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: 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: The Variants of the MGSS Methods for Complex Symmetric Linear System of Equations
items: On the Long-Time $H^1$-Stability of the Linearly Extrapolated BDF2 Time-Stepping Scheme for Coupled Multiphysics Flow Problems
items: Functional Solutions, Multi Line Solitons and Multiple Pole Solutions of the Generalized (2+1)-Dimensional Kaup-Kupershmidt Equation
items: Homoclinic Solutions for a Class of Hamiltonian Systems with Small External Perturbations
items: Equivariant Cohomology and Deformation for Associative Algebras with a Derivation
items: Continuous Spectrum for a Class of Smooth Mixing CMV Matrices
items: Reduced Basis Method Based on Fourier Transform for Time-Dependent Parameterized Nonlocal Problems
items: Numerical Ergodicity of Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise
items: Commutators of Complex Symmetric Operators
items: Degenerations of Nilalgebras
items: When the Gromov-Hausdorff Distance Between Finite-Dimensional Space and Its Subset Is Finite?
items: The m-CMP Inverse in Minkowski Space
items: Uniqueness of Transonic Shock Solutions to Euler-Poisson System with Varying Background Charges
items: Eigenvalue Problem for a Class of Quasilinear Elliptic Operators with Mixed Boundary Value Condition in a Variable Exponent Sobolev Space
items: A High-Order Meshless Energy-Preserving Algorithm for the Beam Equation
items: On Some Cardinal Properties of the Space $P_f(X)$ of Probability Measures
items: Stochastic Chebyshev-Picard Iteration Method for Nonlinear Differential Equations with Random Inputs
items: The General Dabrowski-Sitarz-Zalecki Type Theorem for Odd Dimensional Manifolds with Boundary II
items: Invariance of Conjugate Normality Under Similarity
items: The AOR-Base Splitting Modified Fixed Point Iteration for Solving Absolute Value Equations
items: Convergence Analysis of Kernel Learning FBSDE Filter
items: Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems

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
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
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
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 LandauGinzburg 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
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
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
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
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

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

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

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

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

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

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

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

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

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

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

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

The m-CMP Inverse in Minkowski Space
type: item
description: In this paper, a new generalized m-CMP inverse is introduced into the Minkowski space, and its related properties are discussed. Secondly, three different expressions of m-CMP inverse are listed, namely Hartwig-Spindelböck decomposition, full-rank decomposition and integral expression. Finally, the application of the inverse m-CMP equation in solving linear equations is given.
subject: Articles
  topic: section
subject: Minkowski space
  topic: keywords
subject: m-CMP inverse
  topic: keywords
subject: Hartwig-Spindelböck decomposition
  topic: keywords
subject: full-rank factorization
  topic: keywords
subject: integral expression
  topic: keywords
subject: system of linear equations.
  topic: keywords
subject: 15A10
  topic: subjects
creator: Ming-Xue Shao
creator: Li-Hua Cheng
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-12-19
publication date: 2024-12-19
starting page: 482
ending page: 498
doi: 10.4208/cmr.2024-0037

Uniqueness of Transonic Shock Solutions to Euler-Poisson System with Varying Background Charges
type: item
description: In this paper, we investigate a one-dimensional Euler-Poisson system with varying background charges, which are two different constants when the flow speed is less than and greater than the sound speed. Using the shock matching method, we derive the properties of the solution trajectories and establish a monotonic relationship between the density at the nozzle exit and the shock position. This relationship demonstrates the existence and uniqueness of a transonic shock solution under suitable boundary conditions.
subject: Articles
  topic: section
subject: Euler-Poisson system
  topic: keywords
subject: transonic shock solution
  topic: keywords
subject: uniqueness
  topic: keywords
subject: varying background charges.
  topic: keywords
subject: 35B35
  topic: subjects
subject: 35L67
  topic: subjects
subject: 35Q81
  topic: subjects
subject: 76H05
  topic: subjects
creator: Haoran Zheng
creator: Jianqiao Zhang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-12-19
publication date: 2024-12-19
starting page: 400
ending page: 412
doi: 10.4208/cmr.2024-0028

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

A High-Order Meshless Energy-Preserving Algorithm for the Beam Equation
type: item
description: In this paper, a meshless energy-preserving algorithm which can be arbitrarily high-order in temporal direction for the beam equation has been proposed. Based on the method of lines, we first use the radial basis function quasi-interpolation method to discretize spatial variable and obtain a semi-discrete Hamiltonian system by using the premultiplication of a diagonal matrix. Then, symplectic Runge-Kutta method that can conserve quadratic invariants exactly has been used to discretize the temporal variable, which yields a fully discrete meshless scheme. Due to the specific quadratic energy expression of the beam equation, the proposed meshless scheme here is not only energy-preserving but also arbitrarily high-order in temporal direction. Besides uniform and nonuniform grids, numerical experiments on random grids are also conducted, which demonstrate the properties of the proposed scheme very well.
subject: Articles
  topic: section
subject: Beam equation
  topic: keywords
subject: meshless scheme
  topic: keywords
subject: energy-preserving method
  topic: keywords
subject: radial basis function
  topic: keywords
subject: symplectic Runge-Kutta method.
  topic: keywords
subject: 37K05
  topic: subjects
subject: 65L06
  topic: subjects
subject: 65L12
  topic: subjects
subject: 65N40
  topic: subjects
creator: Jialing Wang
creator: Yali He
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-12-19
publication date: 2024-12-19
starting page: 413
ending page: 436
doi: 10.4208/cmr.2024-0018

On Some Cardinal Properties of the Space $P_f(X)$ of Probability Measures
type: item
description: In this paper, given a compact space $X,$ we construct a continuous mapping from the space $P_f(X)$ of all probability measures on $X$ into the space of maximal linked systems of closed subsets of $X.$ Also, we investigate the behavior of the functional tightness and the local density of topological spaces under the action of the functor $P_f$: Comp→Comp. We show that the functor $P_f$ does not change the functional tightness and the local density of compact spaces.
subject: Articles
  topic: section
subject: Probability measure
  topic: keywords
subject: functional tightness
  topic: keywords
subject: local density
  topic: keywords
subject: normal functor
  topic: keywords
subject: superextension.
  topic: keywords
subject: 18A22
  topic: subjects
subject: 18B30
  topic: subjects
subject: 46E27
  topic: subjects
subject: 46J10
  topic: subjects
subject: 54C35
  topic: subjects
creator: Shavkat Ayupov
creator: Nodirbek Mamadaliev
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-12-19
publication date: 2024-12-19
starting page: 383
ending page: 399
doi: 10.4208/cmr.2024-0036

Stochastic Chebyshev-Picard Iteration Method for Nonlinear Differential Equations with Random Inputs
type: item
description: This work presents a stochastic Chebyshev-Picard iteration method to efficiently solve nonlinear differential equations with random inputs. If the nonlinear problem involves uncertainty, we need to characterize the uncertainty by using a few random variables. The nonlinear stochastic problems require solving the nonlinear system for a large number of samples in the stochastic space to quantify the statistics of the system of response and explore the uncertainty quantification. The computational cost is very expensive. To overcome the difficulty, a low rank approximation is introduced to the solution of the corresponding nonlinear problem and admits a variable-separation form in terms of stochastic basis functions and deterministic basis functions. No iteration is performed at each enrichment step. These basis functions are model-oriented and involve offline computation. To efficiently identify the stochastic basis functions, we utilize the greedy algorithm to select some optimal samples. Then the modified Chebyshev-Picard iteration method is used to solve the nonlinear system at the selected optimal samples, the solutions of which are used to train the deterministic basis functions. With the deterministic basis functions, we can obtain the corresponding stochastic basis functions by solving linear differential systems. The computation of the stochastic Chebyshev-Picard method decomposes into an offline phase and an online phase. This is very desirable for scientific computation. Several examples are presented to illustrate the efficacy of the proposed method for different nonlinear differential equations.
subject: Articles
  topic: section
subject: Nonlinear ordinary differential equation
  topic: keywords
subject: Chebyshev-Picard iteration method
  topic: keywords
subject: variable-separation method
  topic: keywords
subject: random inputs.
  topic: keywords
subject: 65C20
  topic: subjects
subject: 65L60
  topic: subjects
subject: 65M70
  topic: subjects
creator: Lingling Ma
creator: Yicheng Liu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-09-09
publication date: 2024-09-09
starting page: 275
ending page: 312
doi: 10.4208/cmr.2024-0011

The General Dabrowski-Sitarz-Zalecki Type Theorem for Odd Dimensional Manifolds with Boundary II
type: item
description: In [T. Wu et al., arXiv2310.09775, 2023], a general Dabrowski-Sitarz-Zalecki type theorem for odd dimensional manifolds with boundary was proved. In this paper, we give the proof of the another general Dabrowski-Sitarz-Zalecki type theorem for the spectral Einstein functional associated with the Dirac operator on odd dimensional manifolds with boundary.
subject: Articles
  topic: section
subject: Dirac operators
  topic: keywords
subject: spectral Einstein functional
  topic: keywords
subject: Dabrowski-Sitarz-Zalecki type theorems.
  topic: keywords
subject: 53C05
  topic: subjects
subject: 53C25
  topic: subjects
subject: 58A05
  topic: subjects
creator: Hongfeng Li
creator: Yong Wang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-09-09
publication date: 2024-09-09
starting page: 343
ending page: 382
doi: 10.4208/cmr.2024-0030

Invariance of Conjugate Normality Under Similarity
type: item
description: An operator $T$ on a separable, infinite dimensional, complex Hilbert space $\mathcal{H}$ is called conjugate normal if $C|T|C = |T^∗|$ for some conjugate linear, isometric involution $C$ on $\mathcal{H}.$ This paper focuses on the invariance of conjugate normality under similarity. Given an operator $T,$ we prove that every operator $A$ similar to $T$ is conjugate normal if and only if there exist complex numbers $λ_1$, $λ_2$ such that $(T−λ_1)(T−λ_2)=0.$
subject: Articles
  topic: section
subject: $C$-normal operators
  topic: keywords
subject: complex symmetric operators
  topic: keywords
subject: similarity.
  topic: keywords
subject: Primary 47B99
  topic: subjects
subject: 47A05
  topic: subjects
subject: Secondary 47A10
  topic: subjects
subject: 47A58
  topic: subjects
creator: Cun Wang
creator: Meng Yu
creator: Minyi Liang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-09-09
publication date: 2024-09-09
starting page: 245
ending page: 260
doi: 10.4208/cmr.2024-0002

The AOR-Base Splitting Modified Fixed Point Iteration for Solving Absolute Value Equations
type: item
description: Recently, Yu et al. presented a modified fixed point iterative (MFPI) method for solving large sparse absolute value equation (AVE). In this paper, we consider using accelerated overrelaxation (AOR) splitting to develop the modified fixed point iteration (denoted by MFPI-JS and MFPI-GSS) methods for solving AVE. Furthermore, the convergence analysis of the MFPI-JS and MFPI-GSS methods for AVE are also studied under suitable restrictions on the iteration parameters, and the functional equation between the parameter $\tau$ and matrix $Q.$ Finally, numerical examples show that the MFPI-JS and MFPI-GSS are efficient iteration methods.
subject: Articles
  topic: section
subject: Absolute value equation
  topic: keywords
subject: modified point iteration
  topic: keywords
subject: convergence analysis
  topic: keywords
subject: numerical experiment.
  topic: keywords
subject: 65H10
  topic: subjects
subject: 65F10
  topic: subjects
creator: Changfeng Ma
creator: Jing Kang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-09-09
publication date: 2024-09-09
starting page: 261
ending page: 274
doi: 10.4208/cmr.2024-0009

Convergence Analysis of Kernel Learning FBSDE Filter
type: item
description: Kernel learning forward backward stochastic differential equations (FBSDE) filter is an iterative and adaptive meshfree approach to solve the nonlinear filtering problem. It builds from forward backward SDE for Fokker-Planker equation, which defines evolving density for the state variable, and employs kernel density estimation (KDE) to approximate density. This algorithm has shown more superior performance than mainstream particle filter method, in both convergence speed and efficiency of solving high dimension problems. However, this method has only been shown to converge empirically. In this paper, we present a rigorous analysis to demonstrate its local and global convergence, and provide theoretical support for its empirical results.
subject: Articles
  topic: section
subject: Forward backward stochastic differential equations
  topic: keywords
subject: kernel density estimation
  topic: keywords
subject: nonlinear filtering problems
  topic: keywords
subject: convergence analysis.
  topic: keywords
subject: 93E11
  topic: subjects
subject: 65B99
  topic: subjects
subject: 65C05
  topic: subjects
subject: 65C35
  topic: subjects
subject: 65C60
  topic: subjects
creator: Yunzheng Lyu
creator: Feng Bao
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-09-09
publication date: 2024-09-09
starting page: 313
ending page: 342
doi: 10.4208/cmr.2024-0017

Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems
type: item
description: In this paper, we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems. One type of the preconditioners are based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We analyze different preconditioners incorporating the exact Schur complements. We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements. These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly. Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions.
subject: Articles
  topic: section
subject: Schur complement
  topic: keywords
subject: block tridiagonal systems
  topic: keywords
subject: positively stable preconditioners
  topic: keywords
subject: Routh-Hurwitz stability criterion.
  topic: keywords
subject: 65F08
  topic: subjects
subject: 65M22
  topic: subjects
subject: 74F20
  topic: subjects
creator: Mingchao Cai
creator: Guoliang Ju
creator: Jingzhi Li
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-05-08
publication date: 2024-05-08
starting page: 214
ending page: 244
doi: 10.4208/cmr.2023-0051