Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, u^￠_n(t) = ?_{j ? \mathbbZ^d} J_n-ju_j(t)-u_n(t)u^{\prime}_{n}(t) = \sum_{{j\in}{{{\mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t) with t ≥ 0 and n ? \mathbbZ^dn \in {\mathbb{Z}}^{d}. We assume that J is nonnegative and verifies ?_{n ? \mathbbZ^d}J_n = 1\sum_{{n \in {\mathbb{Z}}}^{d}}J_{n}= 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.     

Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, with t ≥ 0 and . We assume that J is nonnegative and verifies . We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.      

Existence of diffusion orbits in a lattice system

We study a model which is a periodic lattice system with nearest neighbors coupling. Using the variational methods, we show the existence of diffusion orbits under a generic perturbation of time periodic.     

The coherent potential method in the diffusion problem on a random substitution lattice

Based on the balance equation, we consider the diffusion problem on a hyperlattice with randomly distributed inaccessible sites. Using diagram methods, we find a self-consistent expression for the configurationally averaged Green’s function in the coherent potential approximation. We show that this approach is applicable in a broad range of concentrations of accessible sites. Using this approximation, we find the exact asymptotic form of the static diffusion coefficient for a low concentration of blocked sites. This allows making good estimates of the percolation threshold in the random-site diffusion problem on an arbitrary hyperlattice. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 252–261, November, 2006.     

Epidemic dynamics and spatial segregation driven by cognitive diffusion and nonlinear incidence

We propose susceptible-infected-susceptible epidemic reaction–diffusion models with cognitive movement and nonlinear incidence $S^q{I}^p$ $(p,q>0)$ in a spatially heterogeneous environment. The cognitive dispersal term takes either random diffusion or symmetric diffusion. Building upon the $L^{\infty }$-estimates of positive solutions under $p,q>0$, we state the asymptotic dynamics for , $q>0$. The numerical results reveal spatial segregation of susceptible and infected populations: (a) the heterogeneous random diffusion can segregate the population and reduce the infection fraction significantly; (b) the segregation phenomenon disappears as the ratio $p/q$ approaches one from below; (c) the disease-free region strengthens the segregation induced by heterogeneous random diffusion; (d) the segregation governed by random diffusion is more sensitive to the incidence mechanism; (e) the distribution of steady states driven by symmetric diffusion is always similar to that by homogeneous diffusion.     

A phase diagram for a stochastic reaction diffusion system

In this paper a stochastic reaction diffusion system is considered, which models the spread of a finite population reacting with a non-renewable resource in the presence of individual based noise. A two-parameter phase diagram is established to describe the large time evolution, distinguishing between certain death or possible life of the population.     

On the motion of a phase interface by surface diffusion

Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullins's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

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Sommario Mullins, in una serie di articoli inerenti la morfologia delle superfici di interfaccia tra fasi, ha sviluppato una dinamica delle superfici la cui evoluzione è governata dal fenomeno di diffusione di massa all'interno dell'interfaccia. Scopo di questo articolo è inscrire la teoria di Mullins in uno schema più generale basato su leggi di bilancio della massa e delle azioni capillari nonchè su una formulazione puramente meccanica del secondo principio della termodinamica, asserente ehe l'incremento di energia libera non possa essere superiore al flusso di energia ed alla potenza fornite all'interfaccia. Viene successivamente sviluppata una appropriata teoria costitutiva, e vengono dedotte le equazioni di evoluzione sia in forma generale che approssimata.

     

The lattice of closure operators on a subgroup lattice

We say a lattice L is a subgroup lattice if there exists a group G such that Sub(G)?L, where Sub(G) is the lattice of subgroups of G, ordered by inclusion. We prove that the lattice of closure operators which act on the subgroup lattice of a finite group G is itself a subgroup lattice if and only if G is cyclic of prime power order.     

Infinite lattice systems of interacting diffusion processes,existence and regularity properties

Summary Generalized stochastic gradient systems for infinite lattice models are investigated. The allowed strength of the interaction depends on the dimension of the lattice. The semigroup of transition probabilities is constructed and its regularity properties are also discussed. Some results of Doss and Royer [2] are improved.     

OnC 0-semigroups of lattice homomorphisms on a Banach lattice

Ohne Zusammenfassung     

All retraction operators on a complete lattice form a complete lattice

H. Crapo raised in 1982 the following problem: IfP is a complete lattice, is Retr (P) a complete lattice? here Retr (P) denotes the set of all retraction operators onP with the pointwise order, that is,f≤g in Retr (P) ifff(x)≤g(x) for everyx inP. An affirmative answer will be given in the present paper.     

Coexistence of a three species predator-prey model with diffusion and density dependent mortality

Rendiconti del Circolo Matematico di Palermo Series 2 - In this paper, we consider a two competitor-one prey model with diffusion in which both competitors exhibit general functional response and...     

Model of deconfinement phase transition in lattice quantum chromodynamics

Institute of Theoretical Physics, Ukrainian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 80, No. 3, pp. 381–390, September, 1989.     

Physical phase space of the lattice Yang-Mills theory and the moduli space of flat connections on a riemann surface

It is shown that the physical phase space of the γ-deformed Hamiltonian lattice in the Yang-Mills theory coincides as a Poisson manifold with the moduli space of flat connections on a Riemann surface with L−V+1 handles and, therefore, with the physical phase space of the corresponding (2+1)-dimensional Chern-Simons model. Here, L and V are, respectively, the total number of links and vertices of the lattice. The deformation parameter γ is identified with 2π/k, where k is an integer appearing in the Chern-Simons action. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 100–111, October, 1997.     

Competing species systems with diffusion and large interactions

We discuss non-negative solutions of a Lotka-Volterra competing species system which includes the effect of diffusion. We discuss when the populations coexist, and secondly the behaviour of the system when the interaction between the systems are large. The limiting problems here raise interesting questions for scalar equations.     

Evolution systems on a lattice

In the framework of the algebraic geometric approach to differential-difference equations, we study symmetries and conservation laws of evolutionary systems on multidimensional lattices. We describe conservation laws in terms of their characteristics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 391–405, December, 2008.