Pettis integrability of Stone transforms

Letf be a bounded Pettis integrable function ranging in a Banach spaceX (the range of the indefinite Pettis integral is separable). We consider Pettis integrability conditions for the Stone transform off and relate this problem to the regular oscillation condition for the family of functions {x ^* fx^*B(X^*)}, whereB(X^*) is the unit ball inX ^*.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 238–253, August, 1996.     

Characterizations of simultaneous farthest point in normed linear spaces with applications

In this paper, we consider a problem of best approximation (simultaneous farthest point) for bounded sets in a real normed linear space X. We study simultaneous farthest point in X by elements of bounded sets, and present various characterizations of simultaneous farthest point of elements by bounded sets in terms of the extremal points of the closed unit ball of X ^*, where X ^* is the dual space of X. We establish the characterizations of simultaneous farthest points for bounded sets in , the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm . It is important to state clearly that the contribution of this paper in relation with the previous works (see, for example, [9, Theorem 1.13]) is a technical method to represent the distance from a bounded set to a compact convex set in X which specifically concentrates on the Hahn-Banach Theorem in X.     

Extended neighborhood: Definition and characterization

We consider neighborhood search defined on combinatorial optimization problems. Suppose that N is a Neighborhood for combinatorial optimization problem X. We say that N is LO-equivalent (locally optimal) to N if for any instance of X, the set of locally optimal solutions with respect to N and N are the same. The union of two LO-equivalent neighborhoods is itself LO-equivalent to the neighborhoods. The largest neighborhood that is LO-equivalent to N is called the extended neighborhood of N, and denoted as N^*. We analyze some basic properties of the extended neighborhood. We provide a geometric characterization of the extended neighborhood N^* when the instances have linear costs defined over a cone. For the TSP, we consider 2-opt^*, the extended neighborhood for the 2-opt (i.e., 2-exchange) neighborhood structure. We show that number of neighbors of each tour T in 2-opt^* is at least (n /2 -2)!. We show that finding the best tour in the 2-opt^* neighborhood is NP-hard. We also show that the extended neighborhood for the graph partition problem is the same as the original neighborhood, regardless of the neighborhood defined. This result extends to the quadratic assignment problem as well. This result on extended neighborhoods relies on a proof that the convex hull of solutions for the graph partition problem has a diameter of 1, that is, every two corner points of this polytope are adjacent.Acknowledgement We thank Ravi Ahuja, Andreas Schulz, and Ozlem Ergun for useful discussions. This research was supported through NSF contract DMI-9820998.     

Dually Degenerate Varieties and the Generalization of a Theorem of Griffiths–Harris

The dual variety X^* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X^* is a hypersurface in the dual space (P^N)^*. If dimX^*<N–1, then the variety X is called dually degenerate. The authors refine these definitions for a variety XP^N with a degenerate Gauss map of rankr. For such a variety, in the general case, the dimension of its dual variety X^* is N–l–1, where l=n–r, and X is dually degenerate if dimX^*<N–l–1. In 1979 Griffiths and Harris proved that a smooth variety XP^N is dually degenerate if and only if all its second fundamental forms are singular. The authors generalize this theorem for a variety XP^N with a degenerate Gauss map of rankr. Mathematics Subject Classification (2000) 53A20.     

Nonmonotone, nonlinear evolution inclusions

In this paper, we consider nonlinear evolution problems, defined on an evolution triple of spaces, driven by a nonmonotone operator, and with a perturbation term which is multivalued. We prove existence theorems for the cases of a convex and of a nonconvex valued perturbation term which is defined on all of T × H or only on T × X with values in H or even in X^* (here X - H - X^* is the evolution triple). Also, we prove the existence of extremal solutions, and for the “monotone” problem we have a strong relaxation theorem. Some examples of nonlinear parabolic problems are presented.     

A Finite Algorithm for Solving Infinite Dimensional Optimization Problems

We consider the general optimization problem (P) of selecting a continuous function x over a -compact Hausdorff space T to a metric space A, from a feasible region X of such functions, so as to minimize a functional c on X. We require that X consist of a closed equicontinuous family of functions lying in the product (over T) of compact subsets Y _t of A. (An important special case is the optimal control problem of finding a continuous time control function x that minimizes its associated discounted cost c(x) over the infinite horizon.) Relative to the uniform-on-compacta topology on the function space C(T,A) of continuous functions from T to A, the feasible region X is compact. Thus optimal solutions x ^* to (P) exist under the assumption that c is continuous. We wish to approximate such an x ^* by optimal solutions to a net {P _i }, iI, of approximating problems of the form min_{xX i} c i(x) for each iI, where (1) the net of sets {X _i } _I converges to X in the sense of Kuratowski and (2) the net {c _i } _I of functions converges to c uniformly on X. We show that for large i, any optimal solution x ^* _i to the approximating problem (P _i ) arbitrarily well approximates some optimal solution x ^* to (P). It follows that if (P) is well-posed, i.e., limsupX _i ^* is a singleton {x ^*}, then any net {x _i ^*} _I of (P _i )-optimal solutions converges in C(T,A) to x ^*. For this case, we construct a finite algorithm with the following property: given any prespecified error and any compact subset Q of T, our algorithm computes an i in I and an associated x _i ^* in X _i ^* which is within of x ^* on Q. We illustrate the theory and algorithm with a problem in continuous time production control over an infinite horizon.     

Simplified Optimality Conditions for Minimizing the Difference of Vector-Valued Functions

This paper deals with an axiomatic approach to certain optimality conditions for the vector nonconvex minimization problem min{g(x)–h(x): xX}, where X is an arbitrary set and g, h are functions defined on X with values in an ordered topological vector space Z.     

On<Emphasis Type="Italic">M</Emphasis>-structure,the asymptotic-norming property and locally uniformly rotund renormings

Letr, s ∈ [0, 1], and letX be a Banach space satisfying theM(r, s)-inequality, that is,

Orbits, weak orbits and local capacity of operators

LetT be an operator on a Banach spaceX. We give a survey of results concerning orbits {T ⁿ x:n=0,1,...} and weak orbits {T ⁿ x,x ^*:n=0,1,...} ofT wherexX andx ^*X ^*. Further we study the local capacity of operators and prove that there is a residual set of pointsxX with the property that the local capacity cap(T, x) is equal to the global capacity capT. This is an analogy to the corresponding result for the local spectral radius.The research was supported by the grant No. A1019801 of AV R.     

Prolongations of Vector Fields and the Invariants-by-Derivation Property

For any given vector field X defined on some open set M ², we characterize the prolongations X _n ^* of X to the nth jet space M ⁽ⁿ⁾, n1, such that a complete system of invariants for X _n ^* can be obtained by derivation of lower-order invariants. This leads to characterizations of C -symmetries and to new procedures for reducing the order of an ordinary differential equation.     

Topological Representation of Sheaf Cohomology of Sites

For a site & (with enough points), we construct a topological space X(&) and a full embedding ^* of the category of sheaves on & into those on X _(&) (i.e., a morphism of toposes :Sh (X_(&)) Sh(&)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms H^*(&,A) = H^*(X_(&), ^*A) for any Abelian sheaf A on &. As a particular case, this will give for any scheme Y a topological space X _(Y) and a functorial isomorphism between the étale cohomology H^*(Y _ét,A) and the ordinary sheaf cohomology H^*(X(_(Y),),^*A), for any sheaf A for the étale topology on Y.     

On intertwining operators

LetB(H) denote the algebra of operators on the Hilbert spaceH into itself. GivenA,BB(H), defineC (A, B) andR (A, B):B(H)B(H) byC (A, B) X=AX–XB andR(A, B) X=AXB–X. Our purpose in this note is a twofold one. we show firstly that ifA andB ^*B (H) are dominant operators such that the pure part ofB has non-trivial kernel, thenC ⁿ (A, B) X=0, n some natural number, implies thatC (A, B)X=C(A ^*,B ^*)X=0. Secondly, it is shown that ifA andB ^* are contractions withC ₀ completely non-unitary parts, thenR ⁿ (A, B) X=0 for some natural numbern implies thatR (A, B) X=R (A ^*,B ^*)X=C (A, B ^*)X=C (A ^*,B) X=0. In the particular case in whichX is of the Hilbert—Schmidt class, it is shown that his result extends to all contractionsA andB.     

Generalized elements of best approximation

It is proved that generalized elements of best approximation (GEBA's) are approximated by minimizing nets in the(X^**, X^*) topology. Criteria for GEBA's are established. GEBA's are considered in ^* where is a 1-normalizing subspace of X^*.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 513–523, April, 1976.     

Transitivity of the norm on Banach spaces¶ having a Jordan structure

We study transitivity conditions on the norm of JB ^*-triples, C ^*-algebras, JB-algebras, and their preduals. We show that, for the predual X of a JBW ^*-triple, each one of the following conditions i) and ii) implies that X is a Hilbert space. i) The closed unit ball of X has some extreme point and the norm of X is convex transitive. ii) The set of all extreme points of the closed unit ball of X is non rare in the unit sphere of X. These results are applied to obtain partial affirmative answers to the open problem whether every JB ^*-triple with transitive norm is a Hilbert space. We extend to arbitrary C ^*-algebras previously known characterizations of transitivity [20] and convex transitivity [36] of the norm on commutative C ^*-algebras. Moreover, we prove that the Calkin algebra has convex transitive norm. We also prove that, if X is a JB-algebra, and if either the norm of X is convex transitive or X has a predual with convex transitive norm, then X is associative. As a consequence, a JB-algebra with almost transitive norm is isomorphic to the field of real numbers. Received: 9 June 1999 / Revised version: 20 February 2000     

A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables

Let {X _i, 1in} be a negatively associated sequence, and let {X* _i , 1in} be a sequence of independent random variables such that X* _i and X _i have the same distribution for each i=1, 2,..., n. It is shown in this paper that Ef( ⁿ _i=1 X _i)Ef( ⁿ _i=1 X* _i ) for any convex function f on R ¹ and that Ef(max_1kn ⁿ _i=k X _i)Ef(max_1kn ^k _i=1 X* _i ) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true for negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population.     

Perturbation analysis of the matrix equation

Consider the nonlinear matrix equation X-A^*X^-pA=Q with 0<p1. This paper shows that there exists a unique positive definite solution to the equation. A perturbation bound and the backward error of an approximate solution to this solution is evaluated. We also obtain explicit expressions of the condition number for the unique positive definite solution. The theoretical results are illustrated by numerical examples.     

The Real Powers of the Convolution of a Gamma Distribution and a Bernoulli Distribution

In this paper, we essentially compute the set of x,y>0 such that the mapping \(z\longmapsto(1-r+re^{z})^{x}(\frac{\lambda}{\lambda-z})^{y}\) is a Laplace transform. If X and Y are two independent random variables which have respectively Bernoulli and Gamma distributions, we denote by μ the distribution of X+Y. The above problem is equivalent to finding the set of x>0 such that μ ^*x exists.     

Multiplizitäten “unendlich-ferner” Spitzen

LetX be the quotient of a bounded symmetric domainD by an arithmetically defined subgroup of all analytic automorphisms ofD and letX ^* be theSatake-compactification ofX. In the present note, the multiplicities of the local rings of the zero-dimensional boundary components ofX ^* will be computed in a completely elementary manner using reduction-theory in selfadjoint homogeneous cones.