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).     

Continuous convolution semigroups integrating a submultiplicative function

Let be a submultiplicative function on a locally compact group G and let S be a convolution semigroup on G with Lévy measure . It is shown that the measures of S integrate if and only if integrates outside some neighbourhood of the identity of G (Theorem 1). Moreover if (X(t);t0) is the G-valued process with independent increments associated with the semigroup S it is shown that the measures of S integrate if and only if the random variable sup { (X(t)):0t1} is integrable (Theorem 2).     

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.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

Best approximations of continuous functions by spline functions

An investigation of the approximation on [0, 1] of functionsf (x) by spline functions s(f,; x) of degree 2r-1 and of deficiency r (r>1) depending on the vector function = ₁ (x),..., _r-1(x) and interpolatingf (x) at fixed points. For the optimal choice of the vector ₀, exact estimates are obtained of the norms f(x)-s (f, ₀; x)_C[0,1] and f (x)-s (f, ₀; x)_{L[0, 1]} on the function classes H Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 41–46, July, 1970.In conclusion we would like to thank N. P. Korneichuk for suggesting this problem and for his valuable advice.     

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.     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.     

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.     

Magnetohydrodynamic Couette flow as affected by wall electrical conductances

Zusammenfassung Die Effekte der Veränderung der elektrischen Leitfähigkeit der Wände auf eine magnetohydrodynamische Couetteströmung werden untersucht. Die Untersuchung wird durchgeführt unter der allgemeinen Voraussetzung verschiedener Dicken und Leitfähigkeiten der beiden Wände. Man findet, dass der Einfluss irgendwelcher Wandungsbedingungen durch die sog. elektrischen Leitfähigkeitsverhältnisse _u und _l wiedergegeben werden kann. Für _u = _l werden die Resultate vonLehnert (1952) wiedergefunden, während für _{u l} stark abweichende Geschwindigkeitsprofile erhalten werden.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

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.

     

Topological transitivity of cylindrical cascades

The existence is proved of a topologically transitive (t.t.) homeomorphism U of the space W = × Z of the formU (, z)=(T,, z+f ()) ( , z Z), where is a complete separable metric space, T is a t.t. homeomorphism of onto itself, Z is a separable banach space, andf is a continuous map: z. For the special case W = S¹×R, T=+ ( is incommensurable with 2) the existence is proved of t.t. homeomorphisms (1) of two types: 1) with zero measure of the set of transitive points, 2) with zero measure of the set of intransitive points. An example is presented of a continuous functionf: S¹R for which the corresponding homeomorphism (1) is t.t. for all incommensurable with 2.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 441–452, September, 1973.The author thanks D. V. Anosov for advice and interest in the work.     

A constructive approach to the quantum (cosh ϕ)2 model. I. The method of the Gel'fand-Levitan-Marchenko equations

In a Hubert , with the aid of the postulated Gel'fand-Levitan-Marchenko quantum equations, one introduces the fields ₁(x) and ₂(x), which are the quantum analogues of the classical fields cosh (x) and sinh (x) in the sinh-Gordon model. It is shown that the fields _j(x) satisfy the Wightman axioms, including the invariance relative to reflections of space-time and mutual local commutativity. In addition, one proves the asymptotic completeness of the theory and one computes explicitly the scattering operator. In the developed approach, no cut-offs are used and, therefore, there are no renormalization effects.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 147–190, 1985.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

Convergence and divergence systems

A generalization of Carleson's theorem concerning series with respect to systems { (nx)} is proved for the case in which (x) can be expanded in an absolutely convergent Fourier series. Systems corresponding to functions(x) which have lacunary Fourier series are studied in detail.Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 253–260, September, 1968.     

L ∞ — Decay estimations of the spherical mean value on symmetric spaces

LetM _t[](x) be the spherical mean value operator applied to a function on a symmetric Riemannian space of the non-compact type.L —decay estimations forM _t [](x) as well as for its derivatives with respect to (t, x) are given, provided that belongs to a Banach space with suitable weighted supremum norm. This leads to estimates of the solutions to the wave equation in certain cases in which Huygens' principle is valid.     

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.     

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.     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

Nonnegative Hall Polynomials

The number of subgroups of type and cotype in a finite abelian p-group of type is a polynomialg with integral coefficients. We prove g has nonnegative coefficients for all partitions and if and only if no two parts of differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections that associate to each subgroup a vector dominated componentwise by . The nonzero components of (H) are the parts of , the type of H; if no two parts of differ by more than one, the nonzero components of – (H) are the parts of , the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.