Asymptotic speeds of spread and traveling waves for monotone semiflows with applications

The theory of asymptotic speeds of spread and monotone traveling waves is established for a class of monotone discrete and continuous‐time semiflows and is applied to a functional differential equation with diffusion, a time‐delayed lattice population model and a reaction‐diffusion equation in an infinite cylinder. © 2005 Wiley Periodicals, Inc.     

Bifurcation and Isochronicity at Infinity in a Class of Cubic Polynomial Vector Fields

In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given.     

Pit your wits against young minds!

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.     

Theory of Non-Equilibrium Stationary States as a Theory of Resonances

We study a small quantum system (e.g., a simplified model for an atom or molecule) interacting with two bosonic or fermionic reservoirs (say, photon or phonon fields). We show that the combined system has a family of stationary states parametrized by two numbers, T ₁ and T ₂ (‘reservoir temperatures’). If T ₁ ≠ T ₂, then these states are non-equilibrium stationary states (NESS). In the latter case we show that they have nonvanishing heat fluxes and positive entropy production and are dynamically asymptotically stable. The latter means that the evolution with an initial condition, normal with respect to any state where the reservoirs are in equilibria at temperatures T ₁ and T ₂, converges to the corresponding NESS. Our results are valid for the temperatures satisfying the bound min (T ₁,T ₂) > g ^{2 + α}, where g is the coupling constant and 0 < α < 1 is a power related to the infra-red behaviour of the coupling functions. Submitted: March 20, 2006. Revised: March 19, 2007. Accepted: May 11, 2007. Marco Merkli: Partly supported by an NSERC PDF, the Institute of Theoretical Physics of ETH Zürich, Switzerland, the Departments of Mathematics of McGill University and the University of Toronto, Canada. Matthias Mück: Supported by DAAD under grant HSP III. Israel Michael Sigal: Supported by NSERC under grant NA7901.     

Exciton reflectivity of GaAs/Ga1−x Al x As multiple quantum wells

Summary The normal incidence reflectivity spectrum of excitons in GaAs/Ga_1−x Al _x As multiple quantum wells is calculated within the local response approximation. It is shown that the reflectivity lineshape strongly depends on the sample geometry. Using realistic parameters of a multiple quantum well structure, we obtain an excellent fit of the experimental reflectivity curve, thus giving exciton energies, oscillator strengths and exciton broadening parameters.

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Riassunto Lo spettro di riflettività ad incidenza normale degli eccitoni nei pozzi quantistici multipli di GaAs/Ga_1−x Al _x As è calcolato nell’ambito dell’approssimazione locale. Si mostra con la forma della linea di riflettività dipende in gran misura dalla forma geometrica del campione. Usando parametri realistici di una struttura a pozzi quantistici multipli, si ottiene un’ottima approssimazione della curva di riflettività sperimentale, che fornisce così energie eccitoniche, forze dell’oscillatore e parameri di ampliamento eccitonico.

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Резюме В рамках приближения линейного отклика вычисляется спекрт отражательной способности экситонов при нормальном падении для множественных квантовых ям в GaAs/Ga_1−x Al _x As. Показывается, что форма линии отражательной способности сильно зависит от геометрии образца. Используя реалистические параметры структуры множественных квантовых ям, мы получаем хорошее соответствие с экспериментальной кривой для отражательной способности. Получаются энергии экситонов, силы осцилляторов и парам⪟тры экситонного уширения.

     

The first <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-cohomology of some groups with one end

Let p be a real number greater than one. In this paper we study the vanishing and nonvanishing of the first L ^p -cohomology space of some groups that have one end. We also make a connection between the first L ^p -cohomolgy space and the Floyd boundary of the Cayley graph of a group. We apply the result about Floyd boundaries to show that there exists a real number p such that the first L ^p -cohomology space of a nonelementary hyperbolic group does not vanish. Received: 4 August 2006 Revised: 2 November 2006     

Time Asymptotic Behaviour for a One-Velocity Transport Operator with Maxwell Boundary Condition

This paper is concerned with the spectral analysis of a one-velocity transport operator with Maxwell boundary condition in L ₁-space. After a detailed spectral analysis it is shown that the associated Cauchy problem is governed by a C ₀-semigroup. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible. Finally, a spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given.