Studies in Applied Mathematics最新文献

{"title":"Bifurcation Analysis of a Resource–Consumer System With Explicit Spatiotemporal Memory","authors":"Luhong Ye,&nbsp;Hao Wang","doi":"10.1111/sapm.70207","DOIUrl":"10.1111/sapm.70207","url":null,"abstract":"<p>In ecological systems, animal movement is often influenced by memory and spatial cognition, especially in advanced species. This paper investigates the dynamics of a diffusive resource–consumer model incorporating explicit spatiotemporal distributed memory, where memory effects are modeled as distributed delays in both time and space. The memory kernel functions represent processes of knowledge accumulation and decay. By using the mean memory delay and the memory-based diffusion coefficient as bifurcation parameters, we analyze the occurrence of Hopf and Turing bifurcations under both weak and strong kernel cases. Notably, in the strong kernel case, rare and complex dynamical behaviors such as double Hopf, Turing–Hopf, and Turing–Turing bifurcations are observed—phenomena that are typically absent in single-species models with distributed delays. The normal form theory is applied to determine the direction and stability of Hopf bifurcations, and numerical simulations validate the theoretical results. We find that when consumers are repelled by resources, distributed memory has minimal effect on system stability. However, when consumers are attracted to resources, memory can induce oscillatory and patterned behaviors. Fast memory decay leads to homogeneous periodic oscillations, while slow decay results in more diverse spatial–temporal patterns. This study provides new insights into the role of distributed memory in shaping animal movement and ecological pattern formation.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 4","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70207","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147668861","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":"Two-Dimensional Solitons in a System of Two Coupled Nonlocal Gross–Pitaevskii Equations","authors":"G. N. Koutsokostas,&nbsp;I. Moseley,&nbsp;G. Huang,&nbsp;T. P. Horikis,&nbsp;D. J. Frantzeskakis","doi":"10.1111/sapm.70205","DOIUrl":"https://doi.org/10.1111/sapm.70205","url":null,"abstract":"<div>\u0000 \u0000 <p>We study a system of two coupled, two-dimensional, and two-component nonlocal nonlinear Schrödinger (or Gross–Pitaevskii) equations. We show that, in the weakly nonlocal regime, and depending on the boundary conditions and the values of the nonlinearity coefficients, this system can be asymptotically reduced—via multiscale expansion methods—to a Kadomtsev–Petviashvili equation or to a Davey–Stewartson equation. In this way, we predict that the considered model supports weak two-dimensional solitons, in the form of lumps or dromions, as well as a variety of dark–antidark line soliton complexes.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 3","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147615287","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":"Inverse Scattering Transform for the Coupled Modified Complex Short Pulse Equation: Multiple Higher Order Poles Case","authors":"Cong Lv,&nbsp;Shoufeng Shen,&nbsp;Q. P. Liu","doi":"10.1111/sapm.70201","DOIUrl":"https://doi.org/10.1111/sapm.70201","url":null,"abstract":"<div>\u0000 \u0000 <p>We develop a Riemann–Hilbert (RH) approach to the inverse scattering transform for the coupled modified complex short pulse (cmcSP) equation when the reflection coefficient has multiple higher order poles. With the help of a generalized cross product defined in four-dimensional vector space, the discrete spectrum is specifically analyzed and the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 <mo>×</mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation>$4times 4$</annotation>\u0000 </semantics></math> RH problem with generalized residue conditions at the <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math> pairs of higher order poles is constructed. In the reflectionless case, the RH problem can be reduced to a linear algebraic system. Consequently, the general formulas for the corresponding multiple higher order pole solutions of the cmcSP equation are obtained. The dynamical behavior of several pole-type solutions is exhibited, including one double-pole solutions (smooth solitons, cuspons, breathers), and several collisions between one double-pole solution and one simple pole solution.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 3","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147615344","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":"Propagation Phenomena of a Time-Periodic Leslie–Gower Predator–Prey System With Nonlocal Dispersal in Shifting Habitats","authors":"Qinhe Fang,&nbsp;Zhixian Yu,&nbsp;Juntao Zhang","doi":"10.1111/sapm.70204","DOIUrl":"https://doi.org/10.1111/sapm.70204","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we investigate the propagation dynamics for a nonlocal dispersal Leslie–Gower predator–prey system in time-periodic shifting habitats. Applying the asymptotic fixed-point theory and upper–lower solution technique, we first establish the existence of three types of time-periodic forced waves that connect from the trivial state to trivial, semi-trivial, and coexistence states, respectively. Long-term behavior of these forced waves will illustrate different dynamics of the species invasion fronts. Next, we study the spreading properties of the time-periodic model. We provide some conditions to guarantee the extinction and persistence for each species individually. Indeed, we point out that the predators can achieve persistent propagation even in the absence of prey, which corresponds to the semi-trivial state. Furthermore, basing on these single-species persistence results and using the persistence theory for dynamical systems, we derive the sufficient conditions for species coexistence.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 3","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147615343","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":"Periodic Dynamics of a Switching Discrete System of Reaction–Diffusion Equations","authors":"Ruibin Jiang,&nbsp;Yunfeng Liu,&nbsp;Yuming Chen,&nbsp;Zhiming Guo","doi":"10.1111/sapm.70202","DOIUrl":"https://doi.org/10.1111/sapm.70202","url":null,"abstract":"<div>\u0000 \u0000 <p>Periodicity is a common phenomenon in biological systems. In this paper, we first characterize it with an abstract switching discrete system of reaction–diffusion equations. Under certain conditions, we obtain the critical thresholds for the time length before switching and temporal period, denoted by <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>N</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <annotation>$N^*$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>T</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <annotation>$T^*$</annotation>\u0000 </semantics></math>, respectively. With the values of the other parameters being fixed, if the time length before switching <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>&gt;</mo>\u0000 <msup>\u0000 <mi>N</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$N&gt;N^*$</annotation>\u0000 </semantics></math> or if the temporal period <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 <mo>&gt;</mo>\u0000 <msup>\u0000 <mi>T</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$T&gt;T^*$</annotation>\u0000 </semantics></math>, then the system admits a unique globally asymptotically stable <span></span><math>\u0000 <semantics>\u0000 <mi>T</mi>\u0000 <annotation>$T$</annotation>\u0000 </semantics></math>-periodic solution; conversely, if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>≤</mo>\u0000 <msup>\u0000 <mi>N</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$Nle N^*$</annotation>\u0000 </semantics></math> or if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 <mo>≤</mo>\u0000 <msup>\u0000 <mi>T</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$Tle T^*$</annotation>\u0000 </semantics></math>, then the trivial equilibrium is globally asymptotically stable. We further demonstrate the applicability of our theoretical framework with two biologically motivated examples: a mosquito population suppression model based on the Sterile Insect Technique and a harvesting model in fishery management. Numerical simulations are also conducted to validate our analytical findings.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 3","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147615140","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":"Discrete-in-Time Data Assimilation for Reaction–Diffusion Models Using the Delay Sparse Data","authors":"Chengyu Jin,&nbsp;Wansheng Wang","doi":"10.1111/sapm.70203","DOIUrl":"https://doi.org/10.1111/sapm.70203","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we propose a new algorithm for predicting future states of the semilinear reaction–diffusion models with unknown or incomplete initial data in science and engineering by solely using the sparse data from the past. This algorithm is based on the discrete approximation of the feedback control (nudging) approach with the measurements given on a coarse mesh by implicit–explicit stage-based interpolation Runge–Kutta (RK) method for time discretization and finite element method for the spatial discretization. The stability of the time semi-discrete and fully discrete data assimilation approximations to the models is first shown under suitable conditions on the nudging parameter by exploring the algebraical stability of the RK methods. The error estimates derived for the state prediction algorithm demonstrate that the time semi-discrete and fully discrete approximations converge to the true state exponentially over time. Several numerical examples are provided to show the efficiency of this proposed algorithm for predicting future states.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 3","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147584994","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":"Modeling Carreau Fluid Flows Through a Very Thin Porous Medium","authors":"María Anguiano,&nbsp;Matthieu Bonnivard,&nbsp;Francisco Javier Suárez-Grau","doi":"10.1111/sapm.70199","DOIUrl":"https://doi.org/10.1111/sapm.70199","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;This study investigates three-dimensional, steady-state, and non-Newtonian flows within a very thin porous medium (VTPM). The medium is modeled as a domain confined between two parallel plates and perforated by solid cylinders that connect the plates and are distributed periodically in perpendicular directions. We denote the order of magnitude of the thickness of the domain by &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;annotation&gt;$varepsilon$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and define the period and order of magnitude of the cylinders' diameter by &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$varepsilon ^ell$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$0&lt;ell &lt;1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is fixed. In other words, we consider the regime &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mo&gt;≪&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$varepsilon ll varepsilon ^ell$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We assume that the viscosity of the non-Newtonian fluid follows Carreau's law and is scaled by a factor of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mi&gt;γ&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$varepsilon ^gamma$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;γ&lt;/mi&gt;\u0000 &lt;annotation&gt;$gamma$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a real number. Using asymptotic techniques with respect to the thickness of the domain, we perform a new, complete study of the asymptotic behavior of the fluid as &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;annotation&gt;$varepsilon$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; tends to zero. From a mathematical perspective, the main novelty of our approach is the use of sharp estimates on the pressure, based on a recent decomposition result for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$L^q$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; functions defined over a thin layer. Depending on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;γ&lt;/mi&gt;\u0000 &lt;annotation&gt;$gamm","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 3","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147615023","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":"Periodic Vibrations of the Nonhomogeneous Strings With Linear and Nonlinear Couplings","authors":"Hui Wei,&nbsp;Shuguan Ji,&nbsp;Yong Li","doi":"10.1111/sapm.70198","DOIUrl":"https://doi.org/10.1111/sapm.70198","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper is concerned with the existence of periodic vibrations of the nonhomogeneous strings with linear and nonlinear couplings under some Sturm–Liouville boundary conditions. Such a model is governed by the variable coefficients wave equations and can also be used to describe the simultaneous propagation of seismic waves in nonisotropic media. In the case of linear couplings, the nonlinear self-interactions exhibit sublinear growth with different powers. In the case of nonlinear coupling, the nonlinear self-interactions are characterized by superlinear growth with the same powers as the coupling terms. We prove the existence of infinitely many periodic solutions for these two classes of problems by variational methods and Galerkin approximations under the same working space. Our results are applicable to the uncoupled problems of either homogeneous or nonhomogeneous strings and the coupled problems of the homogeneous strings.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 3","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147566831","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":"Numerical Study of a Nonlocal Nonlinear Schrödinger Equation (MMT Model)","authors":"Amin Esfahani,&nbsp;Gulcin M. Muslu","doi":"10.1111/sapm.70197","DOIUrl":"https://doi.org/10.1111/sapm.70197","url":null,"abstract":"<p>In this paper, we study a nonlocal nonlinear Schrödinger equation (MMT model). We investigate the effect of the nonlocal operator appearing in the nonlinearity on the long-term behavior of solutions, and we identify the conditions under which the solutions of the Cauchy problem associated with this equation are bounded globally in time in the energy space. We also explore the dynamical behavior of standing wave solutions. Therefore, we first numerically generate standing wave solutions of nonlocal nonlinear Schrödinger equation by using the Petviashvili's iteration method and their stability is investigated by the split-step Fourier method. This equation also has a two-parameter family of standing wave solutions. In a second step, we meticulously concern with the construction and stability of a two-parameter family of standing wave solutions numerically. Finally, we investigate the semiclassical limit of the nonlocal nonlinear Schrödinger equation in both focusing and defocusing cases.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 3","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70197","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147565900","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":"First-Order Continuum Models for Nonlinear Dispersive Waves in the Granular Crystal Lattice","authors":"Su Yang,&nbsp;Gino Biondini,&nbsp;Christopher Chong,&nbsp;Panayotis G. Kevrekidis","doi":"10.1111/sapm.70190","DOIUrl":"https://doi.org/10.1111/sapm.70190","url":null,"abstract":"<div>\u0000 \u0000 <p>We derive and analyze, theoretically and numerically, two first-order continuum models to approximate the nonlinear dynamics of granular crystal lattices, focusing specifically on solitary waves, periodic waves, and dispersive shock waves. The dispersive shock waves predicted by the two continuum models are studied using modulation theory, DSW fitting techniques, and direct numerical simulations. The PDE-based predictions show good agreement with the DSWs generated by the discrete model simulation of the granular lattice itself, even in cases where no precompression is present and the lattice is purely nonlinear. Such an effective description could prove useful for future, more analytically amenable approximations of the original lattice system.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 3","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147565347","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}