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25816 results in Abstract analysis

Page 941 of 1291

  • 1 - Analytic Preparations

  • 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 1-16

  • Summary

    We review here some of the analytic concepts and facts that will be used in later chapters. Most of this material forms part of the standard textbook literature on real analysis or functional analysis and it is not necessary to repeat here the pertinent proofs. However, a few facts of a more special character and not generally known will be formulated as lemmas and proved.

    Throughout this book we let Ed denote the Euclidean d-dimensional space. If x is a point of Ed the coordinates of x will be denoted by xi; hence, x = (x1, …, xd). The letter o denotes the origin (0, …, 0) of Ed. If u, v ∈ Ed we let u · v denote the inner product, and |u| the Euclidean norm. Of course, for points in E1, that is, for real numbers, | · | is the ordinary absolute value. The Lebesgue measure of a subset S of Ed will usually be called the volume of S and denoted by v(S). We write Bd(p, r) for the closed ball in Ed of radius r centered at p, and Bd = Bd(o, 1) for the closed unit ball in Ed centered at o. Furthermore, we let Sd–1 denote the boundary of Bd, that is, the unit sphere in Ed. The spherical Lebesgue measure on Sd–1 will be denoted by σ, the volume of Bd by κd, and the surface area of Bd by σd.

  • 4 - Geometric Applications of Fourier Series

  • 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 133-180

  • Summary

    This chapter concerns various geometric applications of Fourier series that either do not have higher dimensional analogues or serve as good illustrations for the methods used in the more complicated d-dimensional case. For a survey of the results discussed here see Groemer (1993c, chapter 2). Although most of these results are relatively old, some of the proofs have been modified to avoid smoothness assumptions quite often present (explicitly or implicitly) in the original literature.

    A Proof of Hurwitz of the Isoperimetric Inequality

    The aim of this section is to present a proof of the isoperimetric inequality (in E2) based on the ideas of the classical paper of Hurwitz (1901). It is remarkable that this proof can be arranged in such a way that no smoothness assumption and not even convexity are required.

    We first discuss a few concepts and known results regarding curves in E2 that will be used here. A curve is defined as a continuous mapping of a closed interval [α, β] into E2 that is not constant on any subinterval of [α, β]. In this connection intervals are always assumed to have positive length. Any two such curves, say Γ1 and Γ2, are considered to be the same if Γ2 is obtained from Γ1 by an admissible change of parameter.

  • Finite groups of outer automorphisms of free groups

  • Bruno Zimmermann
  • Journal:
    Glasgow Mathematical Journal / Volume 38 / Issue 3 / September 1996
    Published online by Cambridge University Press:
    18 May 2009, pp. 275-282
    Print publication:
    September 1996
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  • Let Fr denote the free group of rank r and Out Fr: = AutFr/Inn Fr the outer automorphism group of Fr (automorphisms modulo inner automorphisms). In [10] we determined the maximal order 2rr! (for r > 2) for finite subgroups of Out Fr as well as the finite subgroup of that order which, for r > 3, is unique up to conjugation. In the present paper we determine all maximal finite subgroups (that is not contained in a larger finite subgroup) of Out F3, up to conjugation (Theorem 2 in Section 3). Here the considered case r = 3 serves as a model case: our method can be applied for other small values of r (in principle for any value of r) but the computations become considerably longer and are more apt for a computer then; the method can also be applied to determine the maximal finite subgroups of the automorphism group Aut Fr of Fr. Note that the canonical projection Aut Fr ⃗ Out Fr is injective on finite subgroups of Aut Fr; however, not all finite subgroups of Out Fr lift to finite subgroups of Aut Fr.

  • Complete submanifolds with parallel mean curvature in a sphere

  • Theodoros Vlachos
  • Journal:
    Glasgow Mathematical Journal / Volume 38 / Issue 3 / September 1996
    Published online by Cambridge University Press:
    18 May 2009, pp. 343-346
    Print publication:
    September 1996
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  • Let Mn be an n-dimensional manifold immersed in an (n+p)-dimensional unit sphere Sn+p, with mean curvature H and second fundamental form B. We put φ(X, Y) = B(X, Y)–(X, Y)H where X and Y are tangent vector fields on Mn. Assume that the mean curvature is parallel in the normal bundle of Mn in Sn+p. Following Alencar and do Carmo [1] we denote by BH the square of the positive root of

  • GMJ volume 38 issue 3 Cover and Front matter

  • Journal:
    Glasgow Mathematical Journal / Volume 38 / Issue 3 / September 1996
    Published online by Cambridge University Press:
    18 May 2009, pp. f1-f8
    Print publication:
    September 1996
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  • A generalized Drazin inverse

  • J. J. Koliha
  • Journal:
    Glasgow Mathematical Journal / Volume 38 / Issue 3 / September 1996
    Published online by Cambridge University Press:
    18 May 2009, pp. 367-381
    Print publication:
    September 1996
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  • The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point ofthe operator. This inverse is useful for instance in the solution of differential equations formulated in a Banach space X. Since the elements of X rarely enter into our considerations, the exposition seems to gain in clarity when the operators are regarded as elements of the Banach algebra L(X).

  • A class of groups rich in finite quotients

  • Vonn Walter
  • Journal:
    Glasgow Mathematical Journal / Volume 38 / Issue 3 / September 1996
    Published online by Cambridge University Press:
    18 May 2009, pp. 263-274
    Print publication:
    September 1996
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  • If X is a class of groups, the class of counter-Xgroups is defined to consist of all groups having no non-trivial X-quotients. The counter-abelian groups are the perfect groups and the counter-counter-abelian groups are the imperfect groups studied by Berrick and Robinson [2]. This paper is concerned with the class of counter-counterfinite groups. It turns out that these are the groups in which any non-trivial quotient has a non-trivial representation over any finitely generated domain (Theorem 1.1), so we shall call these groups highly representable or HR-groups.

  • A problem of Hooley in Diophantine approximation

  • Glyn Harman
  • Journal:
    Glasgow Mathematical Journal / Volume 38 / Issue 3 / September 1996
    Published online by Cambridge University Press:
    18 May 2009, pp. 299-308
    Print publication:
    September 1996
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  • In [5] Professor Hooley announced without proof the following result which is a variant of well-known work by Heilbronn [4]and Danicic [3] (see [1]).

    Let k≥2 be an integer, b a fixed non-zero integer, and a an irrational real number. Then, for any ɛ> 0, there are infinitely many solutions to the inequality

    Here

Page 941 of 1291