Open Mathematics最新文献

{"title":"Classification of positive solutions for a weighted integral system on the half-space","authors":"Qiuping Liao, Haofeng Wang, Yingying Xiao","doi":"10.1515/math-2024-0058","DOIUrl":"https://doi.org/10.1515/math-2024-0058","url":null,"abstract":"In this article, we study the following weighted integral system: <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0058_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <m:mfenced open=\"{\" close=\"\"> <m:mrow> <m:mtable displaystyle=\"true\"> <m:mtr> <m:mtd columnalign=\"left\"> <m:mi>u</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:munder> <m:mrow> <m:mrow> <m:mstyle displaystyle=\"true\"> <m:mo>∫</m:mo> </m:mstyle> </m:mrow> </m:mrow> <m:mrow> <m:msubsup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:munder> <m:mfrac> <m:mrow> <m:msubsup> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>β</m:mi> </m:mrow> </m:msubsup> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mrow> <m:mrow> <m:mo>(</m:mo> </m:mrow> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>v</m:mi> <m:mrow> <m:mrow> <m:mo>(</m:mo> </m:mrow> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>x</m:mi> <m:mo>−</m:mo> <m:mi>y</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>λ</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mi mathvariant=\"normal\">d</m:mi> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mspace width=\"1em\"/> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:mspace width=\"1.0em\"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"left\"> <m:mi>v</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:munder> <m:mrow> <m:mrow> <m:mstyle displaystyle=\"true\"> <m:mo>∫</m:mo> </m:mstyle> </m:mrow> </m:mrow> <m:mrow> <m:msubsup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:munder> <m:mfrac> <m:mrow> <m:msubsup> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>β</m:mi> </m:mrow> </m:msubsup> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mrow> <m:mrow> <m:mo>(</m:mo> </m:mrow> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>v</m:mi> <m:mrow> <m:mrow> <m:mo>(</m:mo> </m:mrow> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>x</m:mi> <m:mo>−</m:mo> <m:mi>y</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"27 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142261359","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}

{"title":"Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions","authors":"Jing Li, Wenchang Chu","doi":"10.1515/math-2024-0052","DOIUrl":"https://doi.org/10.1515/math-2024-0052","url":null,"abstract":"Three classes of trigonometric integrals involving an integer parameter are evaluated by the contour integration and the residue theorem. The resulting formulae are expressed in terms of Riemann zeta function and Dirichlet beta function. Several remarkable integral identities are presented.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"23 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142261360","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}

{"title":"Note on stability estimation of stochastic difference equations","authors":"Evgueni Gordienko, Juan Ruiz de Chavez","doi":"10.1515/math-2024-0041","DOIUrl":"https://doi.org/10.1515/math-2024-0041","url":null,"abstract":"Stability estimates are proposed for two variants of Markov processes defined by stochastic difference equations: uncontrolled and controlled. Processes of this type are widely used in applications where their “governing distributions” are known only approximately, for example, as statistical estimates obtained from real data. Therefore, the problem of estimating deviations of output characteristics arises. The Kantorovich metric is used to measure the variations of probability distributions that govern the processes. In the uncontrolled case, the Kantorovich distance between the stationary distributions of the initial process and its perturbation is evaluated. On the other hand, the control processes being compared are endowed with an expected total discounted cost, and the inequality for the corresponding stability index is obtained. The stability index measures the increase in costs when using the control policy optimal for the “approximating process.”","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191374","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}

{"title":"Construction of a class of half-discrete Hilbert-type inequalities in the whole plane","authors":"Minghui You","doi":"10.1515/math-2024-0044","DOIUrl":"https://doi.org/10.1515/math-2024-0044","url":null,"abstract":"In this work, we first define two special sets of real numbers, and then, we construct a half-discrete kernel function where the variables are defined in the whole plane, and the parameters in the kernel function are limited to the newly constructed special sets. Estimate the kernel function in the whole plane by converting it to the first quadrant, and then, a class of new Hilbert-type inequality is established. Additionally, it is proved that the constant factor of the newly established inequality is the best possible. Furthermore, assigning special values to the parameters and using rational fraction expansion of cosecant function, some special results are presented at the end of this article.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"2 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191375","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}

{"title":"Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods","authors":"Keyan Wang","doi":"10.1515/math-2024-0048","DOIUrl":"https://doi.org/10.1515/math-2024-0048","url":null,"abstract":"In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0048_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>H</m:mi> </m:math> <jats:tex-math>H</jats:tex-math> </jats:alternatives> </jats:inline-formula> and the fine grid size <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0048_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>h</m:mi> </m:math> <jats:tex-math>h</jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfy <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0048_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>h</m:mi> <m:mo>=</m:mo> <m:mi mathvariant=\"script\">O</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>⁄</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>h={mathcal{O}}left({H}^{left(2k+1)/left(k+1)})</jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0048_eq_004.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:math> <jats:tex-math>kge 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_math-2024-0048_eq_005.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> </m:math> <jats:tex-math>k</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"11 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191376","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}

{"title":"On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating","authors":"Angela Iannelli","doi":"10.1515/math-2024-0032","DOIUrl":"https://doi.org/10.1515/math-2024-0032","url":null,"abstract":"This article surveys some results in the study of Iannelli [<jats:italic>Su un modello di Oberbeck-Boussinesq relativo al moto di un fluido viscoso soggetto a riscaldamento</jats:italic>, Fisica Matematica, Istituto Lombardo (rend. Sc.) A 121 (1987), 145–191], in which the motion of a viscous, compressible fluid in a two-dimensional domain, subject to heating at the walls, is studied. A global existence and uniqueness theorem for the time-dependent problem is given, and also, under more stringent assumptions, an existence and uniqueness theorem in the stationary case is given. A theorem on the asymptotic behavior for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0032_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>t</m:mi> <m:mo>→</m:mo> <m:mi>∞</m:mi> </m:math> <jats:tex-math>tto infty </jats:tex-math> </jats:alternatives> </jats:inline-formula> of the time-dependent solutions is proved.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"15 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191379","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}

{"title":"An extension of Schweitzer's inequality to Riemann-Liouville fractional integral","authors":"Thabet Abdeljawad, Badreddine Meftah, Abdelghani Lakhdari, Manar A. Alqudah","doi":"10.1515/math-2024-0043","DOIUrl":"https://doi.org/10.1515/math-2024-0043","url":null,"abstract":"This note focuses on establishing a fractional version akin to the Schweitzer inequality, specifically tailored to accommodate the left-sided Riemann-Liouville fractional integral operator. The Schweitzer inequality is a fundamental mathematical expression, and extending it to the fractional realm holds significance in advancing our understanding and applications of fractional calculus.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"81 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191377","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}

{"title":"Strong laws for weighted sums of widely orthant dependent random variables and applications","authors":"Yong Zhu, Wei Wang, Kan Chen","doi":"10.1515/math-2024-0027","DOIUrl":"https://doi.org/10.1515/math-2024-0027","url":null,"abstract":"In this study, the strong law of large numbers and the convergence rate for weighted sums of non-identically distributed widely orthant dependent random variables are established. As applications, the strong consistency for weighted estimator in nonparametric regression model and the rate of strong consistency for least-squares estimator in multiple linear regression model are obtained. Some numerical simulations are also provided to verify the validity of the theoretical results.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"2 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191378","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}

{"title":"Small values and functional laws of the iterated logarithm for operator fractional Brownian motion","authors":"Wensheng Wang, Jingshuang Dong","doi":"10.1515/math-2024-0045","DOIUrl":"https://doi.org/10.1515/math-2024-0045","url":null,"abstract":"The multivariate Gaussian random fields with matrix-based scaling laws are widely used for inference in statistics and many applied areas. In such contexts, interests are often Hölder regularities of spatial surfaces in any given direction. This article analyzes the almost sure sample function behavior for operator fractional Brownian motion, including multivariate fractional Brownian motion. We obtain the estimations of small ball probability and the strongly locally nondeterministic for operator fractional Brownian motion in any given direction. By applying these estimates, we obtain Chung type laws of the iterated logarithm for operator fractional Brownian motion. Our results show that the precise Hölder regularities of these spatial surfaces are completely determined by the real parts of the eigenvalues of self-similarity exponent and the covariance matrix at time point 1.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"31 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191381","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}

{"title":"The limit theorems on extreme order statistics and partial sums of i.i.d. random variables","authors":"Gaoyu Li, Chengxiu Ling, Zhongquan Tan","doi":"10.1515/math-2024-0047","DOIUrl":"https://doi.org/10.1515/math-2024-0047","url":null,"abstract":"This article proves several weak limit theorems for the joint version of extreme order statistics and partial sums of independently and identically distributed random variables. The results are also extended to almost sure limit version.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"128 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191386","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}