We investigate the hyperuniformity of marked Gibbs point processes that have weak dependencies among distant points whilst the interactions of close points are kept arbitrary. Various stability and range assumptions are imposed on the Papangelou intensity in order to prove that the resulting point process is not hyperuniform. The scope of our results covers many frequently used models, including Gibbs point processes with a superstable, lower-regular, integrable pair potential, as well as the Widom–Rowlinson model with random radii and Gibbs point processes with interactions based on Voronoi tessellations and nearest-neighbour graphs.
{"title":"Non-hyperuniformity of Gibbs point processes with short-range interactions","authors":"David Dereudre, Daniela Flimmel","doi":"10.1017/jpr.2024.21","DOIUrl":"https://doi.org/10.1017/jpr.2024.21","url":null,"abstract":"We investigate the hyperuniformity of marked Gibbs point processes that have weak dependencies among distant points whilst the interactions of close points are kept arbitrary. Various stability and range assumptions are imposed on the Papangelou intensity in order to prove that the resulting point process is not hyperuniform. The scope of our results covers many frequently used models, including Gibbs point processes with a superstable, lower-regular, integrable pair potential, as well as the Widom–Rowlinson model with random radii and Gibbs point processes with interactions based on Voronoi tessellations and nearest-neighbour graphs.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885314","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 manuscript, we address open questions raised by Dieker and Yakir (2014), who proposed a novel method of estimating (discrete) Pickands constants $mathcal{H}^delta_alpha$ using a family of estimators $xi^delta_alpha(T)$ , $T>0$ , where $alphain(0,2]$ is the Hurst parameter, and $deltageq0$ is the step size of the regular discretization grid. We derive an upper bound for the discretization error $mathcal{H}_alpha^0 - mathcal{H}_alpha^delta$ , whose rate of convergence agrees with Conjecture 1 of Dieker and Yakir (2014) in the case $alphain(0,1]$ and agrees up to logarithmic terms for $alphain(1,2)$ . Moreover, we show that all moments of $xi_alpha^delta(T)$ are uniformly bounded and the bias of the estimator decays no slower than
{"title":"On the speed of convergence of discrete Pickands constants to continuous ones","authors":"Krzysztof Bisewski, Grigori Jasnovidov","doi":"10.1017/jpr.2024.37","DOIUrl":"https://doi.org/10.1017/jpr.2024.37","url":null,"abstract":"In this manuscript, we address open questions raised by Dieker and Yakir (2014), who proposed a novel method of estimating (discrete) Pickands constants <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline1.png\"/> <jats:tex-math> $mathcal{H}^delta_alpha$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> using a family of estimators <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline2.png\"/> <jats:tex-math> $xi^delta_alpha(T)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline3.png\"/> <jats:tex-math> $T>0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline4.png\"/> <jats:tex-math> $alphain(0,2]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the Hurst parameter, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline5.png\"/> <jats:tex-math> $deltageq0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the step size of the regular discretization grid. We derive an upper bound for the discretization error <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline6.png\"/> <jats:tex-math> $mathcal{H}_alpha^0 - mathcal{H}_alpha^delta$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, whose rate of convergence agrees with Conjecture 1 of Dieker and Yakir (2014) in the case <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline7.png\"/> <jats:tex-math> $alphain(0,1]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and agrees up to logarithmic terms for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline8.png\"/> <jats:tex-math> $alphain(1,2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Moreover, we show that all moments of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline9.png\"/> <jats:tex-math> $xi_alpha^delta(T)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are uniformly bounded and the bias of the estimator decays no slower than <jats:inline-formula> <jats:alte","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141873190","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 proved that for families of stochastic operators on a countable tensor product, depending smoothly on parameters, any spectral projection persists smoothly, where smoothness is defined using norms based on ideas of Dobrushin. A rigorous perturbation theory for families of stochastic operators with spectral gap is thereby created. It is illustrated by deriving an effective slow two-state dynamics for a three-state probabilistic cellular automaton.
{"title":"Persistence of spectral projections for stochastic operators on large tensor products","authors":"Robert S. Mackay","doi":"10.1017/jpr.2024.34","DOIUrl":"https://doi.org/10.1017/jpr.2024.34","url":null,"abstract":"<p>It is proved that for families of stochastic operators on a countable tensor product, depending smoothly on parameters, any spectral projection persists smoothly, where smoothness is defined using norms based on ideas of Dobrushin. A rigorous perturbation theory for families of stochastic operators with spectral gap is thereby created. It is illustrated by deriving an effective slow two-state dynamics for a three-state probabilistic cellular automaton.</p>","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141257973","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}
We consider a Poisson autoregressive process whose parameters depend on the past of the trajectory. We allow these parameters to take negative values, modelling inhibition. More precisely, the model is the stochastic process $(X_n)_{nge0}$ with parameters $a_1,ldots,a_p in mathbb{R}$ , $pinmathbb{N}$ , and $lambda ge 0$ , such that, for all $nge p$ , conditioned on $X_0,ldots,X_{n-1}$ , $X_n$ is Poisson distributed with parameter $(a_1 X_{n-1} + cdots + a_p X_{n-p} + lambda)_+$ . This process can be regarded as a discrete-time Hawkes process with inhibition and a memory of length p. In this paper we initiate the study of necessary and sufficient conditions of stability for these processes, which seems to be a hard problem in general. We consider specifically the case $p = 2$ , for which we are able to classify the asymptotic behavior of the process for the whole range of parameters, except for boundary cases. In particular, we show that the process remains stochastically bounded whenever the solution to the linear recurrence equation $x_n = a_1x_{n-1} + a_2x_{n-2} + lambda$ remains bounded, but the converse is not true. Furthermore, the criterion for stochastic boundedness is not symmetric in $a_1$ and $a_2$ , in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.
{"title":"(Almost) complete characterization of the stability of a discrete-time Hawkes process with inhibition and memory of length two","authors":"Manon Costa, Pascal Maillard, Anthony Muraro","doi":"10.1017/jpr.2024.28","DOIUrl":"https://doi.org/10.1017/jpr.2024.28","url":null,"abstract":"\u0000 We consider a Poisson autoregressive process whose parameters depend on the past of the trajectory. We allow these parameters to take negative values, modelling inhibition. More precisely, the model is the stochastic process \u0000 \u0000 \u0000 \u0000$(X_n)_{nge0}$\u0000\u0000 \u0000 with parameters \u0000 \u0000 \u0000 \u0000$a_1,ldots,a_p in mathbb{R}$\u0000\u0000 \u0000 , \u0000 \u0000 \u0000 \u0000$pinmathbb{N}$\u0000\u0000 \u0000 , and \u0000 \u0000 \u0000 \u0000$lambda ge 0$\u0000\u0000 \u0000 , such that, for all \u0000 \u0000 \u0000 \u0000$nge p$\u0000\u0000 \u0000 , conditioned on \u0000 \u0000 \u0000 \u0000$X_0,ldots,X_{n-1}$\u0000\u0000 \u0000 , \u0000 \u0000 \u0000 \u0000$X_n$\u0000\u0000 \u0000 is Poisson distributed with parameter \u0000 \u0000 \u0000 \u0000$(a_1 X_{n-1} + cdots + a_p X_{n-p} + lambda)_+$\u0000\u0000 \u0000 . This process can be regarded as a discrete-time Hawkes process with inhibition and a memory of length p. In this paper we initiate the study of necessary and sufficient conditions of stability for these processes, which seems to be a hard problem in general. We consider specifically the case \u0000 \u0000 \u0000 \u0000$p = 2$\u0000\u0000 \u0000 , for which we are able to classify the asymptotic behavior of the process for the whole range of parameters, except for boundary cases. In particular, we show that the process remains stochastically bounded whenever the solution to the linear recurrence equation \u0000 \u0000 \u0000 \u0000$x_n = a_1x_{n-1} + a_2x_{n-2} + lambda$\u0000\u0000 \u0000 remains bounded, but the converse is not true. Furthermore, the criterion for stochastic boundedness is not symmetric in \u0000 \u0000 \u0000 \u0000$a_1$\u0000\u0000 \u0000 and \u0000 \u0000 \u0000 \u0000$a_2$\u0000\u0000 \u0000 , in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141100378","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}
Ritik Soni, Ashok Kumar Pathak, Antonio Di Crescenzo, Alessandra Meoli
We introduce a bivariate tempered space-fractional Poisson process (BTSFPP) by time-changing the bivariate Poisson process with an independent tempered $alpha$ -stable subordinator. We study its distributional properties and its connection to differential equations. The Lévy measure for the BTSFPP is also derived. A bivariate competing risks and shock model based on the BTSFPP for predicting the failure times of items that undergo two random shocks is also explored. The system is supposed to break when the sum of two types of shock reaches a certain random threshold. Various results related to reliability, such as reliability function, hazard rates, failure density, and the probability that failure occurs due to a certain type of shock, are studied. We show that for a general Lévy subordinator, the failure time of the system is exponentially distributed with mean depending on the Laplace exponent of the Lévy subordinator when the threshold has a geometric distribution. Some special cases and several typical examples are also demonstrated.
{"title":"Bivariate tempered space-fractional Poisson process and shock models","authors":"Ritik Soni, Ashok Kumar Pathak, Antonio Di Crescenzo, Alessandra Meoli","doi":"10.1017/jpr.2024.30","DOIUrl":"https://doi.org/10.1017/jpr.2024.30","url":null,"abstract":"We introduce a bivariate tempered space-fractional Poisson process (BTSFPP) by time-changing the bivariate Poisson process with an independent tempered <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000305_inline1.png\"/> <jats:tex-math> $alpha$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-stable subordinator. We study its distributional properties and its connection to differential equations. The Lévy measure for the BTSFPP is also derived. A bivariate competing risks and shock model based on the BTSFPP for predicting the failure times of items that undergo two random shocks is also explored. The system is supposed to break when the sum of two types of shock reaches a certain random threshold. Various results related to reliability, such as reliability function, hazard rates, failure density, and the probability that failure occurs due to a certain type of shock, are studied. We show that for a general Lévy subordinator, the failure time of the system is exponentially distributed with mean depending on the Laplace exponent of the Lévy subordinator when the threshold has a geometric distribution. Some special cases and several typical examples are also demonstrated.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152771","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 $mathcal{V}$ and $mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on $mathbb{R}^d$ . A graph $mathcal{G}_mathcal{V}$ with vertex set $mathcal{V}$ is constructed by first connecting pairs of points (v, u) with $vinmathcal{V}$ and $uinmathcal{U}$ independently with probability $g(v-u)$ , where g is a non-increasing radial function, and then connecting two points $v_1,v_2inmathcal{V}$ if and only if they have a joint neighbor $uinmathcal{U}$
As a generalization of random recursive trees and preferential attachment trees, we consider random recursive metric spaces. These spaces are constructed from random blocks, each a metric space equipped with a probability measure, containing a labelled point called a hook, and assigned a weight. Random recursive metric spaces are equipped with a probability measure made up of a weighted sum of the probability measures assigned to its constituent blocks. At each step in the growth of a random recursive metric space, a point called a latch is chosen at random according to the equipped probability measure, and a new block is chosen at random and attached to the space by joining together the latch and the hook of the block. We use martingale theory to prove a law of large numbers and a central limit theorem for the insertion depth, the distance from the master hook to the latch chosen. We also apply our results to further generalizations of random trees, hooking networks, and continuous spaces constructed from line segments.
{"title":"Depths in random recursive metric spaces","authors":"Colin Desmarais","doi":"10.1017/jpr.2024.32","DOIUrl":"https://doi.org/10.1017/jpr.2024.32","url":null,"abstract":"As a generalization of random recursive trees and preferential attachment trees, we consider random recursive metric spaces. These spaces are constructed from random blocks, each a metric space equipped with a probability measure, containing a labelled point called a hook, and assigned a weight. Random recursive metric spaces are equipped with a probability measure made up of a weighted sum of the probability measures assigned to its constituent blocks. At each step in the growth of a random recursive metric space, a point called a latch is chosen at random according to the equipped probability measure, and a new block is chosen at random and attached to the space by joining together the latch and the hook of the block. We use martingale theory to prove a law of large numbers and a central limit theorem for the insertion depth, the distance from the master hook to the latch chosen. We also apply our results to further generalizations of random trees, hooking networks, and continuous spaces constructed from line segments.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152709","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}
We establish power-series expansions for the asymptotic expectations of the vertex number and missed area of random disc-polygons in planar convex bodies with $C^{k+1}_+$ -smooth boundaries. These results extend asymptotic formulas proved in Fodor et al. (2014).
{"title":"Series expansions for random disc-polygons in smooth plane convex bodies","authors":"F. Fodor, Nicolás A. Montenegro Pinzón","doi":"10.1017/jpr.2024.27","DOIUrl":"https://doi.org/10.1017/jpr.2024.27","url":null,"abstract":"\u0000 We establish power-series expansions for the asymptotic expectations of the vertex number and missed area of random disc-polygons in planar convex bodies with \u0000 \u0000 \u0000 \u0000$C^{k+1}_+$\u0000\u0000 \u0000 -smooth boundaries. These results extend asymptotic formulas proved in Fodor et al. (2014).","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140968604","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}
Ming Cheng, Dimitrios G. Konstantinides, Dingcheng Wang
Multivariate regular variation is a key concept that has been applied in finance, insurance, and risk management. This paper proposes a new dependence assumption via a framework of multivariate regular variation. Under the condition that financial and insurance risks satisfy our assumption, we conduct asymptotic analyses for multidimensional ruin probabilities in the discrete-time and continuous-time cases. Also, we present a two-dimensional numerical example satisfying our assumption, through which we show the accuracy of the asymptotic result for the discrete-time multidimensional insurance risk model.
{"title":"Multivariate regularly varying insurance and financial risks in multidimensional risk models","authors":"Ming Cheng, Dimitrios G. Konstantinides, Dingcheng Wang","doi":"10.1017/jpr.2024.23","DOIUrl":"https://doi.org/10.1017/jpr.2024.23","url":null,"abstract":"Multivariate regular variation is a key concept that has been applied in finance, insurance, and risk management. This paper proposes a new dependence assumption via a framework of multivariate regular variation. Under the condition that financial and insurance risks satisfy our assumption, we conduct asymptotic analyses for multidimensional ruin probabilities in the discrete-time and continuous-time cases. Also, we present a two-dimensional numerical example satisfying our assumption, through which we show the accuracy of the asymptotic result for the discrete-time multidimensional insurance risk model.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930075","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}
By the technique of augmented truncations, we obtain the perturbation bounds on the distance of the finite-time state distributions of two continuous-time Markov chains (CTMCs) in a type of weaker norm than the V-norm. We derive the estimates for strongly and exponentially ergodic CTMCs. In particular, we apply these results to get the bounds for CTMCs satisfying Doeblin or stochastically monotone conditions. Some examples are presented to illustrate the limitation of the V-norm in perturbation analysis and to show the quality of the weak norm.
通过增强截断技术，我们得到了两个连续时间马尔可夫链（CTMC）的有限时间状态分布距离的扰动边界，其规范类型比 V 规范更弱。我们推导了强遍历和指数遍历 CTMC 的估计值。特别是，我们应用这些结果得到了满足多布林或随机单调条件的 CTMC 的边界。我们列举了一些例子来说明 V 准则在扰动分析中的局限性，并展示了弱准则的质量。
{"title":"Perturbation analysis for continuous-time Markov chains in a weak sense","authors":"Na Lin, Yuanyuan Liu","doi":"10.1017/jpr.2024.20","DOIUrl":"https://doi.org/10.1017/jpr.2024.20","url":null,"abstract":"By the technique of augmented truncations, we obtain the perturbation bounds on the distance of the finite-time state distributions of two continuous-time Markov chains (CTMCs) in a type of weaker norm than the <jats:italic>V</jats:italic>-norm. We derive the estimates for strongly and exponentially ergodic CTMCs. In particular, we apply these results to get the bounds for CTMCs satisfying Doeblin or stochastically monotone conditions. Some examples are presented to illustrate the limitation of the <jats:italic>V</jats:italic>-norm in perturbation analysis and to show the quality of the weak norm.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140929800","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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