Dirac Operators on Hypersurfaces of Manifolds with Negative Scalar Curvature

We give a sharp extrinsic lower bound for the first eigenvaluesof the intrinsic Dirac operator of certain hypersurfaces boundinga compact domain in a spin manifold of negative scalar curvature.Limiting-cases are characterized by the existence, on the domain,of imaginary Killing spinors. Some geometrical applications, as anAlexandrov type theorem, are given.     

Structure of Regular Solutions of the Three-Body Schrödinger Equation near the Pair Impact Point

We investigate the six-dimensional Schrödinger equation for a three-body system with central pair interactions of a more general form than Coulomb interactions. Regular general and special physical solutions of this equation are represented by infinite asymptotic series in integer powers of the distance between two particles and in the sought functions of the other three-body coordinates. Constructing such functions in angular bases composed of spherical and bispherical harmonics or symmetrized Wigner D-functions is reduced to solving simple recursive algebraic equations. For projections of physical solutions on the angular bases functions, we derive boundary conditions at the pair impact point.     

On a Construction of Fast Direct Solvers

Fast direct solvers for the Poisson equation with homogeneous Dirichlet and Neumann boundary conditions on special triangles and tetrahedra are constructed. The domain given is extended by symmetrization or skew symmetrization onto a rectangle or a rectangular parallelepiped and a fast direct solver is used there. All extendable domains are found. Eigenproblems are also considered.     

On the Global Stability of Solutions of Moment Systems in Nonequilibrium Thermodynamics

In this paper, we study the linearization of the Cauchy problem and the mixed problem for the system of Grad--Hermite moments in nonequilibrium thermodynamics in the neighborhood of the equilibrium state. Stability conditions for solutions of the Cauchy problem are proved as a generalization of the classical Hermite--Biller theorem on stable polynomials. For the mixed problem, we prove an analog of the Vishik--Lyusternik theorem on small singular perturbations of general elliptic problems. The last observation allows us to introduce the Shapiro--Lopatinskii condition, which implies the well-posedness of the mixed problem.     

On Solvability of a Boundary Value Problem for Singular Integral Equations

This paper deals with the solvability of boundary value problems for singular integral equations of the form (i)-(ii).By an algebraic method we reduce the problem (i)-(ii) to a system of linear algebraic equations which gives all solutions in a closed form.AMS Subject Classification: 47G05, 45GO5, 45E05     

Blind Channel Identification Based on Noisy Observation by Stochastic Approximation Method

A stochastic approximation algorithm for estimating multichannel coefficients is proposed, and the estimate is proved to converge to the true parameters a.s. up-to a constant scaling factor. The estimate is updated after receiving each new observation, so the output data need not be collected in advance. The input signal is allowed to be dependent and the observation is allowed to be corrupted by noise, but no noise statistics are used in the estimation algorithm.     

Galerkin Approximations for the Linear Parabolic Equation with the Third Boundary Condition

We solve a linear parabolic equation in ^d , d 1, with the third nonhomogeneous boundary condition using the finite element method for discretization in space, and the -method for discretization in time. The convergence of both, the semidiscrete approximations and the fully discretized ones, is analysed. The proofs are based on a generalization of the idea of the elliptic projection. The rate of convergence is derived also for variable time step-sizes.     

Degenerate nonlinear hyperbolic conservation laws: Non classical geometrical theory

Kinetic selection principles have been shown to be useful and physically reasonable in nonlinear hyperbolic problems with large amplitude phase transitions. We refer to Abeyaratne and Knowles, [A-K], for discussion on the subject. They also have been used for degenerate nonlinear problems, where the genuine nonlinearity property of Lax is violated. This is the framework of Liu, [Li], in the kinetic situation of Hayes and Le Floch, [H-L], where their so-called non classical shocks may be seen as small amplitude phase transitions. Here, we describe the local geometry generated by the generic non genuinely nonlinear assumption. A geometric kinetic criterion can be used to select indeterminate simple waves and obtain the well-posedness of the local Riemann problem. A particular case is the entropic kinetic criterion of Hayes and Le Floch.

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Sunto Principi di selezione cinetica sono stati mostrati utili e fisicamente ragionevoli in problemi iperbolici con transizioni di fase di grande ampiezza. Ci riferiamo ad Abeyaratne e Knowles, [A-K], per discussioni sull'argomento. Detti principi sono stati usati per problemi nonlineari degeneri, dove la proprietà di genuina nonlinearità di Lax è violata. Questo è la prospettiva di Liu, [Li], nella situazione cinetica di Hayes e Le Floch, [H-L], dove i loro così detti urti non classici possono essere considerati come transizioni di fase di piccola ampiezza. Qui, descriviamo la geometria locale generata dall'ipotesi generica genuinamente nonlineare. Un criterio di geometria cinetica può essere usato per selezionare onde semplici indeterminate ed ottenere la buona posizione per il problema locale di Riemann. Un caso particolare è il criterio di entropia cinetico di Hayes e Le Floc.

     

Invariant Euler–Lagrange Equations and the Invariant Variational Bicomplex

In this paper, we derive an explicit group-invariant formula for the Euler–Lagrange equations associated with an invariant variational problem. The method relies on a group-invariant version of the variational bicomplex induced by a general equivariant moving frame construction, and is of independent interest.     

矩母函数:拟对称概率测度

本文用凸函数的方法研究了概率测度的矩母函数和由C.J.Stone提出的拟对称性,并刻画了Martin边界的法向量.拟对称性在随机游动的比例极限定理中是一个重要的概念.这些结果可应用于Levy过程的研究.