scholarly journals Random Symmetrizations of Convex Bodies

2014 ◽  
Vol 46 (3) ◽  
pp. 603-621 ◽  
Author(s):  
D. Coupier ◽  
Yu. Davydov
Keyword(s):  
Asymptotic Behavior ◽  
Convex Bodies ◽  
Almost Sure Convergence ◽  
Exponential Rate ◽  
Equivalence Result

In this paper we investigate the asymptotic behavior of sequences of successive Steiner and Minkowski symmetrizations. We state an equivalence result between the convergences of those sequences for Minkowski and Steiner symmetrizations. Moreover, in the case of independent (and not necessarily identically distributed) directions, we prove the almost-sure convergence of successive symmetrizations at exponential rate for Minkowski, and at rate with c > 0 for Steiner.

scholarly journals Random Symmetrizations of Convex Bodies

2014 ◽  
Vol 46 (03) ◽  
pp. 603-621 ◽  
Author(s):  
D. Coupier ◽  
Yu. Davydov
Keyword(s):  
Asymptotic Behavior ◽  
Convex Bodies ◽  
Almost Sure Convergence ◽  
Exponential Rate ◽  
Equivalence Result

In this paper we investigate the asymptotic behavior of sequences of successive Steiner and Minkowski symmetrizations. We state an equivalence result between the convergences of those sequences for Minkowski and Steiner symmetrizations. Moreover, in the case of independent (and not necessarily identically distributed) directions, we prove the almost-sure convergence of successive symmetrizations at exponential rate for Minkowski, and at rate with c > 0 for Steiner.

First-passage percolation on the square lattice. III

1978 ◽  
Vol 10 (01) ◽  
pp. 155-171 ◽  
Author(s):  
R. T. Smythe ◽  
John C. Wierman
Keyword(s):  
Asymptotic Behavior ◽  
Almost Sure Convergence ◽  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage ◽  
Underlying Distribution ◽  
Connectivity Constant ◽  
Route Length ◽  
Percolation Processes ◽  
Negative Time

The principal results of this paper concern the asymptotic behavior of the number of arcs in the optimal routes of first-passage percolation processes on the square lattice. Assuming that the underlying distribution has an atom at zero less than λ–1, where λ is the connectivity constant, Lp and (in some cases) almost sure convergence theorems are proved for the normalized route length processes. The proofs involve the extension of much of the existing theory of first-passage percolation to the case where negative time coordinates are permitted.

scholarly journals Asymptotic Behavior of a Tumor Angiogenesis Model with Haptotaxis

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 664
Author(s):  
Chi Xu ◽  
Yifu Wang
Keyword(s):  
Asymptotic Behavior ◽  
Tumor Angiogenesis ◽  
Smooth Domain ◽  
Asymptotic Behavior Of Solutions ◽  
Exponential Rate ◽  
Bounded Smooth Domain ◽  
System P ◽  
Bounded Smooth ◽  
Angiogenesis Model

This paper considers the existence and asymptotic behavior of solutions to the angiogenesis system p t = Δ p − ρ ∇ · ( p ∇ w ) + λ p ( 1 − p ) , w t = − γ p w β in a bounded smooth domain Ω ⊂ R N ( N = 1 , 2 ) , where ρ , λ , γ > 0 and β ≥ 1 . More precisely, it is shown that the corresponding solution ( p , w ) converges to ( 1 , 0 ) with an explicit exponential rate if β = 1 , and polynomial rate if β > 1 as t → ∞ , respectively, in L ∞ -norm.

First-passage percolation on the square lattice. III

1978 ◽  
Vol 10 (1) ◽  
pp. 155-171 ◽  
Author(s):  
R. T. Smythe ◽  
John C. Wierman
Keyword(s):  
Asymptotic Behavior ◽  
Almost Sure Convergence ◽  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage ◽  
Underlying Distribution ◽  
Connectivity Constant ◽  
Route Length ◽  
Percolation Processes ◽  
Negative Time

The principal results of this paper concern the asymptotic behavior of the number of arcs in the optimal routes of first-passage percolation processes on the square lattice. Assuming that the underlying distribution has an atom at zero less than λ–1, where λ is the connectivity constant, Lp and (in some cases) almost sure convergence theorems are proved for the normalized route length processes. The proofs involve the extension of much of the existing theory of first-passage percolation to the case where negative time coordinates are permitted.

scholarly journals Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks

2010 ◽  
Vol 47 (2) ◽  
pp. 513-525 ◽  
Author(s):  
Alexander Iksanov ◽  
Matthias Meiners
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Branching Random Walk ◽  
Exponential Rate ◽  
Branching Random Walks

We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. We include in particular the case of Galton-Watson processes so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering (1983). As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be of interest in themselves and which correct, generalize, and simplify some earlier works.

scholarly journals Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks

2010 ◽  
Vol 47 (02) ◽  
pp. 513-525 ◽  
Author(s):  
Alexander Iksanov ◽  
Matthias Meiners
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Branching Random Walk ◽  
Exponential Rate ◽  
Branching Random Walks

We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. We include in particular the case of Galton-Watson processes so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering (1983). As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be of interest in themselves and which correct, generalize, and simplify some earlier works.

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