Dually discrete spaces

A neighbourhood assignment in a space X is a family O = {O-x: x is an element of X} of open subsets of X such that X is an element of O-x for any x is an element of X. A set Y subset of X is a kernel of O if O(Y) = U{O-x: x is an element of Y} = X. We obtain some new results concerning dually discrete spaces, being those spaces for which every neighbourhood assignment has a discrete kernel. This is a strictly larger class than the class of D-spaces of [E.K. van Douwen, W.F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81 (2) (1979) 371-377]. (c) 2008 Elsevier B.V. All rights reserved.

Finding the Relationship between a Search Algorithm and a Class of Functions on Discrete Space by Exhaustive Search

* The work is supported by RFBR, grant 04-01-00858-a.

Ecological dynamics of extinct species in empty habitat networks. 1. The role of habitat pattern and quantity, stochasticity and dispersal

We examined a remnant host plant (Primula veris L.) habitat network that was last inhabited by the rare butterfly Hamearis lucina L. in north Wales in 1943, to assess the relative contribution of several spatial parameters to its regional extinction. We first examined relationships between P. veris characteristics and H. lucina eggs in surviving H. lucina populations, and used these to predict the suitability and potential carrying capacity of the habitat network in north Wales. This resulted in an estimate of roughly 4500 eggs (ca 227 adults). We developed a discrete space, discrete time metapopulation model to evaluate the relative contribution of dispersal distance, habitat and environmental stochasticity as possible causes of extinction. We simulated the potential persistence of the butterfly in the current network as well as in three artificial (historical and present) habitat networks that differed in the quantity (current and X3) and fragmentation of the habitat (current and aggregated). We identified that reduced habitat quantity and increased isolation would have increased the probability of regional extinction, in conjunction with environmental stochasticity and H. lucina's dispersal distance. This general trend did not change in a qualitative manner when we modified the ability of dispersing females to stay in, and find suitable habitats (by changing the size of the grid cells used in the model). Contrary to most metapopulation model predictions, system persistence declined with increasing migration rate, suggesting that the mortality of migrating individuals in fragmented landscapes may pose significant risks to system-wide persistence. Based on model predictions for the present landscape we argue that a major programme of habitat restoration would be required for a re-established metapopulation to persist for > 100 years.

Invariant discretizations of partial differential equations

Un algorithme permettant de discrétiser les équations aux dérivées partielles (EDP) tout en préservant leurs symétries de Lie est élaboré. Ceci est rendu possible grâce à l'utilisation de dérivées partielles discrètes se transformant comme les dérivées partielles continues sous l'action de groupes de Lie locaux. Dans les applications, beaucoup d'EDP sont invariantes sous l'action de transformations ponctuelles de Lie de dimension infinie qui font partie de ce que l'on désigne comme des pseudo-groupes de Lie. Afin d'étendre la méthode de discrétisation préservant les symétries à ces équations, une discrétisation des pseudo-groupes est proposée. Cette discrétisation a pour effet de transformer les symétries ponctuelles en symétries généralisées dans l'espace discret. Des schémas invariants sont ensuite créés pour un certain nombre d'EDP. Dans tous les cas, des tests numériques montrent que les schémas invariants approximent mieux leur équivalent continu que les différences finies standard.

Fronts from complex two-dimensional dispersal kernels: theory and application to Reid's paradox

Bimodal dispersal probability distributions with characteristic distances differing by several orders of magnitude have been derived and favorably compared to observations by Nathan [Nature (London) 418, 409 (2002)]. For such bimodal kernels, we show that two-dimensional molecular dynamics computer simulations are unable to yield accurate front speeds. Analytically, the usual continuous-space random walks (CSRWs) are applied to two dimensions. We also introduce discrete-space random walks and use them to check the CSRW results (because of the inefficiency of the numerical simulations). The physical results reported are shown to predict front speeds high enough to possibly explain Reid's paradox of rapid tree migration. We also show that, for a time-ordered evolution equation, fronts are always slower in two dimensions than in one dimension and that this difference is important both for unimodal and for bimodal kernels

Implementation of an interior point source in the ultra weak variational formulation through source extraction

The Ultra Weak Variational Formulation (UWVF) is a powerful numerical method for the approximation of acoustic, elastic and electromagnetic waves in the time-harmonic regime. The use of Trefftz-type basis functions incorporates the known wave-like behaviour of the solution in the discrete space, allowing large reductions in the required number of degrees of freedom for a given accuracy, when compared to standard finite element methods. However, the UWVF is not well disposed to the accurate approximation of singular sources in the interior of the computational domain. We propose an adjustment to the UWVF for seismic imaging applications, which we call the Source Extraction UWVF. Differing fields are solved for in subdomains around the source, and matched on the inter-domain boundaries. Numerical results are presented for a domain of constant wavenumber and for a domain of varying sound speed in a model used for seismic imaging.

Un algoritmo de replanificación en tiempo real basado en un índice de estabilidad de Lyapunov para líneas de metro

(SPA) En este trabajo, se propone un nuevo índice basado en el método directo de Lyapunov para el diseño de un algoritmo de reprogramación en tiempo real para líneas de metro. En este estudio se utiliza una versión modificada de un modelo de espacio de estados en tiempo real discreto, que considera los efectos de saturación en la línea de metro. Una vez que el modelo de espacio de estados se ha obtenido, el método directo de Lyapunov se aplica con el fin de analizar la estabilidad del sistema de la línea de metro. Como resultado de este análisis no sólo se propone un nuevo índice de estabilidad, sino también la creación de tres zonas de estabilidad para indicar el estado actual del sistema. Finalmente, se presenta un nuevo algoritmo que permite la reprogramación del calendario de los trenes en tiempo real en presencia de perturbaciones medianas. (ENG) A new Lyapunov-based index for designing a rescheduling algorithm in real time for metro lines has been proposed in this paper. A modified real time discrete space state model which considers saturation effects in the metro line has been utilized in this study. Once the space state model has been obtained, the direct method of Lyapunov is applied in order to analyze the stability of the metro line system. As a result of this analysis not only a new stability index is proposed, but also the establishment of three stability zones to indicate the current state of the system. Finally, a new algorithm which allows the rescheduling of the timetable in the real time of the trains under presence of medium disturbances has been presented.

Fundamental solution and discrete random walk model for time-space fractional diffusion equation

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

Quantum discrete phase space dynamics and its continuous limit

The main aspects of a discrete phase space formalism are presented and the discrete dynamical bracket, suitable for the description of time evolution in finite-dimensional spaces, is discussed. A set of operator bases is defined in such a way that the Weyl-Wigner formalism is shown to be obtained as a limiting case. In the same form, the Moyal bracket is shown to be the limiting case of the discrete dynamical bracket. The dynamics in quantum discrete phase spaces is shown not to be attained from discretization of the continuous case.

Time evolution of the Wigner function in discrete quantum phase space for a soluble quasi-spin model

The discrete phase space approach to quantum mechanics of degrees of freedom without classical counterparts is applied to the many-fermions/quasi-spin Lipkin model. The Wi:ner function is written for some chosen states associated to discrete angle and angular momentum variables, and the rime evolution is numerically calculated using the discrete von Neumnnn-Liouville equation. Direct evidences in the lime evolution of the Wigner function are extracted that identify a tunnelling effect. A connection with a SU(2)-based semiclassical continuous approach to the Lipkin model is also presented.

A discrete finite-dimensional phase space approach for the description of Fe8 magnetic clusters: Wigner and Husimi functions

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