Acta Mathematica Vietnamica
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Issue 3, Volume 45, September 2020
2020
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Baum-Katz’s Type Theorems for Pairwise Independent Random Elements in Certain Metric Spaces Nguyen Tran Thuan , Nguyen Van Quang PDF555-570
▸ Abstract
In this study, some Baum-Katz’s type theorems for pairwise independent random elements are extended to a metric space endowed with a convex combination operation. Our results are considered in the cases of identically distributed and non-identically distributed random elements. Some illustrative examples are provided to sharpen the results. -
On the Existence, Uniqueness, and Stability of $\beta$-Viscosity Solutions to a Class of Hamilton-Jacobi Equations in Banach Spaces Bang Van Tran , Tien Trong Phan PDF571-590
▸ Abstract
This paper is concerned with the qualitative properties of viscosity solutions to a class of Hamilton-Jacobi equations (HJEs) in Banach spaces. Specifically, based on the concept of $\beta$-derivative (Deville et al. 1993), we establish the existence, uniqueness and stability of $\beta$-viscosity solutions for a class of HJEs in the form $u+H(x,u,Du)= 0$. The obtained results in this paper extend earlier works in the literature, for example, Crandall and Lions (J. Funct. Anal. 62, 379–398, 1985, J. Funct. Anal., 65, 368–405, 1986) and Deville et al. (1993).
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On Entire Functions That Share Two Small Functions with Their Derivative Pulak Sahoo , Samar Halder PDF591-609
▸ Abstract
In this paper, we study the relationship between two entire functions sharing two pairs of small functions IM with their derivatives. For such a function $f$ and a positive integer $k$, we establish a linear relationship between $f$ and $f^k$ which generalizes a result due to Chen (Abstr. Appl. Anal., Art. 601507, 1–9, 2014). Examples have also been given in support of the existence of the result and the sharpness of some conditions involved in it. Furthermore, we also study a uniqueness result concerning $f$ and $f^k$ sharing two small functions IM.
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Blow-up Profile of the Focusing Gross-Pitaevskii Minimizer Under Self-Gravitating Effect Thanh Viet Phan PDF611-634
▸ Abstract
We consider a Bose-Einstein condensate in a 2D dilute Bose gas, with an external potential and an interaction potential containing both the short-range attractive self-interaction and the long-range self-gravitating effect. We prove the existence of minimizers and analyze their behavior when the strength of the attractive interaction converges to a critical value. The universal blow-up profile is the unique optimizer of a Gagliardo-Nirenberg interpolation inequality.
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Robust Equilibrium in Transportation Networks Nguyen Ba Minh , Truong Thi Thanh Phuong PDF635-650
▸ Abstract
We introduce the concept of robust equilibrium in a multi-criteria transportation network and obtain a formula to compute the radius of robustness together with an algorithm to find robust equilibrium flows.
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Linear Quadratic Nash Differential Games of Stochastic Singular Systems with Markovian Jumps Bin Liu , Xin Wang PDF651-660
▸ Abstract
This paper investigates Nash games for stochastic singular systems with Markovian jumps. Based on the generalized Itô’s formula, the corresponding linear quadratic optimal control problem is studied for the first time. Then, we establish the existence of Nash strategies by means of generalized coupled Riccati algebraic equations. As an application, the stochastic $H_2/H_{\infty}$ control with state, control, and external disturbance-dependent noise is discussed.
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A Posteriori Error Estimates for Hypersingular Integral Equation on Spheres with Spherical Splines Duong Thanh Pham , Tung Le PDF661-692
▸ Abstract
A posteriori residual and hierarchical upper bounds for the error estimates are proved when solving the hypersingular integral equation on the unit sphere by using the Galerkin method with spherical splines. Based on these a posteriori error estimates, adaptive mesh refining procedures are used to reduce complexity and computational cost of the discrete problems. Numerical experiments illustrate our theoretical results.
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Energy Space Approaches to the Cauchy Problem for Poisson’s Equation Phan Xuan Thanh , Olaf Steinbach PDF693-707
▸ Abstract
Variational methods coupled with Tikhonov’s regularization for solving the Cauchy problem for Poisson’s equation are suggested and studied. The novel idea is to use the Tikhonov regularization term in $H^{1/2}$ norm rather than in $L_2$ norm. The penalty term is evaluated by some appropriate boundary integral operators. The optimality condition in the form of boundary integral equations is derived and then discretized by the Galerkin boundary element method. The error estimates for the discretized problems are proved for noisy data. Some numerical examples and comparisons with the $L_2$ setting are presented for showing the efficiency of our approaches. -
Implicit Monotone Difference Methods for Scalar Conservation Laws with Source Terms Michael Breuß , Andreas Kleefeld PDF709-738
▸ Abstract
In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author (Breuß SIAM J. Numer. Anal. 43(3), 970–986 2005). Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests.
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Algebraic Dependences of Three Meromorphic Mappings Sharing Few Moving Hyperplanes Si Duc Quang , Ha Huong Giang PDF739-748
▸ Abstract
In this article, some algebraic dependence theorems for three meromorphic mappings into a projective space sharing few moving hyperplanes are given. Our results are improvements of many previous results in this topic.
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A Mollification Method for Backward Time-Fractional Heat Equation Nguyen Van Duc , Pham Quy Muoi , Nguyen Van Thang PDF749-766
▸ Abstract
In this paper, we study the ill-posed fractional backward heat equation $$\left\{\begin{array}{ll} \frac{\partial^{\gamma} u}{\partial t^{\gamma}}={\varDelta} u,\quad x\in \mathbb{R}^{n},t\in(0,T),\\ u(x,T)=\varphi(x), \quad x\in\mathbb{R}^{n}, \end{array}\right. $$ where $\varphi $ is unknown exact data and only noisy data $\varphi^{\varepsilon}$ with $$\|\varphi^{\varepsilon}(\cdot)-\varphi(\cdot)\|_{L_{2}(\mathbb{R}^{n})}\leqslant\varepsilon $$ is available. The problem is regularized by the well-posed mollified problem $$\left\{\begin{array}{ll} \frac{\partial^{\gamma} v^{\nu}}{\partial t^{\gamma}}={\varDelta} v^{\nu},\quad x\in \mathbb{R}^{n},t\in(0,T),\\ v^{\nu}(x,T)=S_{\nu}(\varphi^{\varepsilon}(x)), \quad x\in \mathbb{R}^{n}, \end{array}\right. $$ where $\nu > 0$ and $S_{\nu}(\varphi^{\varepsilon}(x))$, a mollification of $\varphi^{\varepsilon}$ defined by the convolution of $\varphi^{\varepsilon}(x)$ with Dirichlet kernel. The error estimates $\|u(\cdot ,t)-v^{\nu }(\cdot ,t)\|_{H^{l}(\mathbb {R}^{n})}, 0\leq l$ are established for $\nu$ chosen a priori and a posteriori.
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New Algorithms for a Class of Accretive Variational Inequalities in Banach Spaces Nguyen Buong , Than Van Dinh , Tran Thi Huong PDF767-781
▸ Abstract
In this paper, for finding a common zero of a finite family of $m$-accretive mappings in uniformly convex Banach spaces with a uniformly Gâteaux differentiable norm, we propose an implicit iteration algorithm and an explicit one, based on a convex combination of the steepest-descent method and a composition of resolvents. We also show that our main algorithm contains some iterative ones in literature as special cases. Finally, we give numerical examples for illustration. -
Characterization of Multipliers on Hypergroups Vishvesh Kumar , N. Shravan Kumar , Ritumoni Sarma PDF783-794
▸ Abstract
Let K be a hypergroup. We characterize translation invariant operators from a vector-valued $L^1$-space to a vector-valued $L^p$-space defined on $K$. Furthermore, for a commutative compact hypergroup $K$, we introduce and study the notion of character convolution transform of a Banach $L^1(K)$-module $M$. Several characterizations of multipliers on $M$ are given. We also prove an analogue of the Schoenberg-Eberlein theorem for the character convolution transform.