Journal of Theoretical Probability最新文献

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On the Local Time of Anisotropic Random Walk on $$mathbb Z^2$$ 各向异性随机行走的局部时间 $$mathbb Z^2$$
4区 数学
Journal of Theoretical Probability Pub Date : 2023-10-31 DOI: 10.1007/s10959-023-01297-y
Endre Csáki, Antónia Földes
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引用次数: 0
A Theory of Singular Values for Finite Free Probability 有限自由概率的奇异值理论
4区 数学
Journal of Theoretical Probability Pub Date : 2023-10-29 DOI: 10.1007/s10959-023-01295-0
Aurelien Gribinski
{"title":"A Theory of Singular Values for Finite Free Probability","authors":"Aurelien Gribinski","doi":"10.1007/s10959-023-01295-0","DOIUrl":"https://doi.org/10.1007/s10959-023-01295-0","url":null,"abstract":"","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136157510","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: 2
Lower Deviation for the Supremum of the Support of Super-Brownian Motion 超布朗运动支持极值的低偏差
4区 数学
Journal of Theoretical Probability Pub Date : 2023-10-19 DOI: 10.1007/s10959-023-01292-3
Yan-Xia Ren, Renming Song, Rui Zhang
{"title":"Lower Deviation for the Supremum of the Support of Super-Brownian Motion","authors":"Yan-Xia Ren, Renming Song, Rui Zhang","doi":"10.1007/s10959-023-01292-3","DOIUrl":"https://doi.org/10.1007/s10959-023-01292-3","url":null,"abstract":"","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135728558","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
Some Properties of Markov chains on the Free Group $${mathbb {F}}_2$$ 自由群上马尔可夫链的一些性质 $${mathbb {F}}_2$$
4区 数学
Journal of Theoretical Probability Pub Date : 2023-10-18 DOI: 10.1007/s10959-023-01294-1
Antoine Goldsborough, Stefanie Zbinden
{"title":"Some Properties of Markov chains on the Free Group $${mathbb {F}}_2$$","authors":"Antoine Goldsborough, Stefanie Zbinden","doi":"10.1007/s10959-023-01294-1","DOIUrl":"https://doi.org/10.1007/s10959-023-01294-1","url":null,"abstract":"","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135825389","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: 1
Cutpoints of (1,2) and (2,1) Random Walks on the Lattice of Positive Half Line (1,2)和(2,1)随机漫步在正半直线格上的截点
4区 数学
Journal of Theoretical Probability Pub Date : 2023-10-13 DOI: 10.1007/s10959-023-01293-2
Lanlan Tang, Hua-Ming Wang
{"title":"Cutpoints of (1,2) and (2,1) Random Walks on the Lattice of Positive Half Line","authors":"Lanlan Tang, Hua-Ming Wang","doi":"10.1007/s10959-023-01293-2","DOIUrl":"https://doi.org/10.1007/s10959-023-01293-2","url":null,"abstract":"","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135854330","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
Green Function for an Asymptotically Stable Random Walk in a Half Space 半空间中渐近稳定随机漫步的格林函数
4区 数学
Journal of Theoretical Probability Pub Date : 2023-09-29 DOI: 10.1007/s10959-023-01283-4
Denis Denisov, Vitali Wachtel
{"title":"Green Function for an Asymptotically Stable Random Walk in a Half Space","authors":"Denis Denisov, Vitali Wachtel","doi":"10.1007/s10959-023-01283-4","DOIUrl":"https://doi.org/10.1007/s10959-023-01283-4","url":null,"abstract":"Abstract We consider an asymptotically stable multidimensional random walk $$S(n)=(S_1(n),ldots , S_d(n) )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . For every vector $$x=(x_1ldots ,x_d)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with $$x_1ge 0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , let $$tau _x:=min {n&gt;0: x_{1}+S_1(n)le 0}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>min</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>n</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:mn>0</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> be the first time the random walk $$x+S(n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> leaves the upper half space. We obtain the asymptotics of $$p_n(x,y):= {textbf{P}}(x+S(n) in y+Delta , tau _x&gt;n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mi>y</mml:mi> <mml:mo>+</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135246621","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
General Mean Reflected Backward Stochastic Differential Equations 一般均值反映后向随机微分方程
4区 数学
Journal of Theoretical Probability Pub Date : 2023-09-25 DOI: 10.1007/s10959-023-01288-z
Ying Hu, Remi Moreau, Falei Wang
{"title":"General Mean Reflected Backward Stochastic Differential Equations","authors":"Ying Hu, Remi Moreau, Falei Wang","doi":"10.1007/s10959-023-01288-z","DOIUrl":"https://doi.org/10.1007/s10959-023-01288-z","url":null,"abstract":"","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135816097","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
The Distributions of the Mean of Random Vectors with Fixed Marginal Distribution 具有固定边际分布的随机向量的均值分布
4区 数学
Journal of Theoretical Probability Pub Date : 2023-09-25 DOI: 10.1007/s10959-023-01277-2
Andrzej Komisarski, Jacques Labuschagne
{"title":"The Distributions of the Mean of Random Vectors with Fixed Marginal Distribution","authors":"Andrzej Komisarski, Jacques Labuschagne","doi":"10.1007/s10959-023-01277-2","DOIUrl":"https://doi.org/10.1007/s10959-023-01277-2","url":null,"abstract":"Abstract Using recent results concerning non-uniqueness of the center of the mix for completely mixable probability distributions, we obtain the following result: For each $$din {mathbb {N}}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;d&lt;/mml:mi&gt; &lt;mml:mo&gt;∈&lt;/mml:mo&gt; &lt;mml:mi&gt;N&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; and each non-empty bounded Borel set $$Bsubset {mathbb {R}}^d$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;B&lt;/mml:mi&gt; &lt;mml:mo&gt;⊂&lt;/mml:mo&gt; &lt;mml:msup&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;R&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mi&gt;d&lt;/mml:mi&gt; &lt;/mml:msup&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; , there exists a d -dimensional probability distribution $$varvec{mu }$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;μ&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; satisfying the following: For each $$nge 3$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;≥&lt;/mml:mo&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; and each probability distribution $$varvec{nu }$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ν&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; on B , there exist d -dimensional random vectors $${textbf{X}}_{varvec{nu },1},{textbf{X}}_{varvec{nu },2},dots ,{textbf{X}}_{varvec{nu },n}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;X&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ν&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;X&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ν&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mo&gt;⋯&lt;/mml:mo&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;X&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ν&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; such that $$frac{1}{n}({textbf{X}}_{varvec{nu },1}+{textbf{X}}_{varvec{nu },2}+dots +{textbf{X}}_{varvec{nu },n})sim varvec{nu }$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mfrac&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;/mml:mfrac&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;X&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ν&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;X&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ν&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:mo&gt;⋯&lt;/mml:mo&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;X&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ν&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;∼&lt;/mml:mo&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ν&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; and $${textbf{X}}_{varvec{nu },i}sim varvec{mu }$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;X&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mro","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135815919","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
Joint Sum-and-Max Limit for a Class of Long-Range Dependent Processes with Heavy Tails 一类具有重尾的长程相关过程的联合和极大极限
4区 数学
Journal of Theoretical Probability Pub Date : 2023-09-25 DOI: 10.1007/s10959-023-01289-y
Shuyang Bai, He Tang
{"title":"Joint Sum-and-Max Limit for a Class of Long-Range Dependent Processes with Heavy Tails","authors":"Shuyang Bai, He Tang","doi":"10.1007/s10959-023-01289-y","DOIUrl":"https://doi.org/10.1007/s10959-023-01289-y","url":null,"abstract":"","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135816101","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
Explicit Approximation of Invariant Measure for Stochastic Delay Differential Equations with the Nonlinear Diffusion Term 具有非线性扩散项的随机时滞微分方程不变测度的显式逼近
4区 数学
Journal of Theoretical Probability Pub Date : 2023-09-22 DOI: 10.1007/s10959-023-01290-5
Xiaoyue Li, Xuerong Mao, Guoting Song
{"title":"Explicit Approximation of Invariant Measure for Stochastic Delay Differential Equations with the Nonlinear Diffusion Term","authors":"Xiaoyue Li, Xuerong Mao, Guoting Song","doi":"10.1007/s10959-023-01290-5","DOIUrl":"https://doi.org/10.1007/s10959-023-01290-5","url":null,"abstract":"","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136016686","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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