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26079 results in Abstract analysis

Page 408 of 1304

Nonlinear Perron–Frobenius Theory
  • Bas Lemmens, Roger Nussbaum
  • Published online:
    05 May 2012
    Print publication:
    03 May 2012
  • In the past several decades the classical Perron–Frobenius theory for nonnegative matrices has been extended to obtain remarkably precise and beautiful results for classes of nonlinear maps. This nonlinear Perron–Frobenius theory has found significant uses in computer science, mathematical biology, game theory and the study of dynamical systems. This is the first comprehensive and unified introduction to nonlinear Perron–Frobenius theory suitable for graduate students and researchers entering the field for the first time. It acquaints the reader with recent developments and provides a guide to challenging open problems. To enhance accessibility, the focus is on finite dimensional nonlinear Perron–Frobenius theory, but pointers are provided to infinite dimensional results. Prerequisites are little more than basic real analysis and topology.

  • Getting a Directed Hamilton Cycle Two Times Faster

  • CHOONGBUM LEE, BENNY SUDAKOV, DAN VILENCHIK
  • Journal:
    Combinatorics, Probability and Computing / Volume 21 / Issue 5 / September 2012
    Published online by Cambridge University Press:
    04 May 2012, pp. 773-801

  • Consider the random graph process where we start with an empty graph on n vertices and, at time t, are given an edge et chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that w.h.p. the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph would have a directed Hamilton cycle w.h.p. only at time (1+o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected case. More precisely, we ask if, at time t, instead of a random direction one is allowed to choose the orientation of et, then whether or not it is possible to make the resulting directed graph Hamiltonian at time earlier than n log n. The main result of our paper answers this question in the strongest possible way, by asserting that one can orient the edges on-line so that w.h.p. the resulting graph has a directed Hamilton cycle exactly at the time at which the underlying graph is Hamiltonian.

  • Appendix B - Classical Perron–Frobenius theory

  • Bas Lemmens, University of Kent, Canterbury, Roger Nussbaum, Rutgers University, New Jersey
  • Book:
    Nonlinear Perron–Frobenius Theory
    Published online:
    05 May 2012
    Print publication:
    03 May 2012, pp 284-299

  • Summary

    In this appendix we provide proofs of most of the results from Section 1.1 concerning classical linear Perron–Frobenius theory. We begin (see Theorem B.1.1) by proving a generalization, valid for general cones, of Perron's theorem which is stated in Theorem 1.1.1. From this result we then derive the finite-dimensional Kreĭn–Rutman theorem (Theorem 1.1.6). We also show that many of the results in the general version of Perron's theorem remain valid for irreducible linear maps, and this yields Theorem 1.1.7. We subsequently give a complete proof of the third assertion in the classical Perron–Frobenius Theorem 1.1.2, which depends on special properties of the cone and is of a qualitatively different nature from the other two assertions (see Proposition B.4.3). We next use this part of the Perron–Frobenius theorem to prove Theorems 1.1.8 and 1.1.9 concerning the peripheral spectrum and iterative behavior of linear maps on polyhedral cones.

    Our treatment here is concise and meant only as an introduction to the linear theory. The reader should consult the books by Bapat and Raghavan [15], Berman and Plemmons [22], Minc [148], and Seneta [202], or the survey paper by Tam [214], for a more thorough discussion of linear Perron–Frobenius theory.

Page 408 of 1304