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.
A High-Order Oscillation-Free Discontinuous Galerkin Scheme for One-Dimensional Compressible Multi-Material Flows
In this paper, a high-order oscillation-free (OF) discontinuous Galerkin (DG) scheme is presented for one-dimensional compressible multi-material flows. For describing the dynamics of fluid mixture, we couple a conservative equation related to the volume-fraction model with the Euler equations. For controlling the oscillations, some damping terms are added into the weak formulation of the system to automatically adjust the high-order terms. There are not any parameters which need to be adjusted artificially in the new damping terms, and the difficulties in solving discontinuous solutions and complexities of designing limiters can be avoided. Our scheme can be applied to the simulation of compressible multi-material flows efficiently with the essentially non-oscillatory property. Moreover, our scheme can be extended to the one with any high order as long as the order of basis functions is increased. In this paper, we only study the third-order OFDG scheme with the basis functions up to the quadratic polynomial. Some examples are tested to demonstrate the third-order accuracy and essentially non-oscillatory property of our scheme.
Copyright (c) 2025 Communications in Mathematical Research
Exploring Exponential Runge-Kutta Methods: A Survey
In this survey, we provide an in-depth investigation of exponential Runge-Kutta methods for the numerical integration of initial-value problems. These methods offer a valuable synthesis between classical Runge-Kutta methods, introduced more than a century ago, and exponential integrators, which date back to the 1960s. This manuscript presents both a historical analysis of the development of these methods up to the present day and several examples aimed at making the topic accessible to a broad audience.
A Symmetric Mountain Pass Lemma and Its Application to an Equation for a Class of Quasilinear Elliptic Operators in a Variable Exponent Sobolev Space
The purpose of this paper is to solve equation for a class of quasilinear elliptic operators containing the $p(·)-$Laplacian and the mean curvature operator with mixed boundary conditions. More precisely, we are concerned with the problem that has the Dirichlet condition in one part of the boundary and the Steklov condition in another. Using a symmetric mountain pass lemma and its corollary, we show the existence of infinitely many weak solutions of the equation and the boundedness of the sequence of solutions, or convergence to zero of the sequence of solutions according to the hypotheses about the data functions.
An ALE-based Numerical Scheme with a Two-Dimensional Riemann Solver for Two-Phase Flows with Non-Equilibrium Pressure
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.
Transfer of Improvement Strategies Between DRS and ADMM: A Unified Classification Framework
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.
alternating direction method of multipliers
Observation-Driven INAR(1) Models with Novel and Flexible Links
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.
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.
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.
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.
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.
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.
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.
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.
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.