Stochastic Processes and their Applications最新文献

{"title":"Linking population-size-dependent and controlled branching processes","authors":"Peter Braunsteins ,&nbsp;Sophie Hautphenne ,&nbsp;James Kerlidis","doi":"10.1016/j.spa.2024.104556","DOIUrl":"10.1016/j.spa.2024.104556","url":null,"abstract":"<div><div>Population-size dependent branching processes (PSDBPs) and controlled branching processes (CBPs) are two classes of branching processes used to model biological populations, including those that exhibit logistic growth. In this paper we develop connections between the two, with the ultimate goal of determining when a population is more appropriately modelled with a PSDBP or a CBP. In particular, we state conditions for the existence of equivalent PSDBPs and CBPs, we then consider the subclass of CBPs with deterministic control functions (DCBPs), stating a necessary and sufficient condition for DCBP–PSDBP equivalence. Finally, we derive an upper bound on the total variation distance between non-equivalent DCBPs and PSDBPs with matching first and second moments and equal initial population size, and show that under certain conditions this bound tends to zero as the initial population size becomes large.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104556"},"PeriodicalIF":1.1,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143127947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Gradient-type estimates for the dynamic φ24-model","authors":"Florian Kunick ,&nbsp;Pavlos Tsatsoulis","doi":"10.1016/j.spa.2024.104548","DOIUrl":"10.1016/j.spa.2024.104548","url":null,"abstract":"<div><div>We prove gradient bounds for the Markov semigroup of the dynamic <span><math><msubsup><mrow><mi>φ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>4</mn></mrow></msubsup></math></span>-model on a torus of fixed size <span><math><mrow><mi>L</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. For sufficiently large mass <span><math><mrow><mi>m</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> these estimates imply exponential contraction of the Markov semigroup. Our method is based on pathwise estimates of the linearized equation. To compensate the lack of exponential integrability of the stochastic drivers we use a stopping time argument in the spirit of Cass–Litterer–Lyons (Cass et al., 2013) and the strong Markov property. Following the classical approach of Bakry-Émery, as a corollary we prove a Poincaré/spectral gap inequality for the <span><math><msubsup><mrow><mi>φ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>4</mn></mrow></msubsup></math></span>-measure of sufficiently large mass <span><math><mrow><mi>m</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> with almost optimal <em>carré du champ</em>.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104548"},"PeriodicalIF":1.1,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128605","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spatial growth-fragmentations and excursions from hyperplanes","authors":"William Da Silva ,&nbsp;Juan Carlos Pardo","doi":"10.1016/j.spa.2024.104551","DOIUrl":"10.1016/j.spa.2024.104551","url":null,"abstract":"<div><div>In this paper, we are interested in the self-similar growth-fragmentation process that shows up when slicing half-space excursions of a <span><math><mi>d</mi></math></span>-dimensional Brownian motion from hyperplanes. Such a family of processes turns out to be a collection of spatial self-similar growth-fragmentation processes driven by an isotropic self-similar Markov process. The former can be seen as multitype growth-fragmentation processes, in the sense of Da Silva and Pardo (2024), where the set of types is <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>2</mn></mrow></msup></math></span>, the <span><math><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>–dimensional unit sphere. In order to characterise such family of processes, we study their spinal description similarly as in the monotype (Bertoin, 2017) and multitype (Da Silva and Pardo, 2024) settings. Finally, we extend our study to the case when the <span><math><mi>d</mi></math></span>-dimensional Brownian motion is replaced by an isotropic Markov process whose first <span><math><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span> coordinates are driven by an isotropic stable Lévy process and the remaining coordinate is an independent standard real-valued Brownian motion.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104551"},"PeriodicalIF":1.1,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Laws of the iterated logarithm for occupation times of Markov processes","authors":"Soobin Cho ,&nbsp;Panki Kim ,&nbsp;Jaehun Lee","doi":"10.1016/j.spa.2024.104552","DOIUrl":"10.1016/j.spa.2024.104552","url":null,"abstract":"<div><div>In this paper, we discuss the laws of the iterated logarithm (LIL) for occupation times of Markov processes <span><math><mi>Y</mi></math></span> in general metric measure space near zero (near infinity, respectively) under minimal assumptions around zero (near infinity, respectively). The LILs near zero in this paper cover the case that the function <span><math><mi>Φ</mi></math></span> in our truncated occupation times <span><math><mrow><mi>r</mi><mo>↦</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mi>Φ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></msubsup><msub><mrow><mi>1</mi></mrow><mrow><mi>B</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></mrow><mi>d</mi><mi>s</mi></mrow></math></span> is spatially dependent on the variable <span><math><mi>x</mi></math></span>. Such function <span><math><mrow><mi>Φ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> is an iterated logarithm of mean exit times of <span><math><mi>Y</mi></math></span> from balls <span><math><mrow><mi>B</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> of radius <span><math><mi>r</mi></math></span>. We first establish LILs of (truncated) occupation times on balls <span><math><mrow><mi>B</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> up to the function <span><math><mrow><mi>Φ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> Our first result on LILs of occupation times covers both near zero and near infinity cases, irrespective of transience and recurrence of the process. Further, we establish a similar LIL for total occupation times <span><math><mrow><mi>r</mi><mo>↦</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup><msub><mrow><mi>1</mi></mrow><mrow><mi>B</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></mrow><mi>d</mi><mi>s</mi></mrow></math></span> when the process is transient. Our second main result addresses large time behaviors of occupation times <span><math><mrow><mi>t</mi><mo>↦</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><msub><mrow><mi>1</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></mrow><mi>d</mi><mi>s</mi></mrow></math></span> under an additional condition that guarantees the recurrence of the process. Our results cover a large class of Feller (Levy-like) processes, random conductance models with long range jumps, jump processes with mixed polynomial local growths and jump processes with singular jumping kernels.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104552"},"PeriodicalIF":1.1,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Local time, upcrossing time and weak cutpoints of a spatially inhomogeneous random walk on the line","authors":"Hua-Ming Wang,&nbsp;Lingyun Wang","doi":"10.1016/j.spa.2024.104550","DOIUrl":"10.1016/j.spa.2024.104550","url":null,"abstract":"<div><div>In this paper, we study a transient spatially inhomogeneous random walk with asymptotically zero drift on the lattice of the positive half line. We give criteria for the finiteness of the number of points having exactly the same local time and/or upcrossing time and weak cutpoints (a point <span><math><mi>x</mi></math></span> is called a weak cutpoint if the walk never returns to <span><math><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span> after its first upcrossing from <span><math><mi>x</mi></math></span> to <span><math><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span>). In addition, for the walk with some special local drift, we also give the order of the expected number of these points in <span><math><mrow><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>]</mo></mrow><mo>.</mo></mrow></math></span> Finally, if the local drift at <span><math><mi>n</mi></math></span> is <span><math><mfrac><mrow><mi>Υ</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></math></span> with <span><math><mrow><mi>Υ</mi><mo>&gt;</mo><mn>1</mn></mrow></math></span> for <span><math><mi>n</mi></math></span> large enough, we show that, when properly scaled the number of these points in <span><math><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>]</mo></mrow></math></span> converges in distribution to a random variable with <em>Gamma</em><span><math><mrow><mo>(</mo><mi>Υ</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span> distribution. Our results answer three conjectures related to the local time, the upcrossing time, and the weak cutpoints posed by E. Csáki, A. Földes, P. Révész [J. Theoret. Probab. 23 (2) (2010) 624-638].</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104550"},"PeriodicalIF":1.1,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143127946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"How the interplay of dormancy and selection affects the wave of advance of an advantageous gene","authors":"Jochen Blath ,&nbsp;Matthias Hammer ,&nbsp;Dave Jacobi ,&nbsp;Florian Nie","doi":"10.1016/j.spa.2024.104537","DOIUrl":"10.1016/j.spa.2024.104537","url":null,"abstract":"<div><div>We investigate the spread of advantageous genes in two variants of the F-KPP model with dormancy. In the first model, dormant individuals do not move in space and instead form ‘localized seed banks’. In the second model, dormant forms of individuals are subject to motion, while the ‘active’ (reproducing) individuals remain spatially static. This can be motivated e.g. by spore dispersal of fungi, where the ‘dormant’ spores are distributed by wind, water or insects, while the ‘active’ fungi are locally fixed. For both models, we establish existence of monotone travelling wave solutions, determine the corresponding critical wave speed in terms of the model parameters, and characterize aspects of the asymptotic shape of the waves depending on the decay properties of the initial condition.</div><div>We find that the slow-down effect of dormancy on the speed of propagation of beneficial alleles is more serious in variant II (the ‘spore model’) than in variant I (the ‘seed bank model’). Mathematically, this can be understood via probabilistic representations of solutions in terms of (two variants of) ‘on/off branching Brownian motion’. A variety of open research questions are briefly discussed at the end of the paper.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104537"},"PeriodicalIF":1.1,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143127944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Monotonicity properties for Bernoulli percolation on layered graphs— A Markov chain approach","authors":"Philipp König,&nbsp;Thomas Richthammer","doi":"10.1016/j.spa.2024.104549","DOIUrl":"10.1016/j.spa.2024.104549","url":null,"abstract":"<div><div>A layered graph <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span> is the Cartesian product of a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> with the linear graph <span><math><mi>Z</mi></math></span>, e.g. <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span> is the 2D square lattice <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. For Bernoulli percolation with parameter <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span> on <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span> one intuitively would expect that <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mrow><mo>(</mo><mi>o</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>↔</mo><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≥</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mrow><mo>(</mo><mi>o</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>↔</mo><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>o</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></math></span>. This is reminiscent of the better known bunkbed conjecture. Here we introduce an approach to the above monotonicity conjecture that makes use of a Markov chain building the percolation pattern layer by layer. In case of finite <span><math><mi>G</mi></math></span> we thus can show that for some <span><math><mrow><mi>N</mi><mo>≥</mo><mn>0</mn></mrow></math></span> the above holds for all <span><math><mrow><mi>n</mi><mo>≥</mo><mi>N</mi></mrow></math></span> <span><math><mrow><mi>o</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></math></span> and <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>. One might hope that this Markov chain approach could be useful for other problems concerning Bernoulli percolation on layered graphs.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104549"},"PeriodicalIF":1.1,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143101899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Time changed spherical Brownian motions with longitudinal drifts","authors":"Giacomo Ascione ,&nbsp;Anna Vidotto","doi":"10.1016/j.spa.2024.104547","DOIUrl":"10.1016/j.spa.2024.104547","url":null,"abstract":"<div><div>We consider a time change of a drifted Brownian motion on the two-dimensional unit sphere. Precisely, we find strong solutions to the related time-nonlocal Kolmogorov equation under suitably regular initial data and we determine the spectral decomposition of its probability density function. Moreover, we study the speed of convergence to the stationary state, proving a non-exponential rate to the equilibrium. Finally, we provide very weak solutions of the same time-nonlocal Kolmogorov equation with general initial data. These results improve the known ones in terms of both the presence of a perturbation and the generality of the initial data.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104547"},"PeriodicalIF":1.1,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143101898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical approximation of SDEs with fractional noise and distributional drift","authors":"Ludovic Goudenège ,&nbsp;El Mehdi Haress ,&nbsp;Alexandre Richard","doi":"10.1016/j.spa.2024.104533","DOIUrl":"10.1016/j.spa.2024.104533","url":null,"abstract":"<div><div>We study the numerical approximation of SDEs with singular drifts (including distributions) driven by a fractional Brownian motion. Under the Catellier–Gubinelli condition that imposes the regularity of the drift to be strictly greater than <span><math><mrow><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mi>H</mi><mo>)</mo></mrow></mrow></math></span>, we obtain an explicit rate of convergence of a tamed Euler scheme towards the SDE, extending results for bounded drifts. Beyond this regime, when the regularity of the drift is <span><math><mrow><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mi>H</mi><mo>)</mo></mrow></mrow></math></span>, we derive a non-explicit rate. As a byproduct, strong well-posedness for these equations is recovered. Proofs use new regularising properties of discrete-time fBm and a new critical Grönwall-type lemma. We present examples and simulations.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104533"},"PeriodicalIF":1.1,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143127945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rate of escape of the conditioned two-dimensional simple random walk","authors":"Orphée Collin ,&nbsp;Serguei Popov","doi":"10.1016/j.spa.2024.104469","DOIUrl":"10.1016/j.spa.2024.104469","url":null,"abstract":"<div><div>We prove sharp asymptotic estimates for the rate of escape of the two-dimensional simple random walk conditioned to avoid a fixed finite set. We derive it from asymptotics available for the continuous analogue of this process (Collin and Comets, 2022), with the help of a KMT-type coupling adapted to this setup.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104469"},"PeriodicalIF":1.1,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}