Variational Inequality with Fuzzy Convex Cone

In this study, we discuss one type of variational inequality problem with a fuzzy convex cone , denoted by VI( , f). Different classes of fuzzy convex cones which are considered in different context of the problems will be discussed. According to the existence theorem, an approach derived from the concepts of multiple objective mathematical programming problems for solving the VI( , f) is proposed. An algorithm is developed to find its fuzzy optimal solution set with complexity analysis.     

Über die Bestimmung der Anzahl dreiachsiger Ellipsoide aus der Anzahl ihrer Schnittellipsen in zufälligen Schnittebenen

Summary A diagram is given for reading off the shape-factor oftri-axial ellipsoids ( = average distance between parallel tangent planes over all orientations).The knowledge of permits the calculation of the number of ellipsoids present per unit volume from the number of elliptical sections observed per unit area.     

Optimal decompositions for theK-functional for a couple of Banach lattices

Letf=g _t+h _t be the optimal decomposition for calculating the exact value of theK-functionalK(t, f; ) of an elementf with respect to a couple =(X ₀ ,X ₁) of Banach lattices of measurable functions. It is shown that this decomposition has a rather simple form in many cases where one of the spacesX ₀ andX ₁ is eitherL ^∞ orL ¹. Many examples are given of couples of lattices for which |g _t| increases monotonically a.e. with respect tot. It is shown that this property implies a sharpened estimate from above for the Brudnyi-KrugljakK-divisibility constant γ( ) for the couple. But it is also shown that certain couples do not have this property. These also provide examples of couples of lattices for which γ( ). Research supported by the Technion V. P. R. Fund.     

Comparison of location models for stochastic processes

Summary Let (Q) be the statistical experiment based on the observation of an unknown function in the presence of an additive noise process with distributionQ. The (possible) loss of information whenQ is replaced by some other noise distributionP is measured by the deficiency of (P) relative to (Q). This deficiency and its relation to the variational distance ofP andQ are studied mainly for Gaussian noise processes. Gaussian diffusion processes and special set-indexed processes are treated in detail.Research supported by a Heisenberg grant of the Deutsche Forschungsgemeinschaft     

Completion of an additive measure to a σ-additive measure by an extension of the space

For an algebra of subsets of a set X there is constructed a set and an algebra of its subsets so that the mapping is a one-to-one correspondence between and and for each additive measure on the measure on defined by the equation is countably additive.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 71–76, January, 1968.The author wishes to express his deep appreciation to S. V. Fomin, under whose guidance this paper was written.     

Positive elements in the algebra of the quantum moment problem

Summary Let denote the extended Weyl algebra, , the Weyl algebra. It is well known that every element of of the formA=B _k ^* B _k is positive. We prove that the converse implication also holds: Every positive elementA in has a quadratic sum factorization for some finite set of elements (B _k ) in . The corresponding result is not true for the subalgebra . We identify states on which do not extend to states on . It follows from a result of Powers (and Arveson) that such states on cannot be completely positive. Our theorem is based on a certain regularity property for the representations which are generated by states on , and this property is not in general shared by representations generated by states defined only on the subalgebra .Work supported in part by the NSF     

Complétude des métriques lorentziennes de\mathbb{T}^2 et difféomorphismes du cercle

Let and the foliations by the null geodesics of some lorentzian metricg on the torus . We analyse how geodesic completeness properties ofg are related to the dynamics of and .     

Regular Unitarily Invariant Spaces on the Complex Sphere

Let K be a compact space, let X be a closed subspace of C(K), and let be a positive measure on K. The triple is said to be regular if, for any positive function and for any , there exists a function such that on K and . The case where K is the unit sphere in and the subspace X is invariant with respect to the unitary group is investigated. Sufficient spectral conditions and a necessary condition for the regularity of a triple are obtained. Connections with compactness of certain Hankel operators and applications to interpolation problems are presented. Bibliography: 16 titles.     

Minimal displacement of branching random walk

Summary Let denote a branching random walk in with mean particle productionm, m>1, and with incremental spatial distributionG, withG({0}) =p andG({1})=1–p. Ifmp=1, then the minimal displacement of behaves asymptotically like log logn/log 2. If the conditionG({1})=1–p is replaced byG((0, ))=1–p, we obtain a similar result.Research was partially supported by the National Science Foundation under grant MCS-7607039     

A Covering Space with No Compact Core

This note describes an example of a compact aspherical 2-complex X with a covering space , such that is finitely presented but has no compact core.I thank John Stallings for being my 'Senior Scientist' in 1996–1997, and for many superimaginative e-mails that led to this relationship.     

On the embedding of core-free projective images of normal subgroups

Sunto Siano G e due gruppi, : GG una proiettività e N un sottogruppo normale di G. Vengono provati alcuni risultati sulle proprietà di immersions in e G rispettivamente, del gruppi , la chiusura normale di N in modulo il nocciolo di N in , e .

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Supported by a C.N.R. grant.     

The Martin boundary and completion of Markov chains

Summary In this paper we treat a time-symmetrical Martin boundary theory for continuous parameter Markov chains. This is done by reversing the time sense of a Markov chainX _t in such a way as to obtain a dual Markov chain , and considering the two chains together. Various relations between the Martin exit boundaries and of these processes are studied. The exit boundary of , is in a sense an entrance boundary forX _t and vice versa. After a natural identification of certain points in and one can topologizeI in such a way thatboth X _t and have standard modifications in this space which are right continuous, have left limits, and are strongly Markov.Research supported in part at Stanford University, Stanford, California under AFOSR 0049.     

On the tail behaviour of the distribution function of multiple stochastic integrals

Summary Let F _n (u) denote the empirical distribution function of a sample of i.i.d. random variables with uniform distribution on [0, 1]. Define , and consider the integrals where f is a bounded measurable function. We give a good upper bound on the probability . An analogous estimate is given for multiple integrals with respect to a Poisson process.     

On the Hellinger type distances for filtered experiments

Summary We study the Hellinger type distances on a filtered space. Herep2 is an arbitrary number andP _T and are two probability measures stopped at a random timeT. We give lower and upper bounds for in predictable terms.     

Path processes and historical superprocesses

Summary A superprocessX over a Markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles.The historical process for is the process whose state at timet is the path of over time interval [0,t]. The superprocess over the historical superprocess over —reflects not only the particle distribution at any fixed time but also the structure of family trees. The principal property of a historical process is that is a function of for alls<t. Every process with this property is calleda path process. We develop a theory of superprocesses over path processes whose core is the integration with respect to measure-functionals. By applying this theory to historical superprocesses we construct the first hitting distributions and prove a special Markov property for superprocesses.Partially supported by National Science Foundation Grant DMS-8802667     

What we know about the second adjunction mapping

Let be an algebraic submanifold of complex projective space ℙ ^N . Assume thatn:=dim and let denote the restriction of the hyperplane section bundle to . The meromorphic map associated to fork≥1 ties together the pluricanonical maps of the surface sections of . Known results show that the behavior of is far better than one would expect from experience with the pluricanonical mappings of algebraic surfaces. In this article we discuss the known results on the structure of the mappings and describe the open problems. Conferenza tenuta il 5 giugno 1997     

Sharp Conditions for the CLT of Linear Processes in a Hilbert Space

In this paper we study the behavior of sums of a linear process associated to a strictly stationary sequence with values in a real separable Hilbert space and are linear operators from H to H. One of the results is that satisfies the CLT provided are i.i.d. centered having finite second moments and . We shall provide an example which shows that the condition on the operators is essentially sharp. Extensions of this result are given for sequences of weak dependent random variables under minimal conditions.     

Isometric representations of some quotients ofH ∞ of an annulus

Let denote an annulus,E a finite subset of with at least three elements, and the ideal of functions in which vanish at the points ofE. The quotient does not have a completely isometric representation on a finite dimensional Hilbert space. This complements a result of [11] which implies that the quotient has an isometric representation on a Hilbert space of dimension twice the cardinality ofE.     

About existence and uniqueness of the restricted total least squares estimation

Summary Total Least Squares (TLS) is an estimation method for the solutiona of the linear system when both data sets and are subject to error. The TLS-method minimizes the functional with weighting parameter . In this paper the TLS-functional is analyzed by the technique of Lagrangian multipliers. The main part of the work deals with the case when the estimatea is restricted by an inequality of the formD a–b0, D a diagonal matrix. It is shown that there exists a unique estimatea if the weighting parameter is chosen sufficiently large.     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.