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Hereditary Saturated Subsets and the Invariant Basis Number Property of the Leavitt Path Algebra of Cartesian Products 笛卡尔积的莱维特路径代数的遗传饱和子集和不变基数特性
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-07-05 DOI: 10.1134/s0001434624030295
Min Li, Huanhuan Li, Yuquan Wen
{"title":"Hereditary Saturated Subsets and the Invariant Basis Number Property of the Leavitt Path Algebra of Cartesian Products","authors":"Min Li, Huanhuan Li, Yuquan Wen","doi":"10.1134/s0001434624030295","DOIUrl":"https://doi.org/10.1134/s0001434624030295","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In this note, first, we describe the (minimal) hereditary saturated subsets of finite acyclic graphs and finite graphs whose cycles have no exits. Then we show that the Cartesian product <span>(C_mtimes L_n)</span> of an <span>(m)</span>-cycle <span>(C_m)</span> by an <span>(n)</span>-line <span>(L_n)</span> has nontrivial hereditary saturated subsets even though the graphs <span>(C_m)</span> and <span>(L_n)</span> themselves have only trivial hereditary saturated subsets. Tomforde (Theorem 5.7 in “Uniqueness theorems and ideal structure for Leavitt path algebras,” J. Algebra <b>318</b> (2007), 270–299) proved that there exists a one-to-one correspondence between the set of graded ideals of the Leavitt path algebra <span>(L(E))</span> of a graph <span>(E)</span> and the set of hereditary saturated subsets of <span>(E^0)</span>. This shows that the algebraic structure of the Leavitt path algebra <span>(L(C_mtimes L_n))</span> of the Cartesian product is plentiful. We also prove that the invariant basis number property of <span>(L(C_mtimes L_n))</span> can be derived from that of <span>(L(C_m))</span>. More generally, we also show that the invariant basis number property of <span>(L(Etimes L_n))</span> can be derived from that of <span>(L(E))</span> if <span>(E)</span> is a finite graph without sinks. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"47 4 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573497","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}
引用次数: 0
Piercing Hyperplane Theorem 穿透超平面定理
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-07-05 DOI: 10.1134/s0001434624030349
Burak Ünveren, Guy Barokas
{"title":"Piercing Hyperplane Theorem","authors":"Burak Ünveren, Guy Barokas","doi":"10.1134/s0001434624030349","DOIUrl":"https://doi.org/10.1134/s0001434624030349","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We prove that any strictly convex and closed set in <span>(mathbb{R}^n)</span> is an affine subspace if it contains a hyperplane as a subset. In other words, no hyperplane fits into a strictly convex and closed set <span>(C)</span> unless <span>(C)</span> is flat. We also present certain applications of this result in economic theory reminiscent of the separating and supporting hyperplane theorems. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"11 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573498","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}
引用次数: 0
On the Additive Complexity of Some Integer Sequences 论某些整数序列的加法复杂性
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-07-05 DOI: 10.1134/s0001434624030106
I. S. Sergeev
{"title":"On the Additive Complexity of Some Integer Sequences","authors":"I. S. Sergeev","doi":"10.1134/s0001434624030106","DOIUrl":"https://doi.org/10.1134/s0001434624030106","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper presents several results concerning the complexity of calculations in the model of vector addition chains. A refinement of N. Pippenger’s upper bound is obtained for the complexity of the class of integer <span>(m times n)</span> matrices with the constraint <span>(q)</span> on the size of the coefficients as <span>(H=mnlog_2 q to infty)</span> up to <span>(min{m,n}log_2 q+(1+o(1))H/log_2 H+n)</span>. Next, we establish an asymptotically tight bound <span>((2+o(1))sqrt n)</span> on the complexity of сomputation of the number <span>(2^n-1)</span> in the base of powers of <span>(2)</span>. Based on generalized Sidon sequences, constructive examples of integer sets of cardinality <span>(n)</span> are constructed: sets, with polynomial size of elements, having the complexity <span>(n+Omega(n^{1-varepsilon}))</span> for any <span>(varepsilon&gt;0)</span> and sets, with the size <span>(n^{O(log n)})</span> of the elements, having the complexity <span>(n+Omega(n))</span>. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"19 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573478","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}
引用次数: 0
Group of Isometries of the Lattice $$K_0(mathbb P_n)$$ K_0(mathbb P_n)$$ 的等距网格群
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-07-05 DOI: 10.1134/s0001434624030222
I. S. Beldiev
{"title":"Group of Isometries of the Lattice $$K_0(mathbb P_n)$$","authors":"I. S. Beldiev","doi":"10.1134/s0001434624030222","DOIUrl":"https://doi.org/10.1134/s0001434624030222","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the isometry group of the Grothendieck group <span>(K_0(mathbb P_n))</span> equipped with a bilinear asymmetric Euler form. We prove several properties of this group; in particular, we show that it is isomorphic to the direct product of <span>(mathbb Z/2mathbb Z)</span> by the free Abelian group of rank <span>([(n+1)/2])</span>. We also explicitly calculate its generators for <span>(nle 6)</span>. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"10 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Periodicity of the Somos Sequences Modulo $$m$$ 关于索莫斯序列模数$m$$$的周期性
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-07-05 DOI: 10.1134/s000143462403012x
A. V. Ustinov
{"title":"On Periodicity of the Somos Sequences Modulo $$m$$","authors":"A. V. Ustinov","doi":"10.1134/s000143462403012x","DOIUrl":"https://doi.org/10.1134/s000143462403012x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We prove the periodicity of finite rank Somos sequences modulo <span>(m)</span>. As an application, we prove the periodicity of the Somos-<span>((6)(mathrm{mod} m))</span> sequence. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"2016 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573481","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}
引用次数: 0
On the Intermediate Values of the Lower Quantization Dimension 关于低量化维度的中间值
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-07-05 DOI: 10.1134/s0001434624030039
A. V. Ivanov
{"title":"On the Intermediate Values of the Lower Quantization Dimension","authors":"A. V. Ivanov","doi":"10.1134/s0001434624030039","DOIUrl":"https://doi.org/10.1134/s0001434624030039","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> It is well known that the lower quantization dimension <span>(underline{D}(mu))</span> of a Borel probability measure <span>(mu)</span> given on a metric compact set <span>((X,rho))</span> does not exceed the lower box dimension <span>(underline{dim}_BX)</span> of <span>(X)</span>. We prove the following intermediate value theorem for the lower quantization dimension of probability measures: for any nonnegative number <span>(a)</span> smaller that the dimension <span>(zunderline{dim}_BX)</span> of the compact set <span>(X)</span>, there exists a probability measure <span>(mu_a)</span> on <span>(X)</span> with support <span>(X)</span> such that <span>(underline{D}(mu_a)=a)</span>. The number <span>(zunderline{dim}_BX)</span> characterizes the asymptotic behavior of the lower box dimension of closed <span>(varepsilon)</span>-neighborhoods of zero-dimensional, in the sense of <span>(dim_B)</span>, closed subsets of <span>(X)</span> as <span>(varepsilonto 0)</span>. For a wide class of metric compact sets, the equality <span>(zunderline{dim}_BX=underline{dim}_BX)</span> holds. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"147 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573359","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}
引用次数: 0
On the Growth Function of $$n$$ -Valued Dynamics 论 $$n$$ 有值动力学的增长函数
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-07-05 DOI: 10.1134/s0001434624030143
M. A. Chirkov
{"title":"On the Growth Function of $$n$$ -Valued Dynamics","authors":"M. A. Chirkov","doi":"10.1134/s0001434624030143","DOIUrl":"https://doi.org/10.1134/s0001434624030143","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> This paper answers the question of V. M. Buchstaber on the growth function in case of certain <span>(n)</span>-valued group. This question is in close relation to specific discrete integrable systems. In the present paper, we find a specific formula for the growth function in the case of prime <span>(n)</span>. We also prove a polynomial asymptotic estimate of the growth function in the general case. At the end, we pose new conjectures and questions regarding growth functions. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"20 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573480","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}
引用次数: 0
Integral Representations of $$mathrm{zeta}(m)$$ $$mathrm{zeta}(m)$$ 的积分表示法
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-07-05 DOI: 10.1134/s0001434624030283
Chunli Li, Wenchang Chu
{"title":"Integral Representations of $$mathrm{zeta}(m)$$","authors":"Chunli Li, Wenchang Chu","doi":"10.1134/s0001434624030283","DOIUrl":"https://doi.org/10.1134/s0001434624030283","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> An open problem about integral representation of <span>(zeta(2n))</span>, proposed recently by Pain (2023), is resolved by integration by parts. More general integrals are examined by manipulating the beta integral and digamma function. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"12 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573494","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}
引用次数: 0
Quasi-Energy Function for Morse–Smale 3-Diffeomorphisms with Fixed Points with Pairwise Distinct Indices 莫尔斯-斯马尔 3-二阶异构的准能量函数，其定点具有成对的不同指数
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-07-05 DOI: 10.1134/s0001434624030301
O. V. Pochinka, E. A. Talanova
{"title":"Quasi-Energy Function for Morse–Smale 3-Diffeomorphisms with Fixed Points with Pairwise Distinct Indices","authors":"O. V. Pochinka, E. A. Talanova","doi":"10.1134/s0001434624030301","DOIUrl":"https://doi.org/10.1134/s0001434624030301","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The present paper is devoted to a lower bound for the number of critical points of the Lyapunov function for Morse–Smale 3-diffeomorphisms with fixed points with pairwise distinct indices. It is known that, in the presence of a single noncompact heteroclinic curve, the supporting manifold of the diffeomorphisms under consideration is a 3-sphere, and the class of topological conjugacy of such a diffeomorphism <span>(f)</span> is completely determined by the equivalence class (there exist infinitely many of them) of the Hopf knot <span>(L_{f})</span>, which is a knot in the generating class of the fundamental group of the manifold <span>(mathbb S^2times mathbb S^1)</span>. </p><p> Moreover, any Hopf knot is realized by some diffeomorphism of the class under consideration. It is known that the diffeomorphisms defined by the standard Hopf knot <span>(L_0={s}times mathbb S^1)</span> have an energy function, which is a Lyapunov function whose set of critical points coincides with the chain recurrent set. However, the set of critical points of any Lyapunov function of a diffeomorphism <span>(f)</span> with a nonstandard Hopf knot is strictly greater than the chain recurrent set of the diffeomorphism. </p><p> In the present paper, for the diffeomorphisms defined by generalized Mazur knots, a quasi-energy function has been constructed, which is a Lyapunov function with a minimum number of critical points. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"5 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573511","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}
引用次数: 0
On a Linear Form in the Ordinates of Zeros of the Riemann Zeta Function 论黎曼 Zeta 函数零点序列中的线性形式
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-04-22 DOI: 10.1134/s0001434624010103
E. D. Yudelevich
{"title":"On a Linear Form in the Ordinates of Zeros of the Riemann Zeta Function","authors":"E. D. Yudelevich","doi":"10.1134/s0001434624010103","DOIUrl":"https://doi.org/10.1134/s0001434624010103","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We obtain an asymptotic formula for the sum </p><span>$$H=sum_{0&lt;gamma_kle T,,1le kle 4}h(gamma_1+gamma_2-gamma_3-gamma_4),$$</span><p> where the <span>(gamma_k)</span> run over the imaginary parts of nontrivial zeros of the Riemann zeta function with multiplicities taken into account and the function <span>(h)</span> belongs to some special class of functions in <span>(L^1(mathbb R))</span>. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"12 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805212","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}
引用次数: 0
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