Open Mathematics最新文献

{"title":"Scattering threshold for the focusing energy-critical generalized Hartree equation","authors":"Saleh Almuthaybiri, Congming Peng, Tarek Saanouni","doi":"10.1515/math-2024-0002","DOIUrl":"https://doi.org/10.1515/math-2024-0002","url":null,"abstract":"This work investigates the asymptotic behavior of energy solutions to the focusing nonlinear Schrödinger equation of Choquard type <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0002_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <m:mi>i</m:mi> <m:msub> <m:mrow> <m:mo>∂</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>I</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msub> <m:mo>*</m:mo> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width=\"1.0em\" /> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> <m:mo>+</m:mo> <m:mi>α</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mfrac> <m:mo>,</m:mo> <m:mspace width=\"1.0em\" /> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> <m:mo>.</m:mo> </m:math> <jats:tex-math>i{partial }_{t}u+Delta u+{| u| }^{p-2}left({I}_{alpha }* {| u| }^{p})u=0,hspace{1.0em}p=1+frac{2+alpha }{N-2},hspace{1.0em}Nge 3.</jats:tex-math> </jats:alternatives> </jats:disp-formula> Indeed, in the energy-critical spherically symmetric regime, one proves a global existence and scattering versus finite time blow-up dichotomy. Precisely, if the data have an energy less than the ground state one, two cases are possible. If the kinetic energy of the radial data is less than the ground state one, then the solution is global and scatters. Otherwise, if the data have a finite variance or is spherically symmetric and have a finite mass, then the solution is nonglobal. The main difficulty is to deal with the nonlocal source term. The argument is the concentration-compactness-rigidity method introduced by Kenig and Merle (<jats:italic>Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case</jats:italic>, Invent. Math. 166 (2006), no. 3, 645–675). This note naturally complements the work by Saanouni (<jats:italic>Scattering theory for a class of defocusing energy-critical Choquard equations</jats:italic>, J. Evol. Equ. 21 (2021), 1551–1571), where the scattering of the defocusing energy-critical generalized Hartree equation was obtained.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"110 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623407","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":"Zariski topology on the secondary-like spectrum of a module","authors":"Saif Salam, Khaldoun Al-Zoubi","doi":"10.1515/math-2024-0005","DOIUrl":"https://doi.org/10.1515/math-2024-0005","url":null,"abstract":"Let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_001.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;ℜ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;Re &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be a commutative ring with unity and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_002.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;ℑ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;Im &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be a left &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_003.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;ℜ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;Re &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-module. We define the secondary-like spectrum of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_004.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;ℑ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;Im &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; to be the set of all secondary submodules &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_005.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;K&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;K&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_006.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;ℑ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;Im &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; such that the annihilator of the socle of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_007.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;K&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;K&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is the radical of the annihilator of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_008.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;K&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;K&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, and we denote it by &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_009.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"13 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140568870","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":"Two results on the value distribution of meromorphic functions","authors":"Degui Yang, Zhiying He, Dan Liu","doi":"10.1515/math-2024-0004","DOIUrl":"https://doi.org/10.1515/math-2024-0004","url":null,"abstract":"In this article, we prove two results on the value distribution of meromorphic functions. Using the theorem of Yamanoi, the first result gives a precise estimation of the relationship between the characteristic function of a meromorphic function and its <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0004_eq_001.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>th derivative in a concise form. This result extends and improves some results of Shan, Singh, Gopalakrishna, Edrei, Weitsman, Yang, Wu and Wu, etc. The second result answers a conjecture posed by C. C. Yang. This conjecture turned to be false by a counter-example, but it will be true with an additional condition.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"55 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140568868","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":"Amitsur's theorem, semicentral idempotents, and additively idempotent semirings","authors":"Martin Rachev, Ivan Trendafilov","doi":"10.1515/math-2023-0180","DOIUrl":"https://doi.org/10.1515/math-2023-0180","url":null,"abstract":"The article explores research findings akin to Amitsur’s theorem, asserting that any derivation within a matrix ring can be expressed as the sum of an inner derivation and a hereditary derivation. In most results related to rings and semirings, Birkenmeier’s semicentral idempotents play a crucial role. This article is intended for PhD students, postdocs, and researchers.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"65 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140301652","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":"Computing the determinant of a signed graph","authors":"Bader Alshamary, Zoran Stanić","doi":"10.1515/math-2023-0188","DOIUrl":"https://doi.org/10.1515/math-2023-0188","url":null,"abstract":"A signed graph is a simple graph in which every edge has a positive or negative sign. In this article, we employ several algebraic techniques to compute the determinant of a signed graph in terms of the spectrum of a vertex-deleted subgraph. Particular cases, including vertex-deleted subgraphs without repeated eigenvalues or singular vertex-deleted subgraphs are considered. As applications, an algorithm for the determinant of a signed graph with pendant edges is established, the determinant of a bicyclic graph and the determinant of a chain graph are computed. In the end, the uniqueness of the polynomial reconstruction for chain graphs is proved.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"12 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140301655","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":"Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions","authors":"Shan Zhou","doi":"10.1515/math-2023-0175","DOIUrl":"https://doi.org/10.1515/math-2023-0175","url":null,"abstract":"In this article, we investigate the following Schrödinger equation: &lt;jats:disp-formula&gt; &lt;jats:alternatives&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0175_eq_001.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:mi&gt;μ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:mfrac&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mspace width=\"1em\" /&gt; &lt;m:mi mathvariant=\"normal\"&gt;in&lt;/m:mi&gt; &lt;m:mspace width=\"0.33em\" /&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;R&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;N&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;-Delta u-frac{mu }{{| x| }^{2}}u=gleft(u)hspace{1em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{N},&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:disp-formula&gt; where &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0175_eq_002.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;N&lt;/m:mi&gt; &lt;m:mo&gt;≥&lt;/m:mo&gt; &lt;m:mn&gt;3&lt;/m:mn&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;Nge 3&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0175_eq_003.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:mi&gt;μ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:mfrac&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;frac{mu }{{| x| }^{2}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is called the Hardy potential and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0175_eq_004.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;g&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; satisfies Berestycki-Lions conditions. If &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0175_eq_005.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;m:mo&gt;&lt;&lt;/m:mo&gt; &lt;m:mi&gt;μ&lt;/m:mi&gt; &lt;m:mo&gt;&lt;&lt;/m:mo&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;N&lt;/m:mi&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;4&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:mfrac&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;0lt mu lt frac{{left(N-2)}^{2}}{4}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, we ","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"23 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140202837","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 quasivarieties generated by finite pointed abelian groups","authors":"Ainur Basheyeva, Svetlana Lutsak","doi":"10.1515/math-2023-0181","DOIUrl":"https://doi.org/10.1515/math-2023-0181","url":null,"abstract":"We prove that a finite pointed abelian group generates a finitely axiomatizable variety that has a finite quasivariety lattice. As a consequence, we obtain that a quasivariety generated by a finite pointed abelian group has a finite basis of quasi-identities. The problems arising from the results obtained are also discussed.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"288 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140202707","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 the generalized exponential sums and their fourth power mean","authors":"Wencong Liu, Shushu Ning","doi":"10.1515/math-2023-0187","DOIUrl":"https://doi.org/10.1515/math-2023-0187","url":null,"abstract":"The main purpose of this article is to study the calculating problem of the fourth power mean of the two-term exponential sums and provide an accurate calculating formula for utilizing analytical methods and character sums’ properties. In the meantime, a result of the fourth power mean of Gauss sums is improved.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"67 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140202831","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 uniqueness of expression for generalized quadratic matrices","authors":"Meixiang Chen, Zhongpeng Yang, Qinghua Chen","doi":"10.1515/math-2023-0186","DOIUrl":"https://doi.org/10.1515/math-2023-0186","url":null,"abstract":"It is shown that the expression as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0186_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mi>α</m:mi> <m:mi>A</m:mi> <m:mo>+</m:mo> <m:mi>β</m:mi> <m:mi>P</m:mi> </m:math> <jats:tex-math>{A}^{2}=alpha A+beta P</jats:tex-math> </jats:alternatives> </jats:inline-formula> for generalized quadratic matrices is not unique by numerical examples. Then it is proven that the uniqueness of expression for generalized quadratic matrices is concerned not only with the properties of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0186_eq_002.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> but also with the rank of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0186_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>. Furthermore, the sufficient and necessary conditions for the uniqueness of generalized quadratic matrices’expression are obtained. Finally, some related discussions about generalized quadratic matrices are also given.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"13 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140202836","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":"A conjecture of Mallows and Sloane with the universal denominator of Hilbert series","authors":"Yang Zhang, Jizhu Nan, Yongsheng Ma","doi":"10.1515/math-2024-0001","DOIUrl":"https://doi.org/10.1515/math-2024-0001","url":null,"abstract":"A conjecture of Mallows and Sloane conveys the dominance of Hilbert series for finding basic invariants of finite linear groups if the Hilbert series of the invariant ring is of a certain explicit canonical form. However, the conjecture does not hold in general by a well-known counterexample of Stanley. In this article, we give a constraint on lower bounds for the degrees of homogeneous system of parameters of rings of invariants of finite linear groups depending on the universal denominator of Hilbert series defined by Derksen. We consider the conjecture with the universal denominator on abelian groups and provide some criteria guaranteeing the existence of homogeneous system of parameters of certain degrees. In this case, Stanley’s counterexample could be avoided, and the homogeneous system of parameters is optimal.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"27 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140167077","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}