The main objective of this article is to study the ordered partial transformations <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0161_eq_001.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">PO</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>{mathcal{PO}}left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula> of a poset <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0161_eq_002.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>. The findings show that the set of all partial transformations of a poset with a pointwise order is not necessarily a pomonoid. Some conditions are implemented to guarantee that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0161_eq_003.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">PO</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>{mathcal{PO}}left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a pomonoid and this pomonoid is denoted by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0161_eq_004.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="script">PO</m:mi> </m:mrow> <m:mrow> <m:mi>↑</m:mi> </m:mrow> </m:msup> <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>{{mathcal{PO}}}^{uparrow }left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Moreover, we determine the necessary conditions in order that the partial order-embedding transformations define the ordered version of the symmetric inverse monoid. The findings show that this set is an inverse pomonoid and we will denote it by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0161_eq_005.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="script">ℐPO</m:mi> </m:mrow> <m:mrow> <m:mi>↑</m:mi> </m:mrow> </m:msup> <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>{{mathcal{ {mathcal I} PO}}}^{uparrow }left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In case the order on the poset <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0161_eq_006.png
本文的主要目的是研究正集 X X 的有序部分变换 PO ( X ) {mathcal{PO}}left(X) 。研究结果表明，poset 的所有有序部分变换的集合并不一定是一个 pomonoid。我们提出了一些条件来保证 PO ( X ) {mathcal{PO}}left(X) 是一个 pomonoid，这个 pomonoid 用 PO ↑ ( X ) {{mathcal{PO}}^{uparrow }left(X) 表示。此外，我们还确定了一些必要条件，以使部分有序嵌入变换定义对称逆单元的有序版本。研究结果表明，这个集合是一个逆单元集，我们将用ℐPO ↑ ( X ) {{mathcal{ {mathcal I} PO}}^{uparrow }left(X) 来表示它。如果集合 X X 上的阶是全阶，我们将探讨 PO ↑ ( X ) {{mathcal{PO}}^{uparrow }/left(X)和ℐPO ↑ ( X ) {{mathcal{ {mathcal I} PO}}^{uparrow }/left(X)的一些性质，包括回归性、单一性和可逆性。
{"title":"On pomonoid of partial transformations of a poset","authors":"Bana Al Subaiei","doi":"10.1515/math-2023-0161","DOIUrl":"https://doi.org/10.1515/math-2023-0161","url":null,"abstract":"The main objective of this article is to study the ordered partial transformations <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0161_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">PO</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>{mathcal{PO}}left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula> of a poset <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0161_eq_002.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>. The findings show that the set of all partial transformations of a poset with a pointwise order is not necessarily a pomonoid. Some conditions are implemented to guarantee that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0161_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">PO</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>{mathcal{PO}}left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a pomonoid and this pomonoid is denoted by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0161_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"script\">PO</m:mi> </m:mrow> <m:mrow> <m:mi>↑</m:mi> </m:mrow> </m:msup> <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>{{mathcal{PO}}}^{uparrow }left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Moreover, we determine the necessary conditions in order that the partial order-embedding transformations define the ordered version of the symmetric inverse monoid. The findings show that this set is an inverse pomonoid and we will denote it by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0161_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"script\">ℐPO</m:mi> </m:mrow> <m:mrow> <m:mi>↑</m:mi> </m:mrow> </m:msup> <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>{{mathcal{ {mathcal I} PO}}}^{uparrow }left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In case the order on the poset <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0161_eq_006.png","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"25 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138687900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
我们已经确定了 (0) 的局部解的存在性和唯一性。1) ∂ t u + ∂ x 5 u - u ∂ x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x <；1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t >；0 ,left{begin{array}{ll}{partial }_{t}u+{partial }_{x}^{5}u-u{partial }_{x}u=0,& 0lt xlt 1,hspace{1.0lt xlt 1,uleft(0,t)={h}_{1}left(t),uleft(1,t)={h}_{2}left(t),hspace{0.33em} {partial }_{x}uleft(1,t)={h}_{3}left(t),& {partial }_{x}uleft(0,t)={h}_{4}left(t),hspace{0.33em}{partial }_{x}^{2}uleft(1,t)={h}_{5}left(t),& tgt 0,end{array}right. in the study of Zhao and Zhang [Non-homogeneous boundary value problem of the fifth-order KdV equations posed on a bounded interval, J. Math.Anal.Appl. 470 (2019)，251-278]。一个问题自然而然地产生了：局部解能否扩展为全局解？本文将探讨这个问题。首先，通过一系列逻辑推导，建立一个全局先验估计，然后将局部解自然扩展为全局解。
In this article, without assuming the compactness of semigroup, we deal with the existence and uniqueness of a mild solution for semilinear impulsive evolution equation with nonlocal condition in a reflexive Banach space by applying the approximate solvability method and Yosida approximations of the infinitesimal generator of C0-semigroup.
在本文中，我们在不假设半群紧凑性的情况下，通过应用近似可解性方法和 C 0 半群无穷小生成器的 Yosida 近似，处理了在反身巴纳赫空间中具有非局部条件的半线性脉冲演化方程的温和解的存在性和唯一性问题。
{"title":"Approximate solvability method for nonlocal impulsive evolution equation","authors":"Weifeng Ma, Yongxiang Li","doi":"10.1515/math-2023-0155","DOIUrl":"https://doi.org/10.1515/math-2023-0155","url":null,"abstract":"In this article, without assuming the compactness of semigroup, we deal with the existence and uniqueness of a mild solution for semilinear impulsive evolution equation with nonlocal condition in a reflexive Banach space by applying the approximate solvability method and Yosida approximations of the infinitesimal generator of <jats:italic>C</jats:italic> <jats:sub>0</jats:sub>-semigroup.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"105 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138687482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we consider the problem of extending Hamilton’s principle to the class of natural mechanical systems with random perturbing forces of white noise type. By the method of moment functions, we construct the functionals taking a stationary value on the solutions of a given stochastic equation of Lagrangian structure.
{"title":"Construction of a functional by a given second-order Ito stochastic equation","authors":"Marat Tleubergenov, Gulmira Vassilina, Shakhmira Ismailova","doi":"10.1515/math-2023-0148","DOIUrl":"https://doi.org/10.1515/math-2023-0148","url":null,"abstract":"In this article, we consider the problem of extending Hamilton’s principle to the class of natural mechanical systems with random perturbing forces of white noise type. By the method of moment functions, we construct the functionals taking a stationary value on the solutions of a given stochastic equation of Lagrangian structure.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"81 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138575883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study the m{mathfrak{m}}-WG∘{}^{circ } inverse which presents a generalization of the m{mathfrak{m}}-WG inverse in the Minkowski space. We first show the existence and the uniqueness of the generalized inverse. Then, we discuss several properties and characterizations of the m{mathfrak{m}}-WG∘{}^{circ } inverse by using the core-EP decomposition. Applying the generalized inverse, we obtain the solutions of some matrix equations in Minkowski space.
在本文中，我们研究了 m {mathfrak{m}} -WG ∘ {}^{circ } 逆，它是 m {mathfrak{m}} 的广义化。 -WG 在闵科夫斯基空间中的逆。我们首先证明了广义逆的存在性和唯一性。然后，我们讨论了 m {mathfrak{m}} -WG ˲Sm_2F2} 的几个性质和特征。 -WG ∘ {}^{circ }逆的几个性质和特征。应用广义逆，我们得到了闵科夫斯基空间中一些矩阵方程的解。
{"title":"The 𝔪-WG° inverse in the Minkowski space","authors":"Xiaoji Liu, Kaiyue Zhang, Hongwei Jin","doi":"10.1515/math-2023-0145","DOIUrl":"https://doi.org/10.1515/math-2023-0145","url":null,"abstract":"In this article, we study the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0145_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"fraktur\">m</m:mi> </m:math> <jats:tex-math>{mathfrak{m}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-WG<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0145_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow /> <m:mrow> <m:mrow> <m:mo>∘</m:mo> </m:mrow> </m:mrow> </m:msup> </m:math> <jats:tex-math>{}^{circ }</jats:tex-math> </jats:alternatives> </jats:inline-formula> inverse which presents a generalization of the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0145_eq_999.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"fraktur\">m</m:mi> </m:math> <jats:tex-math>{mathfrak{m}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-WG inverse in the Minkowski space. We first show the existence and the uniqueness of the generalized inverse. Then, we discuss several properties and characterizations of the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0145_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"fraktur\">m</m:mi> </m:math> <jats:tex-math>{mathfrak{m}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-WG<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0145_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow /> <m:mrow> <m:mrow> <m:mo>∘</m:mo> </m:mrow> </m:mrow> </m:msup> </m:math> <jats:tex-math>{}^{circ }</jats:tex-math> </jats:alternatives> </jats:inline-formula> inverse by using the core-EP decomposition. Applying the generalized inverse, we obtain the solutions of some matrix equations in Minkowski space.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"18 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138560953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, by virtue of a derivative formula for the ratio of two differentiable functions and with aid of a monotonicity rule, the authors expand a logarithmic expression involving the cosine function into the Maclaurin power series in terms of specific determinants and prove a decreasing property of the ratio of two logarithmic expressions containing the cosine function.
{"title":"A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine","authors":"Yan-Fang Li, Feng Qi","doi":"10.1515/math-2023-0159","DOIUrl":"https://doi.org/10.1515/math-2023-0159","url":null,"abstract":"In this study, by virtue of a derivative formula for the ratio of two differentiable functions and with aid of a monotonicity rule, the authors expand a logarithmic expression involving the cosine function into the Maclaurin power series in terms of specific determinants and prove a decreasing property of the ratio of two logarithmic expressions containing the cosine function.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"16 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138560775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abdelbaki Choucha, Salah Mahmoud Boulaaras, Rashid Jan
The Lord Shulman swelling porous thermo-elastic soil system with the presence of a distributed delay term is studied in this work. We will establish the well-posedness of the system and the exponential stability of the system is derived.
本文研究了存在分布式延迟项的 Lord Shulman 膨胀多孔热弹性土壤系统。我们将建立系统的良好拟合性，并推导出系统的指数稳定性。
{"title":"Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term","authors":"Abdelbaki Choucha, Salah Mahmoud Boulaaras, Rashid Jan","doi":"10.1515/math-2023-0165","DOIUrl":"https://doi.org/10.1515/math-2023-0165","url":null,"abstract":"The Lord Shulman swelling porous thermo-elastic soil system with the presence of a distributed delay term is studied in this work. We will establish the well-posedness of the system and the exponential stability of the system is derived.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"195 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138560947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present article, we consider a double-phase eigenvalue problem with large exponents. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0138_eq_001.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:math> <jats:tex-math>{lambda }_{left({p}_{n},{q}_{n})}^{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the first eigenvalues and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0138_eq_002.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{u}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the first eigenfunctions, normalized by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0138_eq_003.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo>‖</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="script">ℋ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> <jats:tex-math>Vert {u}_{n}{Vert }_{{{mathcal{ {mathcal H} }}}_{n}}=1</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Under some assumptions on the exponents <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0138_eq_004.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{p}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0138_eq_005.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{q}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0138_eq_006.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mr
在本文中，我们考虑一个具有大指数的双相特征值问题。设 λ ( p n , q n ) 1 {lambda }_{left({p}_{n},{q}_{n})}^{1} 为第一特征值，u n {u}_{n} 为第一特征函数，归一化为 ‖ u n ‖ ℋ n = 1 Vert {u}_{n} {}Vert }_{{mathcal{ {mathcal H} }}_{n}}=1 .}}}_{n}}=1 .在对指数 p n {p}_{n} 和 q n {q}_{n} 有一些假设的情况下 我们证明 λ ( p n , q n ) 1 {lambda }_{left({p}_{n}、{q}_{n})}^{1} 收敛到 Λ ∞ {Lambda }_{infty }，并且 u n {u}_{n} 收敛到 u ∞ {u}_{infty }，在空间 C α ( Ω ) {C}^{alpha }left(Omega ) 中均匀分布、且 u ∞ {u}_{infty } 是一个 Dirichlet ∞ infty -Laplacian 问题的非微观粘性解。
{"title":"A double-phase eigenvalue problem with large exponents","authors":"Lujuan Yu","doi":"10.1515/math-2023-0138","DOIUrl":"https://doi.org/10.1515/math-2023-0138","url":null,"abstract":"In the present article, we consider a double-phase eigenvalue problem with large exponents. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0138_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:math> <jats:tex-math>{lambda }_{left({p}_{n},{q}_{n})}^{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the first eigenvalues and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0138_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{u}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the first eigenfunctions, normalized by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0138_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mo>‖</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant=\"script\">ℋ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> <jats:tex-math>Vert {u}_{n}{Vert }_{{{mathcal{ {mathcal H} }}}_{n}}=1</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Under some assumptions on the exponents <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0138_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{p}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0138_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{q}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0138_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mr","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"108 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138547194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we define evolutoids and pedaloids of frontals on timelike surfaces in Minkowski 3-space. The evolutoids of frontals on timelike surfaces are not only the generalization of evolutoids of curves in the Minkowski plane but also the generalization of caustics in Minkowski 3-space. As an application of the singularity theory, we classify the singularities of evolutoids and reveal the relationships between the singularities and geometric invariants of frontals. Furthermore, we find that there exists a close connection between the pedaloids of frontals and the pedal surfaces of evolutoids. Finally, we give some examples to demonstrate the results.
{"title":"Evolutoids and pedaloids of frontals on timelike surfaces","authors":"Yongqiao Wang, Lin Yang, Yuan Chang, Haiming Liu","doi":"10.1515/math-2023-0149","DOIUrl":"https://doi.org/10.1515/math-2023-0149","url":null,"abstract":"In this article, we define evolutoids and pedaloids of frontals on timelike surfaces in Minkowski 3-space. The evolutoids of frontals on timelike surfaces are not only the generalization of evolutoids of curves in the Minkowski plane but also the generalization of caustics in Minkowski 3-space. As an application of the singularity theory, we classify the singularities of evolutoids and reveal the relationships between the singularities and geometric invariants of frontals. Furthermore, we find that there exists a close connection between the pedaloids of frontals and the pedal surfaces of evolutoids. Finally, we give some examples to demonstrate the results.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"10 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138547139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let GG be a graph. A perfect matching of GG is a regular spanning subgraph of degree one. Enumeration of perfect matchings of a (molecule) graph is interest in chemistry, physics, and mathematics. But the enumeration problem of perfect matchings for general graphs (even in bipartite graphs) is non-deterministic polynomial (NP)-hard. Xiao et al. [C. Xiao, H. Chen, L. Liu, Perfect matchings in random pentagonal chains, J. Math. Chem. 55 (2017), 1878–1886] have studied the problem of perfect matchings for random odd-polygonal chain (i.e., with odd polygons). In this article, we further present simple counting formulae for the expected value of the number of perfect matchings in random even-polygonal chains (i.e., with even polygons). Based on these formulae, we obtain the average values of the number for perfect matchings with respect to the set of all even-polygonal chains with nn polygons.
设 G G 是一个图。G G 的完美匹配是阶数为 1 的正则遍历子图。枚举（分子）图的完全匹配是化学、物理和数学领域的兴趣所在。但对于一般图（即使是二方图）来说，完美匹配的枚举问题是非确定性多项式（NP）困难的。Xiao et al.Xiao, H. Chen, L. Liu, Perfect matchings in random pentagonal chains, J. Math.Chem.55 (2017), 1878-1886] 研究了随机奇多边形链（即奇多边形）的完全匹配问题。在本文中，我们进一步提出了随机偶多边形链（即偶数多边形）中完全匹配次数期望值的简单计数公式。根据这些计算公式，我们可以得到具有 n n 个多边形的所有偶多边形链的完全匹配数的平均值。
{"title":"On the number of perfect matchings in random polygonal chains","authors":"Shouliu Wei, Yongde Feng, Xiaoling Ke, Jianwu Huang","doi":"10.1515/math-2023-0146","DOIUrl":"https://doi.org/10.1515/math-2023-0146","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0146_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a graph. A perfect matching of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0146_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a regular spanning subgraph of degree one. Enumeration of perfect matchings of a (molecule) graph is interest in chemistry, physics, and mathematics. But the enumeration problem of perfect matchings for general graphs (even in bipartite graphs) is non-deterministic polynomial (NP)-hard. Xiao et al. [C. Xiao, H. Chen, L. Liu, <jats:italic>Perfect matchings in random pentagonal chains</jats:italic>, J. Math. Chem. 55 (2017), 1878–1886] have studied the problem of perfect matchings for random odd-polygonal chain (i.e., with odd polygons). In this article, we further present simple counting formulae for the expected value of the number of perfect matchings in random even-polygonal chains (i.e., with even polygons). Based on these formulae, we obtain the average values of the number for perfect matchings with respect to the set of all even-polygonal chains with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0146_eq_003.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> polygons.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"109 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138548569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：