We use a new method via $p$-Wasserstein bounds to prove Cram'er-type moderate deviations in (multivariate) normal approximations. In the classical setting that $W$ is a standardized sum of $n$ independent and identically distributed (i.i.d.) random variables with sub-exponential tails, our method recovers the optimal range of $0leq x=o(n^{1/6})$ and the near optimal error rate $O(1)(1+x)(log n+x^2)/sqrt{n}$ for $P(W>x)/(1-Phi(x))to 1$, where $Phi$ is the standard normal distribution function. Our method also works for dependent random variables (vectors) and we give applications to the combinatorial central limit theorem, Wiener chaos, homogeneous sums and local dependence. The key step of our method is to show that the $p$-Wasserstein distance between the distribution of the random variable (vector) of interest and a normal distribution grows like $O(p^alpha Delta)$, $1leq pleq p_0$, for some constants $alpha, Delta$ and $p_0$. In the above i.i.d. setting, $alpha=1, Delta=1/sqrt{n}, p_0=n^{1/3}$. For this purpose, we obtain general $p$-Wasserstein bounds in (multivariate) normal approximations using Stein's method.
{"title":"From p-Wasserstein bounds to moderate deviations","authors":"Xiao Fang, Yuta Koike","doi":"10.1214/23-ejp976","DOIUrl":"https://doi.org/10.1214/23-ejp976","url":null,"abstract":"We use a new method via $p$-Wasserstein bounds to prove Cram'er-type moderate deviations in (multivariate) normal approximations. In the classical setting that $W$ is a standardized sum of $n$ independent and identically distributed (i.i.d.) random variables with sub-exponential tails, our method recovers the optimal range of $0leq x=o(n^{1/6})$ and the near optimal error rate $O(1)(1+x)(log n+x^2)/sqrt{n}$ for $P(W>x)/(1-Phi(x))to 1$, where $Phi$ is the standard normal distribution function. Our method also works for dependent random variables (vectors) and we give applications to the combinatorial central limit theorem, Wiener chaos, homogeneous sums and local dependence. The key step of our method is to show that the $p$-Wasserstein distance between the distribution of the random variable (vector) of interest and a normal distribution grows like $O(p^alpha Delta)$, $1leq pleq p_0$, for some constants $alpha, Delta$ and $p_0$. In the above i.i.d. setting, $alpha=1, Delta=1/sqrt{n}, p_0=n^{1/3}$. For this purpose, we obtain general $p$-Wasserstein bounds in (multivariate) normal approximations using Stein's method.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45057308","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}
We consider the spatially inhomogeneous Moran model with seed-banks introduced in den Hollander and Nandan (2021). Populations comprising $active$ and $dormant$ individuals are structured in colonies labelled by $mathbb{Z}^d,~dgeq 1$. The population sizes are drawn from an ergodic, translation-invariant, uniformly elliptic field that form a random environment. Individuals carry one of two types: $heartsuit$, $spadesuit$. Dormant individual resides in what is called a seed-bank. Active individuals exchange type from seed-bank of their own colony and resample type by choosing parent from the active populations according to a symmetric migration kernel. In den Hollander and Nandan (2021) by using a dual (an interacting coalescing particle system), we showed that the spatial system exhibits a dichotomy between $clustering$ (mono-type equilibrium) and $coexistence$ (multi-type equilibrium). In this paper we identify the domain of attraction for each mono-type equilibrium in the clustering regime for a $fixed$ environment. We also show that when the migration kernel is $recurrent$, for a.e. realization of the environment, the system with an initially $consistent$ type distribution converges weakly to a mono-type equilibrium in which the fixation probability to type-$heartsuit$ configuration does not depend on the environment. A formula for the fixation probability is given in terms of an annealed average of type-$heartsuit$ densities in dormant and active population biased by ratio of the two population sizes at the target colony. For the proofs, we use duality and environment seen by particle introduced in Dolgopyat and Goldsheid (2019) for RWRE on a strip. A spectral analysis of Markov operator yields quenched weak convergence of the environment process associated with single-particle dual to a reversible ergodic distribution which we transfer to the spatial system by using duality.
{"title":"Spatial populations with seed-banks in random environment: III. Convergence towards mono-type equilibrium","authors":"S. Nandan","doi":"10.1214/23-EJP922","DOIUrl":"https://doi.org/10.1214/23-EJP922","url":null,"abstract":"We consider the spatially inhomogeneous Moran model with seed-banks introduced in den Hollander and Nandan (2021). Populations comprising $active$ and $dormant$ individuals are structured in colonies labelled by $mathbb{Z}^d,~dgeq 1$. The population sizes are drawn from an ergodic, translation-invariant, uniformly elliptic field that form a random environment. Individuals carry one of two types: $heartsuit$, $spadesuit$. Dormant individual resides in what is called a seed-bank. Active individuals exchange type from seed-bank of their own colony and resample type by choosing parent from the active populations according to a symmetric migration kernel. In den Hollander and Nandan (2021) by using a dual (an interacting coalescing particle system), we showed that the spatial system exhibits a dichotomy between $clustering$ (mono-type equilibrium) and $coexistence$ (multi-type equilibrium). In this paper we identify the domain of attraction for each mono-type equilibrium in the clustering regime for a $fixed$ environment. We also show that when the migration kernel is $recurrent$, for a.e. realization of the environment, the system with an initially $consistent$ type distribution converges weakly to a mono-type equilibrium in which the fixation probability to type-$heartsuit$ configuration does not depend on the environment. A formula for the fixation probability is given in terms of an annealed average of type-$heartsuit$ densities in dormant and active population biased by ratio of the two population sizes at the target colony. For the proofs, we use duality and environment seen by particle introduced in Dolgopyat and Goldsheid (2019) for RWRE on a strip. A spectral analysis of Markov operator yields quenched weak convergence of the environment process associated with single-particle dual to a reversible ergodic distribution which we transfer to the spatial system by using duality.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49533918","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}
For a second-order particle system in $mathbb R^d$ subject to locally-in-space pairwise annihilation, we prove a scaling limit for its empirical measure on position and velocity towards a degenerate elliptic partial differential equation. Crucial ingredients are Green's function estimates for the associated hypoelliptic operator and an It^o-Tanaka trick.
{"title":"Scaling limit for a second-order particle system with local annihilation","authors":"Ruojun Huang","doi":"10.1214/23-ejp973","DOIUrl":"https://doi.org/10.1214/23-ejp973","url":null,"abstract":"For a second-order particle system in $mathbb R^d$ subject to locally-in-space pairwise annihilation, we prove a scaling limit for its empirical measure on position and velocity towards a degenerate elliptic partial differential equation. Crucial ingredients are Green's function estimates for the associated hypoelliptic operator and an It^o-Tanaka trick.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47846970","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}
A L'evy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph ${(t,f(t)):tge0}$ of any continuous, non increasing function $f$ such that $f(0)>0$, we give an expression of the probability that a bivariate subordinator $(Y,Z)$ issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components $Y$ and $Z$. We apply this result to the creeping probability of any real L'evy process through the graph of any continuous, non increasing function at a time where the process also reaches its past supremum. This probability involves the density of the renewal function of the bivariate upward ladder process as well as its drift coefficients. We also investigate the case of L'evy processes conditioned to stay positive creeping at their last passage time below the graph of a function. Then we provide some examples and we give an application to the probability of creeping through fixed levels by stable Ornstein-Uhlenbeck processes. We also raise a couple of open questions along the text.
{"title":"Creeping of Lévy processes through curves","authors":"L. Chaumont, Thomas Pellas","doi":"10.1214/23-ejp942","DOIUrl":"https://doi.org/10.1214/23-ejp942","url":null,"abstract":"A L'evy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph ${(t,f(t)):tge0}$ of any continuous, non increasing function $f$ such that $f(0)>0$, we give an expression of the probability that a bivariate subordinator $(Y,Z)$ issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components $Y$ and $Z$. We apply this result to the creeping probability of any real L'evy process through the graph of any continuous, non increasing function at a time where the process also reaches its past supremum. This probability involves the density of the renewal function of the bivariate upward ladder process as well as its drift coefficients. We also investigate the case of L'evy processes conditioned to stay positive creeping at their last passage time below the graph of a function. Then we provide some examples and we give an application to the probability of creeping through fixed levels by stable Ornstein-Uhlenbeck processes. We also raise a couple of open questions along the text.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48030132","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}
We prove that the SLE$_kappa$ loop measure arises naturally from the conformal welding of two $gamma$-Liouville quantum gravity (LQG) disks for $gamma^2 = kappa in (0,4)$. The proof relies on our companion work on conformal welding of LQG disks and uses as an essential tool the concept of uniform embedding of LQG surfaces. Combining our result with work of Gwynne and Miller, we get that random quadrangulations decorated by a self-avoiding polygon converge in the scaling limit to the LQG sphere decorated by the SLE$_{8/3}$ loop. Our result is also a key input to recent work of the first and third coauthors on the integrability of the conformal loop ensemble.
{"title":"The SLE loop via conformal welding of quantum disks","authors":"M. Ang, N. Holden, Xin Sun","doi":"10.1214/23-ejp914","DOIUrl":"https://doi.org/10.1214/23-ejp914","url":null,"abstract":"We prove that the SLE$_kappa$ loop measure arises naturally from the conformal welding of two $gamma$-Liouville quantum gravity (LQG) disks for $gamma^2 = kappa in (0,4)$. The proof relies on our companion work on conformal welding of LQG disks and uses as an essential tool the concept of uniform embedding of LQG surfaces. Combining our result with work of Gwynne and Miller, we get that random quadrangulations decorated by a self-avoiding polygon converge in the scaling limit to the LQG sphere decorated by the SLE$_{8/3}$ loop. Our result is also a key input to recent work of the first and third coauthors on the integrability of the conformal loop ensemble.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46023105","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}
We study a two-dimensional stochastic differential equation that has a unique weak solution but no strong solution. We show that this SDE shares notable properties with Tsirelson's example of a one-dimensional SDE with no strong solution. In contrast to Tsirelson's equation, which has a non-Markovian drift, we consider a strong Markov martingale with Markovian diffusion coefficient. We show that there is no strong solution of the SDE and that the natural filtration of the weak solution is generated by a Brownian motion. We also discuss an application of our results to a stochastic control problem for martingales with fixed quadratic variation in a radially symmetric environment.
{"title":"SDEs with no strong solution arising from a problem of stochastic control","authors":"A. Cox, Benjamin A. Robinson","doi":"10.1214/23-ejp995","DOIUrl":"https://doi.org/10.1214/23-ejp995","url":null,"abstract":"We study a two-dimensional stochastic differential equation that has a unique weak solution but no strong solution. We show that this SDE shares notable properties with Tsirelson's example of a one-dimensional SDE with no strong solution. In contrast to Tsirelson's equation, which has a non-Markovian drift, we consider a strong Markov martingale with Markovian diffusion coefficient. We show that there is no strong solution of the SDE and that the natural filtration of the weak solution is generated by a Brownian motion. We also discuss an application of our results to a stochastic control problem for martingales with fixed quadratic variation in a radially symmetric environment.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42349346","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}
We show the equivalence of three properties for an infinitely divisible distribution: the subexponentiality of the density, the subexponentiality of the density of its L'evy measure and the tail equivalence between the density and its L'evy measure density, under monotonic-type assumptions on the L'evy measure density. The key assumption is that tail of the L'evy measure density is asymptotic to a non-increasing function or is eventually non-increasing. Our conditions are novel and cover a rather wide class of infinitely divisible distributions. Several significant properties for analyzing the subexponentiality of densities have been derived such as closure properties of [ convolution, convolution roots and asymptotic equivalence ] and the factorization property. Moreover, we illustrate that the results are applicable for developing the statistical inference of subexponential infinitely divisible distributions which are absolutely continuous.
{"title":"Subexponentialiy of densities of infinitely divisible distributions","authors":"Muneya Matsui","doi":"10.1214/23-ejp928","DOIUrl":"https://doi.org/10.1214/23-ejp928","url":null,"abstract":"We show the equivalence of three properties for an infinitely divisible distribution: the subexponentiality of the density, the subexponentiality of the density of its L'evy measure and the tail equivalence between the density and its L'evy measure density, under monotonic-type assumptions on the L'evy measure density. The key assumption is that tail of the L'evy measure density is asymptotic to a non-increasing function or is eventually non-increasing. Our conditions are novel and cover a rather wide class of infinitely divisible distributions. Several significant properties for analyzing the subexponentiality of densities have been derived such as closure properties of [ convolution, convolution roots and asymptotic equivalence ] and the factorization property. Moreover, we illustrate that the results are applicable for developing the statistical inference of subexponential infinitely divisible distributions which are absolutely continuous.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46992491","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}
We study the following model for a diploid population of constant size $N$: Every individual carries a random number of (genetic) elements. Upon a reproduction event each of the two parents passes each element independently with probability $tfrac 12$ on to the offspring. We study the process $X^N = (X^N(1), X^N(2),...)$, where $X_t^N(k)$ is the frequency of individuals at time $t$ that carry $k$ elements, and prove convergence (in some weak sense) of $X^N$ jointly with its empirical first moment $Z^N$ to the ``slow-fast'' system $(Z,X)$, where $X_t = text{Poi}(Z_t)$ and $Z$ evolves according to a critical Feller branching process. We discuss heuristics explaining this finding and some extensions and limitations.
{"title":"A diploid population model for copy number variation of genetic elements","authors":"P. Pfaffelhuber, A. Wakolbinger","doi":"10.1214/23-ejp934","DOIUrl":"https://doi.org/10.1214/23-ejp934","url":null,"abstract":"We study the following model for a diploid population of constant size $N$: Every individual carries a random number of (genetic) elements. Upon a reproduction event each of the two parents passes each element independently with probability $tfrac 12$ on to the offspring. We study the process $X^N = (X^N(1), X^N(2),...)$, where $X_t^N(k)$ is the frequency of individuals at time $t$ that carry $k$ elements, and prove convergence (in some weak sense) of $X^N$ jointly with its empirical first moment $Z^N$ to the ``slow-fast'' system $(Z,X)$, where $X_t = text{Poi}(Z_t)$ and $Z$ evolves according to a critical Feller branching process. We discuss heuristics explaining this finding and some extensions and limitations.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44545357","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}
In this article we study a relatively novel way of constructing chaotic sequences of probability measures supported on Kac's sphere, which are obtained as the law of a vector of $N$ i.i.d. variables after it is rescaled to have unit average energy. We show that, as $N$ increases, this sequence is chaotic in the sense of Kac, with respect to the Wasserstein distance, in $L^1$, in the entropic sense, and in the Fisher information sense. For many of these results, we provide explicit rates of polynomial order in $N$. In the process, we improve a quantitative entropic chaos result of Haurey and Mischler by relaxing the finite moment requirement on the densities from order $6$ to $4+epsilon$.
本文研究了一种较为新颖的构造Kac球上支持的概率测度混沌序列的方法，该混沌序列是由$N$ i. id个变量的向量在重新标度为具有单位平均能量后的定律得到的。我们证明，随着N的增加，这个序列在Kac意义上是混沌的，关于Wasserstein距离，在L^1意义上，在熵意义上，在Fisher信息意义上。对于其中的许多结果，我们给出了N$中多项式阶的显式速率。在此过程中，我们通过将密度的有限矩要求从$6$放宽到$4+epsilon$，改进了Haurey和Mischler的定量熵混沌结果。
{"title":"Chaos for rescaled measures on Kac’s sphere","authors":"R. Cortez, H. Tossounian","doi":"10.1214/23-ejp967","DOIUrl":"https://doi.org/10.1214/23-ejp967","url":null,"abstract":"In this article we study a relatively novel way of constructing chaotic sequences of probability measures supported on Kac's sphere, which are obtained as the law of a vector of $N$ i.i.d. variables after it is rescaled to have unit average energy. We show that, as $N$ increases, this sequence is chaotic in the sense of Kac, with respect to the Wasserstein distance, in $L^1$, in the entropic sense, and in the Fisher information sense. For many of these results, we provide explicit rates of polynomial order in $N$. In the process, we improve a quantitative entropic chaos result of Haurey and Mischler by relaxing the finite moment requirement on the densities from order $6$ to $4+epsilon$.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48325605","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}
In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. Curien and Le Gall have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere.
{"title":"Random cubic planar graphs converge to the Brownian sphere","authors":"M. Albenque, 'Eric Fusy, Thomas Leh'ericy","doi":"10.1214/23-EJP912","DOIUrl":"https://doi.org/10.1214/23-EJP912","url":null,"abstract":"In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. Curien and Le Gall have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46631414","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}
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