Radial equivalence for the two basic nonlinear degenerate diffusion equations

In this paper we prove that there exists an explicit correspondence between the radially symmetric solutions of two well-known models of nonlinear diffusion, the porous medium equation and the p-Laplacian equation. We establish exact correspondence formulas between these solutions. We also study in detail the application of the results in the important case of self-similar solutions. In particular, we derive the existence of new self-similar solutions for the evolution p-Laplacian equation.     

Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type

We establish symmetrization results for the solutions of the linear fractional diffusion equation _∂tu+(−Δ)^σ/2u=f${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma /2}u=f$ and its elliptic counterpart hv+(−Δ)^σ/2v=f$hv+{(-\mathrm{\Delta })}^{\sigma /2}v=f$, h>0$h>0$, using the concept of comparison of concentrations. The results extend to the nonlinear version, _∂tu+(−Δ)^σ/2A(u)=f${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma /2}A(u)=f$, but only when the nondecreasing function A:_R+→_R+$A:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is concave. In the elliptic case, complete symmetrization results are proved for B(v)+(−Δ)^σ/2v=f$B(v)+{(-\mathrm{\Delta })}^{\sigma /2}v=f$ when B(v)$B(v)$ is a convex nonnegative function for v>0$v>0$ with B(0)=0$B(0)=0$, and partial results hold when B is concave. Remarkable counterexamples are constructed for the parabolic equation when A is convex, resp. for the elliptic equation when B   is concave. Such counterexamples do not exist in the standard diffusion case σ=2$\sigma =2$.     

Improving the efficiency of DC global optimization methods by improving the DC representation of the objective function

There are infinitely many ways of representing a d.c. function as a difference of convex functions. In this paper we analyze how the computational efficiency of a d.c.optimization algorithm depends on the representation we choose for the objective function, and we address the problem of characterizing and obtaining a computationally optimal representation. We introduce some theoretical concepts which are necessary for this analysis and report some numerical experiments.      

Weakly closed Jordan ideals in nest algebras on Banach spaces

We show that weakly closed Jordan ideals in nest algebras on Banach spaces are associative ideals. The decomposability of finite-rank operators in Jordan ideals and the commutants of bimodules are also investigated. Author’s address: J. Li and F. Lu, Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China This research was supported by NNSFC (No. 10771154) and PNSFJ (NO. BK2007049).     

From Dyck Paths to Standard Young Tableaux

Annals of Combinatorics - We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we...     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

On the non-commutative fractional Wishart process

We investigate the process of eigenvalues of a fractional Wishart process defined by $N=B^{?}B$, where B is the matrix fractional Brownian motion recently studied in [18]. Using stochastic calculus with respect to the Young integral we show that, with probability one, the eigenvalues do not collide at any time. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter $H\in (1/2,1)$, we derive a stochastic differential equation in the Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally, a functional limit theorem for the empirical measure-valued process of eigenvalues of a fractional Wishart process is obtained. The limit is characterized and referred to as the non-commutative fractional Wishart process, which constitutes the family of fractional dilations of the free Poisson distribution.     

一类非线性发展方程的精确解

利用hirota双线性法,得到(3+1)维孤子方程、(3+1)维KP-Boussinesq方程、(2+1)维修正Caudrey-Dodd-Gibbon-Kotera-S awada方程、Hirota-Satsuma浅水波方程的精确解,并做出一部分解的图形,进一步研究解的结构和性质.     

Strategic simulation models as a new methodological approach: an application to information technologies integration,lean/just-in-time and lead-time

Central European Journal of Operations Research - Hypothesis contrast using statistical models with Structural Equations is a technique widely used in Supply Chain Management research. However,...     

A three level linearized compact difference scheme for the Cahn-Hilliard equation

This article is devoted to the study of high order accuracy difference methods for the Cahn-Hilliard equation.A three level linearized compact difference scheme is derived.The unique solvability and unconditional convergence of the difference solution are proved.The convergence order is O(τ 2 + h 4 ) in the maximum norm.The mass conservation and the non-increase of the total energy are also verified.Some numerical examples are given to demonstrate the theoretical results.