Probability Theory

Bauer, Heinz

doi:10.1515/9783110814668

Cited by 145 publications

(85 citation statements)

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“…Remark: Once again, the restriction to P(W ) is not essential, and the result holds for sequences in M m + (H) as well, compare [Bau,Thm. 23.8], but we do not need the stronger result here.…”

Section: Concrete Examplesmentioning

confidence: 87%

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Self-similar measures for quasicrystals

Baake

Moody

2000

Directions in Mathematical Quasicrystals

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Section: Concrete Examplesmentioning

confidence: 87%

“…Remark: With little extra complication, the result can be extended to vaguely convergent sequences of measures µ n ∈ M m + (H), see [Bau,Lemma 23.7] for a proof that can easily be adapted to this case. Theorem A.5.…”

Section: Concrete Examplesmentioning

confidence: 99%

Self-similar measures for quasicrystals

Baake

Moody

2000

Directions in Mathematical Quasicrystals

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“…, L N t , whereas G t is not A t -measurable as it partially depends on L t+1 . In [1], one can find a comprehensive introduction into σ-algebras and filtrations. Informally, A t captures the information of the underlying stochastic process, that is available at time t, i. e., a random variable is A t -measurable, iff its value is determined at time t and for a random variable V , E[V | A t ] is the expectation of V under taking the information of every time t ≤ t into account.…”

Section: Definitionsmentioning

confidence: 99%

“…A detailed introduction into σ-algebras can be found in [1]. Only a small portion of the introduced variables are needed, but if this huge set of random variables is not defined, random variables would have been used which exist only under certain conditions, which is formally not possible.…”

Section: Unlimited Stagnation Phasesmentioning

confidence: 99%

Explanation of Stagnation at Points that are not Local Optima in Particle Swarm Optimization by Potential Analysis

Raß

Schmitt

Wanka

2015

Proceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation

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Particle Swarm Optimization (PSO) is a nature-inspired meta-heuristic for solving continuous optimization problems. In [16,18], the potential of the particles of a swarm has been used to show that slightly modified PSO guarantees convergence to local optima. Here we show that under specific circumstances the unmodified PSO, even with swarm parameters known (from the literature) to be "good", almost surely does not yield convergence to a local optimum is provided. This undesirable phenomenon is called stagnation. For this purpose, the particles' potential in each dimension is analyzed mathematically. Additionally, some reasonable assumptions on the behavior of the particles' potential are made. Depending on the objective function and, interestingly, the number of particles, the potential in some dimensions may decrease much faster than in other dimensions. Therefore, these dimensions lose relevance, i. e., the contribution of their entries to the decisions about attractor updates becomes insignificant and, with positive probability, they never regain relevance. If Brownian Motion is assumed to be an approximation of the time-dependent drop of potential, practical, i. e., large values for this probability are calculated. Finally, on chosen multidimensional polynomials of degree two, experiments are provided showing that the required circumstances occur quite frequently. Furthermore, experiments are provided showing that even when the very simple sphere function is processed the described stagnation phenomenon occurs. Consequently, unmodified PSO does not converge to any local optimum of the chosen functions for tested parameter settings.

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“…. .. Then the spectrum of A ε consists of the eigenvalues 5) with corresponding eigenfunctions ψ k . These eigenvalues are bounded above by 6) and asymptotically as ε → 0, the largest eigenvalue grows like λ max ε .…”

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confidence: 99%

Complex transient patterns on the disk

Desi¹,

Sander²,

Wanner³

2006

Discrete &Amp; Continuous Dynamical Systems - A

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Abstract. This paper studies spinodal decomposition in the Cahn-Hilliard model on the unit disk. It has previously been shown that starting at initial conditions near a homogeneous equilibrium on a rectangular domain, solutions to the linearized and the nonlinear Cahn-Hilliard equation behave indistinguishably up to large distances from the homogeneous state. In this paper we demonstrate how these results can be extended to nonrectangular domains. Particular emphasis is put on the case of the unit disk, for which interesting new phenomena can be observed. Our proof is based on vector-valued extensions of probabilistic methods used in Wanner [37]. These are the first results of this kind for domains more general than rectangular.

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Probability Theory

Cited by 145 publications

References 0 publications

Self-similar measures for quasicrystals

Self-similar measures for quasicrystals

Explanation of Stagnation at Points that are not Local Optima in Particle Swarm Optimization by Potential Analysis

Complex transient patterns on the disk

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