Convergence in Distribution of the One-Dimensional Kohonen Algorithms when the Stimuli are not Uniform

1994 ◽  
Vol 26 (1) ◽  
pp. 80-103 ◽  
Author(s):  
Catherine Bouton ◽  
Gilles Pagès
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Absolute Continuity ◽  
Invariant Probability Measure ◽  
Convergence In Distribution ◽  
One Dimensional ◽  
Open Set ◽  
Recurrent Markov Chain ◽  
The One

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

Convergence in Distribution of the One-Dimensional Kohonen Algorithms when the Stimuli are not Uniform

1994 ◽  
Vol 26 (01) ◽  
pp. 80-103 ◽  
Author(s):  
Catherine Bouton ◽  
Gilles Pagès
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Absolute Continuity ◽  
Invariant Probability Measure ◽  
Convergence In Distribution ◽  
One Dimensional ◽  
Open Set ◽  
Recurrent Markov Chain ◽  
The One

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

scholarly journals Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector

2019 ◽  
Vol 23 ◽  
pp. 797-802
Author(s):  
Raphaël Cerf ◽  
Joseba Dalmau
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Branching Process ◽  
Invariant Probability Measure ◽  
Classical Formula ◽  
Multitype Branching Process ◽  
Primitive Matrix ◽  
Watson Process

Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

On the minimizers of the Ginzburg–Landau energy for high kappa: the one-dimensional case

1997 ◽  
Vol 8 (4) ◽  
pp. 331-345 ◽  
Author(s):  
AMANDINE AFTALION
Keyword(s):  
Magnetic Field ◽  
Asymptotic Behaviour ◽  
Dimensional Case ◽  
Numerical Computations ◽  
Ginzburg Landau ◽  
Landau Model ◽  
One Dimensional ◽  
Landau Parameter ◽  
Convergence Results ◽  
The One

The Ginzburg–Landau model for superconductivity is examined in the one-dimensional case. First, putting the Ginzburg–Landau parameter κ formally equal to infinity, the existence of a minimizer of this reduced Ginzburg–Landau energy is proved. Then asymptotic behaviour for large κ of minimizers of the full Ginzburg–Landau energy is analysed and different convergence results are obtained, according to the exterior magnetic field. Numerical computations illustrate the various behaviours.

scholarly journals Asymptotics for a parabolic double obstacle problem

1994 ◽  
Vol 444 (1922) ◽  
pp. 429-445 ◽  
Keyword(s):  
Asymptotic Behaviour ◽  
Obstacle Problem ◽  
Thin Strip ◽  
Normal Velocity ◽  
One Dimensional ◽  
Cahn Equation ◽  
Small Constant ◽  
The One ◽  
Short Time ◽  
Double Well Potential

We consider a parabolic double obstacle problem which is a version of the Allen-Cahn equation u t = Δ u — ϵ -2 ψ '( u ) in Ω x (0, ∞), where Ω is a bounded domain, ϵ is a small constant, and ψ is a double well potential; here we take ψ such that ψ ( u ) = (1 — u 2 ) when | u | ≤ 1 and ψ ( u ) = ∞ when | u | > 1. We study the asymptotic behaviour, as ϵ → 0, of the solution of the double obstacle problem. Under some natural restrictions on the initial data, we show that after a short time (of order ϵ 2 |ln ϵ |), the solution takes value 1 in a region Ω + t and value — 1 in Ω - t , where the region Ω ( Ω + t U Ω - t ) is a thin strip and is contained in either a O ( ϵ |ln ϵ |) or O ( ϵ ) neighbourhood of a hypersurface Γ t which moves with normal velocity equal to its mean curvature. We also study the asymptotic behaviour, as t → ∞, of the solution in the one-dimensional case. In particular, we prove that the ω -limit set consists of a singleton.

scholarly journals A MULTIDIMENSIONAL SUBDIFFUSION MODEL WITH CHEMOTAXIS

2019 ◽  
Vol 11 (2) ◽  
pp. 1
Author(s):  
Bambang Hendriya Guswanto
Keyword(s):  
Mathematical Model ◽  
Probability Measure ◽  
Random Walks ◽  
Unit Ball ◽  
Dimensional Case ◽  
One Dimensional ◽  
The One ◽  
The Mathematical Model

The mathematical model for subdiffusion process with chemotaxis proposed by Langlands and Henry [1] for the one-dimensional case is extended to the multi-dimensional case. The model is derived from random walks process using a probability measure on a n-multidimensional unit ball $S^{n-1}$.

One-dimensional branching random walks in a Markovian random environment

2000 ◽  
Vol 37 (4) ◽  
pp. 1157-1163 ◽  
Author(s):  
F. P. Machado ◽  
S. Yu. Popov
Keyword(s):  
Markov Chain ◽  
Random Environment ◽  
Branching Random Walk ◽  
Countable State Space ◽  
One Dimensional ◽  
Branching Random Walks ◽  
Countable State ◽  
Markovian Random Environment ◽  
Recurrent Markov Chain ◽  
Step Transition

We study a one-dimensional supercritical branching random walk in a non-i.i.d. random environment, which considers both the branching mechanism and the step transition. This random environment is constructed using a recurrent Markov chain on a finite or countable state space. Criteria of (strong) recurrence and transience are presented for this model.

scholarly journals Rayleigh-Ritz variational method with suitable asymptotic behaviour

Open Physics ◽  
2014 ◽  
Vol 12 (8) ◽  
Author(s):  
Francisco Fernández ◽  
Javier Garcia
Keyword(s):  
Variational Method ◽  
Asymptotic Behaviour ◽  
Orthogonal Polynomial ◽  
Rate Of Convergence ◽  
Orthogonal Basis ◽  
Basis Sets ◽  
One Dimensional ◽  
Variational Calculations ◽  
Dimensional Models ◽  
The One

AbstractThis paper considers the Rayleigh-Ritz variational calculations with non-orthogonal basis sets that exhibit the correct asymptotic behaviour. This approach is illustrated by constructing suitable basis sets for one-dimensional models such as the two double-well oscillators recently considered by other authors. The rate of convergence of the variational method proves to be considerably greater than the one exhibited by the recently developed orthogonal polynomial projection quantization.

MULTIPLICATIVE STOCHASTIC PERTURBATIONS OF ONE-DIMENSIONAL MAPS

2013 ◽  
Vol 13 (02) ◽  
pp. 1250020 ◽  
Author(s):  
YUKIKO IWATA
Keyword(s):  
Markov Process ◽  
Probability Measure ◽  
Noise Level ◽  
Random Perturbation ◽  
Invariant Probability Measure ◽  
Random Perturbations ◽  
Stochastic Perturbations ◽  
One Dimensional ◽  
One Dimensional Maps

We consider random perturbations of some one-dimensional map S : [0, 1] → [0, 1] such that [Formula: see text] parametrized by 0 < ε < 1, where {Cn} is an i.i.d. sequence. We prove that this random perturbation is small with respect to the noise level 0 < ε < 1 and give a class of one-dimensional maps for which there always exists a smooth invariant probability measure for the Markov process {Xn}n≥0.

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