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25847 results in Abstract analysis

Page 880 of 1293

  • NEIGHBORHOODS OF STRATA IN MANIFOLD STRATIFIED SPACES

  • BRUCE HUGHES
  • Journal:
    Glasgow Mathematical Journal / Volume 46 / Issue 1 / January 2004
    Published online by Cambridge University Press:
    15 January 2004, pp. 1-28
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  • Strata in manifold stratified spaces are shown to have neighborhoods that are teardrops of manifold stratified approximate fibrations (under dimension and compactness assumptions). This is the best possible version of the tubular neighborhood theorem for strata in the topological setting. Applications are given to replacement of singularities, to the structure of neighborhoods of points in manifold stratified spaces, and to spaces of manifold stratified approximate fibrations.

  • ON QUASISIMILARITY FOR LOG-HYPONORMAL OPERATORS

  • I. H. JEON, K. TANAHASHI, A. UCHIYAMA
  • Journal:
    Glasgow Mathematical Journal / Volume 46 / Issue 1 / January 2004
    Published online by Cambridge University Press:
    15 January 2004, pp. 169-176
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  • In this paper we show that the normal parts of quasisimilar log-hyponormal operators are unitarily equivalent. A Fuglede-Putnam type theorem for log-hyponormal operators is proved. Also, it is shown that a log-hyponormal operator that is quasisimilar to an isometry is unitary and that a log-hyponormal spectral operator is normal.

  • TENSOR PRODUCTS OF LOG-HYPONORMAL AND OF CLASS $A(s,t)$ OPERATORS

  • KÔTARÔ TANAHASHI, MUNEO CHŌ
  • Journal:
    Glasgow Mathematical Journal / Volume 46 / Issue 1 / January 2004
    Published online by Cambridge University Press:
    15 January 2004, pp. 91-95
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  • Let $A$ (resp. $B$) be a bounded linear operator on a complex Hilbert space $ {\mathcal H}$ (resp. $ {\mathcal K}$). We show that the tensor product $ A \otimes B $ is log-hyponormal if and only if $A$ and $B$ are log-hyponormal, and that a similar result holds for class $A(s,t)$ operators.

  • GROUPS IN WHICH ALL SUBGROUPS OF INFINITE RANK ARE SUBNORMAL

  • LEONID A. KURDACHENKO, HOWARD SMITH
  • Journal:
    Glasgow Mathematical Journal / Volume 46 / Issue 1 / January 2004
    Published online by Cambridge University Press:
    15 January 2004, pp. 83-89
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  • Let $G$ be a locally soluble-by-finite group in which every non-subnormal subgroup has finite rank. It is proved that either $G$ has finite rank or $G$ is soluble and locally nilpotent (and even a Baer group). On the other hand, a group $G$ is constructed that has infinite rank and satisfies the given hypothesis, but does not have every subgroup subnormal.

  • ON A PROPERTY OF BASES IN A HILBERT SPACE

  • JAMES R. HOLUB
  • Journal:
    Glasgow Mathematical Journal / Volume 46 / Issue 1 / January 2004
    Published online by Cambridge University Press:
    15 January 2004, pp. 177-180
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  • In this paper we study a seemingly unnoticed property of bases in a Hilbert space that falls in the general area of constructing new bases from old, yet is quite atypical of others in this regard. Namely, if $\{x_n\}$ is any normalized basis for a Hilbert space $ H $ and $ \{\,f_n\}$ the associated basis of coefficient functionals, then the sequence $\{x_n+f_n\} $ is again a basis for $ H $. The unusual aspect of this observation is that the basis $ \{x_n+f_n\} $ obtained in this way from $\{x_n\} $ and $ \{\,f_n\} $ need not be equivalent to either, in contrast to the standard techniques of constructing new bases from given ones by means of an isomorphism on $ H $. In this paper we study bases of this form and their relation to the component bases $ \{x_n\} $ and $ \{\,f_n\}$.

  • 38 - What next?

  • James C. Robinson, University of Warwick
  • Book:
    An Introduction to Ordinary Differential Equations
    Published online:
    05 September 2012
    Print publication:
    08 January 2004, pp 373-378

  • Summary

    In this book we have covered all of the basic methods for finding the explicit solutions of simple first and second order differential equations, along with some qualitative methods for coupled nonlinear equations. We have also discussed difference equations, and seen how complicated the dynamics of even very simple iterated nonlinear maps can become.

    There are two ways in which to proceed further with the material developed here. One arises from turning first to the study of partial differential equations, while the other essentially continues from where we have left off.

    Partial differential equations and boundary value problems

    Partial differential equations model systems that have spatial as well as temporal structure, for example the temperature throughout an object, the vibrations of a string or a drum, or the velocity of a fluid.

    In general linear partial differential equations are easier to solve. By using the technique known as ‘separation of variables’ it is possible to convert such a problem into an ordinary differential equation. This was touched on briefly in Exercise 20.10, and the exercises in this chapter apply this method in more detail for the example of the vibrating string.

Page 880 of 1293