Issue 2, Volume 50, June 2025

2025

1. The Dirichlet Problem Associated to Operators Defined on the Nuclear Algebra of Entire Functions Sonia Chaari , Afef Ben Farah PDF

   ▸ Abstract

   This paper is devoted to the Dirichlet problem associated to operators defined on the nuclear algebra of entire functions. Firstly, we give a probabilistic representation of the solution of the Dirichlet problem associated to the extended $K$-Gross Laplacian in terms of $K$-Wiener process. Secondly, we prove that the Dirichlet problem associated to the operator $\mathcal {F}_{t(-K),I}$-transform has an explicit solution. Finally, an application to the large deviation principle is given.

   159-172
2. Well-posedness, Regularity of Solutions and the $\theta$-Euler-Maruyama Scheme for Stochastic Volterra Integral Equations with General Singular Kernels and Jumps Phan Thi Huong , Hoang-Long Ngo , Peter Kloeden PDF

   ▸ Abstract

   In this paper, we consider a class of stochastic Volterra integral equations with general singular kernels, driven by a Brownian motion and a pure jump Lévy process. We first show that these equations have a unique strong solution under certain regular conditions on their coefficients. Furthermore, the solutions of this equation depend continuously on the initial value and on the kernels $k$, $k_B$, and $k_Z$. We will then show the regularity of solutions for these equations. Finally, we propose a $\theta$-Euler-Maruyama approximation scheme for these equations and demonstrate its convergence at a certain rate in the $L^2$-norm. Some numerical simulations is also presented to support for the theoretical results.

   173-195
3. Large Deviation for a 2D Cahn-Hilliard-Oldroyd Model of Order One Under Random Influences Salvador Awo Kougang , Calvin Tadmon , Gabriel Deugoué PDF

   ▸ Abstract

   This paper establishes a large deviation principle for a stochastic 2D Cahn-Hilliard-Oldroyd model of order one under random influences. The model used in the analysis consists of the stochastic Oldroyd model of order one, coupled with a Cahn-Hilliard equation for the order parameter. The approach used is the weak convergence method with the main idea laid on a variational representation formula for the Laplace transform of continuous and bounded functionals.

   197-231
4. Enhanced Adaptive RBF-FD Methods for 2D Elliptic Equations with New Thresholding Strategies Oanh Thi Dang PDF

   ▸ Abstract

   In this paper, we present strategies for determining the error indicator threshold value at each refinement step. The error indicator threshold is not directly computed from the maximum error indicator, improving refinement efficiency. Moreover, to enhance refinement efficiency, we propose three new candidate center structures, which include a greater number of centers and may double the distribution density. Additionally, we improve the variable support size algorithm for making it more flexible. Moreover, we introduce a measure to evaluate the average recursive preprocessing cost per new center generated by the adaptive RBF-FD (Radial Basis Function-Finite Difference) meshless method. This metric is used to assess the preprocessing cost efficiency of the adaptive RBF-FD meshless method and to compare it with our previously published adaptive RBF-FD methods for solving 2D elliptic equations. The results demonstrate that the numerical solutions obtained using the methods proposed in this paper are significantly more accurate, more stable, and more effective in terms of refinement than those from our earlier studies and the adaptive FEM (Finite Element Method), while achieving the lowest computational cost.

   233-261
5. Spanning Trees of $K_{1,4}$-free Graphs with a Bounded Number of Leaves and Branch Vertices Pham Hoang Ha PDF

   ▸ Abstract

   Let $T$ be a tree. A vertex of degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$. A graph is said to be $K_{1,4}$-free if it does not contain $K_{1,4}$ as an induced subgraph. In this paper, we study the spanning trees with a bounded number of leaves and branch vertices of $K_{1,4}$-free graphs. Applying the main results, we also give some improvements of previous results on the spanning tree with few branch vertices for the case of $K_{1,4}$-free graphs.

   263-271
6. Hamilton Type Gradient Estimates for a Heat Equation under the Ricci-harmonic Flow Liang-Chu Chang , Nguyen Thac Dung , Chiung-Jue Anna Sung PDF

   ▸ Abstract

   Our aim in this paper is to study the linear heat equation on a Riemannian manifold evolving by the Ricci-harmonic flow. We first show a Hamilton type gradient estimate for the positive solution of the heat equation, which allows us to derive a Harnack inequality. It is worthy to note that comparing with the Li-Yau type estimate by Bailesteanu [1], our gradient estimate can be obtained without any assumption on the harmonic quantity in the flow.

   273-283
7. Explicit Solutions in Isotropic Planar Elastostatics Andreas Granath , Per Åhag , Antti Perälä , Rafał Czyż PDF

   ▸ Abstract

   Addressing the intricate challenges in plane elasticity, especially with non-vanishing traction and complex geometries, requires innovative methods. This paper offers a novel approach, drawing inspiration from the Neumann problem for the inhomogeneous Cauchy-Riemann equations. Our method applies to domains conformally equivalent to a unit disk or an annulus, focusing on deriving explicit solutions for the displacement field rather than the stress tensor, which distinguishes it from most traditional approaches. We explore solutions for specific classical cases to demonstrate its efficacy, such as a cardioid domain, a ring domain with a shifted hole, and a gear-like structure. This work enhances the toolkit for researchers and practitioners tackling isotropic planar elastostatic challenges with a unified and flexible approach.

   285-301
8. Correction to: Generalized Semi-infinite Polynomial Optimization and Semidefinite Programming Relaxations Liguo Jiao , Jae Hyoung Lee , Tiến-Sơn Phạm PDF

   ▸ Abstract

   Correction to: Acta Mathematica Vietnamica (2024) 49:441–457 https://doi.org/10.1007/s40306-024-00551-7 The original online version of the paper has been updated as it included the wrong received/revised/published dates.

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