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.
The Exponential Decaying Noncommutative Differential Forms and Convolution Algebras
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.
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A Positivity-Preserving Fourier Spectral Moving Mesh Method for the Keller-Segel Chemotaxis Model
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.
Constraint-Preserving High-Order Compact OEDG Method for Spherically Symmetric Einstein-Euler System
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.
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Asymptotic Estimates for the Ruin Probability of a Multidimensional Delay-Claim Risk Model with Dependent Claims
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.
Continuous Subsonic-Sonic Flows in Curved Convergent Nozzles with a Fixed Point
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
Image-Based Rock Typing Using Specific Surface and Iterative Convolution-Thresholding Method
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.
Error Bounds for Linear Complementarity Problem of $SD{D_k}$ Matrices
$SD{D_k}$ 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.
Some Complexes and Modules Induced by Strongly FP-Injective Modules
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.
strongly Gorenstein FP-injective modules
An Explicit Construction of Orthogonal Basis in $\mathcal{p}$-adic Fields
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.
A Schur-Horn Type Theorem for Symplectic Matrices
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.
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A Note on Cohomological Mirror Symmetry of Toric Manifolds
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.
Dynamical Stability of Transonic Shock Solutions to Non-Isentropic Euler Equations
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.
The Spectral Einstein Functional for the Nonminimal de Rham-Hodge Operator
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.
non-self-adjoint spectral Einstein functional
Logarithmic Upper Bound for Solutions of Degenerate Parabolic Equation
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.
Uniqueness in Inverse Electromagnetic Scattering by Locally Rough Surfaces with Phaseless Near-Field Data
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.
Functional Solutions, Multi Line Solitons and Multiple Pole Solutions of the Generalized (2+1)-Dimensional Kaup-Kupershmidt Equation
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.
Homoclinic Solutions for a Class of Hamiltonian Systems with Small External Perturbations
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.
On the Long-Time $H^1$-Stability of the Linearly Extrapolated BDF2 Time-Stepping Scheme for Coupled Multiphysics Flow Problems
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.
linearly extrapolated BDF2 timestepping scheme
The Variants of the MGSS Methods for Complex Symmetric Linear System of Equations
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.
Reduced Basis Method Based on Fourier Transform for Time-Dependent Parameterized Nonlocal Problems
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.
Commutators of Complex Symmetric Operators
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.
Continuous Spectrum for a Class of Smooth Mixing CMV Matrices
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.
Numerical Ergodicity of Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise
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.
Equivariant Cohomology and Deformation for Associative Algebras with a Derivation
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.
All complex 3-dimensional nilalgebras were described. As a corollary, all degenerations in the variety of complex 3-dimensional nilalgebras were obtained.
When the Gromov-Hausdorff Distance Between Finite-Dimensional Space and Its Subset Is Finite?
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.
Uniqueness of Transonic Shock Solutions to Euler-Poisson System with Varying Background Charges
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.
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.
Eigenvalue Problem for a Class of Quasilinear Elliptic Operators with Mixed Boundary Value Condition in a Variable Exponent Sobolev Space
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.
A High-Order Meshless Energy-Preserving Algorithm for the Beam Equation
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.