m4.7b" xref="m4.7.cmml"><mi id="m4.1" xref="m4.1.cmml">F</mi><mo id="m4.7c" xref="m4.7.cmml"></mo><mrow id="m4.7.1" xref="m4.7.cmml"><mo stretchy="false" id="m4.7" xref="m4.7.cmml">(</mo><mi id="m4.2" xref="m4.2.cmml">x</mi><mo stretchy="false" id="m4.7a" xref="m4.7.cmml">)</mo></mrow></mrow><mo id="m4.8" xref="m4.8.cmml"></mo><mrow id="m4.11" xref="m4.11.cmml"><mo rspace="0pt" id="m4.9" xref="m4.9.cmml">𝑑</mo><mi id="m4.10" xref="m4.10.cmml">x</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="m4b"><apply id="m4.13.cmml" xref="m4.13"><apply id="m4.6.cmml" xref="m4.6"><csymbol cd="ambiguous" id="m4.6.1.cmml" xref="m4.6">superscript</csymbol><apply id="m4.6.3.cmml" xref="m4.6"><csymbol cd="ambiguous" id="m4.6.2.cmml" xref="m4.6">subscript</csymbol><int id="m4.3.cmml" xref="m4.3"/><ci id="m4.4.cmml" xref="m4.4">𝑎</ci></apply><ci id="m4.5.cmml" xref="m4.5">𝑏</ci></apply><apply id="m4.12.cmml" xref="m4.12"><times id="m4.8.cmml" xref="m4.8"/><apply id="m4.7.cmml" xref="m4.7b"><ci id="m4.1.cmml" xref="m4.1">𝐹</ci><ci id="m4.2.cmml" xref="m4.2">𝑥</ci></apply><apply id="m4.11.cmml" xref="m4.11"><csymbol cd="latexml" id="m4.9.cmml" xref="m4.9">differential-d</csymbol><ci id="m4.10.cmml" xref="m4.10">𝑥</ci></apply></apply></apply></annotation-xml></semantics></math></td><td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr>
</table>
<table id="S0.Ex5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="m5" class="ltx_Math" alttext="a+F(a,b)" display="block"><semantics id="m5a"><mrow id="m5.7" xref="m5.7.cmml"><mi id="m5.4" xref="m5.4.cmml">a</mi><mo id="m5.5" xref="m5.5.cmml">+</mo><mrow id="m5.6c" xref="m5.6.cmml"><mi id="m5.1" xref="m5.1.cmml">F</mi><mo id="m5.6d" xref="m5.6.cmml"></mo><mrow id="m5.6.1" xref="m5.6.cmml"><mo stretchy="false" id="m5.6" xref="m5.6.cmml">(</mo><mi id="m5.2" xref="m5.2.cmml">a</mi><mo id="m5.6a" xref="m5.6.cmml">,</mo><mi id="m5.3" xref="m5.3.cmml">b</mi><mo stretchy="false" id="m5.6b" xref="m5.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="m5b"><apply id="m5.7.cmml" xref="m5.7"><plus id="m5.5.cmml" xref="m5.5"/><ci id="m5.4.cmml" xref="m5.4">𝑎</ci><apply id="m5.6.cmml" xref="m5.6c"><ci id="m5.1.cmml" xref="m5.1">𝐹</ci><ci id="m5.2.cmml" xref="m5.2">𝑎</ci><ci id="m5.3.cmml" xref="m5.3">𝑏</ci></apply></apply></annotation-xml></semantics></math></td><td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr>
</table>
<table id="S0.Ex6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="m6" class="ltx_Math" alttext="F(a,b)" display="block"><semantics id="m6a"><mrow id="m6.4c" xref="m6.4.cmml"><mi id="m6.1" xref="m6.1.cmml">F</mi><mo id="m6.4d" xref="m6.4.cmml"></mo><mrow id="m6.4.1" xref="m6.4.cmml"><mo stretchy="false" id="m6.4" xref="m6.4.cmml">(</mo><mi id="m6.2" xref="m6.2.cmml">a</mi><mo id="m6.4a" xref="m6.4.cmml">,</mo><mi id="m6.3" xref="m6.3.cmml">b</mi><mo stretchy="false" id="m6.4b" xref="m6.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="m6b"><apply id="m6.4.cmml" xref="m6.4c"><ci id="m6.1.cmml" xref="m6.1">𝐹</ci><ci id="m6.2.cmml" xref="m6.2">𝑎</ci><ci id="m6.3.cmml" xref="m6.3">𝑏</ci></apply></annotation-xml></semantics></math></td><td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr>
</table>
<table id="S0.Ex7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="m7" class="ltx_Math" alttext="F(a)+\binom{a}{b}" display="block"><semantics id="m7a"><mrow id="m7.10" xref="m7.10.cmml"><mrow id="m7.6b" xref="m7.6.cmml"><mi id="m7.4" xref="m7.4.cmml">F</mi><mo id="m7.6c" xref="m7.6.cmml"></mo><mrow id="m7.6.1" xref="m7.6.cmml"><mo stretchy="false" id="m7.6" xref="m7.6.cmml">(</mo><mi id="m7.5" xref="m7.5.cmml">a</mi><mo stretchy="false" id="m7.6a" xref="m7.6.cmml">)</mo></mrow></mrow><mo id="m7.7" xref="m7.7.cmml">+</mo><mrow id="m7.9" xref="m7.9.cmml"><mo id="m7.8" xref="m7.8.cmml">(</mo><mfrac linethickness="0pt" id="m7.3" xref="m7.9.cmml"><mi id="m7.1" xref="m7.1.cmml">a</mi><mi id="m7.2" xref="m7.2.cmml">b</mi></mfrac><mo id="m7.8a" xref="m7.8.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="m7b"><apply id="m7.10.cmml" xref="m7.10"><plus id="m7.7.cmml" xref="m7.7"/><apply id="m7.6.cmml" xref="m7.6b"><ci id="m7.4.cmml" xref="m7.4">𝐹</ci><ci id="m7.5.cmml" xref="m7.5">𝑎</ci></apply><apply id="m7.9.cmml" xref="m7.9"><csymbol cd="latexml" id="m7.8.cmml" xref="m7.8">binomial</csymbol><ci id="m7.1.cmml" xref="m7.1">𝑎</ci><ci id="m7.2.cmml" xref="m7.2">𝑏</ci></apply></apply></annotation-xml></semantics></math></td><td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr>
</table>
<table id="S0.Ex8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="m8" class="ltx_Math" alttext="\left\langle\Psi\middle|\mathcal{H}\middle|\Phi\right\rangle+\defint{a}{b}{F(x% )}{x}" display="block"><semantics id="m8a"><mrow id="m8.16" xref="m8.16.cmml"><mrow id="m8.9" xref="m8.9.cmml"><mo id="m8.8" xref="m8.8.cmml">⟨</mo><mi mathvariant="normal" id="m8.5" xref="m8.5.cmml">Ψ</mi><mo fence="true" id="m8.8a" xref="m8.8.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="m8.6" xref="m8.6.cmml">ℋ</mi><mo fence="true" id="m8.8b" xref="m8.8.cmml">|</mo><mi mathvariant="normal" id="m8.7" xref="m8.7.cmml">Φ</mi><mo id="m8.8c" xref="m8.8.cmml">⟩</mo></mrow><mo id="m8.10" xref="m8.10.cmml">+</mo><mrow id="m8.15" xref="m8.15.cmml"><msubsup id="m8.12" xref="m8.15.cmml"><mo largeop="true" symmetric="true" id="m8.11" xref="m8.11.cmml">∫</mo><mi id="m8.1" xref="m8.1.cmml">a</mi><mi id="m8.2" xref="m8.2.cmml">b</mi></msubsup><mrow id="m8.14" xref="m8.15.cmml"><mrow id="m8.3b" xref="m8.3.cmml"><mi id="m8.3.1" xref="m8.3.1.cmml">F</mi><mo id="m8.3c" xref="m8.3.cmml"></mo><mrow id="m8.3.3" xref="m8.3.cmml"><mo stretchy="false" id="m8.3" xref="m8.3.cmml">(</mo><mi id="m8.3.2" xref="m8.3.2.cmml">x</mi><mo stretchy="false" id="m8.3a" xref="m8.3.cmml">)</mo></mrow></mrow><mo id="m8.11a" xref="m8.11.cmml"></mo><mrow id="m8.13" xref="m8.15.cmml"><mo rspace="0pt" id="m8.11b" xref="m8.11.cmml">𝑑</mo><mi id="m8.4" xref="m8.4.cmml">x</mi></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="m8b"><apply id="m8.16.cmml" xref="m8.16"><plus id="m8.10.cmml" xref="m8.10"/><apply id="m8.9.cmml" xref="m8.9"><csymbol cd="latexml" id="m8.8.cmml" xref="m8.8">quantum-operator-product</csymbol><ci id="m8.5.cmml" xref="m8.5">Ψ</ci><ci id="m8.6.cmml" xref="m8.6">ℋ</ci><ci id="m8.7.cmml" xref="m8.7">Φ</ci></apply><apply id="m8.15.cmml" xref="m8.15"><int id="m8.11.cmml" xref="m8.11"/><bvar id="m8.15a.cmml" xref="m8.15"><ci id="m8.4.cmml" xref="m8.4">𝑥</ci></bvar><lowlimit id="m8.15b.cmml" xref="m8.15"><ci id="m8.1.cmml" xref="m8.1">𝑎</ci></lowlimit><uplimit id="m8.15c.cmml" xref="m8.15"><ci id="m8.2.cmml" xref="m8.2">𝑏</ci></uplimit><apply id="m8.3.cmml" xref="m8.3b"><ci id="m8.3.1.cmml" xref="m8.3.1">𝐹</ci><ci id="m8.3.2.cmml" xref="m8.3.2">𝑥</ci></apply></apply></apply></annotation-xml></semantics></math></td><td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr>
</table>
<table id="S0.Ex9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="m9" class="ltx_Math" alttext="\baddefint{a}{b}{F(x)}{x}" display="block"><semantics id="m9a"><mrow id="m9.9" xref="m9.9.cmml"><msubsup id="m9.6" xref="m9.9.cmml"><mo largeop="true" symmetric="true" id="m9.5" xref="m9.5.cmml">∫</mo><mi id="m9.1" xref="m9.1.cmml">a</mi><mi id="m9.2" xref="m9.2.cmml">b</mi></msubsup><mrow id="m9.8" xref="m9.9.cmml"><mrow id="m9.3b" xref="m9.3.cmml"><mi id="m9.3.1" xref="m9.3.1.cmml">F</mi><mo id="m9.3c" xref="m9.3.cmml"></mo><mrow id="m9.3.3" xref="m9.3.cmml"><mo stretchy="false" id="m9.3" xref="m9.3.cmml">(</mo><mi id="m9.3.2" xref="m9.3.2.cmml">x</mi><mo stretchy="false" id="m9.3a" xref="m9.3.cmml">)</mo></mrow></mrow><mo id="m9.5a" xref="m9.5.cmml"></mo><mrow id="m9.7" xref="m9.9.cmml"><mo rspace="0pt" id="m9.5b" xref="m9.5.cmml">𝑑</mo><mi id="m9.4" xref="m9.4.cmml">x</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="m9b"><apply id="m9.9.cmml" xref="m9.9"><int id="m9.5.cmml" xref="m9.5"/><bvar id="m9.9a.cmml" xref="m9.9"><ci id="m9.4.cmml" xref="m9.4">𝑥</ci></bvar><lowlimit id="m9.9b.cmml" xref="m9.9"><ci id="m9.1.cmml" xref="m9.1">𝑎</ci></lowlimit><uplimit id="m9.9c.cmml" xref="m9.9"><ci id="m9.2.cmml" xref="m9.2">𝑏</ci></uplimit><apply id="m9.3.cmml" xref="m9.3b"><ci id="m9.3.1.cmml" xref="m9.3.1">𝐹</ci><ci id="m9.3.2.cmml" xref="m9.3.2">𝑥</ci></apply></apply></annotation-xml></semantics></math></td><td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr>
</table>
<table id="S0.Ex10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="m10" class="ltx_Math" alttext="\binom{F(a)}{a+b}" display="block"><semantics id="m10a"><mrow id="m10.5" xref="m10.5.cmml"><mo id="m10.4" xref="m10.4.cmml">(</mo><mfrac linethickness="0pt" id="m10.3" xref="m10.5.cmml"><mrow id="m10.1b" xref="m10.1.cmml"><mi id="m10.1.1" xref="m10.1.1.cmml">F</mi><mo id="m10.1c" xref="m10.1.cmml"></mo><mrow id="m10.1.3" xref="m10.1.cmml"><mo stretchy="false" id="m10.1" xref="m10.1.cmml">(</mo><mi id="m10.1.2" xref="m10.1.2.cmml">a</mi><mo stretchy="false" id="m10.1a" xref="m10.1.cmml">)</mo></mrow></mrow><mrow id="m10.2" xref="m10.2.cmml"><mi id="m10.2.1" xref="m10.2.1.cmml">a</mi><mo id="m10.2.2" xref="m10.2.2.cmml">+</mo><mi id="m10.2.3" xref="m10.2.3.cmml">b</mi></mrow></mfrac><mo id="m10.4a" xref="m10.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="m10b"><apply id="m10.5.cmml" xref="m10.5"><csymbol cd="latexml" id="m10.4.cmml" xref="m10.4">binomial</csymbol><apply id="m10.1.cmml" xref="m10.1b"><ci id="m10.1.1.cmml" xref="m10.1.1">𝐹</ci><ci id="m10.1.2.cmml" xref="m10.1.2">𝑎</ci></apply><apply id="m10.2.cmml" xref="m10.2"><plus id="m10.2.2.cmml" xref="m10.2.2"/><ci id="m10.2.1.cmml" xref="m10.2.1">𝑎</ci><ci id="m10.2.3.cmml" xref="m10.2.3">𝑏</ci></apply></apply></annotation-xml></semantics></math></td><td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr>
</table>
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