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Generalized Semi-infinite Polynomial Optimization and Semidefinite Programming Relaxations 广义半无限多项式优化和半无限编程松弛
IF 0.3
Acta Mathematica Vietnamica Pub Date : 2024-10-10 DOI: 10.1007/s40306-024-00551-7
Liguo Jiao, Jae Hyoung Lee, Tiến-Sơn Phạm
{"title":"Generalized Semi-infinite Polynomial Optimization and Semidefinite Programming Relaxations","authors":"Liguo Jiao,&nbsp;Jae Hyoung Lee,&nbsp;Tiến-Sơn Phạm","doi":"10.1007/s40306-024-00551-7","DOIUrl":"10.1007/s40306-024-00551-7","url":null,"abstract":"<div><p>This paper focuses on the study of a generalized semi-infinite programming, where the objective and the constraint functions are all real polynomials. We present a method for finding its global minimizers and global minimum using a hierarchy of semidefinite programming relaxations and prove the convergence result for the method. Numerical experiments are presented to show the efficiency of the proposed algorithm.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":"49 3","pages":"441 - 457"},"PeriodicalIF":0.3,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Normalization of Singular Contact Forms and Primitive 1-forms 奇异接触形式和原始 1-形式的规范化
IF 0.3
Acta Mathematica Vietnamica Pub Date : 2024-10-09 DOI: 10.1007/s40306-024-00545-5
Kai Jiang, Hong Minh Truong, Nguyen Tien Zung
{"title":"Normalization of Singular Contact Forms and Primitive 1-forms","authors":"Kai Jiang,&nbsp;Hong Minh Truong,&nbsp;Nguyen Tien Zung","doi":"10.1007/s40306-024-00545-5","DOIUrl":"10.1007/s40306-024-00545-5","url":null,"abstract":"<div><p>A differential 1-form <span>(alpha )</span> on a manifold of odd dimension <span>(2n+1)</span>, which satisfies the contact condition <span>(alpha wedge (dalpha )^n ne 0)</span> almost everywhere, but which vanishes at a point <i>O</i>, i.e., <span>(alpha (O) = 0)</span>, is called a <i>singular contact form</i> at <i>O</i>. The aim of this paper is to study local normal forms (formal, analytic and smooth) of such singular contact forms. Our study leads naturally to the study of normal forms of singular primitive 1-forms of a symplectic form <span>(omega )</span> in dimension 2<i>n</i>, i.e., differential 1-forms <span>(gamma )</span> which vanish at a point and such that <span>(dgamma = omega )</span>, and their corresponding conformal vector fields. Our results are an extension and improvement of previous results obtained by other authors, in particular Lychagin (1975), Webster (Amer. J. Math. <b>109</b>, 807–832 (1987)) and Zhitomirskii (1986, 1992). Besides the classical normalization techniques, such as the step-by step normalization methods based on the cohomological equations and the Moser path method, we also use the toric approach to the normalization problem for dynamical systems (Jiang et al. 2019; Zung, Ann. Math. <b>161</b>, 141–156 2005; Zung 2016; Zung, Arch. Rational Mech. Anal. <b>229</b>, 789–833 (2018)).</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":"49 3","pages":"327 - 345"},"PeriodicalIF":0.3,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40306-024-00545-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Degenerate Forward-backward Problem Involving the Spectral Dirichlet Laplacian 涉及谱Dirichlet拉普拉斯算子的退化正向后问题
IF 0.3
Acta Mathematica Vietnamica Pub Date : 2024-10-02 DOI: 10.1007/s40306-024-00555-3
Nguyen Ngoc Trong, Bui Le Trong Thanh, Tan Duc Do
{"title":"A Degenerate Forward-backward Problem Involving the Spectral Dirichlet Laplacian","authors":"Nguyen Ngoc Trong,&nbsp;Bui Le Trong Thanh,&nbsp;Tan Duc Do","doi":"10.1007/s40306-024-00555-3","DOIUrl":"10.1007/s40306-024-00555-3","url":null,"abstract":"<div><p>Let <span>(varOmega )</span> be an open bounded subset of <span>({mathbb {R}})</span>, <span>(s in (frac{1}{2},1))</span> and <span>(epsilon &gt; 0)</span>. We investigate the problem </p><div><div><span>$$begin{aligned} (P_epsilon ) quad left{ begin{array}{ll} {partial }_t u = -(-Delta )^s big ( varphi (u) + epsilon , {partial }_t(psi (u)) big ) &amp; text { in } varOmega times (0,T], varphi (u) + epsilon , {partial }_t(psi (u)) = 0 &amp; text { on } {partial }varOmega times (0,T], u = u_0 &amp; text { in } varOmega times {0}, end{array}right. end{aligned}$$</span></div></div><p>where <span>(varphi , psi in C^infty ({mathbb {R}}))</span> and <span>(u_0 in {mathcal {M}}^+(varOmega ))</span> satisfy certain assumptions. Here <span>((-Delta )^s)</span> denotes the spectral Dirichlet Laplacian and <span>({mathcal {M}}^+(varOmega ))</span> is the set of positive Radon measures on <span>(varOmega )</span>. We show that <span>((P_epsilon ))</span> has a unique weak solution.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":"49 4","pages":"691 - 718"},"PeriodicalIF":0.3,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142762002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Note on the Linear Independence of Alternating Multiple Zeta Values in Positive Characteristic 关于正特征交替多重泽塔值线性无关性的说明
IF 0.3
Acta Mathematica Vietnamica Pub Date : 2024-09-10 DOI: 10.1007/s40306-024-00554-4
Bo-Hae Im, Hojin Kim, Khac Nhuan Le, Tuan Ngo Dac, Lan Huong Pham
{"title":"Note on the Linear Independence of Alternating Multiple Zeta Values in Positive Characteristic","authors":"Bo-Hae Im,&nbsp;Hojin Kim,&nbsp;Khac Nhuan Le,&nbsp;Tuan Ngo Dac,&nbsp;Lan Huong Pham","doi":"10.1007/s40306-024-00554-4","DOIUrl":"10.1007/s40306-024-00554-4","url":null,"abstract":"<div><p>We discuss certain results related to the linear independence of alternating multiple zeta values introduced by Harada in 2021.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":"49 3","pages":"485 - 521"},"PeriodicalIF":0.3,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Weak Lefschetz Property of Artinian Algebras Associated to Paths and Cycles 与路径和循环相关的阿尔丁代数的弱勒夫谢茨性质
IF 0.3
Acta Mathematica Vietnamica Pub Date : 2024-08-14 DOI: 10.1007/s40306-024-00549-1
Hop D. Nguyen, Quang Hoa Tran
{"title":"The Weak Lefschetz Property of Artinian Algebras Associated to Paths and Cycles","authors":"Hop D. Nguyen,&nbsp;Quang Hoa Tran","doi":"10.1007/s40306-024-00549-1","DOIUrl":"10.1007/s40306-024-00549-1","url":null,"abstract":"<div><p>Given a base field <span>(Bbbk )</span> of characteristic zero, for each graph <i>G</i>, we associate the artinian algebra <i>A</i>(<i>G</i>) defined by the edge ideal of <i>G</i> and the squares of the variables. We study the weak Lefschetz property of <i>A</i>(<i>G</i>). We classify some classes of graphs with relatively few edges, including paths and cycles, such that its associated artinian ring has the weak Lefschetz property.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":"49 3","pages":"523 - 544"},"PeriodicalIF":0.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
New Algorithms for Solving the Split Equality Problems in Hilbert Spaces 求解Hilbert空间中分裂等式问题的新算法
IF 0.3
Acta Mathematica Vietnamica Pub Date : 2024-08-13 DOI: 10.1007/s40306-024-00552-6
Nguyen Song Ha, Truong Minh Tuyen
{"title":"New Algorithms for Solving the Split Equality Problems in Hilbert Spaces","authors":"Nguyen Song Ha,&nbsp;Truong Minh Tuyen","doi":"10.1007/s40306-024-00552-6","DOIUrl":"10.1007/s40306-024-00552-6","url":null,"abstract":"<div><p>We introduce a new approach by using unconstrained optimization to find a solution to the system of the split equality problems in real Hilbert spaces. Our new algorithms do not depend on the norm of the transfer mappings. We also give the relaxed iterative algorithms corresponding to the proposed algorithms. Finally, we present some numerical experiments to demonstrate the performance of the main results.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":"49 4","pages":"667 - 689"},"PeriodicalIF":0.3,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142761949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Minimal Balanced Neighborly Polynomials 最小平衡邻接多项式
IF 0.3
Acta Mathematica Vietnamica Pub Date : 2024-08-06 DOI: 10.1007/s40306-024-00547-3
Satoshi Murai, Nguyen Thi Thanh Tam
{"title":"Minimal Balanced Neighborly Polynomials","authors":"Satoshi Murai,&nbsp;Nguyen Thi Thanh Tam","doi":"10.1007/s40306-024-00547-3","DOIUrl":"10.1007/s40306-024-00547-3","url":null,"abstract":"<div><p>In this paper we introduce minimal balanced neighborly polynomials and show some methods to construct such polynomials. In particular, using this notion, we prove the existence of balanced neighborly polynomials of the following types: (i) type <span>((p,dots ,p))</span> for most prime numbers <i>p</i>, (ii) types <span>((d-1,d,d,d))</span>, <span>((d-1,d-1,d,d))</span> and <span>((d-1,d-1,d-1,d))</span> when <i>d</i> is odd or is divisible by 4. We also construct balanced neighborly simplicial spheres of type <span>((2,4k-1,4k-1,4k-1))</span>.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":"49 3","pages":"459 - 484"},"PeriodicalIF":0.3,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Vanishing and Non-negativity of the First Normal Hilbert Coefficient 第一正态希尔伯特系数的消失和非负性
IF 0.3
Acta Mathematica Vietnamica Pub Date : 2024-08-06 DOI: 10.1007/s40306-024-00548-2
Linquan Ma, Pham Hung Quy
{"title":"Vanishing and Non-negativity of the First Normal Hilbert Coefficient","authors":"Linquan Ma,&nbsp;Pham Hung Quy","doi":"10.1007/s40306-024-00548-2","DOIUrl":"10.1007/s40306-024-00548-2","url":null,"abstract":"<div><p>Let <span>((R,mathfrak {m}))</span> be a Noetherian local ring such that <span>(widehat{R})</span> is reduced. We prove that, when <span>(widehat{R})</span> is <span>(S_2)</span>, if there exists a parameter ideal <span>(Qsubseteq R)</span> such that <span>(bar{e}_1(Q)=0)</span>, then <i>R</i> is regular and <span>(nu (mathfrak {m}/Q)le 1)</span>. This leads to an affirmative answer to a problem raised by Goto-Hong-Mandal [Goto, S., Hong, J., Mandal, M.: The positivity of the first coefficients of normal Hilbert polynomials. Proc. Amer. Math. Soc. <b>139</b>(7), 2399–2406 \u0000(2011)]. We also give an alternative proof (in fact a strengthening) of their main result. In particular, we show that if <span>(widehat{R})</span> is equidimensional, then <span>(bar{e}_1(Q)ge 0)</span> for all parameter ideals <span>(Qsubseteq R)</span>, and in characteristic <span>(p&gt;0)</span>, we actually have <span>(e_1^*(Q)ge 0)</span>. Our proofs rely on the existence of big Cohen-Macaulay algebras.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":"49 3","pages":"311 - 325"},"PeriodicalIF":0.3,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Refinements and Extensions of Some Strong Duality Theorems in Conic Linear Programming 圆锥线性规划中若干强对偶定理的完善与扩展
IF 0.3
Acta Mathematica Vietnamica Pub Date : 2024-08-06 DOI: 10.1007/s40306-024-00543-7
Nguyen Ngoc Luan, Nguyen Dong Yen
{"title":"Refinements and Extensions of Some Strong Duality Theorems in Conic Linear Programming","authors":"Nguyen Ngoc Luan,&nbsp;Nguyen Dong Yen","doi":"10.1007/s40306-024-00543-7","DOIUrl":"10.1007/s40306-024-00543-7","url":null,"abstract":"<div><p>In this paper, we establish a series of new results on strong duality and solution existence for conic linear programs in locally convex Hausdorff topological vector spaces and finite-dimensional Euclidean spaces. Namely, under certain regularity conditions based on quasi-relative interiors of convex sets, we prove that if one problem in the dual pair consisting of a primal program and its dual has a solution, then the other problem also has a solution, and the optimal values of the problems are equal. In addition, we show that if the cones are generalized polyhedral convex, then the regularity conditions can be omitted. Moreover, if the spaces are finite-dimensional and the ordering cones are closed convex, then instead of the solution existence condition, it suffices to require the finiteness of the optimal value. The present paper complements our recent research work [Luan, N.N., Yen, N.D.: <i>Strong duality and solution existence under minimal assumptions in conic linear programming</i>. J. Optim. Theory Appl. (https://doi.org/10.1007/s10957-023-02318-w)].</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":"49 3","pages":"545 - 561"},"PeriodicalIF":0.3,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Betti Numbers of the Tangent Cones of Monomial Space Curves 单项式空间曲线切锥的贝蒂数
IF 0.3
Acta Mathematica Vietnamica Pub Date : 2024-08-06 DOI: 10.1007/s40306-024-00546-4
Nguyen P. H. Lan, Nguyen Chanh Tu, Thanh Vu
{"title":"Betti Numbers of the Tangent Cones of Monomial Space Curves","authors":"Nguyen P. H. Lan,&nbsp;Nguyen Chanh Tu,&nbsp;Thanh Vu","doi":"10.1007/s40306-024-00546-4","DOIUrl":"10.1007/s40306-024-00546-4","url":null,"abstract":"<div><p>Let <span>(H = langle n_1, n_2,n_3rangle )</span> be a numerical semigroup. Let <span>(widetilde{H})</span> be the interval completion of <i>H</i>, namely the semigroup generated by the interval <span>(langle n_1,n_1+1, ldots , n_3rangle )</span>. Let <i>K</i> be a field and <i>K</i>[<i>H</i>] the semigroup ring generated by <i>H</i>. Let <span>(I_H^{*})</span> be the defining ideal of the tangent cone of <i>K</i>[<i>H</i>]. In this paper, we describe the defining equations of <span>(I_H^{*})</span>. From that, we prove the Herzog-Stamate conjecture for monomial space curves stating that <span>(beta _i(I_H^{*}) le beta _i(I_{widetilde{H}}^{*}))</span> for all <i>i</i>, where <span>(beta _i(I_H^{*}))</span> and <span>(beta _i(I_{widetilde{H}}^{*}))</span> are the <i>i</i>th Betti numbers of <span>(I_H^{*})</span> and <span>(I_{widetilde{H}}^{*})</span> respectively.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":"49 3","pages":"347 - 365"},"PeriodicalIF":0.3,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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