On Global Hypoellipticity

We consider a first order linear partial differential operator of principal type on a closed connected orientable two-dimensional manifold sending sections of one complex line bundle to sections of another. We prove that the assumption of global hypoellipticity of the operator implies a relation between the degrees of the line bundles and the Euler characteristic of the manifold.     

The Volterra Property of Some Problems with the Bitsadze-Samarskii-Type Conditions for a Mixed Parabolic-Hyperbolic Equation

We consider problems with the Bitsadze-Samarskii-type conditions for a mixed parabolic-hyperbolic equation with noncharacteristic type change curve. We prove theorems on the unique existence of regular and strong solutions and the Volterra property for the problems under consideration.Original Russian Text Copyright © 2005 Berdyshev A. S.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 500–510, May–June, 2005.     

Solvability of F 4 Quantum Integrable Systems

It is shown that the F ₄ rational and trigonometric integrable systems are exactly solvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions are obtained by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are a certain invariants of the F ₄ Weyl group. Both Hamiltonians preserve the same (minimal) flag of spaces of polynomials, which is found explicitly.     

THE SOLVABILITY FOR HEAT CONDUCTIVE PROBLEM WITH BOUNDARY PERTURBATION

In this paper, the heat conductive problem with boundary perturbations is considered. Using the solvability of the problem, two eigenfunctions corresponding to two eigenvalues for the nm-th modular form are obtained.     

一类三阶奇摄动特征值问题

将带有边界条件的三阶非齐次线性方程的可解性条件应用到退化特征值问题上,得出了奇摄动问题的解的渐近表示式.     

Finite Difference Method for Reaction-Diffusion Equation with Nonlocal Boundary Conditions

In this paper, we present a numerical approach to a class of nonlinear reaction-diffusion equations with nonlocal Robin type boundary conditions by finite difference methods. A second-order accurate difference scheme is derived by the method of reduction of order. Moreover, we prove that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. A simple numerical example is given to illustrate the efficiency of the proposed method.     

变系数的一般型发展方程的色散估计

本文研究的是带变系数的一般型线性发展方程.首先建立了其基本解的一系列色散估计:Kato光滑型估计,极大函数估计及Strichartz估计.最后应用这些估计研究了一些非自治非线性色散方程的初值问题在H~s(R)空间中的局部可解性.     

SOLVABILITY FOR NONLINEAR ELLIPTIC EQUATION WITH BOUNDARY PERTURBATION

The solvability of nonlinear elliptic equation with boundary perturbation is consid- ered.The perturbed solution of original problem is obtained and the uniformly valid expansion of solution is proved.     

Global Solvability and Simultaneously Approximable Vectors

We consider a class of sum of squares operators on a torus and we prove that global solvability is equivalent to an algebraic condition involving simultaneously approximable vectors.     

On classification of four-dimensional nilpotent Leibniz algebras

Leibniz algebras are certain generalization of Lie algebras. In this paper, we give the classification of four-dimensional non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of bilinear forms and some other techniques to obtain our result.