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Truong Trung Tuyen

Publications and source records attributed to Truong Trung Tuyen.

9 recordsLinked to original sources
arXiv2008

Determination of the body force of a two-dimensional isotropic elastic body

Let $Ω$ represent a two$-$dimensional isotropic elastic body. We consider the problem of determining the body force $F$ whose form $ϕ(t)(f_1(x),f_2(x))$ with $ϕ$ be given inexactly. The problem is nonlinear and ill-posed. Using the Fourier transform, the methods of Tikhonov's regularization and truncated integration, we construct a regularized solution from the data given inexactly and derive the explicitly error estimate. Numerical part is given

math.AP↗
arXiv2008

Determine the source term of a two-dimensional heat equation

Let $Ω$ be a two-dimensional heat conduction body. We consider the problem of determining the heat source $F(x,t)=φ(t)f(x,y)$ with $φ$ be given inexactly and $f$ be unknown. The problem is nonlinear and ill-posed. By a specific form of Fourier transforms, we shall show that the heat source is determined uniquely by the minimum boundary condition and the temperature distribution in $Ω$ at the initial time $t=0$ and at the final time $t=1$. Using the methods of Tikhonov's regularization and truncated integration, we construct the regularized solutions. Numerical part is given.

math.AP↗
arXiv2007

Sets non-thin at \infty in \Bbb C ^m, and the growth of sequences of entire functions of genus zero

In this paper we define the notion of non-thin at $\infty$ as follows: Let $E$ be a subset of $\Bbb C^m$. For any $R>0$ define $E_R=E\cap \{z\in \Bbb C ^m :|z|\leq R\}$. We say that $E$ is non-thin at $\infty$ if \lim_{R\to\infty}V_{E_R}(z)=0 for all $z\in \Bbb C^m$, where $V_E$ is the pluricomplex Green function of $E$. This definition of non-thin at $\infty$ has good properties: If $E\subset \Bbb C^m$ is non-thin at $\infty$ and $A$ is pluripolar then $E\backslash A$ is non-thin at $\infty$, if $E\subset \Bbb C^m$ and $F\subset \Bbb C^n$ are closed sets non-thin at $\infty$ then $E\times F\subset \Bbb C^m\times \Bbb C^n$ is non-thin at $\infty$ (see Lemma \ref{Lem1}). Then we explore the properties of non-thin at $\infty$ sets and apply this to extend the results in \cite{mul-yav} and \cite{trong-tuyen}.

math.CV↗
arXiv2007

The growth at infinity of a sequence of entire functions of bounded orders

In this paper we shall consider the growth at infinity of a sequence $(P_n)$ of entire functions of bounded orders. Our results extend the results in \cite{trong-tuyen2} for the growth of entire functions of genus zero. Given a sequence of entire functions of bounded orders $P_n(z)$, we found a nearly optimal condition, given in terms of zeros of $P_n$, for which $(k_n)$ that we have \begin{eqnarray*} \limsup_{n\to\infty}|P_n(z)|^{1/k_n}\leq 1 \end{eqnarray*} for all $z\in \mathbb C$ (see Theorem \ref{theo5}). Exploring the growth of a sequence of entire functions of bounded orders lead naturally to an extremal function which is similar to the Siciak's extremal function (See Section 6).

math.CV↗
arXiv2007

Error of Tikhonov's regularization for integral convolution equations

Let $ϕ$ be a nontrivial function of $L^1(\RR)$. For each $s\geq 0$ we put \begin{eqnarray*} p(s)=-\log \int_{|t|\geq s}|ϕ(t)|dt. \end{eqnarray*} If $ϕ$ satisfies \begin{equation} \lim_{s\to \infty}\frac{p(s)}{s}=\infty ,\label{170506.1} \end{equation} we obtain asymptotic estimates of the size of small-valued sets $B_ε=\{x\in\RR : |\hatϕ(x)|\leq ε, |x|\leq R_ε\}$ of Fourier transform \begin{eqnarray*} \hatϕ(x)=\int_{-\infty}^{\infty}e^{-ixt}ϕ(t)dt, x\in \RR, \end{eqnarray*} in terms of $p(s)$ or in terms of its Young dual function \begin{eqnarray*} p^{*}(t)=\sup_{s\geq 0}[st-p(s)], t\geq 0. \end{eqnarray*} Applying these results, we give an explicit estimate for the error of Tikhonov's regularization for the solution $f$ of the integral convolution equation \begin{eqnarray*} \int_{-\infty}^{\infty}f(t-s)ϕ(s)ds =g(t), \end{eqnarray*} where $f,g \in L^2(\RR)$ and $ϕ$ is a nontrivial function of $L^1(\RR)$ satisfying condition (\ref{170506.1}), and $g,ϕ$ are known non-exactly. Also, our results extend some results of \cite{tld} and \cite{tqd}.

math.NA↗
arXiv2007

Approximate solutions to the Dirichlet problem for harmonic maps between hyperbolic spaces

Our main result in this paper is the following: Given $H^m, H^n$ hyperbolic spaces of dimensional $m$ and $n$ corresponding, and given a Holder function $f=(s^1,...,f^{n-1}):\partial H^m\to \partial H^n$ between geometric boundaries of $H^m$ and $H^n$. Then for each $ε>0$ there exists a harmonic map $u:H^m\to H^n$ which is continuous up to the boundary (in the sense of Euclidean) and $u|_{\partial H^m}=(f^1,...,f^{n-1},ε)$.

math.DG↗
arXiv2007

Stabiliti and Identity of analytic functions of Hardy classes

Let $E$ be a subset of the unit disc $U$ of the complex plane $\CC$. Recall that $H^p(U)$ is the space of all holomorphic functions $g$ on $U$ for which $\|g\|_{H^p}$ $<$ $\infty$. Put \begin{equation} C_p(ε, R) = \sup \{\sup_{|z| \leq R}|g(z)|: g\in H^p, \|g\|_p\leq 1, |g(ζ)| \leq ε\forall ζ\in E\}, \end{equation} for positive $ε$ and $R$ in $(0, 1)$. It can be seen that $C_p(ε, R)$ is bounded from above by $(1-R^2)^{-1/p}$. \begin{theorem} If $\bar E$ $\subset$ $U$ then there exists $ε_0>0$ such that for $0<ε<ε_0$ there is correspondingly a finite Blaschke product $B_ε(z)$ whose zeros are in $\bar E$ satisfying \begin{eqnarray*} \max_{|z|\leq R}|B_ε(z)|\leq C_p(ε, R)\leq C\max_{|z|\leq R}|B_ε(z)|^{1/2}, \end{eqnarray*} where $C$ is a positive constant that depends only on $R$ and $p$. Moreover we have \begin{eqnarray*} \sup_{z\in E}|B_ε(z)|\leq ε. \end{eqnarray*} \end{theorem}

math.CV↗
arXiv2006

The growth of entire functions of genus zero

In this paper we shall consider the assymptotic growth of $|P_n(z)|^{1/k_n}$ where $P_n(z)$ is a sequence of entire functions of genus zero. Our results extend a result of J. Muller and A. Yavrian. We shall prove that if the sequence of entire functions has a geometric growth at each point in a set $E$ being non-thin at $\infty$ then it has a geometric growth in $\CC$ also. Moreover, if $E$ has some more properties, a similar result also holds for a more general kind of growth. Even in the case where $P_n$ are polynomials, our results are new in the sense that it does not require $k_n\succeq deg(P_n)$ as usually required.

math.CV↗