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.
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
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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 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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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.
On Some Cardinal Properties of the Space $P_f(X)$ of Probability Measures
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.
Stochastic Chebyshev-Picard Iteration Method for Nonlinear Differential Equations with Random Inputs
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.
Nonlinear ordinary differential equation
The General Dabrowski-Sitarz-Zalecki Type Theorem for Odd Dimensional Manifolds with Boundary II
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.
Invariance of Conjugate Normality Under Similarity
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 $(T−λ_1)(T−λ_2)=0.$
The AOR-Base Splitting Modified Fixed Point Iteration for Solving Absolute Value Equations
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.
Convergence Analysis of Kernel Learning FBSDE Filter
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.
Forward backward stochastic differential equations
Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems
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.