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26077 results in Abstract analysis

Page 1024 of 1304

  • Lattice packing of spheres and the Wulff-Shape

  • Part of
  • J. M. Wills
  • Journal:
    Mathematika / Volume 43 / Issue 2 / December 1996
    Published online by Cambridge University Press:
    26 February 2010, pp. 229-236
    Print publication:
    December 1996

  • The shape of large densest sphere packings in a lattice L ⊂ Ed (d ≥ 2), measured by parametric density, tends asymptotically not to a sphere but to a polytope, the Wulff-shape, which depends only on L and the parameter. This is proved via the density deviation, derived from parametric density and diophantine approximation. In crystallography the Wulff-shape describes the shape of ideal crystals. So the result further indicates that the shape of ideal crystals can be described by dense lattice packings of spheres in E3.

  • A pair correlation hypothesis and the exceptional set in Goldbach's problem

  • Part of
  • A. Languasco, A. Perelli
  • Journal:
    Mathematika / Volume 43 / Issue 2 / December 1996
    Published online by Cambridge University Press:
    26 February 2010, pp. 349-361
    Print publication:
    December 1996

  • In 1973, Montgomery [12] introduced, in order to study the vertical distribution of the zeros of the Riemann zeta function, the pair correlation function

    where w(u) = 4/(4 + u2) and γjj = 1, 2, run over the imaginary part of the nontrivial zeros of ζ(s). It is easy to see that, for T → ∞,

    uniformly in X, and Montgomery [12], see also Goldston-Montgomery [7], proved that under the Riemann Hypothesis (RH)

    uniformly for X ≤ T ≤ XA, for any fixed A > 1. He also conjectured, under RH, that (1) holds uniformly for Xε ≤ T ≤ X, for every fixed ε > 0. We denote by MC the above conjecture.

  • Volumes of projection bodies

  • Part of
  • Noah Samuel Brannen
  • Journal:
    Mathematika / Volume 43 / Issue 2 / December 1996
    Published online by Cambridge University Press:
    26 February 2010, pp. 255-264
    Print publication:
    December 1996

  • C. M. Petty has conjectured the minimum value for a certain affine-invariant functional denned on the class of convex bodies. We give sharp bounds for this functional on a certain subclass of convex bodies, and we give a counterexample to an upper bound proposed by R. Schneider for the class of centrally symmetric convex bodies. We conjecture that the simplex provides the maximum on the class of all convex bodies, while the largest centrally symmetric subset of a simplex gives a sharp upper bound on the class of all centrally symmetric convex bodies.

  • Preface

  • Helmut Groemer, University of Arizona
  • Book:
    Geometric Applications of Fourier Series and Spherical Harmonics
    Published online:
    11 September 2009
    Print publication:
    13 September 1996, pp ix-xii

  • Summary

    In 1901 Adolf Hurwitz published a short note showing that Fourier series can be used to prove the isoperimetric inequality for domains in the Euclidean plane, and in a subsequent article he showed how spherical harmonics can be utilized to prove an analogous inequality for three-dimensional convex bodies. A few years later Hermann Minkowski used spherical harmonics to prove an interesting characterization of (three-dimensional) convex bodies of constant width. The work of Hurwitz and Minkowski has convincingly shown that a study of this interplay of analysis and geometry, in particular of Fourier series and spherical harmonics on the one hand, and the theory of convex bodies on the other hand, can lead to interesting geometric results. Since then many articles have appeared that explored the possibilities of such methods.

    The aim of the present book is to provide a fairly comprehensive exposition of geometric results, more specifically, of results in the theory of convex sets, that have been proved by the use of Fourier series or spherical harmonics. Almost all theorems that are stated are also proved. Furthermore, to make the book more self-contained, all results from the theory of spherical harmonics that are used are also proved. Thus the only prerequisite for reading this book is some familiarity with the basic facts of the theory of (finite dimensional) convex sets and the theory of functions of real variables.

Page 1024 of 1304