Studies in Applied Mathematics最新文献

{"title":"On the Controllability and Stabilization of the Intermediate Long Wave Equation on a Periodic Domain","authors":"Lionel Rosier,&nbsp;Francisco J. Vielma Leal","doi":"10.1111/sapm.70274","DOIUrl":"https://doi.org/10.1111/sapm.70274","url":null,"abstract":"<div>\u0000 \u0000 <p>In this work, we show that the intermediate long wave equation with periodic boundary conditions is exactly controllable and exponentially stabilizable. A feedback law in the form of a localized damping is incorporated into the equation. A smoothing effect established with the aid of a propagation of the regularity property is used to prove the semiglobal stabilization in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>L</mi>\u0000 <mn>0</mn>\u0000 <mn>2</mn>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>T</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$L_{0}^{2}(mathbb {T})$</annotation>\u0000 </semantics></math> of weak solutions obtained by the method of vanishing viscosity. The local well-posedness and local exponential stability in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>H</mi>\u0000 <mn>0</mn>\u0000 <mi>s</mi>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>T</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$H_{0}^{s}(mathbb {T})$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <mo>&gt;</mo>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation>$s&gt;frac{1}{2}$</annotation>\u0000 </semantics></math> is established using the contraction mapping theorem. Finally, the local exact controllability is derived in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>H</mi>\u0000 <mn>0</mn>\u0000 <mi>s</mi>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>T</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$H_{0}^{s}(mathbb {T})$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <mo>&gt;</mo>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation>$s&gt;frac{1}{2}$</annotation>\u0000 </semantics></math> by combining the above feedback law with some open loop control.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148616310","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":"Predator Resource Tracking and Advection–Driven Coexistence in Heterogeneous Environments","authors":"Qian Qin,&nbsp;Lei Li,&nbsp;Hua Nie","doi":"10.1111/sapm.70269","DOIUrl":"https://doi.org/10.1111/sapm.70269","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper investigates a reaction–diffusion–advection predator–prey model in heterogeneous environments, where the predator combines random diffusion with directed advection along prey resource gradients. Using principal eigenvalue analysis, we examine the impact of predator mortality, diffusion, and advection on invasion outcomes. A novel advection-driven mechanism for predator–prey coexistence is identified, which is absent in models without advection. Moderate advection promotes predator invasion by enhancing resource tracking, while excessive advection hinders predator distribution and foraging, resulting in a nonmonotonic relationship between advection intensity and invasion success. This contrasts with traditional models and reveals how advection shapes predator–prey spatial distributions and modulates interaction dynamics, offering a fresh framework for ecological modeling.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148616327","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":"Hadamard-Type Formulas for Real Eigenvalues of Canonically Symplectic Operators","authors":"Mitchell Curran,&nbsp;Selim Sukhtaiev","doi":"10.1111/sapm.70276","DOIUrl":"https://doi.org/10.1111/sapm.70276","url":null,"abstract":"<div>\u0000 \u0000 <p>We give first-order asymptotic expansions for the resolvent and Hadamard-type formulas for the eigenvalue curves of one-parameter families of canonically symplectic operators. We allow for parameter dependence in the boundary conditions, bounded perturbations and trace operators associated with each off-diagonal operator and give formulas for derivatives of eigenvalue curves emanating from the discrete eigenvalue of the unperturbed operator in terms of Maslov crossing forms. We derive the Hadamard-type formulas using two different methods: via a symplectic resolvent difference formula and asymptotic expansions of the resolvent, and using Lyapunov–Schmidt reduction and the implicit function theorem. The latter approach facilitates derivative formulas when the eigenvalue curves are viewed as functions of the spectral parameter. We apply our abstract results to derive a spectral index theorem for the linearized operator associated with a standing wave in the nonlinear Schrödinger equation on a compact star graph.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148616328","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":"Optimal Preconditioning Techniques for Finite Volume Approximation of Three-Dimensional Conservative Space-Fractional Diffusion Equations","authors":"Wei Qu,&nbsp;Siu-Long Lei,&nbsp;Sean Y. Hon,&nbsp;Yuan-Yuan Huang","doi":"10.1111/sapm.70271","DOIUrl":"https://doi.org/10.1111/sapm.70271","url":null,"abstract":"<div>\u0000 \u0000 <p>A Crank–Nicolson finite volume approximation for three-dimensional conservative space-fractional diffusion equation results in large and dense three-level Toeplitz discrete linear systems. Preconditioned Krylov subspace methods with sine transform–based preconditioners are developed to solve these systems, including the preconditioned conjugate gradient (PCG) method for the symmetric case and the preconditioned generalized minimal residual (PGMRES) method for the nonsymmetric case. Moreover, we provide detailed analysis of the convergence of these Krylov subspace methods. Specifically, for the symmetric case, we prove the spectra of the preconditioned matrices are uniformly bounded in the open interval <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>3</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(1/2, 3/2)$</annotation>\u0000 </semantics></math>, which results in a linear convergence rate of the PCG method. For the nonsymmetric case, we demonstrate that the PGMRES method also achieves a linear convergence rate independent of discretization stepsizes from the residual point of view. These results imply that the iteration counts of the PCG and PGMRES methods are uniformly bounded and independent of the matrix sizes. Numerical experiments in both symmetric and nonsymmetric cases in two- and three-dimensions are conducted to confirm the optimal performance of the proposed preconditioned Krylov subspace methods.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148615709","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":"Far-Field Compressible Ocean Response From Two Opposite Moving Rectangular Seafloor Blocks","authors":"Ritika Das,&nbsp;Santu Das","doi":"10.1111/sapm.70270","DOIUrl":"https://doi.org/10.1111/sapm.70270","url":null,"abstract":"<div>\u0000 \u0000 <p>Acoustic-gravity wave (AGW) and ocean surface wave profiles produced by the simultaneous movement of two adjacent rectangular seafloor blocks of unequal size in opposite directions, imitating a two-fault submarine earthquake, have been analytically studied using linearized water wave theory. Utilizing the Fourier transformation method coupled with the stationary phase approximation, the far-field solutions of the generating surface wave and AGW profiles are deduced. The model is validated with the existing literature. Different expressions of far-field profiles induced by asymmetry of the moving blocks are emphasized. The surface amplitudes and pressure impressions of our two-fault system are found to be higher than those of an equivalent single rectangular moving block. The spectral energy densities (gravitational potential, kinetic, and elastic) are derived and computed. While gravitational potential energy density typically dominates for pure gravity waves, it remains negligible for AGWs, where the kinetic and elastic energies are nearly equally partitioned, reflecting a fundamental energy balance inherent to compressible fluid dynamics. The derived solutions for the potential function generalize the earlier model of a single moving block and provide insight into the wave dynamics generated by more complex multifault seafloor displacements, where the present solution would be crucial for applying the superposition of wave profiles.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148534476","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":"Pattern Formation Induced by Chemotaxis for the Weak Competition System","authors":"Liping Si,&nbsp;Jianhua Wu,&nbsp;Hongqiang Yu,&nbsp;Zhiguo Wang,&nbsp;Jie Xing","doi":"10.1111/sapm.70265","DOIUrl":"https://doi.org/10.1111/sapm.70265","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper investigates the coexistence and stability induced by chemotaxis for a weak competition model. Such a system may be used to describe one of the competing species avoids encounters with rivals by means of chemorepulsion. First, a sufficient condition is derived to estimate the stability of the positive equilibrium. The result shows that the large chemotaxis coefficient destabilizes the positive constant equilibrium, and hence chemotaxis may give rise to patterns that cannot form at its absence. Moreover, the existence, multiplicity, and stability of nonconstant positive solutions are established by taking the chemotaxis coefficient as the bifurcation parameter. Finally, numerical simulations further validate and complement the theoretical results. Our results suggest that a larger chemotaxis coefficient makes system more likely to exhibit spatial patterns and induces interspecific segregation.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148534477","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":"Time-Periodic Solutions of Stochastic Discounted Hamilton–Jacobi Equations","authors":"F. Xinze Zhang,&nbsp;S. Yong Li,&nbsp;T. Kaizhi Wang","doi":"10.1111/sapm.70267","DOIUrl":"https://doi.org/10.1111/sapm.70267","url":null,"abstract":"<div>\u0000 \u0000 <p>We study time-periodic solutions to stochastic discounted Hamilton–Jacobi equations driven by a cylindrical Wiener process in the sense of most probable paths. We first analyze the existence and regularity of solutions. Then, within a spectral Galerkin limit framework, we derive the Onsager–Machlup functional using a stopping-time localization argument and <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$L^{2}$</annotation>\u0000 </semantics></math>-type techniques. Finally, based on a minimization principle for this action functional, we characterize the system's most probable paths. By further combining weak KAM theory, we prove that there exist most probable paths with a time-periodic structure. This result extends the classical theory of time-periodic viscosity solutions for deterministic discounted Hamilton–Jacobi equations to the stochastic setting for the first time, and offers a new perspective on stochastic discounted Hamilton–Jacobi equations.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148533697","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":"Robustness of mKdV Breathers: A Numerical Perspective","authors":"Chandler Haight,&nbsp;Svetlana Roudenko,&nbsp;Diana Son,&nbsp;Kai Yang","doi":"10.1111/sapm.70263","DOIUrl":"https://doi.org/10.1111/sapm.70263","url":null,"abstract":"<div>\u0000 \u0000 <p>We systematically investigate breather solutions in the modified Korteweg–de Vries (mKdV) equation via numerical simulations. We show that the breather solutions are stable under a variety of perturbations, including amplitude changes, modifications of internal parameters, and perturbations to the nonlinearity of the equation. Our results are not only consistent with the known analytical studies on the stability of breathers in integrable systems, extending them further, but also provide numerical evidence that such breather-type structures can persist and remain stable over the simulated time scales in some nonintegrable settings with symmetric potentials.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148533492","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":"An Algebraic Study of Parametric Stokes Phenomena","authors":"Inês Aniceto,&nbsp;Samuel Crew","doi":"10.1111/sapm.70262","DOIUrl":"https://doi.org/10.1111/sapm.70262","url":null,"abstract":"<p>We investigate geometric aspects of co-equational parametric resurgence, by studying physical problems whose formal asymptotic solutions give rise to Borel transforms lying on an algebraic curve. This perspective allows us to elucidate concepts unique to parametric resurgence such as singularity structures, (virtual) turning points, and the higher order Stokes phenomenon. We construct examples as solutions to Borel plane partial differential equations using an algebraic curve ansatz before turning to the general analytic structure of co-equational resurgence problems, where we provide a systematic description of analytic continuation and Stokes constants through a Borel plane inner–outer matching procedure.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70262","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148533477","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":"On the Instability and Stability of Couette Flows for Boussinesq Equations With Partial Dissipation and Rotation","authors":"Zhibo Hou,&nbsp;Quan Wang,&nbsp;Chao Xing","doi":"10.1111/sapm.70261","DOIUrl":"https://doi.org/10.1111/sapm.70261","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;This paper investigates the stability and instability of the rotating Boussinesq equation with a general gravitational potential on the unbounded domain &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathbb {T}times mathbb {R}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. First, we analyze the linear stability and instability of the steady-state solution &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ρ&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(rho _{s},0,v_{s},0,p_{s})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Specifically, for a steady state satisfying &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;∇&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ρ&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;δ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;z&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;∇&lt;/mo&gt;\u0000 &lt;mi&gt;ψ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$nabla rho _{s}=delta (x,z)nabla psi$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$v_{s}=a_{0}x+a_{1}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, we establish linear instability when &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;δ&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;z","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148533476","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}