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Max






Mathematica Notation

Traditional Notation









Elementary Functions > Max[x1 ,x2,...] > Summation





http://functions.wolfram.com/01.34.23.0001.01









  


  










Input Form





Sum[Max[k/m, l/n], {k, 0, m - 1}, {l, 0, n - 1}] == (2 m n)/3 - (m + n + 1)/4 - (m^2 + n^2 - GCD[m, n]^2)/(12 m n) /; Element[m, Integers] && m > 0 && Element[n, Integers] && n > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["Max", "[", RowBox[List[FractionBox["k", "m"], ",", FractionBox["l", "n"]]], "]"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["2", "m", " ", "n"]], "3"], "-", FractionBox[RowBox[List["m", "+", "n", "+", "1"]], "4"], "-", FractionBox[RowBox[List[SuperscriptBox["m", "2"], "+", SuperscriptBox["n", "2"], "-", SuperscriptBox[RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]], "2"]]], RowBox[List["12", " ", "m", " ", "n"]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> max </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> k </mi> <mi> m </mi> </mfrac> <mo> , </mo> <mfrac> <mi> l </mi> <mi> n </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mn> 3 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> l </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <max /> <apply> <times /> <ci> k </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> l </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> <ci> n </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <gcd /> <ci> m </ci> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> m </ci> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], RowBox[List["m_", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l_", "=", "0"]], RowBox[List["n_", "-", "1"]]], RowBox[List["Max", "[", RowBox[List[FractionBox["k_", "m_"], ",", FractionBox["l_", "n_"]]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", " ", "m", " ", "n"]], "3"], "-", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["m", "+", "n", "+", "1"]], ")"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["m", "2"], "+", SuperscriptBox["n", "2"], "-", SuperscriptBox[RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]], "2"]]], RowBox[List["12", " ", "m", " ", "n"]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2002-12-18





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