Eigenvalues of convex processes and convergence properties of differential inclusions

In this paper, we study (real) eigenvalues and eigenvectors of convex processes, and provide conditions for the existence of eigenvectors in a given convex coneK ⁿ . It is established that the maximal eigenvalue ofG(·) inK is expressed by (whereK ⁰ is the polar cone ofK) provided that the minimum is attained in intK ⁰. This result is applied to study the asymptotic behaviour of certain differential inclusions{G(x(t)). We extend some known results for the von Neumann-Gale model to our more general framework. We prove that ifx ₀ is the unique eigenvector corresponding to the maximal eigenvalue ₀ ofG(·) inK, then the nonexistence of solutions of a certain special trigonometric form is necessary and sufficient for every viable solutionx(·) to satisfy^- ₀ ^t x(t)cx ₀ ast for somec0. Our method is to study the family of convex conesW =cl{v–x :xK,vG(x) where is any real number. We characterize the maximal eigenvalue ₀ as the minimal for whichW can be separated fromK.The research was supported in part by a grant from the ministry of science and the Maagara special project for the absorption of new immigrants in the Department of Mathematics at Technion.     

Ein Störungssatz für abgeschlossene,lineare Operatoren

Zusammenfassung In vorliegender Note wird ein Satz von Kato [7] über die Störung eines abgeschlossenen, normal auflösbaren OperatorsT mit endlichem Null-defekt (T) durch einen streng singulären Operator verallgemeinert. Zu diesem Zweck wird für jedes 0 mit Hilfe des Kuratowskischen Nichtkompaktheitsmaßes eine KlasseC von beschränkten, linearen Operatoren eingeführt, welche sowohl die streng singulären Operatoren als auch die OperatorenS mit S enthält.Das erzielte Resultat steht in engem Zusammenhang mit den Untersuchungen von Gol'denteinn, Gohberg und Markus [5] und von Gol'denteienn und Markus [6].     

A sandwich theorem and Hyers—Ulam stability of affine functions

It is shown that two real functionsf andg, defined on a real intervalI, satisfy the inequalitiesf(x + (1 – )y) g(x) + (1 – )g(y) andg(x + (1 – )y) f(x) + (1 – )f(y) for allx, y I and [0, 1], iff there exists an affine functionh: I such thatf h g. As a consequence we obtain a stability result of Hyers—Ulam type for affine functions.     

On extremal perturbations of semi-Fredholm operators

Given a bounded linear operatorA in an infinite dimensional Banach space and a compact subset of a connected component of its semi-Fredholm domain, we construct a finite rank operatorF such that –A+F is bounded below (or surjective) for each ,F ²=0 and rankF=max min{dimN(–A), codimR(–A)}, if ind(–A)0 (or ind(–A)0, respectively) for each .     

Geometrical Characterization of Strict Suns in ℓ∞(n)

A subset M of a normed linear space X is called a strict sun if, for any x X\M, the set of its nearest points from M is nonempty and for any point y M which is nearest to x, the point y is a nearest point from M to any point of the ray {x + (1 - )y | > 0\}. We give an intrinsic geometrical characterization of strict suns in the space (n).     

C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers

We prove that if aC ¹ smooth change of variable : generates a bounded composition operatorff° in the spaceA _p()=L ^p ,p2, then is linear (affine).We also prove that for a nonlinearC ¹ mapping , the norms of exponentialse ⁱ as Fourier multipliers inL ^p () tend to infinity (,||). In both results the condition C ¹ is sharp, it cannot be replaced by the Lipschitz condition.     

Completeness of the plans of sequential estimation for Wiener spaces with a drift and some uniqueness theorems

One considers a family of n-dimensional Wiener processess x(t) =(t) +t, depending on a drift parameter , where is the standard Wiener process. Let be a closed subset of the space of trajectories ⁿ×₊ (plan) and assume that the measure , defined by the first occurrence of the trajectory of the process x in the set , is a probability measure. One gives conditions which the plan has to satisfy in order that from the equality for any there should follow that f 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 69–72, 1983.The authors are grateful to V. P.Khavin for discussions on certain questions of potential theory.     

On dominant operators

This paper is devoted to the study of dominant operators with an emphasis on their spectral properties. In particular the equation (T–)f() x (T a dominant or hyponormal operator on the Hilbert space ,x andf a function from the open setU to ) is investigated in an effort to discover necessary and/or sufficient conditions for the analyticity off.Supported in part by the National Science Foundation.     

Subdivision algorithms with nonnegative masks generally converge

In this paper, a subdivision scheme consists of an operator froml () tol () determined by a doubly infinite sequence, called the mask. This operator convolutes, in a certain sense, sequences l () with the mask, thus producing a new sequence inl (). Moreover, this new sequence is placed on a finer grid. If we iterate this process with a positive mask infinitely many times, it is known that this process will produce a continuous function, which we callf . In this paper, we consider the extent to which non-negative masks yield similar results. An important application of subdivision schemes in computer graphics is the generation of curves and surfaces from an initial sequence.     

On a property of the entire dirichlet series with decreasing coefficients

The classS ^* (A) of the entire Dirichlet series is studied, which is defined for a fixed sequence by the conditions 0 _n + and _n (1n⁺(1/a _n )) imposed on the parameters _n, where is a positive continuous function on (0, +) such that (x) + and x/(x) + asx + . In this class, the necessary and sufficient conditions are given for the relation (InM(,F))(In (,F)) to hold as +, where , and is a positive continuous function increasing to + on (0,+), forwhich ln (x) is a concave function and(lnx) is a slowly increasing function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 843–853, June. 1993.     

Extensions and selections in subspaces ofC(K)

LetK be a compact Hausdorff space and letFK be a peak interpolation set for a function algebraAC(K). Let be a map fromK to the family of all convex subsets of such that the set {(z, x)zK, x(z)} is open inK×C and such thatg(z)(z) (zK) for somegA. We prove that everyfC(F) satisfyingf(s)(s) (sF) (f(s)closure (s) (sF)) admits an extensionfAA} satisfyingf(z)(z) (zK) (f(z))}closure (z) (zK), respectively). We prove a more general theorem of this kind and present various applications which generalize known dominated interpolation theorems for subspaces ofC(K).     

Generalised Holley-Preston inequalities on measure spaces and their products

Summary It is shown that if (X, ) is a product of totally ordered measure spaces andf _j (j=1,2,3,4) are measurable non-negative functions onX satisfyingf ₁(x)f₂(y)f₃(xy)f₄(xy), where (, ) are the lattice operations onX, then (f ₁ d)(f ₂ d)(f ₃ d)(f ₄ d). This generalises results of Ahlswede and Daykin (for counting measure on finite sets) and Preston (for special choices off _j).     

Minimum Divergence Estimators Based on Grouped Data

The paper considers statistical models with real-valued observations i.i.d. by F(x, ₀) from a family of distribution functions (F(x, ); ), R ^s , s 1. For random quantizations defined by sample quantiles (F _n ^–1 (₁),, F _n ^–1 ( _m–1)) of arbitrary fixed orders 0 < ₁ < _m-1 < 1, there are studied estimators _,n of ₀ which minimize -divergences of the theoretical and empirical probabilities. Under an appropriate regularity, all these estimators are shown to be as efficient (first order, in the sense of Rao) as the MLE in the model quantified nonrandomly by (F ^–1 (₁,₀),, F ^–1 ( _m–1, ₀)). Moreover, the Fisher information matrix I _m (₀, ) of the latter model with the equidistant orders = ( _j = j/m : 1 j m – 1) arbitrarily closely approximates the Fisher information J(₀) of the original model when m is appropriately large. Thus the random binning by a large number of quantiles of equidistant orders leads to appropriate estimates of the above considered type.     

A Residue Calculus for Root Systems

Let V be a finite-dimensional real vector space on which a root system is given. Consider a meromorphic function on V =V+iV, the singular locus of which is a locally finite union of hyperplanes of the form V , = s, , s . Assume is of suitable decay in the imaginary directions, so that integrals of the form _+iV , d make sense for generic V. A residue calculus is developed that allows shifting . This residue calculus can be used to obtain Plancherel and Paley–Wiener theorems on semisimple symmetric spaces.     

Reverse Functional Inequalities and Their Applications to Nonlinear Elliptic Boundary Value Problems

We establish some reverse inequalities. We give applications to nonlinear elliptic boundary value problems containing a parameter which have two branches of solutions u (0) and U (>0) of which the first is continuous at the origin and the second increases indefinitely as 0.     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

Distribution of Values of Additive Functions with Respect to the Logarithmic Frequency

We study the weak convergence of distribution functions _x(n x: f _x (n) < u). Here _x denotes the logarithmic frequency and f _x , x 6, is a set of integer-valued strongly additive functions. The method of factorial moments is basic in the proofs.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 546–557, October–December, 2004.     

An inversion theorem for integral transforms related to singular Sturm-Liouville problems on the half line

We extend the classical theory of singular Sturm-Liouville boundary value problems on the half line, as developed by Titshmarsh and Levitan to generalized functions in order to obtain a general approach to handle many integral transforms, such as the sine, cosine, Weber, Hankel, and the K-transforms, in a unified way. This approach will lead to an inversion formula that holds in the sense of generalized functions. More precisely, for [0,) and 0<, let (x,) be a solution of the Sturm-Liouville equation

Non-Uniformly Elliptic Equations with Natural Growth in the Gradient

We prove the existence of bounded solutions for a class of nonlinear elliptic problems of type–div(a(x,u,Du))=H(x,u,Du)+f, uW ^1,p ₀()L (),where a(x,,)b(||)|| ^p , b is a continuous monotone decreasing function and |H(x,,)| k()|| ^p , k is a continuous monotone increasing function.