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The Brown-Halmos theorems on the Fock-Sobolev space
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2025-03-27 DOI: 10.1016/j.jmaa.2025.129532
Jie Qin
{"title":"The Brown-Halmos theorems on the Fock-Sobolev space","authors":"Jie Qin","doi":"10.1016/j.jmaa.2025.129532","DOIUrl":"10.1016/j.jmaa.2025.129532","url":null,"abstract":"<div><div>In this paper, we generalize the Brown-Halmos theorems to the Fock-Sobolev space. We obtain that the Brown-Halmos theorems hold true on the Fock-Sobolev space for Toeplitz operators with harmonic symbols. We completely explain the difference between the geometries of the Fock and Fock-Sobolev space by using the Berezin transform.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 1","pages":"Article 129532"},"PeriodicalIF":1.2,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143760161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Evaluation of reciprocal sums of hyperbolic functions using quasimodular forms
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2025-03-27 DOI: 10.1016/j.jmaa.2025.129526
Wei Wang
{"title":"Evaluation of reciprocal sums of hyperbolic functions using quasimodular forms","authors":"Wei Wang","doi":"10.1016/j.jmaa.2025.129526","DOIUrl":"10.1016/j.jmaa.2025.129526","url":null,"abstract":"<div><div>This paper studies eight families of infinite series involving hyperbolic functions. Under some conditions, these series are linear combinations of derivatives of Eisenstein series. Using complex multiplication theory, the structure of the rings of modular forms and quasimodular forms, and certain differential operators defined on these rings, this paper gives a systematic method for computing the values of these series at CM points. This paper also expresses the generalized reciprocal sums of Fibonacci numbers as the special values of the series mentioned above. Thus it gives some algebraic independence results about the generalized reciprocal sums of Fibonacci numbers.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 2","pages":"Article 129526"},"PeriodicalIF":1.2,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 Zhang-Yang's open question about iterative roots of PM functions of height 1 (II): Decreasing case
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2025-03-26 DOI: 10.1016/j.jmaa.2025.129517
Siyi Zhao , Liu Liu , Weinian Zhang
{"title":"On Zhang-Yang's open question about iterative roots of PM functions of height 1 (II): Decreasing case","authors":"Siyi Zhao ,&nbsp;Liu Liu ,&nbsp;Weinian Zhang","doi":"10.1016/j.jmaa.2025.129517","DOIUrl":"10.1016/j.jmaa.2025.129517","url":null,"abstract":"<div><div>A Zhang-Yang's open question reads: Does a PM function <em>F</em> with height <span><math><mi>H</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> have an iterative root <em>f</em> of order <span><math><mi>n</mi><mo>≤</mo><mi>N</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> if the ‘characteristic endpoints condition’ is not satisfied? This question was answered in the case that <em>F</em> is strictly increasing on its characteristic interval <span><math><mi>K</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span>. However, a more difficult case is that <em>F</em> is strictly decreasing on <span><math><mi>K</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span>. In this paper we discuss the decreasing case, giving existence of <em>f</em> of order <span><math><mi>n</mi><mo>&lt;</mo><mi>N</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span> with <span><math><mi>H</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>=</mo><mi>n</mi></math></span> and of order <span><math><mi>n</mi><mo>≤</mo><mi>N</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> with <span><math><mi>H</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"550 1","pages":"Article 129517"},"PeriodicalIF":1.2,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Asymptotic analysis of the nonsteady micropolar fluid flow through a system of thin pipes revisited: Boundary-layer-in-time effects 重新审视流经细管系统的非稳态微极性流体流动的渐近分析：边界层-时间效应
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2025-03-26 DOI: 10.1016/j.jmaa.2025.129519
Grigory Panasenko , Igor Pažanin , Borja Rukavina
{"title":"Asymptotic analysis of the nonsteady micropolar fluid flow through a system of thin pipes revisited: Boundary-layer-in-time effects","authors":"Grigory Panasenko ,&nbsp;Igor Pažanin ,&nbsp;Borja Rukavina","doi":"10.1016/j.jmaa.2025.129519","DOIUrl":"10.1016/j.jmaa.2025.129519","url":null,"abstract":"<div><div>In this paper, we revisit the problem of the time-dependent micropolar fluid flow in a thin pipe system considered in Pažanin et al. (2024) <span><span>[29]</span></span>. We remove the restriction that the inflow/outflow and the external source functions vanish for small values of time and extend the analysis to non-homogeneous initial conditions. This requires the construction of the boundary-layer-in-time and the boundary-layer-in-space-and-in-time correctors in the asymptotic expansion of the solution. Consequently, we propose a new asymptotic approximation of higher order of accuracy for a general case with strong coupling between velocity and microrotation. The error estimates are also proved justifying the use of the derived effective model and indicating its range of applicability.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"550 1","pages":"Article 129519"},"PeriodicalIF":1.2,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 asymptotic analysis of lazy reinforced random walks: A martingale approach
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2025-03-26 DOI: 10.1016/j.jmaa.2025.129520
Manuel González-Navarrete , Rodrigo Lambert , Víctor Hugo Vázquez-Guevara
{"title":"On the asymptotic analysis of lazy reinforced random walks: A martingale approach","authors":"Manuel González-Navarrete ,&nbsp;Rodrigo Lambert ,&nbsp;Víctor Hugo Vázquez-Guevara","doi":"10.1016/j.jmaa.2025.129520","DOIUrl":"10.1016/j.jmaa.2025.129520","url":null,"abstract":"<div><div>We provide a comprehensive characterization of the limiting behavior of lazy reinforced random walks (LRRW's). These random walks exhibit three distinct phases: diffusive, critical, and superdiffusive. Using a martingale theory approach, we establish proper versions of the law of large numbers, the almost sure convergence to even moments of Gaussian distribution, the law of the iterated logarithm, the almost sure central limit theorem, and the functional central limit theorem for the diffusive and critical regimes. In the superdiffusive regime, we demonstrate strong convergence to a random variable, as well as a central limit theorem and a law of the iterated logarithm for the fluctuations.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 2","pages":"Article 129520"},"PeriodicalIF":1.2,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Most of the minimization problems have a unique solution
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2025-03-26 DOI: 10.1016/j.jmaa.2025.129523
Ľubica Holá
{"title":"Most of the minimization problems have a unique solution","authors":"Ľubica Holá","doi":"10.1016/j.jmaa.2025.129523","DOIUrl":"10.1016/j.jmaa.2025.129523","url":null,"abstract":"<div><div>Let <em>X</em> be a Tychonoff topological space, <span><math><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> be the space of continuous real-valued functions defined on <em>X</em> and <span><math><mi>K</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> be the space of all nonempty compact subsets of <em>X</em>. Define the multifunction argmin<span><math><mo>:</mo><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>×</mo><mi>K</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><mi>X</mi></math></span> as follows: argmin <span><math><mo>(</mo><mi>f</mi><mo>,</mo><mi>K</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>x</mi><mo>∈</mo><mi>K</mi><mo>:</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>:</mo><mi>y</mi><mo>∈</mo><mi>K</mi><mo>}</mo><mo>}</mo></math></span>. Let <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mi>U</mi></mrow></msub></math></span> be the topology of uniform convergence on <span><math><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mi>V</mi></mrow></msub></math></span> the Vietoris topology on <span><math><mi>K</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. We prove that argmin<span><math><mo>:</mo><mo>(</mo><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>U</mi></mrow></msub><mo>)</mo><mo>×</mo><mo>(</mo><mi>K</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>V</mi></mrow></msub><mo>)</mo><mo>→</mo><mi>X</mi></math></span> is minimal usco and extend Kenderov's generic optimization theorem to Tychonoff almost Čech-complete spaces.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 1","pages":"Article 129523"},"PeriodicalIF":1.2,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143760164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Unique solutions to power-transformed affine systems
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2025-03-25 DOI: 10.1016/j.jmaa.2025.129515
John Stachurski , Ole Wilms , Junnan Zhang
{"title":"Unique solutions to power-transformed affine systems","authors":"John Stachurski ,&nbsp;Ole Wilms ,&nbsp;Junnan Zhang","doi":"10.1016/j.jmaa.2025.129515","DOIUrl":"10.1016/j.jmaa.2025.129515","url":null,"abstract":"<div><div>Systems of the form <span><math><mi>x</mi><mo>=</mo><msup><mrow><mo>(</mo><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mi>s</mi></mrow></msup><mo>+</mo><mi>b</mi></math></span> arise in a range of economic and financial applications, where <em>A</em> is a linear operator acting on a space of real-valued functions (or vectors) and <em>s</em> is a nonzero real value. In these applications, attention is focused on positive solutions. We provide a simple characterization of existence and uniqueness of positive solutions when <em>b</em> is positive and <em>A</em> is irreducible.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"550 1","pages":"Article 129515"},"PeriodicalIF":1.2,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Riesz transforms for bi-Schrödinger operators on weighted Lebesgue spaces
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2025-03-24 DOI: 10.1016/j.jmaa.2025.129516
Nguyen Ngoc Trong , Le Xuan Truong , Tan Duc Do
{"title":"Riesz transforms for bi-Schrödinger operators on weighted Lebesgue spaces","authors":"Nguyen Ngoc Trong ,&nbsp;Le Xuan Truong ,&nbsp;Tan Duc Do","doi":"10.1016/j.jmaa.2025.129516","DOIUrl":"10.1016/j.jmaa.2025.129516","url":null,"abstract":"<div><div>Let <span><math><mi>d</mi><mo>∈</mo><mo>{</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>,</mo><mo>…</mo><mo>}</mo></math></span> and a weight <span><math><mi>w</mi><mo>∈</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>ρ</mi></mrow></msubsup></math></span>. We consider the fourth-order Riesz transform <span><math><mi>T</mi><mo>=</mo><msup><mrow><mi>∇</mi></mrow><mrow><mn>4</mn></mrow></msup><mspace></mspace><msup><mrow><mi>L</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> associated with the bi-Schrödinger operator <span><math><mi>L</mi><mo>=</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, where <span><math><mi>V</mi><mo>∈</mo><mi>R</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>σ</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> with <span><math><mi>σ</mi><mo>&gt;</mo><mfrac><mrow><mn>2</mn><mi>d</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> stands for a Gaussian class of potentials. We show that <em>T</em> is bounded on <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>w</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> for all <em>p</em> in a suitable range depending on <em>σ</em>. If more conditions are imposed on either <em>σ</em> or <em>V</em>, the range for <em>p</em> can be extended to <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"548 2","pages":"Article 129516"},"PeriodicalIF":1.2,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143697914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Identities for the product of two Dirichlet series satisfying Hecke's functional equation
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2025-03-24 DOI: 10.1016/j.jmaa.2025.129514
Bruce C. Berndt, Likun Xie
{"title":"Identities for the product of two Dirichlet series satisfying Hecke's functional equation","authors":"Bruce C. Berndt,&nbsp;Likun Xie","doi":"10.1016/j.jmaa.2025.129514","DOIUrl":"10.1016/j.jmaa.2025.129514","url":null,"abstract":"<div><div>We derive a general formula for the product of two Dirichlet series that satisfy Hecke's functional equation. Several examples are provided to demonstrate the applicability of the formula. In addition, we discuss prior work on similar products and clarify certain issues arising in the existing literature.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"550 1","pages":"Article 129514"},"PeriodicalIF":1.2,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On asymptotics of oscillatory solutions to nth-order delay differential equations
IF 1.2 3区 数学
Journal of Mathematical Analysis and Applications Pub Date : 2025-03-24 DOI: 10.1016/j.jmaa.2025.129507
Elena Braverman , Alexander Domoshnitsky , John Ioannis Stavroulakis
{"title":"On asymptotics of oscillatory solutions to nth-order delay differential equations","authors":"Elena Braverman ,&nbsp;Alexander Domoshnitsky ,&nbsp;John Ioannis Stavroulakis","doi":"10.1016/j.jmaa.2025.129507","DOIUrl":"10.1016/j.jmaa.2025.129507","url":null,"abstract":"<div><div>We study bounded and decaying to zero solutions of the delay differential equation<span><span><span><math><msup><mrow><mi>x</mi></mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mi>x</mi><mo>(</mo><mi>t</mi><mo>−</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn><mspace></mspace><mtext>for</mtext><mspace></mspace><mi>t</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>(</mo><mi>ξ</mi><mo>)</mo><mo>=</mo><mi>φ</mi><mo>(</mo><mi>ξ</mi><mo>)</mo><mspace></mspace><mtext>for </mtext><mspace></mspace><mi>ξ</mi><mo>&lt;</mo><mn>0</mn><mo>.</mo></math></span></span></span> Kondrat'ev and Kiguradze introduced and defined principles of asymptotic behavior for its solution in the sense of the trichotomy: oscillatory, non-oscillatory with absolute values monotonically decaying to zero or monotonically increasing to ∞. Expanding upon such studies, we estimate the oscillation amplitudes of solutions. Decay to zero is established through fast oscillation: once distances between zeros are small enough, the Grönwall inequality growth estimate implies the amplitudes decrease to zero as <span><math><mi>t</mi><mo>→</mo><mo>∞</mo></math></span>. Exact growth estimates and calculation of these distances between zeros are proposed through evaluation for the spectral radii of some compact operators associated with the Green's function for an <em>n</em>-point problem.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"549 1","pages":"Article 129507"},"PeriodicalIF":1.2,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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