1.7: Difference between revisions

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{{DISPLAYTITLE:Algebraic and Analytic Methods - 1.7 Inequalities}}
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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/1.7.E1 1.7.E1] || [[Item:Q242|<math>\left(\sum^{n}_{j=1}a_{j}b_{j}\right)^{2} \leq \left(\sum^{n}_{j=1}a_{j}^{2}\right)\left(\sum^{n}_{j=1}b_{j}^{2}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\left(\sum^{n}_{j=1}a_{j}b_{j}\right)^{2} \leq \left(\sum^{n}_{j=1}a_{j}^{2}\right)\left(\sum^{n}_{j=1}b_{j}^{2}\right)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(sum(a[j]*b[j], j = 1..n))^(2) <= (sum((a[j])^(2), j = 1..n))*(sum((b[j])^(2), j = 1..n))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((Sum[Subscript[a, j]*Subscript[b, j], {j, 1, n}, GenerateConditions->None]))^(2) <= (Sum[(Subscript[a, j])^(2), {j, 1, n}, GenerateConditions->None])*(Sum[(Subscript[b, j])^(2), {j, 1, n}, GenerateConditions->None])</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.7.E1 1.7.E1] || <math qid="Q242">\left(\sum^{n}_{j=1}a_{j}b_{j}\right)^{2} \leq \left(\sum^{n}_{j=1}a_{j}^{2}\right)\left(\sum^{n}_{j=1}b_{j}^{2}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\left(\sum^{n}_{j=1}a_{j}b_{j}\right)^{2} \leq \left(\sum^{n}_{j=1}a_{j}^{2}\right)\left(\sum^{n}_{j=1}b_{j}^{2}\right)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(sum(a[j]*b[j], j = 1..n))^(2) <= (sum((a[j])^(2), j = 1..n))*(sum((b[j])^(2), j = 1..n))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((Sum[Subscript[a, j]*Subscript[b, j], {j, 1, n}, GenerateConditions->None]))^(2) <= (Sum[(Subscript[a, j])^(2), {j, 1, n}, GenerateConditions->None])*(Sum[(Subscript[b, j])^(2), {j, 1, n}, GenerateConditions->None])</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.7.E2 1.7.E2] || [[Item:Q243|<math>\sum^{n}_{j=1}a_{j}b_{j} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}\left(\sum^{n}_{j=1}b_{j}^{q}\right)^{1/q}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum^{n}_{j=1}a_{j}b_{j} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}\left(\sum^{n}_{j=1}b_{j}^{q}\right)^{1/q}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum(a[j]*b[j], j = 1..n) <= (sum((a[j])^(p), j = 1..n))^(1/p)*(sum((b[j])^(q), j = 1..n))^(1/q)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[Subscript[a, j]*Subscript[b, j], {j, 1, n}, GenerateConditions->None] <= ((Sum[(Subscript[a, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p)*((Sum[(Subscript[b, j])^(q), {j, 1, n}, GenerateConditions->None]))^(1/q)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.7.E2 1.7.E2] || <math qid="Q243">\sum^{n}_{j=1}a_{j}b_{j} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}\left(\sum^{n}_{j=1}b_{j}^{q}\right)^{1/q}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum^{n}_{j=1}a_{j}b_{j} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}\left(\sum^{n}_{j=1}b_{j}^{q}\right)^{1/q}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum(a[j]*b[j], j = 1..n) <= (sum((a[j])^(p), j = 1..n))^(1/p)*(sum((b[j])^(q), j = 1..n))^(1/q)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[Subscript[a, j]*Subscript[b, j], {j, 1, n}, GenerateConditions->None] <= ((Sum[(Subscript[a, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p)*((Sum[(Subscript[b, j])^(q), {j, 1, n}, GenerateConditions->None]))^(1/q)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/1.7.E3 1.7.E3] || [[Item:Q244|<math>\left(\sum^{n}_{j=1}(a_{j}+b_{j})^{p}\right)^{1/p} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}+\left(\sum^{n}_{j=1}b_{j}^{p}\right)^{1/p}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\left(\sum^{n}_{j=1}(a_{j}+b_{j})^{p}\right)^{1/p} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}+\left(\sum^{n}_{j=1}b_{j}^{p}\right)^{1/p}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(sum((a[j]+ b[j])^(p), j = 1..n))^(1/p) <= (sum((a[j])^(p), j = 1..n))^(1/p)+(sum((b[j])^(p), j = 1..n))^(1/p)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((Sum[(Subscript[a, j]+ Subscript[b, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p) <= ((Sum[(Subscript[a, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p)+((Sum[(Subscript[b, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.7.E3 1.7.E3] || <math qid="Q244">\left(\sum^{n}_{j=1}(a_{j}+b_{j})^{p}\right)^{1/p} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}+\left(\sum^{n}_{j=1}b_{j}^{p}\right)^{1/p}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\left(\sum^{n}_{j=1}(a_{j}+b_{j})^{p}\right)^{1/p} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}+\left(\sum^{n}_{j=1}b_{j}^{p}\right)^{1/p}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(sum((a[j]+ b[j])^(p), j = 1..n))^(1/p) <= (sum((a[j])^(p), j = 1..n))^(1/p)+(sum((b[j])^(p), j = 1..n))^(1/p)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((Sum[(Subscript[a, j]+ Subscript[b, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p) <= ((Sum[(Subscript[a, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p)+((Sum[(Subscript[b, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.7.E7 1.7.E7] || [[Item:Q248|<math>H \leq G</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>H \leq G</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">H <= ((a[1]*a[2] .. a[n])^(1/n))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">H <= ((Subscript[a, 1]*Subscript[a, 2] \[Ellipsis]Subscript[a, n])^(1/n))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.7.E7 1.7.E7] || <math qid="Q248">H \leq G</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>H \leq G</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">H <= ((a[1]*a[2] .. a[n])^(1/n))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">H <= ((Subscript[a, 1]*Subscript[a, 2] \[Ellipsis]Subscript[a, n])^(1/n))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.7.E9 1.7.E9] || [[Item:Q250|<math>M(r) \leq M(s)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>M(r) \leq M(s)</syntaxhighlight> || <math>r < s</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((p[1]*(a[1])^(r)+ p[2]*(a[2])^(r)+ .. + p[n]*(a[n])^(r))^(1/r)) <= M(s)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((Subscript[p, 1]*(Subscript[a, 1])^(r)+ Subscript[p, 2]*(Subscript[a, 2])^(r)+ \[Ellipsis]+ Subscript[p, n]*(Subscript[a, n])^(r))^(1/r)) <= M[s]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.7.E9 1.7.E9] || <math qid="Q250">M(r) \leq M(s)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>M(r) \leq M(s)</syntaxhighlight> || <math>r < s</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((p[1]*(a[1])^(r)+ p[2]*(a[2])^(r)+ .. + p[n]*(a[n])^(r))^(1/r)) <= M(s)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((Subscript[p, 1]*(Subscript[a, 1])^(r)+ Subscript[p, 2]*(Subscript[a, 2])^(r)+ \[Ellipsis]+ Subscript[p, n]*(Subscript[a, n])^(r))^(1/r)) <= M[s]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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Latest revision as of 10:59, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
1.7.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(\sum^{n}_{j=1}a_{j}b_{j}\right)^{2} \leq \left(\sum^{n}_{j=1}a_{j}^{2}\right)\left(\sum^{n}_{j=1}b_{j}^{2}\right)}
\left(\sum^{n}_{j=1}a_{j}b_{j}\right)^{2} \leq \left(\sum^{n}_{j=1}a_{j}^{2}\right)\left(\sum^{n}_{j=1}b_{j}^{2}\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(sum(a[j]*b[j], j = 1..n))^(2) <= (sum((a[j])^(2), j = 1..n))*(sum((b[j])^(2), j = 1..n))
 
((Sum[Subscript[a, j]*Subscript[b, j], {j, 1, n}, GenerateConditions->None]))^(2) <= (Sum[(Subscript[a, j])^(2), {j, 1, n}, GenerateConditions->None])*(Sum[(Subscript[b, j])^(2), {j, 1, n}, GenerateConditions->None])
 Skipped - no semantic math Skipped - no semantic math - -
1.7.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum^{n}_{j=1}a_{j}b_{j} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}\left(\sum^{n}_{j=1}b_{j}^{q}\right)^{1/q}}
\sum^{n}_{j=1}a_{j}b_{j} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}\left(\sum^{n}_{j=1}b_{j}^{q}\right)^{1/q}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
sum(a[j]*b[j], j = 1..n) <= (sum((a[j])^(p), j = 1..n))^(1/p)*(sum((b[j])^(q), j = 1..n))^(1/q)
 
Sum[Subscript[a, j]*Subscript[b, j], {j, 1, n}, GenerateConditions->None] <= ((Sum[(Subscript[a, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p)*((Sum[(Subscript[b, j])^(q), {j, 1, n}, GenerateConditions->None]))^(1/q)
 Skipped - no semantic math Skipped - no semantic math - -
1.7.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(\sum^{n}_{j=1}(a_{j}+b_{j})^{p}\right)^{1/p} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}+\left(\sum^{n}_{j=1}b_{j}^{p}\right)^{1/p}}
\left(\sum^{n}_{j=1}(a_{j}+b_{j})^{p}\right)^{1/p} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}+\left(\sum^{n}_{j=1}b_{j}^{p}\right)^{1/p}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(sum((a[j]+ b[j])^(p), j = 1..n))^(1/p) <= (sum((a[j])^(p), j = 1..n))^(1/p)+(sum((b[j])^(p), j = 1..n))^(1/p)
 
((Sum[(Subscript[a, j]+ Subscript[b, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p) <= ((Sum[(Subscript[a, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p)+((Sum[(Subscript[b, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p)
 Skipped - no semantic math Skipped - no semantic math - -
1.7.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle H \leq G}
H \leq G		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
H <= ((a[1]*a[2] .. a[n])^(1/n))
 
H <= ((Subscript[a, 1]*Subscript[a, 2] \[Ellipsis]Subscript[a, n])^(1/n))
 Skipped - no semantic math Skipped - no semantic math - -
1.7.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle M(r) \leq M(s)}
M(r) \leq M(s)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle r < s} 
((p[1]*(a[1])^(r)+ p[2]*(a[2])^(r)+ .. + p[n]*(a[n])^(r))^(1/r)) <= M(s)
 
((Subscript[p, 1]*(Subscript[a, 1])^(r)+ Subscript[p, 2]*(Subscript[a, 2])^(r)+ \[Ellipsis]+ Subscript[p, n]*(Subscript[a, n])^(r))^(1/r)) <= M[s]
 Skipped - no semantic math Skipped - no semantic math - -
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