Quelques resultats d'entropie sur l'espace de Wiener

One studies three problems related to entropy phenomenon in the classical Wiener space. In particular, the minoration of the Wiener measure for the set {xX/(x)} is given where is a Sobolev norm in the Wiener spaceX.

     

Commutative Jacobi fields in Fock space

A jacobi field is understood to be a family (Ã()) of commuting selfadjoint operatorsÃ() acting in a Fock space, having a Jacobi structure, and depending linearly on the test functions . In this article, we give a spectral representation of such a family and outline its applications to the theory of distributions on an infinite dimensional space.This article is dedicated to the memory of my dear teacher Mark G. KreinThe work is partially supported by Fundamental Research Foundation of Ukraine, grant 1.4/62.     

Jensen inequalities for functions with higher monotonicities

Summary We investigate generalizations of the classical Jensen and Chebyshev inequalities. On one hand, we restrict the class of functions and on the other we enlarge the class of measures which are allowed. As an example, consider the inequality (J)(f(x) d) A (f(x) d, d d = 1. Iff is an arbitrary nonnegativeL _x function, this holds if 0, is convex andA = 1. Iff is monotone the measure need not be positive for (J) to hold for all convex withA = 1. If has higher monotonicity, e.g., is also convex, then we get a version of (J) withA < 1 and measures that need not be positive.     

The asymptotics of some evolution equations

We consider the contributions to the asymptotics of the function exp (T), t± , caused by the spectral singularity of a nonselfadjoint Sturm-Liouville operator T in the space L²(0, ). The potential is assumed to decrease like a power of the variable, and the element belongs to the domain of definition of both the operator T and a certain extension of it to increasing functions.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 75–85.     

A Class of Operators on L h 2 . II

Let D be the open unit disk in C, and L _h ² the space of quadratic integrable harmonic functions defined on D. Let be a function in L(D) with the property that (b) = lim _x b,xD (x) for all b D. Define the operator C on L _h ² as follows: C(f) = Q( · f), where Q is the orthogonal projection of L²(D) onto L _h ² . In this paper it is shown that if C is Fredholm, then is bounded away from zero on a neighborhood of D. Also, if C is compact, then |D 0, and the commutator ideal of (D) is K(D), where (D) denotes the norm closed subalgebra of the algebra of all bounded operators on L _h ² generated by , and K(D) is the ideal of compact operators on L _h ² . Finally, the spectrum of classes of operators defined on L _h ² is characterized.     

A lower closure theorem for abstract control problems withL p-bounded controls

A lower closure theorem for an abstract control problem is proved. The functional isJ(,u)= _G f ⁰(t, (M)(t),u(t))dt and the state equations areN(t)=f(t, (M)(t),u(t)). It is shown that, if {( _k ,u _k)} is a sequence of admissible controlsu _k and corre-sponding trajectories _k such that lim infJ( _k ,u _k)<+ and such that _k weakly,M _k M strongly,N _k N weakly, and {u _k} is bounded in someL _p norm, then there is a controlu such that (,u) is admissible and lim infJ( _k ,u _k)J(,u).Dedicated to Professor M. R. HestenesThis research was supported by the National Science Foundation, Grant No. GP-33551X.     

Bounded projections and duality on spaces of holomorphic functions in the unit ball ofℂ n

In this paper three Banach spacesA ₀(),A andA ¹() of functions holomorphic in the unit ballB of ⁿ are defined. We exhibit bounded projections fromC ₀(B) ontoA ₀(), fromL ¹(B) ontoA ¹(), and fromL(B) ontoA(). Using these projections, we show thatA ₀()* A ¹() andA ¹()* A().Supported in part by the National Natural Science Foundation of China.     

Wave operators and similarity of certain non-normal one-parameter groups

This paper starts from a self-adjoint Schrödinger operator H(0) for three particles. If the interaction is dilation-analytic, H(0) has an analytic continuation H() (>0). G(t,) (–(±,a,) defined as strong limits, when t±, of t-dependent operators. The wave operators establish transformations under which the subgroups are similar to unitary groups. The scattering matrix determined by G(t,) is diagonal with respect to a.This work was supported in part by the National Science Foundation under grant DMS-8301096.     

Über eine Verallgemeinerung der Nicoletti-Aufgabe für Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1) (t)=F(t,) fort[0,+) together with a generalized Nicoletti condition (2)H()=. The functionF: [0,+)×C ⁰[0,+)B is given (whereB denotes the Banach space) and the value ofF (t, ) may depend on the values of (t) fort[0,+);H: C ⁰[0,+)B is a given linear operator and B. Under suitable assumptions we show that when the solution :[0,+)B satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).     

Laplacian eigenvalues and the maximum cut problem

We introduce and study an eigenvalue upper bound(G) on the maximum cut mc (G) of a weighted graph. The function(G) has several interesting properties that resemble the behaviour of mc (G). The following results are presented.We show that is subadditive with respect to amalgam, and additive with respect to disjoint sum and 1-sum. We prove that(G) is never worse that 1.131 mc(G) for a planar, or more generally, a weakly bipartite graph with nonnegative edge weights. We give a dual characterization of(G), and show that(G) is computable in polynomial time with an arbitrary precision.The research has been partially done when the second author visited LRI in September 1989.     

Global bifurcation of fixed points and the Poincaré translation operator on manifolds

Let M be a differentiable manifold and [0, )×MM be a C¹ map satisfying the condition (0, p)=p for all pM. Among other results, we prove that when the degree (also called Hopf index or Euler characteristic) of the tangent vector field wMTM, given by w(p)=(/)(0, p), is well defined and nonzero, then the set (of nontrivial pairs) admits a connected subset whose closure is not compact and meets the slice {0}×M of [0, )×M. This extends known results regarding the existence of harmonic solutions of periodic ordinary differential equations on manifolds.     

Eine Charakterisierung unendlich oft teilbarer Verteilungen auf [0, ∞)

A function : [0, )(-,0] is the logarithm of the Laplace transform of an infinitely divisible probability measure on [0,) if and only if (0)=0, is continuous and conditionally positiv semi-definite.     

On the Existence and Regularity of Mass-Minimizing Currents with an Elastic Boundary

We study the following variational problem. For a compact manifold S₀ embedded in the Euclidean space we consider deformations of S₀. They are represented by Lipschitz continuous homeomorphisms of S₀ whose images are embedded manifolds. We introduce an energy of a deformation which depends on the first derivative of the curvature of (S₀) and the mass of a mass minimizing current which is bounded by (S₀). In this paper it is shown that an energy minimizing deformation of (S₀) exists. Moreover, in the case that S₀ has codimension 1, (S₀) is an embedded C^3a -submanifold, if is of the class C^2,1.     

High order approximation of implicitly defined maps

Summary Approximations for the function implicitly defined by (u)=(u, (u)) are obtained via the iterative scheme _n(u)=(u, _n–1(u)). In this paper the uniform convergence of high order derivatives of _n to the corresponding derivatives of is proved. This result yields a high order approximation theorem for the input-output map generated by a nonlinear control system, using linear combinations of iterated integrals of the control.Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Fundamental equation of information revisited

Summary This paper presents a new, shorter and more direct proof of the following result of J. Aczél and C. T. Ng: IfM: J R (J =]0, 1[ ^k ) is both multiplicative and additive, then the general solution: J R of(x) + M(1 – x)(y/1 – x) = (y) + M(1 – y)(x/1 – y) (x, y, x + y J) is given by(x) = ifM = 0,(x) = M(x)[L(x) + ] + M(1 – x)L(1 – x) ifM 0,where is an arbitrary constant andL: J R is an arbitrary solution of the logarithmic functional equationL(xy) = L(x) + L(y) (x, y J). Also, some extensions of this result to fields more general than the reals are given.     

Fubini theorem for anticipating stochastic integrals in Hilbert space

Let (X,l,) be a measure space, letW be a cylindrical Hilbert-Wiener process, and let be an anticipating integrable process-valued function onX. We prove, under natural assumptions on, that there exists a measurable version Y_x,x X, of the anticipating integral of(x) such that the integral _x Y_x(dx) is a version of the anticipating integral of _X (x)(dx). We apply this anticipating Fubini theorem to study solutions of a class of stochastic evolution equations in Hilbert space.     

Contact metric manifolds

In this paper we study contact metric manifoldsM ²ⁿ⁺¹(, , ,g) with characteristic vector field belonging to thek-nullity distribution. Moreover we prove that there exist i) nonK-contact, contact metric manifolds of dimension greater than 3 with Ricci operator commuting with and ii) 3-dimensional contact metric manifolds with non-zero constant -sectional curvature.     

On Interpolation of the Fourier Maximal Operator in Orlicz Spaces

Let and be positive increasing convex functions defined on [0, ). Suppose satisfies the 2-condition, that is, (t)2 (C1t) for sufficiently large t, and has some nice properties. If -1(u)log(u+1) C2-1(u) for sufficiently large uthen we have S*(f) L CfL for all f L ([-, ])where S*(f) is the majorant function of partial sums of trigonometric Fourier series and fL is the Orlicz norm of f. This result is sharp.     

Theorem on a convergence condition in the spaces

For a given -function (u), a condition on a -function (u) is found such that it is necessary and sufficient for the following to hold: if f_n(x) f(x) and f _n (x)M (n=1, 2, ...) where M>0 is an absolute constant, then f _n (x)–f(x)0(n). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.     

Local partial regularity theorems for suitable weak solutions of a class of degenerate systems

A partial regularity theorem is established for a particular class of weak solutions to the systemu/t– div(K(u)u)=(u)¦¦², div((u))=0 on a bounded domain inR ^N . Under our assumptions, (u) may exhibit exponential decay, and thus the system may be degenerate. Our proof is based upon a blow-up argument.This work was supported in part by NSF Grant DMS9424448.