The relationship between differential equations with piecewise constant argument and the associated discrete equations, via dichotomic maps

Dichotomic maps are considered by means of the stability and asymptotic stability of the null solution of a class of differential equations with argument [t] via associated discrete equations, where [.] designates the greatest integer function.

SEMICLASSICAL THEORY OF MAGNETIZATION FOR A 2-DIMENSIONAL NONINTERACTING ELECTRON-GAS

We compute the semiclassical magnetization and susceptibility of non-interacting electrons, confined by a smooth two-dimensional potential and subjected to a uniform perpendicular magnetic field, in the general case when their classical motion is chaotic. It is demonstrated that the magnetization per particle m(B) is directly related to the staircase function N(E), which counts the single-particle levels up to energy E. Using Gutzwiller's trace formula for N, we derive a semiclassical expression for m. Our results show that the magnetization has a non-zero average, which arises from quantum corrections to the leading-order Weyl approximation to the mean staircase and which is independent of whether the classical motion is chaotic or not. Fluctuations about the average are due to classical periodic orbits and do represent a signature of chaos. This behaviour is confirmed by numerical computations for a specific system.

Conjectures and theorems in the theory of entire functions

Motivated by the recent solution of Karlin's conjecture, properties of functions in the Laguerre-Polya class are investigated. The main result of this paper establishes new moment inequalities fur a class of entire functions represented by Fourier transforms. The paper concludes with several conjectures and open problems involving the Laguerre-Polya class and the Riemann xi -function.

Dynamics in discrete phase spaces and time interval operators

The von Neumann-Liouville time evolution equation is represented in a discrete quantum phase space. The mapped Liouville operator and the corresponding Wigner function are explicitly written for the problem of a magnetic moment interacting with a magnetic field and the precessing solution is found. The propagator is also discussed and a time interval operator, associated to a unitary operator which shifts the energy levels in the Zeeman spectrum, is introduced. This operator is associated to the particular dynamical process and is not the continuous parameter describing the time evolution. The pair of unitary operators which shifts the time and energy is shown to obey the Weyl-Schwinger algebra. (C) 1999 Elsevier B.V. B.V. All rights reserved.

A homological version of the implicit function theorem

Our objective in this paper is to prove an Implicit Function Theorem for general topological spaces. As a consequence, we show that, under certain conditions, the set of the invertible elements of a topological monoid X is an open topological group in X and we use the classical topological group theory to conclude that this set is a Lie group.

A Note on the Prior Distributions of Weibull Parameters for the Reliability Function

In Bayesian Inference it is often desirable to have a posterior density reflecting mainly the information from sample data. To achieve this purpose it is important to employ prior densities which add little information to the sample. We have in the literature many such prior densities, for example, Jeffreys (1967), Lindley (1956); (1961), Hartigan (1964), Bernardo (1979), Zellner (1984), Tibshirani (1989), etc. In the present article, we compare the posterior densities of the reliability function by using Jeffreys, the maximal data information (Zellner, 1984), Tibshirani's, and reference priors for the reliability function R(t) in a Weibull distribution.

GENERALIZED HEAVISIDE FUNCTIONS IN THE COLOMBEAU THEORY CONTEXT

We defined generalized Heaviside functions for a variable x in R-n, and for variables (x, t) in R-n x R-m. Then study properties such as: composition, invertibility, and association relation (the weak equality). This work is developed in the Colombeau generalized functions context.

Toda lattice field theories, discrete W algebras, Toda lattice hierarchies and quantum groups

In analogy with the Liouville case we study the sl3 Toda theory on the lattice and define the relevant quadratic algebra and out of it we recover the discrete W3 algebra. We define an integrable system with respect to the latter and establish the relation with the Toda lattice hierarchy. We compute the relevant continuum limits. Finally we find the quantum version of the quadratic algebra.

Radiative corrections for the gauged Thirring model in causal perturbation theory

We evaluate the one-loop fermion self-energy for the gauged Thirring model in (2+1) dimensions. with one massive fermion flavor. We do this in the framework of the causal perturbation theory. In contrast to QED3, the corresponding two-point function turns out to be infrared finite on the mass shell. Then, by means of a Ward identity, we derive the on-shell vertex correction and discuss the role played by causality for non-renormalizable theories.

Composition for a class of generalized functions in Colombeau's theory

In Colombeau's theory, given an open subset Ω of ℝn, there is a differential algebra G(Ω) of generalized functions which contains in a natural way the space D′(Ω) of distributions as a vector subspace. There is also a simpler version of the algebra G,(Ω). Although this subalgebra does not contain, in canonical way, the space D′(Ω) is enough for most applications. This work is developed in the simplified generalized functions framework. In several applications it is necessary to compute higher intrinsic derivatives of generalized functions, and since these derivatives are multilinear maps, it is necessary to define the space of generalized functions in Banach spaces. In this article we introduce the composite function for a special class of generalized mappings (defined in open subsets of Banach spaces with values in Banach spaces) and we compute the higher intrinsic derivative of this composite function.

Discrete-time sliding mode control of input-delay systems applied on a power generation system

This paper presents two discrete sliding mode control (SMC) design. The first one is a discrete-time SMC design that doesn't take into account the time-delay. The second one is a discrete-time SMC design, which takes in consideration the time-delay. The proposed techniques aim at the accomplishment simplicity and robustness for an uncertainty class. Simulations results are shown and the effectiveness of the used techniques is analyzed. © 2006 IEEE.

3-Dimensional hopf bifurcation via averaging theory

We consider the Lorenz system ẋ = σ(y - x), ẏ = rx - y - xz and ż = -bz + xy; and the Rössler system ẋ = -(y + z), ẏ = x + ay and ż = b - cz + xz. Here, we study the Hopf bifurcation which takes place at q± = (±√br - b,±√br - b, r - 1), in the Lorenz case, and at s± = (c+√c2-4ab/2, -c+√c2-4ab/2a, c±√c2-4ab/2a) in the Rössler case. As usual this Hopf bifurcation is in the sense that an one-parameter family in ε of limit cycles bifurcates from the singular point when ε = 0. Moreover, we can determine the kind of stability of these limit cycles. In fact, for both systems we can prove that all the bifurcated limit cycles in a neighborhood of the singular point are either a local attractor, or a local repeller, or they have two invariant manifolds, one stable and the other unstable, which locally are formed by two 2-dimensional cylinders. These results are proved using averaging theory. The method of studying the Hopf bifurcation using the averaging theory is relatively general and can be applied to other 3- or n-dimensional differential systems.

Born-infeld corrections to the entropy function of heterotic black holes

We use the black hole entropy function to study the effect of Born-Infeld terms on the entropy of extremal black holes in heterotic string theory in four dimensions. We find, that after adding a set of higher curvature terms to the effective action, attractor mechanism, works and Born-Infeld terms contribute to the stretching of near horizon geometry. In the α′ → 0 limit, the solutions of attractor equations for moduli, fields and the resulting entropy, are in conformity with the ones for standard two charge black holes.

Podolsky's electromagnetic theory on the null-plane

We have analyzed the null-plane canonical structure of Podolsky's electromagnetic theory. As a theory that contains higher order derivatives in the Lagrangian function, it was necessary to redefine the canonical momenta related to the field variables. We were able to find a set of first and second-class constraints, and also to derive the field equations of the system. Copyright © owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.

On stability of differential equations with piecewise constant argument using dichotomic maps

This paper deals with the study of the basic theory of existence, uniqueness and continuation of solutions of di®erential equations with piecewise constant argument. Results about asymptotic stability of the equation x(t) =-bx(t) + f(x([t])) with argu- ment [t], where [t] designates the greatest integer function, are established by means of dichotomic maps. Other example is given to illustrate the application of the method. Copyright © 2011 Watam Press.