Issue 4, Volume 50, December 2025

2025

1. Relative Cohen-Macaulay Modules under Ring Homomorphisms Parisa Pourghobadian , Kamran Divaani-Aazar , Ahad Rahimi PDF

   ▸ Abstract

   Let $R$ be a commutative Noetherian ring with identity (not necessarily local), and let $\mathfrak {a}$ be a proper ideal of $R$. We investigate the invariance of certain classes of $\mathfrak {a}$-relative Cohen-Macaulay modules under pure ring homomorphisms and ring homomorphisms of finite flat dimension. Our results generalize several existing results in the literature on homological modules.

   449-466
2. Non-uniform Bounds for Non-normal Approximation via Stein’s Method with Applications to the Curie–Weiss Model and the Imitative Monomer-dimer Model Lê Vǎn Thành , Nguyen Ngoc Tu PDF

   ▸ Abstract

   This paper establishes a non-uniform Berry–Esseen bound for non-normal approximation using Stein’s method. The main theorem generalizes the result of the authors in [Comptes Rendus Mathématique, 2024] to the context of non-normal approximation. As applications of the main result, we derive non-uniform Berry–Esseen bounds in non-central limit theorems for the magnetization in the Curie–Weiss model and the imitative monomer-dimer model. These extend some existing results in the literature, including Theorem 2.1 of Chatterjee and Shao [Ann. Appl. Probab., 2011] and Theorem 1 of Chen [J. Math. Physics, 2016].

   467-486
3. Pertubations of Invariants of Plane Curve Singularities Hong Duc Nguyen PDF

   ▸ Abstract

   We introduce, using resolution of singularities, a new invariant $\lambda (f)$ of a plane curve singularity $f$ which can be easily computed and study its relation to the problem of the stability under small perturbations. More percisely, we prove that the number of braches and delta invatiants of plane curve singularities are stable under small perturbations of degree at least $\lambda (f)$.

   487-493
4. A First Approach to the Burchnall-Chaundy Theory for Quadratic Algebras Having PBW Bases Arturo Niño , María Camila Ramírez , Armando Reyes PDF

   ▸ Abstract

   In this paper, we present a first approach toward a Burchnall-Chaundy theory for the skew Ore polynomials of higher order generated by quadratic relations defined by Golovashkin and Maksimov (Russian Math. Surv. 53(2), 384–386 1998).

   495-519
5. Products of Reflections and Commutators of Reflections Over Division Algebras M. H. Bien , N. T. T. Ha , T. N. N. Hung , D. T. Toan PDF

   ▸ Abstract

   Let $D$ be a division algebra and let $n$ be a positive integer greater than 1. Assume that the commutator width $\omega (D)$ of $D^{*}$ is positive and finite. First, we revisit a question posed by F. S. Cater concerning the decomposition of matrices into a product of reflections over a division ring. Among results, we show that every matrix $A$ in the special linear group $\textrm{SL}_n(D)$ can be expressed as a product of at most $\textrm{rank} (A-\mathrm I_n)+4\omega (D)$ reflections. Next, we study the decomposition of matrices in $\textrm{SL}_n(D)$ into products of commutators of reflections in the general linear group $\textrm{GL}_n(D)$.

   521-539
6. Solving Variational Inequalities Related to the Split Common Fixed Point Problem with Multiple Output Sets Constraints Tran Viet Anh PDF

   ▸ Abstract

   In this paper, we explore the task of addressing strongly monotone variational inequalities over the solution set of the split common fixed point problem with multiple output sets in real Hilbert spaces. We propose a new iterative algorithm designed specifically for this task, which incorporates dynamic step sizes that adapt based on information from previous iterations. This approach ensures strong convergence without the need for prior knowledge of the norm of the bounded linear operator involved. Additionally, our method does not require information about the Lipschitz constants or the strongly monotone constants of the mappings. We also present several corollaries derived from our main result. Finally, we present an application of the split common fixed point problem with multiple output sets to supply chain planning, and provide numerical experiments to evaluate the performance of the proposed algorithm in comparison with existing methods.

   541-562
7. The Dirichlet Problem for Nonsymmetric Augmented Hessian Quotient Type Equations Van-Bang Tran , Tien-Ngoan Ha , Huu-Tho Nguyen , Trong-Tien Phan , Hong-Quang Dinh PDF

   ▸ Abstract

   We deal with the Dirichlet problem for nonsymmetric augmented Hessian quotient type equations. First, we look for an admissible solution to the problem for corresponding symmetric augmented Hessian quotient type equations. Then we apply the Banach fixed point theorem to prove the existence of a $\delta$-admissible solution in $C^{2,\alpha }$ to the problem by assuming that the augmented skew-symmetric matrix is sufficiently small in a certain sense. We also give a necessary condition for the existence and sufficient conditions for uniqueness of this kind of $\delta$-admissible solutions.

   563-594
8. Stability in a Two-dimensional Keller-Segel-Navier-Stokes System Qiong Chen , Qian Zhang PDF

   ▸ Abstract

   We study the initial-boundary value problem for a Keller-Segel-Navier-Stokes model that includes a logistic source term. The system is considered in a two-dimensional bounded domain with a smooth boundary as follows $$\begin{aligned}\,\, \left\{ \begin{aligned}&n_{t}+u\cdot \nabla n=\Delta n-\nabla \cdot (n\nabla c)+\mu n-\kappa n^{2},\\&c_{t}+u\cdot \nabla c=\Delta c -nc,\\&v_{t}+u\cdot \nabla v=\Delta v -\gamma v+n,\\&u_{t}+(u\cdot \nabla ) u+\nabla \pi =\Delta u-nf,\\&\nabla \cdot u=0. \end{aligned} \right. \end{aligned}$$ This system characterizes the interaction of chemotactic microorganisms with an incompressible fluid. Under the conditions of positive $\mu$ and non-negative $\kappa$, sufficiently regular initial data $(n_{0}, c_{0}, v_{0}, u_{0})$ with $n_{0} \not \equiv 0$ yield a uniformly bounded global classical solution. Moreover, when $\mu =0$, the solution obeys $$\begin{aligned} n(\cdot ,t)\rightarrow 0,\quad c(\cdot ,t)\rightarrow 0,\quad v(\cdot ,t)\rightarrow 0\quad \textrm{and} \quad u(\cdot ,t)\rightarrow 0\qquad \textrm{in} \ L^{\infty }(\Omega ) \end{aligned}$$ as $t\rightarrow \infty$.

   595-622