Verifying DLMF with Maple and Mathematica: Difference between revisions

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By clicking on a chapter of the [https://dlmf.nist.gov/ DLMF] below, you will see a large table that looks like this:
By clicking on a chapter of the [https://dlmf.nist.gov/ DLMF] below, you will see a large table that looks like this:


{| class="wikitable sortable"
<div style="width: 100%; height: 75vh; overflow: auto;">
{| class="wikitable sortable" style="margin: 0;"
|-
|-
! DLMF !! Formula !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | DLMF  
! scope="col" style="position: sticky; top: 0;" | Formula  
! scope="col" style="position: sticky; top: 0;" | Constraints
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
|-
|-
| Link to DLMF || Formula || <code>Translation to Maple</code> || <code>Translation to Mathematica</code> || Symbolic Result of Maple || Symbolic Result of Mathematica || Numeric Result of Maple || Numeric Result of Mathematica
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/4.12.E1 4.12.E1] || [[Item:Q1629|<math>\phi(x+1) = e^{\phi(x)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\phi(x+1) = e^{\phi(x)}</syntaxhighlight> || <math>-1 < x, x < \infty</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">phi(x + 1) = exp(phi(x))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Phi][x + 1] == Exp[\[Phi][x]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
|-
| [https://dlmf.nist.gov/11.5.E2 11.5.E2] || [[Item:Q3969|<math>\StruveK{\nu}@{z} = \frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StruveK{\nu}@{z} = \frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{2}}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0, \realpart@@{(\nu+\tfrac{1}{2})} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0, \realpart@@{(n+\nu+\tfrac{3}{2})} > 0</math> || <syntaxhighlight lang=mathematica>StruveH(nu, z) - BesselY(nu, z) = (2*((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(1)/(2)))*int(exp(- z*t)*(1 + (t)^(2))^(nu -(1)/(2)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StruveH[\[Nu], z] - BesselY[\[Nu], z] == Divide[2*(Divide[1,2]*z)^\[Nu],Sqrt[Pi]*Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*t]*(1 + (t)^(2))^(\[Nu]-Divide[1,2]), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [15 / 25]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.9495382353861556, -0.46093572348323536]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.7706973036767981, -0.20650772012904173]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
|-
| [https://dlmf.nist.gov/18.5.E8 18.5.E8] || [[Item:Q5516|<math>\JacobipolyP{\alpha}{\beta}{n}@{x} = 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\JacobipolyP{\alpha}{\beta}{n}@{x} = 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiP(n, alpha, beta, x) = (2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiP[n, \[Alpha], \[Beta], x] == (2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 81] || Successful [Tested: 81]
|-
| [https://dlmf.nist.gov/18.5.E8 18.5.E8] || [[Item:Q5516|<math>2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\left(\frac{x+1}{2}\right)^{n}\genhyperF{2}{1}@@{-n,-n-\beta}{\alpha+1}{\frac{x-1}{x+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\left(\frac{x+1}{2}\right)^{n}\genhyperF{2}{1}@@{-n,-n-\beta}{\alpha+1}{\frac{x-1}{x+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n) = (pochhammer(alpha + 1, n))/(factorial(n))*((x + 1)/(2))^(n)* hypergeom([- n , - n - beta], [alpha + 1], (x - 1)/(x + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(Divide[x + 1,2])^(n)* HypergeometricPFQ[{- n , - n - \[Beta]}, {\[Alpha]+ 1}, Divide[x - 1,x + 1]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 81] || Successful [Tested: 81]
|}
|}
</div>


The result tables do not contain every equation with a label in the [https://dlmf.nist.gov/ DLMF] since quite a few equations were skipped (see explanations in the paper).
The result tables do not contain every equation with a label in the [https://dlmf.nist.gov/ DLMF] since quite a few equations were skipped (see explanations in the paper).
Line 144: Line 163:
||  7 || 14.9% ||  33 || [  5 /  28] ||  1  
||  7 || 14.9% ||  33 || [  5 /  28] ||  1  
| style="border-right: solid 2px #aaa;" |  5  
| style="border-right: solid 2px #aaa;" |  5  
||  4 ||  6.8% ||  38 || [  6 /  32] ||   7 ||  9  
||  4 ||  6.8% ||  38 || [  6 /  32] || {| class="wikitable sortable"
|-
! DLMF !! Formula !! Constraints !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
|-  7 ||  9  
|-
|-
! scope="row" style="text-align: left; border-right: solid 1px #000"|  3. [[Results of Numerical Methods|NM]]  
! scope="row" style="text-align: left; border-right: solid 1px #000"|  3. [[Results of Numerical Methods|NM]]  

Revision as of 11:12, 22 May 2021

This page presents the results of the publication: Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems.

Bug Reports

You can find a PDF with commands that illustrate the encountered errors in Mathematica here: File:Mathematica Bugs Overview.pdf

We provide the same file in the Wolfram system notebook format (NB) here: File:Mathematica Bugs Notebook File.nb

DLMF Translations and Results

In the following, we present the translations of the DLMF equations to the CAS Maple and Mathematica.

DLMF Formula Translations
Maple		 Translations
Mathematica		 Symbolic Evaluation
Maple		 Symbolic Evaluation
Mathematica		 Numeric Evaluation
Maple		 Numeric Evaluation
Mathematica
DLMF 6,623 4,114 (62.1%) 4,713 (71.2%) 1,084 (26.3%) 1,235 (26.2%) 698 (26.7%) 784 (22.6%)

By clicking on a chapter of the DLMF below, you will see a large table that looks like this:

DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.12.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phi(x+1) = e^{\phi(x)}}
\phi(x+1) = e^{\phi(x)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -1 < x, x < \infty} 
phi(x + 1) = exp(phi(x))
 
\[Phi][x + 1] == Exp[\[Phi][x]]
 Skipped - no semantic math Skipped - no semantic math - -
11.5.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StruveK{\nu}@{z} = \frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{2}}\diff{t}}
\StruveK{\nu}@{z} = \frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{2}}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{z} > 0, \realpart@@{(\nu+\tfrac{1}{2})} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0, \realpart@@{(n+\nu+\tfrac{3}{2})} > 0} 
StruveH(nu, z) - BesselY(nu, z) = (2*((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(1)/(2)))*int(exp(- z*t)*(1 + (t)^(2))^(nu -(1)/(2)), t = 0..infinity)

StruveH[\[Nu], z] - BesselY[\[Nu], z] == Divide[2*(Divide[1,2]*z)^\[Nu],Sqrt[Pi]*Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*t]*(1 + (t)^(2))^(\[Nu]-Divide[1,2]), {t, 0, Infinity}, GenerateConditions->None]
Successful Successful -
Failed [15 / 25]
Result: Complex[0.9495382353861556, -0.46093572348323536]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 1.5]}


Result: Complex[0.7706973036767981, -0.20650772012904173]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 0.5]}

... skip entries to safe data
18.5.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \JacobipolyP{\alpha}{\beta}{n}@{x} = 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}}
\JacobipolyP{\alpha}{\beta}{n}@{x} = 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
JacobiP(n, alpha, beta, x) = (2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n)

JacobiP[n, \[Alpha], \[Beta], x] == (2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]
Failure Failure Successful [Tested: 81] Successful [Tested: 81]
18.5.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\left(\frac{x+1}{2}\right)^{n}\genhyperF{2}{1}@@{-n,-n-\beta}{\alpha+1}{\frac{x-1}{x+1}}}
2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\left(\frac{x+1}{2}\right)^{n}\genhyperF{2}{1}@@{-n,-n-\beta}{\alpha+1}{\frac{x-1}{x+1}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n) = (pochhammer(alpha + 1, n))/(factorial(n))*((x + 1)/(2))^(n)* hypergeom([- n , - n - beta], [alpha + 1], (x - 1)/(x + 1))

(2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(Divide[x + 1,2])^(n)* HypergeometricPFQ[{- n , - n - \[Beta]}, {\[Alpha]+ 1}, Divide[x - 1,x + 1]]
Failure Failure Successful [Tested: 81] Successful [Tested: 81]

The result tables do not contain every equation with a label in the DLMF since quite a few equations were skipped (see explanations in the paper).

Translations and Evaluations of the Digital Library of Mathematical Functions

  1. Algebraic and Analytic Methods
  2. Asymptotic Approximations
  3. Numerical Methods
  4. Elementary Functions
  5. Gamma Function
  6. Exponential, Logarithmic, Sine, and Cosine Integrals
  7. Error Functions, Dawson’s and Fresnel Integrals
  8. Incomplete Gamma and Related Functions
  9. Airy and Related Functions
  10. Bessel Functions I & Bessel Functions II
  11. Struve and Related Functions
  12. Parabolic Cylinder Functions
  13. Confluent Hypergeometric Functions
  14. Legendre and Related Functions
  15. Hypergeometric Function
  16. Generalized Hypergeometric Functions and Meijer G-Function
  17. q-Hypergeometric and Related Functions
  18. Orthogonal Polynomials
  19. Elliptic Integrals I & Elliptic Integrals II
  20. Theta Functions
  21. Multidimensional Theta Functions
  22. Jacobian Elliptic Functions
  23. Weierstrass Elliptic and Modular Functions
  24. Bernoulli and Euler Polynomials
  25. Zeta and Related Functions
  26. Combinatorial Analysis
  27. Functions of Number Theory
  28. Mathieu Functions and Hill’s Equation
  29. Lamé Functions
  30. Spheroidal Wave Functions
  31. Heun Functions
  32. Painlevé Transcendents
  33. Coulomb Functions
  34. 3j,6j,9j Symbols
  35. Functions of Matrix Argument
  36. Integrals with Coalescing Saddles

Translations and Evaluations Overview Table

Meaning
2C Chapter Code
S Successful
% Percentage
F Fail
P/T Partially / Totally Failed
A Aborted
E Errors
Base The baseline performance of the translator
Maple The CAS Maple 2020
Mathematica The CAS Mathematica
Symbolic Numeric
Formulae Translations Maple Mathematica Maple Mathematica
2C Total Base Maple Math S % F S % F S % F [P/T] A E S % F [P/T] A E
1. AL 227 60 102 103 36 35.3% 60 34 33.0% 69 14 23.3% 35 [ 12 / 23] 7 4 14 20.3% 40 [ 9 / 31] 11 4
2. AS 136 33 65 65 6 9.2% 47 6 9.2% 59 7 14.9% 33 [ 5 / 28] 1 5 4 6.8% 38 [ 6 / 32] class="wikitable sortable"
DLMF Formula Constraints Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
3. NM 53 36 40 40 6 15.0% 31 5 12.5% 35 1 3.2% 27 [ 9 / 18] 0 2 0 0.0% 29 [ 8 / 21] 6 0
4. EF 569 353 494 564 270 54.7% 221 304 53.9% 260 88 39.8% 126 [ 64 / 62] 0 6 110 42.3% 146 [ 55 / 91] 2 0
5. GA 144 38 130 139 41 31.5% 76 65 46.8% 74 39 51.3% 25 [ 12 / 13] 4 8 30 40.5% 20 [ 9 / 11] 13 9
6. EX 107 21 56 77 13 23.2% 43 18 23.4% 59 10 23.2% 31 [ 13 / 18] 0 2 23 39.0% 32 [ 6 / 26] 4 0
7. ER 149 35 101 120 52 51.5% 47 45 37.5% 75 21 44.7% 23 [ 10 / 13] 2 1 21 28.0% 43 [ 13 / 30] 9 1
8. IG 204 84 160 163 51 31.9% 102 65 39.9% 98 27 26.5% 61 [ 20 / 41] 9 5 22 22.4% 44 [ 19 / 25] 16 15
9. AI 235 36 180 179 54 30.0% 124 69 38.5% 110 34 27.4% 75 [ 41 / 34] 4 8 30 27.3% 58 [ 38 / 20] 14 7
10. BS I & II 653 143 392 486 80 20.4% 209 115 23.7% 371 86 41.1% 59 [ 41 / 18] 52 12 90 24.2% 151 [ 57 / 94] 92 18
11. ST 124 48 121 112 39 32.2% 73 36 32.1% 76 25 34.2% 40 [ 14 / 26] 3 5 21 27.6% 33 [ 8 / 25] 10 11
12. PC 106 33 79 90 25 31.6% 50 18 20.0% 72 15 30.0% 24 [ 15 / 9] 11 0 13 18.0% 43 [ 15 / 28] 12 3
13. CH 260 126 252 254 75 29.8% 143 69 27.2% 185 14 9.8% 90 [ 55 / 35] 37 2 23 12.4% 95 [ 59 / 36] 45 21
14. LE 238 166 230 229 30 13.0% 163 30 13.1% 199 40 24.5% 93 [ 57 / 36] 18 12 59 29.6% 92 [ 54 / 38] 41 5
15. HY 206 148 198 197 46 23.2% 115 53 26.9% 144 17 14.8% 52 [ 34 / 18] 23 23 23 16.0% 77 [ 52 / 25] 29 6
16. GH 53 20 23 25 3 13.0% 16 2 8.0% 23 1 6.2% 9 [ 8 / 1] 6 0 1 4.3% 10 [ 7 / 3] 9 2
17. QH 175 1 53 124 23 43.4% 24 6 4.8% 118 0 0.0% 0 [ 0 / 0] 1 23 13 11.0% 57 [ 52 / 5] 39 5
18. OP 468 132 235 288 65 27.6% 148 101 35.1% 185 67 45.3% 50 [ 32 / 18] 14 17 45 24.3% 68 [ 31 / 37] 52 12
19. EL I & II 516 103 252 416 39 15.5% 192 51 12.2% 365 18 9.4% 123 [ 44 / 79] 34 17 18 4.9% 264 [ 49 / 215] 61 15
20. TH 128 52 98 98 10 10.2% 68 1 1.0% 97 0 0.0% 32 [ 17 / 15] 20 16 33 34.0% 40 [ 25 / 15] 24 0
21. MT - - - - - - - - - - - - - - - - - - - - - -
22. JA 264 115 232 238 46 19.8% 176 30 12.6% 206 20 11.4% 116 [ 25 / 91] 36 4 22 10.7% 131 [ 39 / 92] 51 0
23. WE 164 7 19 34 1 5.3% 16 4 11.8% 30 0 0.0% 14 [ 2 / 12] 1 1 2 6.7% 23 [ 9 / 14] 2 3
24. BP 175 31 117 148 15 12.8% 101 23 15.5% 125 67 66.3% 32 [ 19 / 13] 1 1 78 62.4% 33 [ 22 / 11] 14 0
25. ZE 154 28 124 120 19 15.3% 90 48 40.0% 72 43 47.8% 29 [ 18 / 11] 10 8 22 30.5% 22 [ 6 / 16] 22 3
26. CM 136 31 78 87 20 25.6% 50 19 21.8% 68 30 60.0% 11 [ 8 / 3] 2 7 44 64.7% 18 [ 10 / 8] 5 1
27. NT 79 5 26 15 3 11.5% 17 6 40.0% 9 2 11.8% 6 [ 3 / 3] 0 8 3 33.3% 6 [ 3 / 3] 0 0
28. MA 267 52 97 110 7 7.2% 80 7 6.4% 103 6 7.5% 32 [ 12 / 20] 26 15 3 2.9% 48 [ 13 / 35] 33 17
29. LA 111 11 23 22 0 0.0% 21 0 0.0% 22 0 0.0% 19 [ 2 / 17] 0 2 0 0.0% 21 [ 1 / 20] 0 1
30. SW 71 14 19 26 0 0.0% 18 0 0.0% 26 0 0.0% 18 [ 5 / 13] 0 0 0 0.0% 19 [ 2 / 17] 5 1
31. HE 35 29 22 15 5 22.7% 13 2 13.3% 13 2 15.4% 10 [ 0 / 10] 0 1 0 0.0% 8 [ 0 / 8] 5 0
32. PT 67 43 57 57 3 5.3% 51 3 5.3% 54 1 2.0% 44 [ 7 / 37] 4 2 0 0.0% 41 [ 2 / 39] 8 5
33. CW 108 21 14 11 1 7.1% 13 0 0.0% 11 0 0.0% 5 [ 2 / 3] 0 8 0 0.0% 11 [ 2 / 9] 0 0
34. TJ 57 0 1 37 0 0.0% 1 0 0.0% 37 0 0.0% 1 [ 0 / 1] 0 0 14 37.8% 10 [ 5 / 5] 13 0
35. FM - - - - - - - - - - - - - - - - - - - - - -
36. IC 106 12 24 24 0 0.0% 19 0 0.0% 24 3 15.8% 12 [ 1 / 11] 3 1 3 12.5% 13 [ 1 / 12] 1 6
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum} 6545 2067 4114 4713 1084 26.3% 2618 1235 26.2% 3474 698 26.7% 1357 [607 / 750] 329 226 784 22.6% 1784 [687 / 1097] 655 180
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