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25994 results in Abstract analysis

Page 146 of 1300

  • Cardinal invariants of Haar null and Haar meager sets

  • Part of
  • Márton Elekes, Márk Poór
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 5 / October 2021
    Published online by Cambridge University Press:
    27 October 2020, pp. 1568-1594
    Print publication:
    October 2021
    • Article
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  • A subset X of a Polish group G is Haar null if there exists a Borel probability measure μ and a Borel set B containing X such that μ(gBh) = 0 for every g, h ∈ G. A set X is Haar meager if there exists a compact metric space K, a continuous function f : K → G and a Borel set B containing X such that f−1(gBh) is meager in K for every g, h ∈ G. We calculate (in ZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb {Z}^\omega$. In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky.

  • The minimum perfect matching in pseudo-dimension 0 < q < 1

  • Part of
  • Joel Larsson
  • Journal:
    Combinatorics, Probability and Computing / Volume 30 / Issue 3 / May 2021
    Published online by Cambridge University Press:
    27 October 2020, pp. 374-397
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  • It is known that for Kn,n equipped with i.i.d. exp (1) edge costs, the minimum total cost of a perfect matching converges to $\zeta(2)=\pi^2/6$ in probability. Similar convergence has been established for all edge cost distributions of pseudo-dimension $q \geq 1$. In this paper we extend those results to all real positive q, confirming the Mézard–Parisi conjecture in the last remaining applicable case.

Potential Theory and Geometry on Lie Groups
  • N. Th. Varopoulos
  • Published online:
    23 October 2020
    Print publication:
    22 October 2020
  • This book provides a complete and reasonably self-contained account of a new classification of connected Lie groups into two classes. The first part describes the use of tools from potential theory to establish the classification and to show that the analytic and algebraic approaches to the classification are equivalent. Part II covers geometric theory of the same classification and a proof that it is equivalent to the algebraic approach. Part III is a new approach to the geometric classification that requires more advanced geometric technology, namely homotopy, homology and the theory of currents. Using these methods, a more direct, but also more sophisticated, approach to the equivalence of the geometric and algebraic classification is made. Background material is introduced gradually to familiarise readers with ideas from areas such as Lie groups, differential topology and probability, in particular, random walks on groups. Numerous open problems inspire students to explore further.

  • JMJ volume 19 Issue 6 Cover and Back matter

  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 6 / November 2020
    Published online by Cambridge University Press:
    23 October 2020, pp. b1-b2
    Print publication:
    November 2020
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  • JMJ volume 19 Issue 6 Cover and Front matter

  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 6 / November 2020
    Published online by Cambridge University Press:
    23 October 2020, pp. f1-f2
    Print publication:
    November 2020
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Page 146 of 1300