POWERS OF REAL SYMMETRIC DIFFERENTIAL EXPRESSIONS WITHOUT SMOOTHNESS ASSUMPTIONS

Abstract

In recent years many authors have studied properties of powers of symmetric (formally self-adjoint) ordinary linear differential expressions. So that these powers can be formed in the classical way these authors have placed heavy smoothness assumptions on the coefficients. Here we show that no differentiability conditions whatsoever are needed on the coefficients in order to form powers of a given expression-provided these powers are formed in the quasi-differential sense rather than the classical one.     

ON THE TITCHMARSH-WEYL THEORY OF ORDINARY SYMMETRIC ODD-ORDER DIFFERENTIAL EXPRESSIONS AND A DIRECT CONVERGENCE THEOREM

Abstract

In this paper the odd-order differential equation M[y] λ wy on the interval (O,∞), associated with the symmetric differential expression M of (2k-1)st order (k ≥ 2) with w a positive weight function and λ a complex number, is shown to possess k-Titchmarsh-Weyl solutions for every non-real λ in the underlying Hilbert space L² _w(O, ∞) having identical representation for every non-real λ. In terms of these solutions the Green's function associated with the singular boundary value problem is shown to possess identical representation for all non-real λ which has been further made use of in the third-order case to establish a direct convergence eigenfunction expansion theorem. The symmetric spectral matrix appearing in the expansion theorem has been characterized in terms of the Titchmarsh-Weyl m-coefficients.     

Nonlinear functionals of wavelet expansions – adaptive reconstruction and fast evaluation

Summary. This paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an adaptive process. In particular, evaluation covers here the computation of inner products of such expressions with wavelets which arise, for instance, in the context of Galerkin or Petrov Galerkin schemes for the solution of differential equations. The central objective is to develop schemes that facilitate such evaluations at a computational expense exceeding the complexity of the given expansion, i.e., the number of nonzero wavelet coefficients, as little as possible. The following issues are addressed. First, motivated by previous treatments of the subject, we discuss the type of regularity assumptions that are appropriate in this context and explain the relevance of Besov norms. The principal strategy is to relate the computation of inner products of wavelets with compositions to approximations of compositions in terms of possibly few dual wavelets. The analysis of these approximations finally leads to a concrete evaluation scheme which is shown to be in a certain sense asymptotically optimal. We conclude with a simple numerical example. Received June 25, 1998 / Revised version received June 5, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Standard symmetric operators in Pontryagin spaces: a generalized von Neumann formula and minimality of boundary coefficients

Certain meromorphic matrix valued functions on , the so-called boundary coefficients, are characterized in terms of a standard symmetric operator S in a Pontryagin space with finite (not necessarily equal) defect numbers, a meromorphic mapping into the defect subspaces of S, and a boundary mapping for S. Under some simple assumptions the boundary coefficients also satisfy a minimality condition. It is shown that these assumptions hold if and only if for S a generalized von Neumann equality is valid.     

Laplacian Operators and Radon Transforms on Grassmann Graphs

Let Ω be a vector space over a finite field with q elements. Let G denote the general linear group of automorphisms of Ω and let us consider the left regular representation associated with the natural action of G on the set X of linear subspaces of Ω. In this paper we study a natural basis B of the algebra End_G(L ₂(X)) of intertwining maps on L ₂(X). By using a Laplacian operator on Grassmann graphs, we identify the kernels in B as solutions of a basic hypergeometric difference equation. This provides two expressions for these kernels. One in terms of the q-Hahn polynomials and the other by means of a Rodrigues type formula. Finally, we obtain a useful product formula for the mappings in B. We give two different proofs. One uses the theory of classical hypergeometric polynomials and the other is supported by a characterization of spherical functions in finite symmetric spaces. Both proofs require the use of certain associated Radon transforms.     

Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients

This paper is devoted to strictly hyperbolic systems and equations with non‐smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of generalised functions. Extending earlier results on symmetric hyperbolic systems, we introduce generalised strict hyperbolicity, construct symmetrisers, prove an appropriate Gårding inequality and establish existence, uniqueness and regularity of generalised solutions. Under additional regularity assumptions on the coefficients, when a classical solution of the Cauchy problem (or of a transmission problem in the piecewise regular case) exists, the generalised solution is shown to be associated with the classical solution (or the piecewise classical solution satisfying the appropriate transmission conditions).     

USE OF LIE SYMMETRIES AND THE PAINLEVÉ ANALYSIS IN COSMOLOGY

Abstract

Two techniques for the analysis of differential equations, the Lie method of extended groups and the Painlevé analysis, are reviewed. The implications for integrability in the case of the existence of symmetries in the former or the possession of the Painlevé property in the latter are explored. By way of an example an equation which arises in shear-free flow of a spherically symmetric perfect fluid with conformal symmetry in general relativity is given an extensive treatment with both techniques.     

An Upper and a Lower Bound Theorem for Combinatorial 4-Manifolds

In this paper we prove two inequalities. The first one gives a lower bound for the Euler characteristic of a tight combinatorial 4-manifold M under the additional assumptions that |M| is 1-connected, that M is a subcomplex of H (M) , and that H (M) is a centrally symmetric and simplicial d -polytope. The second inequality relates the Euler characteristic with the number of vertices of a combinatorial 4-manifold admitting a fixed-point free involution. Furthermore, we construct a new and highly symmetric 12-vertex triangulation of S ² x S ² realizing equality in each of these inequalities. Received January 23, 1996, and in revised form September 13, 1996.     

ON J-SYMMETRIC QUASI-DIFFERENTIAL EXPRESSIONS WITH MATRIX-VALUED COEFFICIENTS

Abstract

A very general class of quasi-differential expressions which generate J-symmetric operators in Hilbert space is defined with the aid of Green's formula and characterized in terms of the coefficients. Furthermore, the limit-point condition is discussed for not necessarily symmetric expressions.     

On center sets of ODEs determined by moments of their coefficients

The classical H. Poincaré Center-Focus problem asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a center. This problem can be reduced to a center problem for some ordinary differential equation whose coefficients are trigonometric polynomials depending polynomially on the coefficients of the field. In this paper we show that the set of centers in the Center-Focus problem can be determined as the set of zeros of some continuous functions from the moments of coefficients of this equation.     

SOME REMARKS ON QUASI-DIFFERENTIAL EXPRESSIONS WITH MATRIX-VALUED COEFFICIENTS

Abstract

This paper is Concerned with the relationship between ordinary linear quasi-differential expressions, defined in terms of locally Lebesgue integrable matrix coefficients given on an interval of the real line and “Classical” differential expressions with smooth (differentiable to certain prescribed orders) matrix coefficients. This relationship wan investigated in a recent paper of Everitt and Race [1] for the case of scalar quasi-differential expressions of Shin-Zettl type. The present work extends the ideas given there, to the more general quasi-differential expressions considered by Frentzen in recent years (see, for example [4,5]) and applies them to products and polynomials of expressions.     

A model of immune system with time-dependent immune reactivity

In this paper we deal with the Marchuk model of an immune system. Among the main parameters of the model are the coefficients which describe the state of infected organism and the rate of production of antibodies. In the classical model these coefficients are constants. We consider the case when these coefficients are time-dependent. In particular, we are interested in the case of periodic coefficients which can describe periodic changes of the immune reactivity due to periodic changes of the environment. We examine the asymptotic behaviour of solutions. Under some assumptions we prove that the solutions tend to periodic functions. We also present the results of numerical simulations to illustrate the behaviour of solutions.     

The Module Decomposition of Super-Symmetric Powers of Matrices

ABSTRACT

Let M be the k  ×  m matrices over ?. The GL ( k ) ×  GL ( m ) decompositions of the symmetric and of the exterior powers of M are described by two classical theorems. We describe a theorem for Lie superalgebras, which implies both of these classical theorems as special cases. The constructions of both the exterior and the symmetric algebras are generalized to a class of algebras defined by partitions. That superalgebra theorem is further generalized to these algebras.     

Idempotent structure of Clifford algebras

Spinor spaces can be represented as minimal left ideals of Clifford algebras and they are generated by primitive idempotents. Primitive idempotents of the Clifford algebras R _{p, q} are shown to be products of mutually nonannihilating commuting idempotent % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqGaaO% qaamaaleaaleaacaaIXaaabaGaaGOmaaaaaaa!3DBD!\[{\textstyle{1 \over 2}}\]2}}\](1+e _T ), where the k=q–r _q–p basis elements e _T satisfy e _T ²=1. The lattice generated by a set of mutually annihilating primitive idempotents is examined. The final result characterizes all Clifford algebras R _{p, q} with an anti-involution such that each symmetric elements is either a nilpotent or then some right multiple of it is a nonzero symmetric idempotent. This happens when p+q<-3 and (p, q)(2, 1).     

Elliptic operators with infinite-dimensional state spaces

Motivated by applications to problems from physics, we study elliptic operators with operator-valued coefficients acting on Banach-space-valued distributions. After giving a definition of ellipticity, normal ellipticity in particular, generalizing the classical concepts, we show that normally elliptic operators are negative generators of analytic semigroups on for 1 and on and , as well as on all Besov spaces of E-valued distributions on , where E is any Banach space. This is true under minimal regularity assumptions for the coefficients, thanks to a point-wise multiplier theorem for E-valued distributions proven in the appendix. Received August 23, 2000; accepted December 12, 2000.     

Small perturbations of canonical systems

We consider a singular two-dimensional canonical systemJy=–zHy on [0, ) such that at Weyl's limit point case holds. HereH is a measurable, real and nonnegative definite matrix function, called Hamiltonian. From results of L. de Branges it follows that the correspondence between canonical systems and their Titchmarsh-Weyl coefficients is a bijection between the class of all Hamiltonians with trH=1 and the class of Nevanlinna functions. In this note we show how the HamiltonianH of a canonical system changes if its Titchmarsh-Weyl coefficient or the corresponding spectral measure undergoes certain small perturbations. This generalizes results of H. Dym and N. Kravitsky for so-called vibrating strings, in particular a generalization of a construction principle of I.M. Gelfand and B.M. Levitan can be shown.     

Multiplicity Free Expansions of Schur <Emphasis Type="Italic">P</Emphasis>-Functions

After deriving inequalities on coefficients arising in the expansion of a Schur P-function in terms of Schur functions we give criteria for when such expansions are multiplicity free. From here we study the multiplicity of an irreducible spin character of the twisted symmetric group in the product of a basic spin character with an irreducible character of the symmetric group, and determine when it is multiplicity free. Received February 28, 2005     

Symmetry of the restricted 4 + 1 body problem with equal masses

We consider the problem of symmetry of the central configurations in the restricted 4 + 1 body problem when the four positive masses are equal and disposed in symmetric configurations, namely, on a line, at the vertices of a square, at the vertices of a equilateral triangle with a mass at the barycenter, and finally, at the vertices of a regular tetrahedron [1–3]. In these situations, we show that in order to form a non collinear central configuration of the restricted 4 + 1 body problem, the null mass must be on an axis of symmetry. In our approach, we will use as the main tool the quadratic forms introduced by A. Albouy and A. Chenciner [4]. Our arguments are general enough, so that we can consider the generalized Newtonian potential and even the logarithmic case. To get our results, we identify some properties of the Newtonian potential (in fact, of the function ϕ(s) = −s ^k, with k < 0) which are crucial in the proof of the symmetry.     

VARIETAL HULLS OF FUNCTORS

Abstract

The well known characterizations of equational classes of algebras with not necessaryly finitary operations by FELSCHER [6.7] and of categories of A-algebras for algebraic theories A in the sense of LINTON [10], esp., by means of their forgetful functors are the foundations of a concept of varietal functors U:K → L over arbitrary basecategories L. They prove to be monadic functors which satisfy an additional HOM-condition [17]. (In the case L = Set this condition is always fulfilled, see LINTON [11].)

Contrary to monadic functors, varietal functors are closed under composition. Pleasent algebraic properties of the base-category L can be ‘lifted’ along varietal functors, such as e.g. factorization properties, (co-) completeness, classical isomorphism theorems, etc.

By means of the well known EILENBERG-MOORE-algebras there is a universal monadic functor U^T:L ^T → L for any functor U: K → L, having a left adjoint F (T: = UF). But, in general, UT is not varietal. Under some suitable conditions, however it is possible, to construct a canonical varietal functor ?:R → L, the varietal hull of U. This hull has much more interesting (algebraic) properties than the EILENBERG-MOORE construction. Moreover, results of BANASCHEWSKI-HERRLICH [2] are extended.     

Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical systemJy=–Hy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The formal defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ) to a trace-normed system on which is in the limit-point case at ±. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.