arXiv subjects
Publications and source records attributed to Dang Vu Giang.
We prove that if an integral equation has a positive solution then all complex roots of the famous Riemann zeta function are distinct and having the real part 1/2. We also prove that the minimal distance between two consecutive real simple roots of the function $Ξ$ in $(0,T)$ is less than $\frac 1{A\ln T}$.
Using the Galois theory over function field, and the holomorphy of algebroids defined via irreducible polynomial at singular points, we prove the injectivity of any kellerian mapping. The famous Jacobian conjecture is true.
We prove among other things that the omega-limit set of a bounded solution of a Hamilton system \[\left\{\begin{aligned} & \mathbf{\dot{p}}=\frac{\partial H}{\partial \mathbf{q}} & \mathbf{\dot{q}}=-\frac{\partial H}{\partial \mathbf{p}} \\ \end{aligned} \right.\] is containing a full-time solution so there are the limits of $\frac 1t\int_0^t {\mathbf p}(s)ds$ and $\frac 1t\int_0^t {\mathbf q}(s)ds$ as $t\to\infty$ for any bounded solution $(\mathbf {p,q})$ of the Hamilton system. These limits are stationary points of the Hamilton system so if a Hamilton system has no stationary point then every solution of this system is unbounded.
We study the asymptotic behavior of a bounded solution of an inhomogeneous delay linear difference equation in a Banach space by using the spectrum of bounded sequences. We get a significant extension of excellent results in [1]. A new simple proof is also found for the famous Gelfand spectral radius theorem. Moreover, among other things we prove that if the spectrum of a bounded sequence $\{x_n\}_n$ is finite then $x_n=c_1\vartheta_1^n+c_2\vartheta_2^n+\cdots+c_k\vartheta_k^n+o(1)$ as $n\to\infty$ where $|\vartheta_1|=|\vartheta_2|=\cdots=|\vartheta_k|=1$.
Several interesting formulas concerning finite Hilbert transform and logarithmic integrals are proved with application in determining equilibrium measures, planar limits of analytic random matrix models with $1-$cut potential and solving singular integral equations.
First, we study the linear equations in general. Second, we focus our attention in periodic sequences over finite fields and de Bruijn directed graph.
We are interested in Beurling spectrum of $\mathbb X-$valued functions with application in functional delay differential equations.
We are interested in computing the spectral measure of Laplacean operators in Paley-Wiener space, the Hilbert space of all square integrable functions having Fourier transforms supported in a compact set $K$, the closure of an open bounded set in $\R^N$. I is well-known that every differential operator is bounded in this space. Among others, we will prove that the spectrum of Laplace operator is the set $$\{-|x|^2: x\in K\}.$$
Using the Gröbner basis of an ideal generated by a family of polynomials we prove that every planar graph is 4-colorable. Here we also use the fact that the complete graph of 5 vertices is not included in any planar graph.
We are interested in Moebius function and related topics!
First, we systemize ealier results the uniform persistence for discrete model $A_{n+1}=A_nF(A_{n-m})$ of population growth, where $F:(0,\infty)\to(0,\infty)$ is continuous and strictly decreasing. Second, we investigation the effect of delay $m$ when $F$ is not monotone. We are mainly using $ω$-limit set of persistent solution, which is discussed in more general by P. Walters, 1982.