Communications in Mathematical Research
type: channel
description: Communications in Mathematical Research (CMR) was established in 1985 by Jilin University, with the title Northeastern Mathematical Journal. Recently the journal was renamed to the current one and publishes articles written in English. In early 2020, a new editorial board is formed aiming to enhance the quality of the journal. CMR publishes original research and survey papers in major areas of mathematics, including pure mathematics, applied mathematics, computational mathematics, and statistics.
publisher: Global Science Press
language: en
publication name: Communications in Mathematical Research
issn: 1674-5647
items: An ALE-based Numerical Scheme with a Two-Dimensional Riemann Solver for Two-Phase Flows with Non-Equilibrium Pressure
items: Transfer of Improvement Strategies Between DRS and ADMM: A Unified Classification Framework
items: Observation-Driven INAR(1) Models with Novel and Flexible Links
items: The Exponential Decaying Noncommutative Differential Forms and Convolution Algebras
items: A Positivity-Preserving Fourier Spectral Moving Mesh Method for the Keller-Segel Chemotaxis Model
items: Constraint-Preserving High-Order Compact OEDG Method for Spherically Symmetric Einstein-Euler System
items: Asymptotic Estimates for the Ruin Probability of a Multidimensional Delay-Claim Risk Model with Dependent Claims
items: Continuous Subsonic-Sonic Flows in Curved Convergent Nozzles with a Fixed Point
items: Image-Based Rock Typing Using Specific Surface and Iterative Convolution-Thresholding Method
items: Error Bounds for Linear Complementarity Problem of $SD{D_k}$ Matrices
items: Some Complexes and Modules Induced by Strongly FP-Injective Modules
items: An Explicit Construction of Orthogonal Basis in $\mathcal{p}$-adic Fields
items: A Schur-Horn Type Theorem for Symplectic Matrices
items: A Note on Cohomological Mirror Symmetry of Toric Manifolds
items: Dynamical Stability of Transonic Shock Solutions to Non-Isentropic Euler Equations
items: The Spectral Einstein Functional for the Nonminimal de Rham-Hodge Operator
items: Logarithmic Upper Bound for Solutions of Degenerate Parabolic Equation
items: Uniqueness in Inverse Electromagnetic Scattering by Locally Rough Surfaces with Phaseless Near-Field Data
items: Homoclinic Solutions for a Class of Hamiltonian Systems with Small External Perturbations
items: On the Long-Time $H^1$-Stability of the Linearly Extrapolated BDF2 Time-Stepping Scheme for Coupled Multiphysics Flow Problems
items: The Variants of the MGSS Methods for Complex Symmetric Linear System of Equations
items: Functional Solutions, Multi Line Solitons and Multiple Pole Solutions of the Generalized (2+1)-Dimensional Kaup-Kupershmidt Equation
items: Continuous Spectrum for a Class of Smooth Mixing CMV Matrices
items: Commutators of Complex Symmetric Operators
items: Reduced Basis Method Based on Fourier Transform for Time-Dependent Parameterized Nonlocal Problems
items: Degenerations of Nilalgebras
items: Numerical Ergodicity of Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise
items: Equivariant Cohomology and Deformation for Associative Algebras with a Derivation
items: When the Gromov-Hausdorff Distance Between Finite-Dimensional Space and Its Subset Is Finite?
items: Eigenvalue Problem for a Class of Quasilinear Elliptic Operators with Mixed Boundary Value Condition in a Variable Exponent Sobolev Space

An ALE-based Numerical Scheme with a Two-Dimensional Riemann Solver for Two-Phase Flows with Non-Equilibrium Pressure
type: item
description: In this work, a homogeneous two-phase flow model in two-dimensional space is proposed for systems without a sharp interface between phases. The model allows the two phases to coexist within the same domain with distinct pressures, while their local proportions are represented by volume fractions. Within an arbitrary Lagrangian-Eulerian (ALE) framework, a two-dimensional Riemann solver is developed to construct the numerical scheme with novel formulations for numerical fluxes and nodal velocities, ensuring consistency on moving control volumes. Furthermore, a new moving-mesh strategy is designed, in which the evolution of the volume fraction is obtained directly from the geometric motion of the mesh and mass conservation, thereby avoiding solving the non-conservative convective equation with source terms. This enables the incorporation of physical drag models and finite relaxation coefficients, allowing the gradual equilibration of velocity and pressure between the two phases to be calculated. Based on the volume-fraction formulation, a discrete representation of the pressure work term is derived. Finally, a series of numerical experiments are presented to demonstrate the accuracy, stability, and physical consistency of the proposed model and numerical scheme.
subject: Articles
  topic: section
subject: Multiphase flows
  topic: keywords
subject: ALE method
  topic: keywords
subject: HLLC-2D
  topic: keywords
subject: moving mesh
  topic: keywords
creator: Jianqiao Zhang
creator: Wei Yan
creator: Xianggui Li
rights: Copyright (c) 2025 Communications in Mathematical Research
date: 2026-03-26
publication date: 2026-03-26
doi: 10.4208/cmr.2026-0028

Transfer of Improvement Strategies Between DRS and ADMM: A Unified Classification Framework
type: item
description: Douglas-Rachford splitting (DRS) and the alternating direction method of multipliers (ADMM) are two fundamental first-order methods for structured convex optimization. Although derived from different viewpoints, ADMM can be interpreted as the application of DRS to the dual problem. Based on this structural equivalence, this paper studies how algorithmic improvement strategies can be transferred between the two methods. We classify transferable strategies into three categories: exact operator-level transfer, parameter-driven transfer, and heuristic transfer. Representative examples including relaxation, metric scaling, adaptive parameter updates, and residual balancing are discussed to illustrate the different levels of transferability. This perspective provides a systematic way to understand the relationship between DRS and ADMM and clarifies how algorithmic ideas developed for one method may inform the design of variants of the other, offering a unified framework that both explains existing variants and guides the design of new ones.
subject: Articles
  topic: section
subject: Douglas-Rachford splitting
  topic: keywords
subject: alternating direction method of multipliers
  topic: keywords
subject: monotone operator splitting
  topic: keywords
subject: convex optimization
  topic: keywords
subject: algorithmic equivalence
  topic: keywords
subject: relaxation techniques
  topic: keywords
subject: adaptive step sizes
  topic: keywords
creator: Shuting Liu
rights: Copyright (c) 2025 Communications in Mathematical Research
date: 2026-03-18
publication date: 2026-03-18
doi: 10.4208/cmr.2026-0004

Observation-Driven INAR(1) Models with Novel and Flexible Links
type: item
description: Observation-driven integer-valued autoregressive models are widely used for modeling count time series exhibiting dynamic dependence, yet their performance critically depends on the way that thinning probabilities are linked to past observations. Most existing specifications rely on the logit link and may respond excessively to large counts. In this paper, we introduce a class of new observation-driven integer-valued autoregressive models using logarithmic and soft-clipping links that attenuate the influence of large observations. The proposed framework allows for stochastic covariates. Estimation is carried out using conditional maximum likelihood and conditional least squares methods. Simulation studies and two real data applications are used to illustrate the proposed models.
subject: Articles
  topic: section
subject: Observation-driven model
  topic: keywords
subject: covariate
  topic: keywords
subject: logarithmic link
  topic: keywords
subject: soft-clipping link
  topic: keywords
subject: conditional maximum likelihood
  topic: keywords
creator: Yiling Dong
creator: Fukang Zhu
rights: Copyright (c) 2025 Communications in Mathematical Research
date: 2026-03-12
publication date: 2026-03-12
starting page: 97
ending page: 120
doi: 10.4208/cmr.2026-0014

The Exponential Decaying Noncommutative Differential Forms and Convolution Algebras
type: item
description: We construct an algebra of rapidly decaying $C_0$ functions on an étale (Lie) groupoid, which extends the standard algebra of compactly supported noncommutative differential forms. In particular, using the theory of bisections, we prove that this algebra is closed under convolution. This construction clarifies the superconnection proof of Gorokhovsky and Lott.
subject: Articles
  topic: section
subject: Noncommutative differential form
  topic: keywords
subject: étale groupoid
  topic: keywords
subject: bisection
  topic: keywords
creator: Wen Zhang
rights: Copyright (c) 2025 Communications in Mathematical Research
date: 2026-02-23
publication date: 2026-02-23
starting page: 82
ending page: 96
doi: 10.4208/cmr.2025-0054

A Positivity-Preserving Fourier Spectral Moving Mesh Method for the Keller-Segel Chemotaxis Model
type: item
description: We develop a numerical method for the Keller-Segel chemotaxis system that is designed to (i) preserve the model’s fundamental structural properties (positivity/bound preservation, mass conservation, and energy dissipation), (ii) efficiently and accurately resolve the near-singular dynamics associated with spike formation and finite-time blow-up. Our approach combines a linear, positivity-preserving scalar auxiliary variable (SAV) scheme (following the framework in [15]) with a Fourier spectral spatial discretization and an moving-mesh PDE-based method. The SAV reformulation provides a convenient platform for stable, linear time stepping while maintaining energy dissipation; the Fourier spectral discretization delivers high accuracy in smooth regions; and the moving-mesh PDE mesh redistribution concentrates collocation points in regions of large gradients so that sharp, localized structures can be resolved without prohibitive cost. We show that the proposed moving mesh SAV scheme inherits positivity preservation, mass conservation, and discrete energy dissipation provided the mesh motion avoids element overlap. Two-dimensional tests demonstrate the method’s ability to capture fine spike profiles and estimate blow-up times with substantially reduced computational effort; the formulation extends straightforwardly to three spatial dimensions. Numerical results show that the proposed method is a practical and effective method for accurate simulation of chemotactic aggregation.
subject: Articles
  topic: section
subject: Keller-Segel chemotaxis model
  topic: keywords
subject: moving mesh PDE
  topic: keywords
subject: scalar auxiliary variable
  topic: keywords
subject: positivity preserving
  topic: keywords
subject: finite-time blowup
  topic: keywords
creator: Yutong Kuang
creator: Zhiwen Zhang
rights: Copyright (c) 2025 Communications in Mathematical Research
date: 2026-02-23
publication date: 2026-02-23
starting page: 57
ending page: 81
doi: 10.4208/cmr.2025-0051

Constraint-Preserving High-Order Compact OEDG Method for Spherically Symmetric Einstein-Euler System
type: item
description: Numerical simulation of the spherically symmetric Einstein-Euler (EE) system faces severe challenges due to the stringent physical admissibility constraints of relativistic fluids and the geometric singularities inherent in metric evolution. This paper proposes a high-order constraint-preserving (CP) compact oscillation-eliminating discontinuous Galerkin (cOEDG) method specifically tailored to address these difficulties. The method integrates a scale-invariant oscillation-eliminating mechanism [M. Peng, Z. Sun, K. Wu, Math. Comp., 94 (2025)] into a compact Runge-Kutta DG framework. By characterizing the convex invariant region of the hydrodynamic subsystem with general barotropic equations of state, we prove that the proposed scheme preserves physical realizability (specifically, positive density and subluminal velocity) directly in terms of conservative variables, thereby eliminating the need for complex primitive-variable checks. To ensure the geometric validity of the spacetime, we introduce a bijective transformation of the metric potentials. Rather than evolving the constrained metric components directly, the scheme advances unconstrained auxiliary variables whose inverse mapping automatically enforces strict positivity and asymptotic bounds without any limiters. The resulting CP compact OEDG (CPcOEDG) method exhibits robustness and the expected accuracy in capturing strong gravity-fluid interactions, as demonstrated by simulations of black hole accretion and relativistic shock waves.
subject: Articles
  topic: section
subject: Einstein-Euler system
  topic: keywords
subject: general relativistic hydrodynamics
  topic: keywords
subject: discontinuous Galerkin method
  topic: keywords
subject: oscillation elimination
  topic: keywords
subject: constraint-preserving
  topic: keywords
creator: Yuchen Huang
creator: Manting Peng
creator: Kailiang Wu
rights: Copyright (c) 2026 Communications in Mathematical Research
date: 2026-01-12
publication date: 2026-01-12
starting page: 370
ending page: 414
doi: 10.4208/cmr.2025-0052

Asymptotic Estimates for the Ruin Probability of a Multidimensional Delay-Claim Risk Model with Dependent Claims
type: item
description: This paper studies a multidimensional delay-claim risk model in which an insurance company operates $d (d≥2)$ lines of business exposed to a common renewal counting process. Each catastrophic event simultaneously produces main and delayed claims across all business lines, where the delayed claims are settled after random delay periods. The surplus process incorporates a geometric Lévy price process to describe investment returns. Assuming that the main and delayed claims follow subexponential distributions and satisfy a conditional linear dependence structure, we derive asymptotic estimates for the finite-time ruin probability. The obtained results extend existing findings on delay-claim models to the multidimensional framework and contribute to a deeper understanding of ruin behavior under dependence and heavy-tailed risks.
subject: Articles
  topic: section
subject: Asymptotics
  topic: keywords
subject: multidimensional risk model
  topic: keywords
subject: dependence structure
  topic: keywords
subject: delayed claim
  topic: keywords
subject: subexponential class
  topic: keywords
creator: Yuchen Sun
creator: Weipeng Sun
rights: Copyright (c) 2025 Communications in Mathematical Research
date: 2026-01-05
publication date: 2026-01-05
starting page: 36
ending page: 56
doi: 10.4208/cmr.2025-0046

Continuous Subsonic-Sonic Flows in Curved Convergent Nozzles with a Fixed Point
type: item
description: This paper concerns continuous subsonic-sonic potential flows in a two-dimensional, convergent nozzle, which is governed by a free boundary problem of a quasilinear degenerate elliptic equation. It is shown that for a given nozzle perturbed from a straight one, a given point on its wall where the curvature is zero, a given inlet which is a perturbation of an arc centered at the vertex, and a given incoming flow angle perturbed from the angle of the inner normal of the inlet, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the sonic curve, which satisfies the slip conditions on the nozzle walls and whose sonic curve intersects the upper wall at the given point. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only $C^{1/2}$ Hölder continuous and the acceleration blows up. The perturbation problem is solved in the potential plane, where the flow is governed by a free boundary problem of a degenerate elliptic equation with three free boundaries and two nonlocal boundary conditions, and the equation is degenerate at one free boundary.
subject: Articles
  topic: section
subject: Degeneracy
  topic: keywords
subject: free boundary
  topic: keywords
subject: continuous subsonic-sonic flow
  topic: keywords
subject: nonlocal boundary condition
  topic: keywords
creator: Yuanyuan Nie
creator: Mingjun Zhou
rights: Copyright (c) 2025 Communications in Mathematical Research
date: 2026-01-01
publication date: 2026-01-01
starting page: 1
ending page: 35
doi: 10.4208/cmr.2025-0049

Image-Based Rock Typing Using Specific Surface and Iterative Convolution-Thresholding Method
type: item
description: Image-based rock typing (IBRT) is an effective way to understand the pore scale heterogeneity of the reservoir samples. IBRT is aimed at segmenting a rock sample’s image into different regions where each region represents a homogeneous porous medium, also known as rock type. Currently, the phase-field rock typing method has attracted more attention due to its impressive performance in classifying the heterogeneous rock images with highly irregular pore structures. In this paper, a modified specific surface CV (SSCV) model is proposed to realize the IBRT. In the SSCV model, the specific surface of a pixel is calculated within a given size neighborhood to distinguish different rock types, and the iterative convolution-thresholding method (ICTM) is applied as the classifier. Compared to the LHFCV method, an existing phase-field rock typing method, the proposed SSCV is capable of processing the images with more than two rock types and can be solved by ICTM which has higher computational efficiency. The proposed SSCV method has demonstrated remarkable performance in the segmentation of various images of both synthetic and natural rock samples.
subject: Articles
  topic: section
subject: Rock typing
  topic: keywords
subject: special surface
  topic: keywords
subject: SSCV
  topic: keywords
subject: ICTM
  topic: keywords
creator: Chaoyu Liu
creator: Zhonghua Qiao
creator: Shuyu Sun
creator: Yuzhu Wang
rights: Copyright (c) 2025 Communications in Mathematical Research
date: 2026-01-01
publication date: 2026-01-01
starting page: 354
ending page: 369
doi: 10.4208/cmr.2025-0047

Error Bounds for Linear Complementarity Problem of $SD{D_k}$ Matrices
type: item
description: $SD{D_k}$ matrices are a subclass of the nonsingular $H$-matrices. The infinity norm of the inverse for $SD{D_k}$ matrices has been given. In the paper, we utilize this result in the context of the linear complementarity problem, and the error bounds of the linear complementarity problem for $SD{D_k}$ matrices are obtained. By the relationship between the $SDD$ matrices and the $SD{D_k}$ matrices, we further obtain the error bounds of the linear complementarity problem for $SDD$ matrices. In addition, it is proved that the bounds presented in this paper are sharper than the well-known bounds under some conditions. Finally, numerical examples are provided to demonstrate the effectiveness of our results.
subject: Articles
  topic: section
subject: Linear complementarity problem
  topic: keywords
subject: error bound
  topic: keywords
subject: $SD{D_k}$ matrices
  topic: keywords
subject: $SDD$ matrices
  topic: keywords
creator: Kexin Zhang
creator: Min Hui
creator: Yaqiang Wang
rights: Copyright (c) 2025 Communications in Mathematical Research
date: 2025-12-12
publication date: 2025-12-12
starting page: 331
ending page: 353
doi: 10.4208/cmr.2025-0044

Some Complexes and Modules Induced by Strongly FP-Injective Modules
type: item
description: Let $R$ be an associative ring with identity. In this paper, we consider generalizations of Gorenstein FP-injective $R$-modules and FP-injective complexes, give the definitions and characterizations of strongly Gorenstein FP-injective $R$-modules and strongly FP-injective complexes, which are induced by strongly FP-injective modules. Then we investigate the strongly FP-injective dimension of complexes.
subject: Articles
  topic: section
subject: Cotorsion pairs
  topic: keywords
subject: strongly Gorenstein FP-injective modules
  topic: keywords
subject: strongly FP-injective complexes
  topic: keywords
subject: strongly FP-injective dimension
  topic: keywords
creator: Donglin He
creator: Yuyan Li
creator: Wujun Pu
rights: Copyright (c) 2025 Communications in Mathematical Research
date: 2025-12-01
publication date: 2025-12-01
starting page: 311
ending page: 330
doi: 10.4208/cmr.2025-0040

An Explicit Construction of Orthogonal Basis in $\mathcal{p}$-adic Fields
type: item
description: In 2021, the $\mathcal{p}$-adic signature scheme and public-key encryption cryptosystem were introduced. These schemes have good efficiency but are shown to be not secure. The attack succeeds because the extension fields used in these schemes are totally ramified. In order to avoid this attack, the extension field should have a large residue degree. In this paper, we propose a method of constructing a kind of specific orthogonal basis in $\mathcal{p}$-adic fields with a large residue degree, which would be helpful to modify the $\mathcal{p}$-adic signature scheme and public-key encryption cryptosystem.
subject: Articles
  topic: section
subject: Local field
  topic: keywords
subject: orthogonal basis
  topic: keywords
subject: $\mathcal{p}$-adic lattice
  topic: keywords
creator: Chi Zhang
creator: Yingpu Deng
rights: Copyright (c) 2025 Communications in Mathematical Research
date: 2025-11-14
publication date: 2025-11-14
starting page: 297
ending page: 310
doi: 10.4208/cmr.2025-0025

A Schur-Horn Type Theorem for Symplectic Matrices
type: item
description: In this note, we establish a new version of Schur-Horn type theorem for symplectic matrices. Meanwhile, we establish a necessary and sufficient condition for the equality to hold in the above result.
subject: Articles
  topic: section
subject: Positive definite matrix
  topic: keywords
subject: symplectic matrix
  topic: keywords
subject: equality
  topic: keywords
subject: doubly stochastic matrix
  topic: keywords
creator: Shaowu Huang
creator: Yimin Wei
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-26
publication date: 2025-09-26
starting page: 287
ending page: 295
doi: 10.4208/cmr.2025-0043

A Note on Cohomological Mirror Symmetry of Toric Manifolds
type: item
description: In this note we describe a logarithmic version of mirror Landau-Ginzburg model for semi-projective toric manifolds and show in an elementary and explicit way that the state space ring of the Landau-Ginzburg mirror is isomorphic to the $\mathbb{C}$-valued cohomology of the toric manifold.
subject: Articles
  topic: section
subject: Mirror symmetry
  topic: keywords
subject: Landau-Ginzburg model
  topic: keywords
subject: toric manifold
  topic: keywords
creator: Hao Wen
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-26
publication date: 2025-09-26
starting page: 271
ending page: 286
doi: 10.4208/cmr.2025-0027

Dynamical Stability of Transonic Shock Solutions to Non-Isentropic Euler Equations
type: item
description: In this paper, we investigate the dynamical stability of transonic shock solutions for the full compressible Euler system in a two dimensional nozzle with a symmetric divergent part. Building upon the existence and uniqueness results for steady symmetric transonic shock solutions to the non-isentropic Euler system established in [Z.P. Xin and H.C. Yin, The transonic shock in a nozzle, 2-D and 3-D complete Euler systems, J. Differential Equations 245 (2008)], we prove the dynamical stability of the transonic shock solutions under small perturbations. More precisely, if the initial unsteady transonic flow is located in the symmetric divergent part of the nozzle and the flow is a symmetric small perturbation of the steady transonic flow, we use the characteristic method to establish the dynamical stability.
subject: Articles
  topic: section
subject: Euler equation
  topic: keywords
subject: transonic shock
  topic: keywords
subject: dynamical stability
  topic: keywords
creator: Ben Duan
creator: Yan Zhou
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-26
publication date: 2025-09-26
starting page: 250
ending page: 270
doi: 10.4208/cmr.2025-0032

The Spectral Einstein Functional for the Nonminimal de Rham-Hodge Operator
type: item
description: In this paper, we give the definitions of the non-self-adjoint spectral triple and its spectral Einstein functional. We compute the spectral Einstein functional associated with the nonminimal de Rham-Hodge operator on even-dimensional compact manifolds without boundary. Finally, several examples of the non-self-adjoint spectral triple are listed.
subject: Articles
  topic: section
subject: The nonminimal de Rham-Hodge operator
  topic: keywords
subject: non-self-adjoint spectral Einstein functional
  topic: keywords
subject: noncommutative residue
  topic: keywords
creator: Hongfeng Li
creator: Yong Wang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-26
publication date: 2025-09-26
starting page: 225
ending page: 249
doi: 10.4208/cmr.2025-0023

Logarithmic Upper Bound for Solutions of Degenerate Parabolic Equation
type: item
description: In this note, we consider the following degenerate parabolic equation studied in [F. Chiarenza and R. Serapioni, Degenerate parabolic equations and Harnack inequality, Ann. Mat. Pura Appl. 137 (1984)] i.e., where $f=(f^1,···,f^n)$ and $Ω$ is a bounded domain in $\mathbb{R}^n$ with Lipschitz boundary, $n≥2$ and $T>0.$ In this paper, we apply Moser iteration argument to build up the explicit relationship among the coefficients $a_{i,j}(x,t)$, $f$ and the maximum norm of the solution. Meanwhile, we also find that the weighed Lebesgue space $L^{2l/(l−1)}$ to which $f$ belongs is essentially sharp in order to establish local boundedness of the solution. Here the definition of $l$ is found in Lemma 2.3. Our results cover the well-known results.
subject: Articles
  topic: section
subject: ${A_p}$ weight
  topic: keywords
subject: degenerate parabolic equations
  topic: keywords
subject: Moser iteration argument
  topic: keywords
creator: Zheng Li
creator: Bin Guo
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-26
publication date: 2025-09-26
starting page: 209
ending page: 224
doi: 10.4208/cmr.2025-0026

Uniqueness in Inverse Electromagnetic Scattering by Locally Rough Surfaces with Phaseless Near-Field Data
type: item
description: In this paper, we consider the reconstruction of a locally rough surface from phaseless near-field data generated by the incident electric dipoles. To obtain the uniqueness for this inverse problem, we use the superposition of point sources as the incident waves. These point sources lie on an admissible surface. The measured data are collected from another admissible surface above this locally rough surface. We derive the Rellich’s lemma and the reciprocity relation for the electric total fields. Based on them, we establish the uniqueness in phaseless inverse electromagnetic scattering by locally rough surfaces.
subject: Articles
  topic: section
subject: Uniqueness
  topic: keywords
subject: inverse electromagnetic scattering
  topic: keywords
subject: phaseless near field
  topic: keywords
subject: locally rough surface
  topic: keywords
subject: reciprocity relation.
  topic: keywords
subject: 78A46
  topic: subjects
subject: 35P25
  topic: subjects
creator: Jiahui Gao
creator: Junliang Lv
creator: Tian Niu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-06-18
publication date: 2025-06-18
starting page: 189
ending page: 208
doi: 10.4208/cmr.2025-0007

Homoclinic Solutions for a Class of Hamiltonian Systems with Small External Perturbations
type: item
description: This paper is concerned with the existence of nontrivial homoclinic solutions for a class of second order Hamiltonian systems with external forcing perturbations $\ddot{q}+A\dot{q}+V_q(t,q)= f(t),$ where $q= (q_1,q_2,···,q_N)∈\mathbb{R}^N,$ $A$ is an antisymmetric constant $N×N$ matrix, $V(t,q) = −K(t,q)+W(t,q)$ with $K,W ∈ C^1 (\mathbb{R},\mathbb{R}^N)$ and satisfying $b_1|q|^2 ≤ K(t,q) ≤ b_2|q|^2$ for some positive constants $b_2 ≥b_1 >0$ and external forcing term $f ∈C(\mathbb{R},\mathbb{R}^N)$ being small enough. Under some new weak superquadratic conditions for $W,$ by using the mountain pass theorem, we obtain the existence of at least one nontrivial homoclinic solution.
subject: Articles
  topic: section
subject: Homoclinic solution
  topic: keywords
subject: Hamiltonian system
  topic: keywords
subject: mountain pass theorem
  topic: keywords
subject: critical point.
  topic: keywords
subject: 58E05
  topic: subjects
subject: 58E50
  topic: subjects
subject: 34C37
  topic: subjects
creator: Wenzhuang Zhu
creator: Chunhui Hu
creator: Shuguan Ji
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-06-18
publication date: 2025-06-18
starting page: 148
ending page: 172
doi: 10.4208/cmr.2025-0016

On the Long-Time $H^1$-Stability of the Linearly Extrapolated BDF2 Time-Stepping Scheme for Coupled Multiphysics Flow Problems
type: item
description: The purpose of the current article is to study the $H^1$-stability for all positive time of the linearly extrapolated BDF2 time-stepping scheme for the magnetohydrodynamics and Boussinesq equations. Specifically, we discretize in time using the linearly backward differentiation formula, and by employing both the discrete Gronwall lemma and the discrete uniform Gronwall lemma, we establish that each numerical scheme is uniformly bounded in the $H^1$-norm.
subject: Articles
  topic: section
subject: Magnetohydrodynamics equations
  topic: keywords
subject: Boussinesq equations
  topic: keywords
subject: linearly extrapolated BDF2 timestepping scheme
  topic: keywords
subject: long-time stability.
  topic: keywords
subject: 35Q56
  topic: subjects
subject: 35Q35
  topic: subjects
creator: Mine Akbas
creator: Cristina Tone
creator: Florentina Tone
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-06-18
publication date: 2025-06-18
starting page: 122
ending page: 147
doi: 10.4208/cmr.2025-0013

The Variants of the MGSS Methods for Complex Symmetric Linear System of Equations
type: item
description: Recently, inspired by a modified generalized shift-splitting iteration method for complex symmetric linear systems, we propose two variants of the modified generalized shift-splitting iteration (MGSS) methods for solving complex symmetric linear systems. One is a parameterized MGSS iteration method and the other is a modified parameterized MGSS iteration method. We prove that the proposed methods are convergent under appropriate constraints on the parameters. In addition, we also give the eigenvalue distributions of different preconditioned matrices to verify the effectiveness of the preconditioners proposed in this paper.
subject: Articles
  topic: section
subject: Complex symmetric linear system
  topic: keywords
subject: variants of MGSS method
  topic: keywords
subject: convergence analysis.
  topic: keywords
subject: 15A24
  topic: subjects
subject: 65F10
  topic: subjects
creator: Changfeng Ma
creator: Xiaojuan Yu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-06-18
publication date: 2025-06-18
starting page: 173
ending page: 188
doi: 10.4208/cmr.2025-0017

Functional Solutions, Multi Line Solitons and Multiple Pole Solutions of the Generalized (2+1)-Dimensional Kaup-Kupershmidt Equation
type: item
description: The (2+1)-dimensional integrable generalization of the Kaup-Kupershmidt (KK) equation is solved by the inverse spectral transform method in this paper. Several new long derivative operators $V_x,$ $V_y$ and $V_t$ and the kernel functions $K$ of $\overline{∂}$-problem are introduced to construct a type of general solution of the KK equation. Based on these, several classes of the new exact solutions, with constant asymptotic values at infinity $u|_{x^2+y^2→∞} →0,$ for the KK equation are constructed via the $\overline{∂}$-dressing method.
subject: Articles
  topic: section
subject: KK equation
  topic: keywords
subject: inverse scattering transform
  topic: keywords
subject: $\overline{∂}$-dressing method
  topic: keywords
subject: exact solution.
  topic: keywords
subject: 35C08
  topic: subjects
subject: 45G15
  topic: subjects
subject: 45Q05
  topic: subjects
creator: Linlin Gui
creator: Yufeng Zhang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-06-18
publication date: 2025-06-18
starting page: 103
ending page: 121
doi: 10.4208/cmr.2025-0008

Continuous Spectrum for a Class of Smooth Mixing CMV Matrices
type: item
description: This note proves that the extended CMV matrices with Verblunsky coefficient that is generated by a smooth volume preserving mixing dynamical system and a Hölder sampling function have almost surely continuous spectrum.
subject: Articles
  topic: section
subject: CMV matrix
  topic: keywords
subject: continuous spectrum
  topic: keywords
subject: Gordon lemma
  topic: keywords
subject: super-recurrent.
  topic: keywords
subject: 37A30
  topic: subjects
subject: 37B36
  topic: subjects
subject: 28A78
  topic: subjects
creator: Yaxin Peng
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-03-31
publication date: 2025-03-31
starting page: 25
ending page: 29
doi: 10.4208/cmr.2024-0022

Commutators of Complex Symmetric Operators
type: item
description: Let $C$ be a conjugation on a separable complex Hilbert space $\mathcal{H}.$ An operator $T$ on $\mathcal{H}$ is said to be $C$-symmetric if $CTC = T^∗,$ and $T$ is said to be $C$-skew symmetric if $CTC=−T^∗$. It is proved in this paper that each $C$-skew symmetric operator can be written as the sum of two commutators of $C$-symmetric operators.
subject: Articles
  topic: section
subject: Complex symmetric operators
  topic: keywords
subject: commutators
  topic: keywords
subject: skew symmetric operators.
  topic: keywords
subject: Primary 47B47
  topic: subjects
subject: 47B99
  topic: subjects
subject: Secondary 47L05
  topic: subjects
creator: Rui Dou
creator: Xiaolong Ruan
creator: Sen Zhu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-03-31
publication date: 2025-03-31
starting page: 59
ending page: 68
doi: 10.4208/cmr.2024-0053

Reduced Basis Method Based on Fourier Transform for Time-Dependent Parameterized Nonlocal Problems
type: item
description: In the paper, a reduced basis (RB) method for time-dependent nonlocal problems with a special parameterized fractional Laplace kernel function is proposed. Because of the lack of sparsity of discretized nonlocal systems compared to corresponding local partial differential equation (PDE) systems, model reduction for nonlocal systems becomes more critical. The method of snapshots and greedy (MOS-greedy) algorithm of RB method is developed for nonlocal problems with random inputs, which provides an efficient and reliable approximation of the solution. A major challenge lies in the excessive influence of the time domain on the model reduction process. To address this, the Fourier transform is applied to convert the original time-dependent parabolic equation into a frequency-dependent elliptic equation, where variable frequencies are independent. This enables parallel computation for approximating the solution in the frequency domain. Finally, the proposed MOS-greedy algorithm is applied to the nonlocal diffusion problems. Numerical results demonstrate that it provides an accurate approximation of the full order problems and significantly improves computational efficiency.
subject: Articles
  topic: section
subject: Nonlocal problems
  topic: keywords
subject: reduced basis method
  topic: keywords
subject: method of snapshots
  topic: keywords
subject: greedy algorithm
  topic: keywords
subject: Fourier transform.
  topic: keywords
subject: 65N99
  topic: subjects
subject: 60H35
  topic: subjects
subject: 35R60
  topic: subjects
creator: Dou Dai
creator: Huailing Song
creator: Yuming Ba
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-03-31
publication date: 2025-03-31
starting page: 69
ending page: 102
doi: 10.4208/cmr.2025-0002

Degenerations of Nilalgebras
type: item
description: All complex 3-dimensional nilalgebras were described. As a corollary, all degenerations in the variety of complex 3-dimensional nilalgebras were obtained.
subject: Articles
  topic: section
subject: Nilalgebra
  topic: keywords
subject: algebraic classification
  topic: keywords
subject: geometric classification
  topic: keywords
subject: degeneration.
  topic: keywords
subject: Primary: 17A30
  topic: subjects
subject: Secondary: 14L30
  topic: subjects
creator: Ivan Kaygorodov
creator: Oleg Shashkov
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-03-31
publication date: 2025-03-31
starting page: 9
ending page: 24
doi: 10.4208/cmr.2024-0033

Numerical Ergodicity of Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise
type: item
description: We establish the unique ergodicity of a fully discrete scheme for monotone SPDEs with polynomial growth drift and bounded diffusion coefficients driven by multiplicative white noise. The main ingredient of our method depends on the satisfaction of a Lyapunov condition followed by a uniform moments’ estimate, combined with the regularity property for the full discretization. We transform the original stochastic equation into an equivalent random equation where the discrete stochastic convolutions are uniformly controlled to derive the desired uniform moments’ estimate. Applying the main result to the stochastic Allen-Cahn equation driven by multiplicative white noise indicates that this full discretization is uniquely ergodic for any interface thickness. Numerical experiments validate our theoretical results.
subject: Articles
  topic: section
subject: Numerical invariant measure
  topic: keywords
subject: numerical ergodicity
  topic: keywords
subject: stochastic Allen-Cahn equation.
  topic: keywords
subject: Primary 60H35
  topic: subjects
subject: Secondary 60H15
  topic: subjects
subject: 65M60
  topic: subjects
creator: Zhihui Liu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-03-31
publication date: 2025-03-31
starting page: 30
ending page: 44
doi: 10.4208/cmr.2024-0042

Equivariant Cohomology and Deformation for Associative Algebras with a Derivation
type: item
description: We introduce finite group action for associative algebras equipped with a derivation (that is, AssDer pairs) and equivariant cohomology for such algebraic object. Next, we discuss equivariant deformation theory and study its relation with the equivariant cohomology.
subject: Articles
  topic: section
subject: Associative algebra
  topic: keywords
subject: derivation
  topic: keywords
subject: group action
  topic: keywords
subject: equivariant cohomology
  topic: keywords
subject: equivariant deformation.
  topic: keywords
subject: 16E40
  topic: subjects
subject: 16S80
  topic: subjects
subject: 17A36
  topic: subjects
subject: 55N91
  topic: subjects
creator: Jingru Ren
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-03-31
publication date: 2025-03-31
starting page: 45
ending page: 58
doi: 10.4208/cmr.2024-0052

When the Gromov-Hausdorff Distance Between Finite-Dimensional Space and Its Subset Is Finite?
type: item
description: In this paper we prove that the Gromov-Hausdorff distance between $\mathbb{R}^n$ and its subset $A$ is finite if and only if $A$ is an $ε$-net in $\mathbb{R}^n$ for some $ε > 0.$ For infinite-dimensional Euclidean spaces this is not true. The proof is essentially based on upper estimate of the Euclidean Gromov-Hausdorff distance by means of the Gromov-Hausdorff distance.
subject: Articles
  topic: section
subject: Metric space
  topic: keywords
subject: $ε$-net
  topic: keywords
subject: Gromov-Hausdorff distance.
  topic: keywords
subject: 46B20
  topic: subjects
subject: 51F99
  topic: subjects
creator: I.N. Mikhailov
creator: A.A. Tuzhilin
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-03-31
publication date: 2025-03-31
starting page: 1
ending page: 8
doi: 10.4208/cmr.2024-0041

Eigenvalue Problem for a Class of Quasilinear Elliptic Operators with Mixed Boundary Value Condition in a Variable Exponent Sobolev Space
type: item
description: In this paper, we consider an eigenvalue problem for a class of nonlinear elliptic operators containing $p(·)$-Laplacian and mean curvature operator with mixed boundary conditions. More precisely, we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on another part of the boundary. Using the Ljusternik-Schnirelmann variational method, we show the existence of infinitely many positive eigenvalues of the equation. Furthermore, under some conditions, we derive that the infimum of the set of all the eigenvalues becomes zero or remains to be positive.
subject: Articles
  topic: section
subject: Eigenvalue problem
  topic: keywords
subject: $p(·)$-Laplacian type equation
  topic: keywords
subject: mean curvature operator
  topic: keywords
subject: mixed boundary value problem.
  topic: keywords
subject: 49R50
  topic: subjects
subject: 35A01
  topic: subjects
subject: 35J62
  topic: subjects
subject: 35J57
  topic: subjects
creator: Junichi Aramaki
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-12-19
publication date: 2024-12-19
starting page: 437
ending page: 481
doi: 10.4208/cmr.2024-0039