Acta Mathematica Vietnamica
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Issue 2, Volume 36, August 2011
2011 · Open access
Dedicated to Tran Duc Van on the occasion of his sixtieth birthday
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Preface Preface
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STABILITY ANALYSIS FOR LINEAR NON-AUTONOMOUS SYSTEMS WITH CONTINUOUSLY DISTRIBUTED MULTIPLE TIME-VARYING DELAYS AND APPLICATIONS T. T. ANH, L. V. HIEN, V. N. PHAT PDF129-143
▸ Abstract
In this paper, the exponential stability problem of a class of linear non-autonomous systems with continuously distributed multiple time-varying delays is studied. Based on the Lyapunov-Krasovskii functional approach, sufficient conditions for the exponential stability of the system are established via the solution of Riccati differential inequalities. By using this generalized result, new sufficient conditions are derived for the robust stability and stabilization of the systems subjected to uncertainties and external controls. Illustrative numerical examples are given to indicate significant improvements of the results.
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SOME PROPERTIES OF ORLICZ-LORENTZ SPACES HA HUY BANG, NGUYEN VAN HOANG, VU NHAT HUY PDF145-167
▸ Abstract
In this paper we study some fundamental properties of Orlicz-Lorentz spaces defined on $\mathbb R $ such as finding their dual spaces, giving best constants for the inequalities between the Orlicz norm and the Luxemburg norm on Orlicz-Lorentz spaces and establishing the Kolmogorov inequality in these spaces.
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ITERATED POLYHARMONIC GREEN FUNCTIONS FOR PLANE DOMAINS HEINRICH BEGEHR PDF169-181
▸ Abstract
Convoluting harmonic Green, Neumann, and Robin functions leads to certain hybrid polyharmonic Green functions and related boundary value problems for partial differential equations with some power of the Laplacian as leading operator. This paper is a survey about hybrid polyharmonic Green functions for plane domains and related results.
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ON SOME ASYMPTOTIC PROPERTIES OF FINITELY GENERATED MODULES NGUYEN TU CUONG, PHAM HUU KHANH PDF183-192
▸ Abstract
Let $I, J$ be a pair of ideals of a Noetherian local ring $R$. The purpose of this paper is to study asymptotic properties such as the asymptotic stability of the depths in dimension $> k$, $\mathrm{depth}_k(I, M_n)$ and $\mathrm{depth}_k(I,M_n/M_{n+1})$ of an ideal $I$ with respect to a $J$-filtration $(M_n)$ of submodules of a finitely generated module $M$; or the asymptotic behaviour of sets of associated prime ideals or of attached prime ideals of certain local cohomology modules determined by $M$ and $I, J$.
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SOME FRIEDRICHS TYPE INEQUALITIES IN THE FULL EUCLIDEAN SPACE JU. A. DUBINSKII PDF193-198
▸ Abstract
In this paper we prove the inequality $$\int_{\mathbb R^n}\mu_R(|x|)|u(x)|^pdx\leqslant M\left[\int_{\mathbb R^n}|\nabla u(x)|^p\omega(|x|)dx+\left|\int_{|x|=R}u(x)ds\right|^p\right],$$ where $\omega(|x|) > 0$ and $\mu_R(|x|) > 0$ are the weight functions, $R > 0$ is an arbitrary number. In doing so, we first show some ”two-sides” Hardy type inequalities.
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ON HARNACK’S INEQUALITY FOR NON-UNIFORMLY P-LAPLACIAN EQUATIONS DUONG MINH DUC, HO SI TUNG LAM, NGUYEN QUANG THANG, DINH CAO DUY THIEN VU PDF199-214
▸ Abstract
We prove the Harnack inequality for non-uniformly $p$-Laplacian equations.
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CONSTRUCTION OF IRREDUCIBLE REPRESENTATIONS OF THE QUANTUM SUPER GROUP $GL_q(3|1)$ NGUYEN THI PHUONG DUNG, PHUNG HO HAI, NGUYEN HUY HUNG PDF215-229
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In this note, we construct all irreducible representations of the quantum general linear super group $GL_q(3|1)$ using the double Koszul complex.
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ON THE EXISTENCE OF SOLUTIONS TO GENERALIZED QUASI-EQUILIBRIUM PROBLEMS OF TYPE II AND RELATED PROBLEMS TRUONG THI THUY DUONG, NGUYEN XUAN TAN PDF231-248
▸ Abstract
The generalized quasi-equilibrium problem of type II is formulated and some sufficient conditions on the existence of its solutions are shown. As special cases, we obtain several results on the existence of solutions of ideal quasivariational inclusion problems, quasivariational relation problems of type II, generalized quasi-KKM theorems etc. As corollaries, we show several results on the existence of solutions to other problems in the vector optimization theory concerning multivalued mappings.
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WEIGHTED ESTIMATES FOR SOLUTIONS OF $\overline{\partial}$-EQUATIONS ON CLOSED POSITIVE $(1, 1)$-CURRENTS LE MAU HAI, NGUYEN XUAN HONG PDF249-263
▸ Abstract
In this paper, we give estimates of Hörmander type and Donnelly-Fefferman type for solutions of the $\overline{\partial}$-equation on a closed positive $(1, 1)$- current $T$ in a pseudoconvex domain of $\mathbb C^{n+1}$.
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MINIMAX VARIATIONAL INEQUALITIES NGUYEN QUANG HUY, NGUYEN DONG YEN PDF265-281
▸ Abstract
We introduce a new notion called minimax variational inequality (MVI). The solution existence of nonmonotone MVIs in Euclidean spaces, pseudomonotone MVIs in reflexive Banach spaces, and strongly monotone MVIs in Hilbert spaces is studied in detail. We show that MVIs can serve as a good tool for studying minimax problems given by convex sets and differentiable functions.
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ON THE CONVOLUTIONS OF FOURIER-TYPE TRANSFORMS NGUYEN MINH KHOA PDF283-298
▸ Abstract
Some convolutions of Fourier-type transforms are introduced. The poly-convolution of the Fourier cosine integral transforms is formulated and its properties are studied. Its application to solving an integral equation and a system of integral equations is presented.
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MULTI-PRODUCT SUPPLY DEMAND NETWORKS WITH ELEMENTARY FLOWS DINH THE LUC PDF299-317
▸ Abstract
In this paper, we study a multi-product multi-criteria supply demand network with capacity constraints. We analyze different concepts of equilibrium and establish some relationships between them. Particular attention is paid on elementary flows and on the construction of variational inequalities which are equivalent to network equilibrium problems.
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TRAVELING WAVE DISPERSAL IN PARTIALLY SEDENTARY AGE-STRUCTURED POPULATIONS THUC MANH LE, FRITHJOF LUTSCHER, NGUYEN VAN MINH PDF319-330
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In this paper we present a thorough study on the existence of traveling waves in a mathematical model of dispersal in a partially sedentary age-structured population. This type of model was first proposed by Veit and Lewis in [Am. Nat. 148 (1996), 255-274]. We choose the fecundity function to be the Beverton-Holt type function. We extend the theory of traveling waves in the population genetics model of Weinberger in [SIAM J. Math. Anal. 13 (1982), 353-396] to the case when migration depends on age groups and a fraction of the population does not migrate.
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ON CHARACTERISTIC SYSTEMS FOR GENERAL MULTIDIMENSIONAL MONGE-AMPÈRE EQUATIONS HA TIEN NGOAN PDF331-344
▸ Abstract
Like the characteristic method in solving the Cauchy problem for a first-order partial differential equation, we reduce the Cauchy problem for a general multidimensional Monge-Ampère equation to that for a normal firstorder system of nonlinear partial differential equations, that could be called characteristic system for a given multidimensional Monge-Amp`ere equation.
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ON A NONLINEAR WAVE EQUATION WITH A NONLOCAL BOUNDARY CONDITION LE THI PHUONG NGOC, TRAN MINH THUYET, PHAM THANH SON, NGUYEN THANH LONG PDF345-374
▸ Abstract
Consider the initial-boundary value problem for the nonlinear wave equation $$\begin{cases} u_{tt}-\mu(t)u_{xx}+K|u|^{p-2}u+\lambda|u_t|^{q-2}u_t=F(x,t), \,\, 0 < x < 1, \,\, 0 < t < T,\\ \mu(t)u_x(0,t)=K_0u(0,t)+\int_0^tk(t-s)u(0,s)ds+g(t),\\ -\mu(t)u_x(1,t)=K_1u(1,t)+\lambda_1|u_t(1,t)|^{\alpha-2}u_t(1,t),\\ u(x,0)=\tilde{u}_0(x),\,\, u_t(x,0)=\tilde{u}_1(x),\end{cases}$$ where $p,q,\alpha \geq 2$; $K_0, K_1, K \geq 0$; $\lambda, \lambda_1 > 0$ are given constants and $\mu, F, g, k, \tilde{u}_0, \tilde{u}_1$ are given functions. First, the existence and uniqueness of a weak solution are proved by using the Galerkin method. Next, with $\alpha =2$, we obtain an asymptotic expansion of the solution up to order $N$ in two small parameters $\lambda, \lambda_1$ with error $(\sqrt{\lambda^2+\lambda_1^2})^{N+\frac 12}$.
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SOLVABILITY OF A SYSTEM OF DUAL INTEGRAL EQUATIONS OF A MIXED BOUNDARY VALUE PROBLEM FOR THE BIHARMONIC EQUATION IN A STRIP NGUYEN VAN NGOC, NGUYEN THI NGAN PDF375-396
▸ Abstract
The aim of the present work is to consider a mixed boundary value problem of the biharmonic equation in a strip. The problem may be interpreted as a deflection surface of a strip plate with the edges $y = 0$, $y = h$ having clamped conditions on intervals $|x| \leq 6$ a and hinged support conditions for $|x| > a$. Using the Fourier transform, the problem is reduced to studying a system of dual integral equations on the edges of the strip. The uniqueness and existence theorems of a solution of the system of dual integral equations are established in appropriate Sobolev spaces. A method for reducing the system of dual integral equations to an infinite system of linear algebraic equations is also proposed. -
ASYMPTOTIC PROPERTIES OF SOLUTIONS OF OPERATOR EQUATIONS VU QUOC PHONG PDF397-417
▸ Abstract
We study properties of solutions of operator equations $\mathcal Du−\mathcal Bu = f\,\, (∗)$ where $\mathcal D$ is the generator of an isometric group $V(t)$ on a Banach space $\mathcal F$ and $\mathcal B$ is a closed operator commuting with $\mathcal D$. We introduce the equation spectrum $\Sigma$ and prove that if $f$ is an almost periodic element (with respect to the group $V(t)$) and $\Sigma$ is countable, then any solution $u$ of $(*)$ is almost periodic, provided either $\mathcal F \not\supset c_0$ or $u$ is totally ergodic. The presented approach when applied to functional-differential equations gives spectral criteria of almost periodicity of bounded uniformly continuous solutions. The discrete version of the results, with applications to properties of solutions of functionaldifference equations, also is described.
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A NEW REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF THE HELMHOLTZ EQUATION WITH NONHOMOGENEOUS CAUCHY DATA PHAM HOANG QUAN, PHAN TRUNG HIEU PDF419-430
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In this paper, we investigate the Cauchy problem for the Helmholtz equation in the infinite strip $\{(x, y) \,\,|\,\, x \in \mathbb R,\,\, 0 < y < 1\}$ with nonhomogeneous Cauchy data given at $y = 0$. The problem is severely ill-posed. We shall use the Fourier transform to get an integral equation and give a regularized solution by directly perturbing this equation in combination with truncating high frequencies. The error estimate between the regularization solution and the exact solution is given. Finally, a numerical example shows the effectiveness of the proposed method.
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RANDOM MATRICES: LOCALIZATION OF THE EIGENVALUES AND THE NECESSITY OF FOUR MOMENTS TERENCE TAO, VAN VU PDF431-449
▸ Abstract
Consider the eigenvalues $\lambda_i(M_n)$ (in increasing order) of a random Hermitian matrix $M_n$ whose upper-triangular entries are independent with mean zero and variance one, and are exponentially decaying. By Wigner’s semicircular law, one expects that $\lambda_i(M_n)$ concentrates around $\gamma_i\sqrt{n}$, where $\int_{-\infty}^{\gamma_i}\rho_{sc}(x)dx = \frac{i}{n}$ and $\rho_{sc}$ is the semicircular function.
In this paper, we show that if the entries have vanishing third moment, then for all $1 \leq i \leq n$ $$\mathbf{E}|\lambda_i(M_n) - \sqrt{n}\gamma_i|^2 = O(\min(n^{−c} \min(i, n + 1 - i)^{−2/3} n^{2/3}, n^{1/{3+\varepsilon}})),$$ for some absolute constant $c > 0$ and any absolute constant $\varepsilon > 0$. In particular, for the eigenvalues in the bulk ($\min\{i, n - i\} = \Theta(n))$, $$\mathbf{E}|\lambda_i(M_n) − \sqrt{n}\gamma_i|^2 = O(n^{−c}).$$ A similar result is achieved for the rate of convergence.
\indent As a corollary, we show that the four moment condition in the Four Moment Theorem is necessary, in the sense that if one allows the fourth moment to change (while keeping the first three moments fixed), then the mean of $\lambda_i(M_n)$ changes by an amount comparable to $n^{−1/2}$ on the average. We make a precise conjecture about how the expectation of the eigenvalues vary with the fourth moment.
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REMARKS ON NONCLASSICAL SHOCK WAVES FOR VAN DER WAALS FLUIDS MAI DUC THANH PDF451-468
▸ Abstract
We consider the Riemann problem for isentropic van der Waals fluids, where LeFloch’s concept of nonclassical shock waves is studied. Motivated by [3, 32], we study all types of nonclassical shocks including shocks that correspond to the traveling waves of an autonomous system of differential equations with four equilibria resulted from a diffusive-dispersive model. Moreover, the range of kinetic relation is extended to the whole admissible nonclassical shock set. Corresponding to each of the two inflection points of the pressure function we can define a kinetic function. The kinetic functions may not be monotone. It is very interesting that there could be nonclassical shocks that satisfy both a kinetic relation and the Lax shock inequalities. It turns out that nonclassical Riemann solutions may form a two-parameter family of solutions. This raises an open question for the study on the selection of a unique nonclassical solution.
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THERMAL INFRARED TECHNIQUE FOR LANDMINE DETECTION: MATHEMATICAL FORMULATION AND METHODS NGUYEN TRUNG THANH, DINH NHO HÀO, HICHEM SAHLI PDF469-504
▸ Abstract
This paper introduces a mathematical formulation of the problem of detection and characterization of shallowly buried landmines (more generally, buried objects) using passive thermal infrared technique. The problem consists of two steps. In the first step, referred to as thermal modeling which aims at predicting the soil temperature provided by the thermal properties of the soil and the buried objects, a parabolic partial differential equation based model is formulated. The proposed model is validated using experimental data. For solving the model, a splitting finite difference scheme is used. In the second step, referred to as inverse problem setting for landmine detection, the forward thermal model and acquired infrared images are used to detect the presence of buried objects and characterize them based on the estimation of their thermal and geometrical properties. Mathematically, this inverse problem is stated as the estimation of a piecewise constant coefficient of the heat transfer equation. To reduce the ill-posedness of this problem, which is due to the lack of spatial information in the measured data, we make use of a parametrization of the coefficient which needs only a small number of unknowns. The problem is then solved by gradient-based optimization methods. Numerical results both validate the proposed thermal model and illustrate the performance of the suggested algorithm for the inverse problem.
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REGULARIZATION OF A BACKWARD HEAT TRANSFER PROBLEM WITH A NONLINEAR SOURCE DANG DUC TRONG, NGUYEN MINH DIEN PDF505-515
▸ Abstract
We consider the problem of finding, from the final data $u(x, T)$, the function $u$ satisfying $$u_t - u_{xx} = f(x, t, u(x, t), u_x(x, t)),\quad (x, t) \in \mathbb R \times (0, T).$$ The problem is ill-posed and we shall use the Fourier transform to get a nonlinear integral equation in the frequency space. By truncating high frequencies, we give a regularized solution. Error estimates are given.
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DETERMINATION OF THE INTERNAL HEAT SOURCE FOR A HALF-BURIED ROD VU KIM TUAN, FADHEL AL-MUSALLAM PDF517-535
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The paper considers a one-dimensional heat transfer in a perfectly lateral insulated uniform rod, which is partially exposed for measurement at one end, and proposes an algorithm that uniquely recovers the heat source proportional coefficient, the coefficient of convective heat transfer from a countable set of temperature recordings at one end of the rod. The length of the rod can be recovered by just one single measurement.
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ATYPICAL VALUES AT INFINITY OF POLYNOMIAL AND RATIONAL FUNCTIONS ON AN ALGEBRAIC SURFACE IN $\mathbb R^n$ HA HUY VUI, NGUYEN THI THAO PDF537-553
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We consider polynomial and rational functions defined on an algebraic surface in $\mathbb R^n$ and we characterize values which are atypical at infinity due to singularities at infinity. This characterization gives an upper bound of the number of atypical values at infinity.