On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Poisson Semigroup Associated with Elliptic Systems

We study the infinitesimal generator of the Poisson semigroup in L ^p associated with homogeneous, second-order, strongly elliptic systems with constant complex coefficients in the upper-half space, which is proved to be the Dirichlet-to-Normal mapping in this setting. Also, its domain is identified as the linear subspace of the L ^p -based Sobolev space of order one on the boundary of the upper-half space consisting of functions for which the Regularity problem is solvable. Moreover, for a class of systems containing the Lamé system, as well as all second-order, scalar elliptic operators, with constant complex coefficients, the action of the infinitesimal generator is explicitly described in terms of singular integral operators whose kernels involve first-order derivatives of the canonical fundamental solution of the given system. Furthermore, arbitrary powers of the infinitesimal generator of the said Poisson semigroup are also described in terms of higher order Sobolev spaces and a higher order Regularity problem for the system in question. Finally, we indicate how our techniques may be adapted to treat the case of higher order systems in graph Lipschitz domains.     

Stability of solutions to differential equations with several variable delays. II

We consider a class of scalar linear differential equations with several variable delays and constant coefficients. We treat coefficients and maximum admissible values of delays as parameters that define a family of equations from the class under consideration. We study domains in the parameter space, where fundamental solutions of all equations of the family are uniformly or exponentially stable and have a fixed sign. We establish explicit necessary and sufficient conditions for the stability and sign-definiteness of the equations family.     

Numerical solution of nonlinear inverse coefficient problems for ordinary differential equations

Parametric identification for a class of nonlinear objects with lumped parameters described by systems of ordinary differential equations is studied. The problem is to recover the coefficients of a dynamical system depending on the phase state. For that purpose, the phase space is subdivided into a finite set of subsets or zones in which the coefficients are assumed to be constant or linear functions of state. Once the coefficients in such a form are obtained, interpolation and approximation can be used to represent the coefficients as functions of the phase variables.     

Realizations of Countable Groups as Fundamental Groups of Compacta

It has been an open question for a long time whether every countable group can be realized as a fundamental group of a compact metric space. Such realizations are not hard to obtain for compact or metric spaces but the combination of both properties turn out to be quite restrictive for the fundamental group. The problem has been studied by many topologists (including Cannon and Conner) but the solution has not been found. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of ${\mathbb{R}^4}$ . According to the theorem of Shelah [10] such space can not be locally path connected if the group is not finitely generated. The theorem is proved by an explicit construction of an appropriate space X _G for every countable group G.     

p-Adic elliptic quadratic forms,parabolic-type pseudodifferential equations with variable coefficients and Markov processes

In this article we study the Cauchy problem for a new class of parabolic-type pseudodifferential equations with variable coefficients for which the fundamental solutions are transition density functions of Markov processes in the four dimensional vector space over the field of p-adic numbers.     

具奇系数发展型p-Laplace不等方程整体解的不存在性

考虑两类带奇性系数发展型p-Laplace不等方程整体非负解的不存在性,也即要证明某些拟线性抛物型方程的非线性Liouville定理．通过先验估计,证明了整体解在某个带参数的函数空间中的不存在性．当只考虑连续整体解时,可选取不带参数的函数空间并证明整体解在其中的不存在性．     

The problem of lacunas and analysis on root systems

A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens' principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and Gårding (1970-73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-Gårding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.

     

Fuzzy Topological Representation Theory For D_(01)

In this paper, we shall introduce the concepts of fuzzy prime ideal, fuzzy Stone space, fuzzy fundamental set, etc. And the representations of a distributive lattice with 0, 1 (D_(01) denotes the equational class of such lattices) by fuzzy sets will be investigated and many useful results are obtained.     

Fundamental Solutions of Some Linear Operator Equations and Applications

A concept of a fundamental solution is introduced for linear operator equations given in some functional spaces. In the case where this fundamental solution does not exist, the representation of the solution is found by the concept of a generalized fundamental solution, which is introduced for operators with nontrivial and generally infinite-dimensional kernels. The fundamental and generalized fundamental solutions are also investigated for a class of Fredholm-type operator equations. Some applications are given for one-dimensional generally nonlocal hyperbolic problems with trivial, finite- and infinite-dimensional kernels. The fundamental and generalized fundamental solutions of such problems are constructed as particular solutions of a system of integral equations or an integral equation. These fundamental solutions become meaningful in a general case when the coefficients are generally nonsmooth functions satisfying only some conditions such as p-integrablity and boundedness.     

Some regularity properties of diffusion processes on abstract Wiener space

Diffusion processes on an abstract Wiener space are constructed from fundamental solutions of second-order parabolic equations with variable coefficients. The transition probabilities of such processes are compared with those of the Wiener process, and continuity of sample paths is established. Several operator semigroups associated with the processes are defined (one locally), and some regularity properties of these semigroups are established.     

Hypersurfaces with constant inner curvature of the second fundamental form,and the non-rigidity of the sphere

We classify the hypersurfaces of revolution in euclidean space whose second fundamental form defines an abstract pseudo-Riemannian metric of constant sectional curvature. In particular we find such piecewise analytic hypersurfaces of class C ² where the second fundamental form defines a complete space of constant positive, zero, or negative curvature. Among them there are closed convex hypersurfaces distinct from spheres, in contrast to a theorem of R. Schneider (Proc. AMS 35, 230–233, (1972)) saying that such a hypersurface of class C ⁴ has to be a round sphere. In particular, the sphere is not II-rigid in the class of all convex C ² -hypersurfaces. Received 11 October 1994; in final form 26 April 1995     

Extrinsically immersed symplectic symmetric spaces

Let (V, Ω) be a symplectic vector space and let \({\phi : M \rightarrow V}\) be a symplectic immersion. We show that \({\phi(M) \subset V}\) is locally an extrinsic symplectic symmetric space (e.s.s.s.) in the sense of Cahen et al. (J Geom Phys 59(4):409f?b-425, 2009) if and only if the second fundamental form of \({\phi}\) is parallel. Furthermore, we show that any symmetric space, which admits an immersion as an e.s.s.s., also admits a full such immersion, i.e., such that \({\phi(M)}\) is not contained in a proper affine subspace of V, and this immersion is unique up to affine equivalence. Moreover, we show that any extrinsic symplectic immersion of M factors through to the full one by a symplectic reduction of the ambient space. In particular, this shows that the full immersion is characterized by having an ambient space V of minimal dimension.     

Towards strong Banach property (T) for SL(3,ℝ)

We prove that SL(3, ?) has Strong Banach property (T) in Lafforgue’s sense with respect to the Banach spaces that are θ > 0 interpolation spaces (for the complex interpolation method) between an arbitrary Banach space and a Banach space with sufficiently good type and cotype. As a consequence, every action of SL(3, ?) or its lattices by affine isometries on such a Banach space X has a fixed point, and the expanders contructed from SL(3, ?) do not admit a coarse embedding into X. We also prove a quantitative decay of matrix coefficients (Howe-Moore property) for representations with small exponential growth of SL(3, ?) on X.     

Randomly generated polytopes for testing mathematical programming algorithms

Randomly generated polytopes are used frequently to test and compare algorithms for a variety of mathematical programming problems. These polytopes are constructed by generating linear inequality constraints with coefficients drawn independently from a distribution such as the uniform or the normal. It is noted that this class of ‘random’ polytopes has a special property: the angles between the hyperplanes, though dependent on the specific distribution used, tend to be equal when the dimension of the space increases. Obviously this structure of ‘random’ polytopes may bias test results.     

Analyticity of Semigroups Generated by a Class of Differential Operators with Matrix Coefficients and Interface

In this work we are interested in studying a class of mixed differential operators with matrix coefficients and the matching interface conditions, the basic question treated in this paper is that such mixed operators generate an analytic semigroup on an appropriate Hilbert space X.     

Hypersurfaces with constant inner curvature of the second fundamental form,and the non-rigidity of the sphere

We classify the hypersurfaces of revolution in euclidean space whose second fundamental form defines an abstract pseudo-Riemannian metric of constant sectional curvature. In particular we find such piecewise analytic hypersurfaces of classC ² where the second fundamental form defines a complete space of constant positive, zero, or negative curvature. Among them there are closed convex hypersurfaces distinct from spheres, in contrast to a theorem of R. Schneider (Proc. AMS 35, 230–233, (1972)) saying that such a hypersurface of classC ⁴ has to be a round sphere. In particular, the sphere is notII-rigid in the class of all convexC ²-hypersurfaces.     

一类具有离散谱的对称微分算子

研究了一类2n-阶线性微分算子在加权Hilbert空间中的谱.通过对加权Sobolev空间H     

Reliable Solution of Parabolic Obstacle Problems with Respect to Uncertain Data

A class of parabolic initial-boundary value problems is considered, where admissible coefficients are given in certain intervals. We are looking for maximal values of the solution with respect to the set of admissible coefficients. We give the abstract general scheme, proposing how to solve such problems with uncertain data. We formulate a general maximization problem and prove its solvability, provided all fundamental assumptions are fulfilled. We apply the theory to certain Fourier obstacle type maximization problem.     

Stochastic optimization theory in Hilbert spaces—1

We study a class of stochastic optimization problems in which the state as well as the observation spaces are permitted to be (Hilbert spaces) of non-finite dimension. Although there have been previous attempts in the Hilbert space setting, our results, techniques, as well as applications, are totally different. We initiate the use of Gauss measure on a Hilbert space even though it is only finitely additive; and an associated theory of white noise, in contrast to the Wiener process theory, which is novel even in the finite dimensional case. We only treat time-invariant systems, but no strong ellipticity or coercivity conditions are used; we exploit the theory of semigroups of operators in contrast to the Lions-Magenes theory. A key result involves a far-reaching generalization of the Factorization theorem of Krein. We apply the results to the problem of boundary observation and control for partial differential equations. By the creation of a special state space, we can apply the theory to problems in which the state equations are finitedimensional but the noise does not have a rational spectrum. In a final section, we present a stochastic theory for inverse problems (System Identification) in the Hilbert space setting. The basic theoretical problem is the calculation of R-N derivatives for finitely additive measures. A fundamental result concerns Identifiability; in particular the identifiability of diffusion coefficients from boundary data is treated here for the first time.     

An application of the Fourier method of separation of variables to constructing exactly solvable deformations of partial differential operators

As is known, in mathematical physics there are differential operators with constant coefficients whose fundamental solutions can be constructed explicitly; such operators are said to be exactly solvable. In this paper, the problem of adding lower-order terms with variable coefficients to exactly solvable operators in such a way that the new operators (deformations) admit constructing fundamental solutions in explicit form is posed. This problem is directly related to Hadamard’s problem of describing differential operators satisfying the Huygens’ principle. On the basis of the Fourier method of separation of variables and the method of gauge-equivalent operators, an effective method for finding exactly solvable deformations depending on one variable is constructed. An application of such deformations to constructing Huygens’ differential operators associated with the cone of real symmetric positive-definite matrices is suggested. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.