We compute an asymptotic formula for the divisor class numbers of real cubic function fields Km=k(m3){K}_{m}=kleft(sqrt[3]{m}), where Fq{{mathbb{F}}}_{q} is a finite field with qq elements, q≡1(mod3)qequiv 1hspace{0.3em}left(mathrm{mod}hspace{0.3em}3), k≔Fq(T)k:= {{mathbb{F}}}_{q}left(T) is the rational function field, and m∈Fq
我们计算了实三次函数域 K m = k ( m 3 ) {K}_{m}=kleft(sqrt[3]{m}) 的因子类数的渐近公式，其中 F q {{mathbb{F}}}_{q} 是具有 q q 个元素的有限域，q ≡ 1 ( mod 3 ) qequiv 1hspace{0.3em}left(mathrm{mod}hspace{0.3em}3) , k ≔ F q ( T ) k:= {{mathbb{F}}}_{q}left(T) 是有理函数域，并且 m ∈ F q [ T ] min {{mathbb{F}}}_{q}left[T] 是无立方多项式；在这种情况下，m m 的阶数可以被 3 整除。为了计算渐近公式，我们要找到 ∣ L ( s , χ ) ∣ 2 {| Lleft(s. chi )| }^^ 的平均值、当 χ chi 经过 F q [ T ] {{mathbb{F}}}_{q}left[T] 的原始立方偶数 Dirichlet 字符时，在 s = 1 s=1 处求值，其中 L ( s , χ ) Lleft(s,chi ) 是相关的 Dirichlet L L - 函数。
{"title":"Average value of the divisor class numbers of real cubic function fields","authors":"Yoonjin Lee, Jungyun Lee, Jinjoo Yoo","doi":"10.1515/math-2023-0160","DOIUrl":"https://doi.org/10.1515/math-2023-0160","url":null,"abstract":"We compute an asymptotic formula for the divisor class numbers of <jats:italic>real</jats:italic> cubic function fields <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_001.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>m</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mi>k</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mroot> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mroot> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{K}_{m}=kleft(sqrt[3]{m})</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-2023-0160_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi mathvariant=\"double-struck\">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{{mathbb{F}}}_{q}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a finite field with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>q</m:mi> </m:math> <jats:tex-math>q</jats:tex-math> </jats:alternatives> </jats:inline-formula> elements, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>q</m:mi> <m:mo>≡</m:mo> <m:mn>1</m:mn> <m:mspace width=\"0.3em\" /> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mi>mod</m:mi> </m:mrow> <m:mspace width=\"0.3em\" /> <m:mn>3</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>qequiv 1hspace{0.3em}left(mathrm{mod}hspace{0.3em}3)</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-2023-0160_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> <m:mo>≔</m:mo> <m:msub> <m:mrow> <m:mi mathvariant=\"double-struck\">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>k:= {{mathbb{F}}}_{q}left(T)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the rational function field, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi mathvariant=\"double-struck\">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139411275","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 investigate a spherically symmetric backward heat conduction problem, starting from the final temperature. This problem is severely ill posed: the solution (if it exists) does not depend continuously on the final data. A conditional stability result of its solution is given. Further, we propose a quasi-boundary value regularization method to solve this ill-posed problem. Two Hölder type error estimates between the approximate solution and its exact solution are obtained under an a priori and an a posteriori regularization parameter choice rule, respectively.
{"title":"A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem","authors":"Wei Cheng, Yi-Liang Liu","doi":"10.1515/math-2023-0171","DOIUrl":"https://doi.org/10.1515/math-2023-0171","url":null,"abstract":"In this article, we investigate a spherically symmetric backward heat conduction problem, starting from the final temperature. This problem is severely ill posed: the solution (if it exists) does not depend continuously on the final data. A conditional stability result of its solution is given. Further, we propose a quasi-boundary value regularization method to solve this ill-posed problem. Two Hölder type error estimates between the approximate solution and its exact solution are obtained under an <jats:italic>a priori</jats:italic> and an <jats:italic>a posteriori</jats:italic> regularization parameter choice rule, respectively.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139411677","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, the main aim is to consider the boundedness of the nonlinear commutator of pp-adic sharp maximal operator ℳp♯{{mathcal{ {mathcal M} }}}_{p}^{sharp } with symbols belonging to the pp-adic Lipschitz spaces in the context of the pp-adic version of (variable) Lebesgue spaces, by which some new characterizations of the Lipschitz spaces are obtained in the pp-adic field context.
本文的主要目的是考虑 p p -adic 尖锐最大算子 ℳ p ♯ {{mathcal{ {mathcal M}}}_{p}^{sharp } 的非线性换向器的有界性。}}}_{p}^{sharp }，符号属于 p p -adic Lipschitz 空间的（可变）Lebesgue 空间的 p p -adic 版本，通过这些符号，可以得到 p p -adic 场背景下 Lipschitz 空间的一些新特征。
{"title":"Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces","authors":"Jianglong Wu, Yunpeng Chang","doi":"10.1515/math-2023-0168","DOIUrl":"https://doi.org/10.1515/math-2023-0168","url":null,"abstract":"In this article, the main aim is to consider the boundedness of the nonlinear commutator of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0168_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>p</m:mi> </m:math> <jats:tex-math>p</jats:tex-math> </jats:alternatives> </jats:inline-formula>-adic sharp maximal operator <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0168_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi mathvariant=\"script\">ℳ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>♯</m:mi> </m:mrow> </m:msubsup> </m:math> <jats:tex-math>{{mathcal{ {mathcal M} }}}_{p}^{sharp }</jats:tex-math> </jats:alternatives> </jats:inline-formula> with symbols belonging to the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0168_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>p</m:mi> </m:math> <jats:tex-math>p</jats:tex-math> </jats:alternatives> </jats:inline-formula>-adic Lipschitz spaces in the context of the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0168_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>p</m:mi> </m:math> <jats:tex-math>p</jats:tex-math> </jats:alternatives> </jats:inline-formula>-adic version of (variable) Lebesgue spaces, by which some new characterizations of the Lipschitz spaces are obtained in the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0168_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>p</m:mi> </m:math> <jats:tex-math>p</jats:tex-math> </jats:alternatives> </jats:inline-formula>-adic field context.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139411718","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}
One of the symbolic parameters to measure the fault tolerance of a network is its connectivity. The HH-structure connectivity and HH-substructure connectivity extend the classical connectivity and are more practical. For a graph GG and its connected subgraph HH, the HH-structure connectivity κ(G;H)kappa left(G;hspace{0.33em}H) (resp. HH-substructure connectivity κs(
网络连通性是衡量网络容错性的符号参数之一。H H -结构连通性和 H H -子结构连通性是经典连通性的扩展，更加实用。对于一个图 G 和它的连通子图 H H，G 的 H H -结构连通性 κ ( G ; H ) kappa left(G;hspace{0.33em}H)（或者 H H -子结构连通性 κ s ( G ; H ) {kappa }^{s}left(G;hspace{0.33em}H) )G G 的最小子图集的心数，使得该集的每个元素都与 G G 中的 H H 同构（即该集的每个元素都与 H H 的连通子图同构），其顶点移除断开 G G 的连通性。本文将研究 n n 维烧饼网络 BP n {{rm{BP}}_{n} 中每个 H∈ { K 1 , K 1 , 1 , ... , K 1 , n - 1 , P 4 , ... , P 7 , C 8 } 的 H H 结构连通性和 H H 子结构连通性。｝ Hleft{{K}_{1},{K}_{1,1},ldots ,{K}_{1,n-1},{P}_{4},ldots ,{P}_{7},{C}_{8}right} .
{"title":"The structure fault tolerance of burnt pancake networks","authors":"Huifen Ge, Chengfu Ye, Shumin Zhang","doi":"10.1515/math-2023-0154","DOIUrl":"https://doi.org/10.1515/math-2023-0154","url":null,"abstract":"One of the symbolic parameters to measure the fault tolerance of a network is its connectivity. The <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0154_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>-structure connectivity and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0154_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>-substructure connectivity extend the classical connectivity and are more practical. For a graph <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0154_eq_003.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> and its connected subgraph <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0154_eq_004.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>, the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0154_eq_005.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>-structure connectivity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0154_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>κ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo>;</m:mo> <m:mspace width=\"0.33em\" /> <m:mi>H</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>kappa left(G;hspace{0.33em}H)</jats:tex-math> </jats:alternatives> </jats:inline-formula> (resp. <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0154_eq_007.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>-substructure connectivity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0154_eq_008.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>κ</m:mi> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo>","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139411278","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}
After introducing a graph connectedness induced by a given set of paths of the same length, we focus on the 2-adjacency graph on the digital line Z{mathbb{Z}} with a certain set of paths of length nn for every positive integer nn. The connectedness in the strong product of three copies of the graph is used to define digital Jordan surfaces. These are obtained as polyhedral surfaces bounding the polyhedra that can be face-to-face tiled with digital tetrahedra.
在介绍了由一组长度相同的给定路径所诱导的图连通性之后，我们将重点放在数字线 Z {mathbb{Z}} 上的 2-相接图上，对于每一个正整数 n n，该图具有一组长度为 n n 的路径。图的三个副本的强积中的连通性被用来定义数字乔丹曲面。这些曲面作为多面体的边界，可以与数字四面体面对面平铺。
{"title":"A digital Jordan surface theorem with respect to a graph connectedness","authors":"Josef Šlapal","doi":"10.1515/math-2023-0172","DOIUrl":"https://doi.org/10.1515/math-2023-0172","url":null,"abstract":"After introducing a graph connectedness induced by a given set of paths of the same length, we focus on the 2-adjacency graph on the digital line <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0172_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"double-struck\">Z</m:mi> </m:math> <jats:tex-math>{mathbb{Z}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with a certain set of paths of length <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0172_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> for every positive integer <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0172_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>. The connectedness in the strong product of three copies of the graph is used to define digital Jordan surfaces. These are obtained as polyhedral surfaces bounding the polyhedra that can be face-to-face tiled with digital tetrahedra.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139375970","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}
Recently, regime-switching option pricing based on fractional diffusion models has been used, which explains many significant empirical facts about financial markets better. There are many methods to solve the problem, but to the best of our knowledge, effective preconditioners for the second-order schemes have not been proposed. Thus, in this article, an implicit numerical scheme is developed for a regime-switching European option pricing problem under a multi-state tempered fractional model. The scheme is proven to be unconditionally stable and converges quadratically in space and linearly in time. Besides, the resulting linear system is solved using an iterative method, and a preconditioner is proposed to accelerate the rate of convergence. The preconditioner is constructed through circulant approximations to the Toeplitz blocks due to the coefficient matrix, which is is a block matrix with Toeplitz blocks. The spectral analysis of the preconditioned matrix is given, which demonstrates that the spectrum of the preconditioned matrix is clustered around 1. Numerical examples show the efficiency of the proposed method, and an empirical study is also provided.
{"title":"A preconditioned iterative method for coupled fractional partial differential equation in European option pricing","authors":"Shuang Wu, Lot-Kei Chou, Xu Chen, Siu-Long Lei","doi":"10.1515/math-2023-0169","DOIUrl":"https://doi.org/10.1515/math-2023-0169","url":null,"abstract":"Recently, regime-switching option pricing based on fractional diffusion models has been used, which explains many significant empirical facts about financial markets better. There are many methods to solve the problem, but to the best of our knowledge, effective preconditioners for the second-order schemes have not been proposed. Thus, in this article, an implicit numerical scheme is developed for a regime-switching European option pricing problem under a multi-state tempered fractional model. The scheme is proven to be unconditionally stable and converges quadratically in space and linearly in time. Besides, the resulting linear system is solved using an iterative method, and a preconditioner is proposed to accelerate the rate of convergence. The preconditioner is constructed through circulant approximations to the Toeplitz blocks due to the coefficient matrix, which is is a block matrix with Toeplitz blocks. The spectral analysis of the preconditioned matrix is given, which demonstrates that the spectrum of the preconditioned matrix is clustered around 1. Numerical examples show the efficiency of the proposed method, and an empirical study is also provided.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139095254","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 researched the existence of the solution to the fractional Navier-Stokes equations with the Coriolis force under initial data, which belong to the Lei-Lin-Gevrey spaces. Moreover, we showed a blow-up criterion, i.e., when the maximal time of existence T*{T}^{* } is finite, we proved that the norm of this same solution, in a specific Lei-Lin-Gevrey space, goes to infinity, as time tends to the maximal time of its existence.
在这篇文章中，我们研究了在初始数据下带科里奥利力的分数纳维-斯托克斯方程的解的存在性，它属于 Lei-Lin-Gevrey 空间。此外，我们还提出了一个炸毁准则，即当最大存在时间 T * {T}^{* } 有限时，我们证明了在特定的 Lei-Lin-Gevrey 空间中，随着时间趋向于最大存在时间，该同一解的常模会趋向于无穷大。
{"title":"On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces","authors":"Xiaochun Sun, Gaoting Xu, Yulian Wu","doi":"10.1515/math-2023-0170","DOIUrl":"https://doi.org/10.1515/math-2023-0170","url":null,"abstract":"In this article, we researched the existence of the solution to the fractional Navier-Stokes equations with the Coriolis force under initial data, which belong to the Lei-Lin-Gevrey spaces. Moreover, we showed a blow-up criterion, i.e., when the maximal time of existence <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0170_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:math> <jats:tex-math>{T}^{* }</jats:tex-math> </jats:alternatives> </jats:inline-formula> is finite, we proved that the norm of this same solution, in a specific Lei-Lin-Gevrey space, goes to infinity, as time tends to the maximal time of its existence.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139083696","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}
Gaussian solitons are important examples in the theory of Riemannian measure space. In the first part, we explicitly characterize the first eigenfunctions of the drift Laplacian in a Gaussian shrinking soliton, which shows that apart from each coordinate function, other first eigenfunctions must involve exponential functions and the so-called error functions. Moreover, the second eigenfunctions are also described. In the second part, we discuss the corresponding issues in Finsler Gaussian shrinking solitons, which is a natural generalization of Gaussian shrinking solitons. For the first eigenfunction, we complement an example to show that if a coordinate function is a first eigenfunction, then the Finsler Gaussian shrinking soliton must be a Euclidean measure space. For the second eigenfunction, we give some necessary and sufficient conditions for these spaces to be Euclidean measure spaces.
{"title":"Eigenfunctions in Finsler Gaussian solitons","authors":"Caiyun Liu, Songting Yin","doi":"10.1515/math-2023-0167","DOIUrl":"https://doi.org/10.1515/math-2023-0167","url":null,"abstract":"Gaussian solitons are important examples in the theory of Riemannian measure space. In the first part, we explicitly characterize the first eigenfunctions of the drift Laplacian in a Gaussian shrinking soliton, which shows that apart from each coordinate function, other first eigenfunctions must involve exponential functions and the so-called error functions. Moreover, the second eigenfunctions are also described. In the second part, we discuss the corresponding issues in Finsler Gaussian shrinking solitons, which is a natural generalization of Gaussian shrinking solitons. For the first eigenfunction, we complement an example to show that if a coordinate function is a first eigenfunction, then the Finsler Gaussian shrinking soliton must be a Euclidean measure space. For the second eigenfunction, we give some necessary and sufficient conditions for these spaces to be Euclidean measure spaces.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139083705","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 develop an efficient Legendre-Galerkin approximation based on a reduced-dimension scheme for the fourth-order equation with singular potential and simply supported plate (SSP) boundary conditions in a circular domain. First, we deduce the equivalent reduced-dimension scheme and essential pole condition associated with the original problem, based on which a class of weighted Sobolev spaces are defined and a weak formulation and its discrete scheme are also established for each reduced one-dimensional problem. Second, the existence and uniqueness of the weak solution and the approximation solutions are given using the Lax-Milgram theorem. Then, we construct a class of projection operators, give their approximation properties, and then prove the error estimates of the approximation solutions. In addition, we construct a set of effective basis functions in approximate space using orthogonal property of Legendre polynomials and derive the equivalent matrix form of the discrete scheme. Finally, a large number of numerical examples are performed, and the numerical results illustrate the validity and high accuracy of our algorithm.
{"title":"An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition","authors":"Shuimu Zou, Jun Zhang","doi":"10.1515/math-2023-0128","DOIUrl":"https://doi.org/10.1515/math-2023-0128","url":null,"abstract":"In this article, we develop an efficient Legendre-Galerkin approximation based on a reduced-dimension scheme for the fourth-order equation with singular potential and simply supported plate (SSP) boundary conditions in a circular domain. First, we deduce the equivalent reduced-dimension scheme and essential pole condition associated with the original problem, based on which a class of weighted Sobolev spaces are defined and a weak formulation and its discrete scheme are also established for each reduced one-dimensional problem. Second, the existence and uniqueness of the weak solution and the approximation solutions are given using the Lax-Milgram theorem. Then, we construct a class of projection operators, give their approximation properties, and then prove the error estimates of the approximation solutions. In addition, we construct a set of effective basis functions in approximate space using orthogonal property of Legendre polynomials and derive the equivalent matrix form of the discrete scheme. Finally, a large number of numerical examples are performed, and the numerical results illustrate the validity and high accuracy of our algorithm.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139027937","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 present new fractional integral inequalities via Euler’s beta function in terms of ss-convex mappings. We develop some new generalizations of fractional trapezoid- and midpoint-type inequalities using the class of differentiable ss-convexity. The results obtained in this study extended other related results reported in the literature.
在本文中，我们以 s s -凸映射为条件，通过欧拉的贝塔函数提出了新的分数积分不等式。我们利用可微 s s -凸性类，对分数梯形不等式和中点不等式进行了一些新的概括。本研究获得的结果扩展了文献中报道的其他相关结果。
{"title":"New fractional integral inequalities via Euler's beta function","authors":"Ohud Bulayhan Almutairi","doi":"10.1515/math-2023-0163","DOIUrl":"https://doi.org/10.1515/math-2023-0163","url":null,"abstract":"In this article, we present new fractional integral inequalities via Euler’s beta function in terms of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0163_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>s</m:mi> </m:math> <jats:tex-math>s</jats:tex-math> </jats:alternatives> </jats:inline-formula>-convex mappings. We develop some new generalizations of fractional trapezoid- and midpoint-type inequalities using the class of differentiable <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0163_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>s</m:mi> </m:math> <jats:tex-math>s</jats:tex-math> </jats:alternatives> </jats:inline-formula>-convexity. The results obtained in this study extended other related results reported in the literature.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139027992","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}
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