This work investigates the asymptotic behavior of energy solutions to the focusing nonlinear Schrödinger equation of Choquard type i∂tu+Δu+∣u∣p−2(Iα*∣u∣p)u=0,p=1+2+αN−2,N≥3.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. 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 (Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645–675). This note naturally complements the work by Saanouni (Scattering theory for a class of defocusing energy-critical Choquard equations, J. Evol. Equ. 21 (2021), 1551–1571), where the scattering of the defocusing energy-critical generalized Hartree equation was obtained.
这项工作研究了乔夸尔型聚焦非线性薛定谔方程能量解的渐近行为 i ∂ t u + Δ u + ∣ u ∣ p - 2 ( I α * ∣ u ∣ p ) u = 0 , p = 1 + 2 + α N - 2 , N ≥ 3 。 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. 事实上，在能量临界球对称体系中，我们可以证明全局存在和散射与有限时间炸毁的二分法。确切地说，如果数据的能量小于基态能量，则可能出现两种情况。如果径向数据的动能小于基态动能，那么解就是全局的，并且会发生散射。否则，如果数据具有有限方差或球面对称且具有有限质量，则解为非全局解。主要困难在于如何处理非局部源项。其论据是 Kenig 和 Merle 引入的集中-紧凑-刚性方法（Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent.Math.166 (2006), no.3, 645-675).本注释自然补充了 Saanouni 的工作（Scattering theory for a class of defocusing energy-critical Choquard equations, J. Evol. Equ.21 (2021), 1551-1571) 的工作的补充，在该论文中，我们得到了离焦能量临界广义哈特里方程的散射理论。
{"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":null,"pages":null},"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}
Let ℜRe be a commutative ring with unity and ℑIm be a left ℜRe -module. We define the secondary-like spectrum of ℑIm to be the set of all secondary submodules KK of ℑIm such that the annihilator of the socle of KK is the radical of the annihilator of KK, and we denote it by
让 ℜ Re 是一个具有统一性的交换环，而 ℑ Im 是一个左 ℜ Re 模块。我们定义ℑ Im 的类二级谱是ℑ Im 的所有二级子模块 K K 的集合，使得 K K 的湮没子是 K K 的湮没子的基，我们用 Spec L ( ℑ ) {{rm{Spec}}}^{L}left(Im ) 表示它。在本研究中，我们在 Spec L ( ℑ ) {{rm{Spec}}^{L}left(Im ) 上引入了一个具有第二谱 Spec s ( ℑ ) {{rm{Spec}}^{s}left(Im ) 上的扎里斯基拓扑的子空间拓扑，并研究了这个拓扑的几个拓扑结构。
{"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 <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℜ</m:mi> </m:math> <jats:tex-math>Re </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a commutative ring with unity and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℑ</m:mi> </m:math> <jats:tex-math>Im </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a left <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℜ</m:mi> </m:math> <jats:tex-math>Re </jats:tex-math> </jats:alternatives> </jats:inline-formula>-module. We define the secondary-like spectrum of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℑ</m:mi> </m:math> <jats:tex-math>Im </jats:tex-math> </jats:alternatives> </jats:inline-formula> to be the set of all secondary submodules <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_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> of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℑ</m:mi> </m:math> <jats:tex-math>Im </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that the annihilator of the socle of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_007.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 radical of the annihilator of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_008.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>, and we denote it by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_009.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"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}
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 kkth 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.
在这篇文章中，我们证明了两个关于分形函数值分布的结果。利用 Yamanoi 定理，第一个结果以简洁的形式给出了分形函数特征函数与其 k k th 导数之间关系的精确估计。这一结果扩展并改进了 Shan、Singh、Gopalakrishna、Edrei、Weitsman、Yang、Wu 和 Wu 等人的一些结果。第二个结果回答了 C. C. Yang 提出的一个猜想。通过一个反例，这个猜想变成了假的，但只要附加一个条件，它就会变成真的。
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
In this article, we investigate the following Schrödinger equation: −Δu−μ∣x∣2u=g(u)inRN,-Delta u-frac{mu }{{| x| }^{2}}u=gleft(u)hspace{1em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{N}, where N≥3Nge 3, μ∣x∣2frac{mu }{{| x| }^{2}} is called the Hardy potential and gg satisfies Berestycki-Lions conditions. If 0<μ<(N−2)240lt mu lt frac{{left(N-2)}^{2}}{4}, we
本文将研究以下薛定谔方程： - Δ u - μ ∣ x ∣ 2 u = g ( u ) in R N , -Delta u-frac{mu }{{| x| }^{2}}u=gleft(u)hspace{1em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{N}，其中 N ≥ 3 Nge 3 ，μ ∣ x∣ 2 frac{mu }{{| x| }^{2}}称为哈代势，g g 满足贝里切基-狮子条件。如果 0 < μ < ( N - 2 ) 2 4 0lt mu lt frac{left(N-2)}^{2}}{4} ，我们将取对称的山形。 我们将采用对称山口法来证明这个问题存在无穷多个解。
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
It is shown that the expression as A2=αA+βP{A}^{2}=alpha A+beta P 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 AA but also with the rank of PP. 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.
通过数字实例证明，广义二次矩阵的表达式 A 2 = α A + β P {A}^{2}=alpha A+beta P 并不是唯一的。然后证明广义二次矩阵表达式的唯一性不仅与 A A 的性质有关，还与 P P 的秩有关。此外，还得到了广义二次矩阵表达式唯一性的充分条件和必要条件。最后，还给出了关于广义二次矩阵的一些相关讨论。
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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