UnitStep[Product[x - Subscript[a, k], {k, 1, n}]] == Sum[(-1)^(n - k) UnitStep[x - Subscript[a, k]], {k, 1, n}] - 2 Floor[(n - 1)/2] + n - 1 /; x != Subscript[a, k] && Subscript[a, k] < Subscript[a, k + 1] && Element[Subscript[a, k], Reals] && 1 <= k <= n

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8800; </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &lt; </mo> <msub> <mi> a </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> UnitStep </ci> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <and /> <apply> <neq /> <ci> x </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> </apply> <apply> <lt /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> <reals /> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <ci> n </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29