Lih Wang's and Dittert's conjectures on permanents release_s5u3ntiac5f67ef5bhjc6n7kae

Abstract

<jats:title>Abstract</jats:title> Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_001.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{\Omega }_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the set of all doubly stochastic matrices of order <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_002.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> <jats:tex-math>n</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Lih and Wang conjectured that for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_003.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> <jats:tex-math>n\ge 3</jats:tex-math> </jats:alternatives> </jats:inline-formula>, per<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_004.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:msub> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mi>t</m:mi> </m:math> <jats:tex-math>\left(t{J}_{n}+\left(1-t)A)\le t</jats:tex-math> </jats:alternatives> </jats:inline-formula>per<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_005.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{J}_{n}+\left(1-t)</jats:tex-math> </jats:alternatives> </jats:inline-formula>per<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_006.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> </m:math> <jats:tex-math>A</jats:tex-math> </jats:alternatives> </jats:inline-formula>, for all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_007.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>A\in {\Omega }_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_008.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0.5</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> <jats:tex-math>t\in \left[0.5,1]</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_009.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{J}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_010.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>×</m:mo> <m:mi>n</m:mi> </m:math> <jats:tex-math>n\times n</jats:tex-math> </jats:alternatives> </jats:inline-formula> matrix with each entry equal to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_011.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:mfrac> </m:math> <jats:tex-math>\frac{1}{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. This conjecture was proved partially for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_012.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mn>5</m:mn> </m:math> <jats:tex-math>n\le 5</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_013.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{K}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the set of nonnegative <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_014.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>×</m:mo> <m:mi>n</m:mi> </m:math> <jats:tex-math>n\times n</jats:tex-math> </jats:alternatives> </jats:inline-formula> matrices whose elements have sum <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_015.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> <jats:tex-math>n</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_016.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ϕ</m:mi> </m:math> <jats:tex-math>\phi </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a real valued function defined on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_017.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{K}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_018.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ϕ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msubsup> <m:mrow> <m:mrow> <m:mo>∏</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> <m:msub> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msubsup> <m:mrow> <m:mrow> <m:mo>∏</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>\phi \left(X)={\prod }_{i=1}^{n}{r}_{i}+{\prod }_{j=1}^{n}{c}_{j}</jats:tex-math> </jats:alternatives> </jats:inline-formula> - per<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_019.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> <jats:tex-math>X</jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_020.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>X\in {K}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with row sum vector <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_021.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\left({r}_{1},{r}_{2},\ldots {r}_{n})</jats:tex-math> </jats:alternatives> </jats:inline-formula> and column sum vector <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_022.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\left({c}_{1},{c}_{2},\ldots {c}_{n})</jats:tex-math> </jats:alternatives> </jats:inline-formula>. A matrix <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_023.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>A\in {K}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is called a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_024.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ϕ</m:mi> </m:math> <jats:tex-math>\phi </jats:tex-math> </jats:alternatives> </jats:inline-formula>-maximizing matrix if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_025.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ϕ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>≥</m:mo> <m:mi>ϕ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\phi \left(A)\ge \phi \left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_026.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>X\in {K}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Dittert conjectured that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_027.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{J}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the unique <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_028.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ϕ</m:mi> </m:math> <jats:tex-math>\phi </jats:tex-math> </jats:alternatives> </jats:inline-formula>-maximizing matrix on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_029.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{K}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Sinkhorn proved the conjecture for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_030.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:math> <jats:tex-math>n=2</jats:tex-math> </jats:alternatives> </jats:inline-formula> and Hwang proved it for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_031.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>3</m:mn> </m:math> <jats:tex-math>n=3</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this article, we prove the Lih and Wang partially for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_032.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>6</m:mn> </m:math> <jats:tex-math>n=6</jats:tex-math> </jats:alternatives> </jats:inline-formula>. It is also proved that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_033.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> </m:math> <jats:tex-math>A</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_034.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ϕ</m:mi> </m:math> <jats:tex-math>\phi </jats:tex-math> </jats:alternatives> </jats:inline-formula>-maximizing matrix on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_035.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:msub> </m:math> <jats:tex-math>{K}_{4}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2024-0006_eq_036.png"/> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> </m:math> <jats:tex-math>A</jats:tex-math> </jats:alternatives> </jats:inline-formula> is fully indecomposable.
In application/xml+jats format

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