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26078 results in Abstract analysis

Page 464 of 1304

  • ESTIMATION OF SOME EXPONENTIAL SUM BY MEANS OF q-DEGREE

  • VALÉRIE GILLOT, PHILIPPE LANGEVIN
  • Journal:
    Glasgow Mathematical Journal / Volume 52 / Issue 2 / May 2010
    Published online by Cambridge University Press:
    29 March 2010, pp. 315-324
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  • In this paper, we improve results of Gillot, Kumar and Moreno to estimate some exponential sums by means of q-degrees. The method consists in applying suitable elementary transformations to see an exponential sum over a finite field as an exponential sum over a product of subfields in order to apply Deligne bound. In particular, we obtain new results on the spectral amplitude of some monomials.

  • EIGENVALUES OF POSITIVE INTEGRAL OPERATORS WITH LAPLACE TRANSFORM-TYPE KERNELS

  • YÜKSEL SOYKAN, GRAHAM LITTLE
  • Journal:
    Glasgow Mathematical Journal / Volume 52 / Issue 2 / May 2010
    Published online by Cambridge University Press:
    29 March 2010, pp. 333-348
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  • The aim of this paper is to prove a theorem concerning asymptotic estimates of eigenvalues of certain positive integral operators with Laplace transform-type kernels.

Operator Algebras in Dynamical Systems
  • Shōichirō Sakai
  • Published online:
    27 March 2010
    Print publication:
    30 August 1991
  • This book is concerned with the theory of unbounded derivations in C*-algebras, a subject whose study was motivated by questions in quantum physics and statistical mechanics, and to which the author has made a considerable contribution. This is an active area of research, and one of the most ambitious aims of the theory is to develop quantum statistical mechanics within the framework of the C*-theory. The presentation, which is based on lectures given in Newcastle upon Tyne and Copenhagen, concentrates on topics involving quantum statistical mechanics and differentiations on manifolds. One of the goals is to formulate the absence theorem of phase transitions in its most general form within the C* setting. For the first time, he globally constructs, within that setting, derivations for a fairly wide class of interacting models, and presents a new axiomatic treatment of the construction of time evolutions and KMS states.

Integration and Harmonic Analysis on Compact Groups
  • R. E. Edwards
  • Published online:
    26 March 2010
    Print publication:
    07 September 1972
  • These notes provide a reasonably self-contained introductory survey of certain aspects of harmonic analysis on compact groups. The first part of the book seeks to give a brief account of integration theory on compact Hausdorff spaces. The second, larger part starts from the existence and essential uniqueness of an invariant integral on every compact Hausdorff group. Topics subsequently outlined include representations, the Peter–Weyl theory, positive definite functions, summability and convergence, spans of translates, closed ideals and invariant subspaces, spectral synthesis problems, the Hausdorff-Young theorem, and lacunarity.

  • CPC volume 19 issue 3 Cover and Back matter

  • Journal:
    Combinatorics, Probability and Computing / Volume 19 / Issue 3 / May 2010
    Published online by Cambridge University Press:
    25 March 2010, pp. b1-b2
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  • CPC volume 19 issue 3 Cover and Front matter

  • Journal:
    Combinatorics, Probability and Computing / Volume 19 / Issue 3 / May 2010
    Published online by Cambridge University Press:
    25 March 2010, pp. f1-f2
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  • 9 - Intersections and self-intersections of Brownian paths

  • Peter Mörters, University of Bath, Yuval Peres
  • Book:
    Brownian Motion
    Published online:
    05 June 2012
    Print publication:
    25 March 2010, pp 255-289

  • Summary

    In this chapter we study multiple points of d-dimensional Brownian motion. We shall see, for example, in which dimensions the Brownian path has double points and explore how many double points there are. This chapter also contains some of the highlights of the book: a proof that planar Brownian motion has points of infinite multiplicity, the intersection equivalence of Brownian motion and percolation limit sets, and the surprising dimension-doubling theorem of Kaufman.

    Intersection of paths: Existence and Hausdorff dimension

    Existence of intersections

    Suppose that {B1 (t): t ≥ 0} and {B2(t): t ≥ 0} are two independent d-dimensional Brownian motions started in arbitrary points. The question we ask in this section is, in which dimensions the ranges, or paths, of the two motions have a nontrivial intersection, in other words whether there exist times t1, t1 > 0 such that B1(t1) = B2(t2). As this question is easy if d = 1 we assume d ≥ 2 throughout this section.

    We have developed the tools to decide this question in Chapter 4 and Chapter 8. Keeping the path {B1(t): t ≥ 0} fixed, we have to decide whether it is a polar set for the second Brownian motion. By Kakutani's theorem, Theorem 8.20, this question depends on its capacity with respect to the potential kernel. As the capacity is again related to Hausdorff measure and dimension, the results of Chapter 4 are crucial in the proof of the following result.

Page 464 of 1304