Showing papers in "Portugaliae Mathematica in 2011"

Null controllability of degenerate parabolic cascade systems

Abstract: . In this paper, we study the null controllability of degenerate semilinear cas-cade parabolic systems with one control force. The key tool is the Carleman estimatesdeveloped recently for degenerate one dimension parabolic equations. We develop a Carle-man estimate for these systems and then an observability inequality for the linear adjointsystem. We conclude by linearization and fixed point arguments. Portugaliae Mathemat-ica .Keywords.semigroups, Carleman estimates, degenerate parabolic equations, Cascade,observability inequality, null controllability.Mathematics Subject Classi cation (2000). 35K05, 35K65, 93C20 1. Introduction This paper is concerned with null controllability for the following coupled degeneratesemilinear cascade parabolic system u t − ( a 1 ( x ) u x ) x + F 1 ( t,x,u ) = h ( t,x )1 ω , ( t,x ) ∈ (0 ,T ) × (0 , 1) , (1) v t − ( a 2 ( x ) v x ) x + F 2 ( t,x,u,v ) = 0 , ( t,x ) ∈ (0 ,T ) × (0 , 1) , (2) u ( t, 0) = u ( t, 1) = v ( t, 0) = v ( t, 1) = 0 , t∈

Euclidean distance matrices, semidefinite programming and sensor network localization

Abstract: The fundamental problem of distance geometry involves the characterization and study of sets of points based only on given values of some or all of the distances between pairs of points. This problem has a wide range of applications in various areas of mathe- matics, physics, chemistry, and engineering. Euclidean distance matrices play an important role in this context by providing elegant and powerful convex relaxations. They play an important role in problems such as graph realization and graph rigidity. Moreover, by relaxing the embedding dimension restriction, these matrices can be used to approximate the hard problems e‰ciently using semidefinite programming. Throughout this survey we emphasize the interplay between these concepts and problems. In addition, we illustrate this interplay in the context of the sensor network localization problem.

Linear motions in a periodically forced Kepler problem

Abstract: is necessary and sufficient for the existence of a 2π-periodic solution. The results by Campos and Torres in [6] are also applicable and the equation has a simple dynamics of saddle type. In particular the periodic solution is unique and unstable (hyperbolic). In both papers the solutions are understood in a classical sense and no collisions are admitted. The purpose of the present paper is to point out that the equation has a rich dynamics of twist type if one admits solutions with collisions. As it is typical in Celestial Mechanics for a binary collision, at an instant where u = 0 the velocity becomes infinity but the energy remains finite and has a well defined limit; that is,

Global existence for two regularized MHD models in three space-dimension

Abstract: The global existence of solutions for the 3D incompressible Euler equations is a major open problem. For the 3D inviscid MHD system, the global existence is an open problem as well. Our main concern in this paper is to understand which kind of regularization, of the form of α-regularization or partial viscous regularization, is capable to provide the global in time solvability for the 3D inviscid MHD system of equations. We consider two different regularized magnetohydrodynamic models for an incompressible fluid. In both cases, we provide a global existence result for the solution of the system. Mathematics Subject Classification (2000). Primary 35Q35; Secondary 76D03.

Stabilization of the Schrodinger equation with a delay term in boundary feedback or internal feedback

Abstract: In this paper, we investigate the eect of time delays in boundary or internal feedback stabilization of the Schrodinger equation. In both cases, under suitable assump- tions, we establish su‰cient conditions on the delay term that guarantee the exponential stability of the solution. These results are obtained by using suitable energy functionals and some observability estimates.

The Fibonacci version of the Brocard–Ramanujan Diophantine equation

Abstract: In this note, we prove that the Fibonacci version of the Brocard-Ramanujan Diophantine equation n!+1 = m, that is, Fn · · ·F1 +1 = F 2 m, has no solution in positive integers n,m. Mathematics Subject Classification (2000). Primary 11Dxx, Secondary 11B39

Newton’s method and secant methods: A longstanding relationship from vectors to matrices

Abstract: Nonlinear matrix equations arise in dierent scientific topics, such as applied statistics, control theory, and financial mathematics, among others. As in many other scientific areas, Newton's method has played an important role when solving these matrix problems. Under standard assumptions, the specialized Newton methods that have been developed for specific problems exhibit local and q-quadratic convergence and require a suitable initial guess. They also require, as usual, a significant amount of computational work per iteration, that in this case involve several matrix factorizations per iterations. As expected, whenever a Newton method can be developed, a secant method can also be developed. Indeed, more recently, secant methods for solving specific nonlinear matrix problems have been developed opening a new line of research. As in previous scenarios, these specialized secant methods exhibit local and q-superlinear convergence, also require a suitable initial guess, and avoid the use of derivatives in the formulation of the schemes. In this review we start by recalling the presence of Newton's method and the secant methods, and also their classical relationship, in dierent and sometimes unexpected scenarios for vector problems. Then we present and describe the state of the art in the use of Newton's method and also the secant method in the space of matrices. A second objec- tive is to present a unified approach for describing the features of these classical schemes, that in the space of matrices represent an interesting research area with special features to be explored.

Cross Varieties of Aperiodic Monoids with Central Idempotents

Abstract: Let A denote the class of all aperiodic monoids with central idempotents. A description of all Cross subvarieties of A, based on identities that they satisfy and monoids that they cannot contain, is given. The two limit subvarieties of A, published by Marcel Jackson in 2005, turn out to be the only finitely generated, almost Cross subvarieties of A. It follows that it is decidable in quartic time if a finite monoid in A generates a Cross variety. Mathematics Subject Classification (2010). 20M07.

On the endomorphism monoid of a profinite semigroup

Abstract: Necessary and sufficient conditions are given for the endomorphism monoid of a profinite semigroup to be profinite. A similar result is established for the automorphism group.

Bi-Lipschitz equivalent metrics on groups, and a problem in additive number theory

Abstract: There is a standard "word length" metric canonically associated to any set of generators for a group. In particular, for any integers a and b greater than 1, the additive group of integers has generating sets {a^i}_{i=0}^{\infty} and {b^j}_{j=0}^{\infty} with associated metrics d_A and d_B, respectively. It is proved that these metrics are bi-Lipschitz equivalent if and only if there exist positive integers m and n such that a^m = b^n.

Special divisors of large dimension on curves with many points over finite fields

Abstract: We prove a non-existence result for special divisors of large dimension on curves over finite fields with many points. We also give a family of examples where such divisors exist under less stringent hypotheses.

Product approximations for solutions to a class of evolution equations in Hilbert space

Abstract: In this article we prove approximation formulae for a class of unitary evolution operators U(t; s)s;t2[0;T ] associated with linear non-autonomous evolution equations of Schrödinger type de…ned in a Hilbert space H. An important feature of the equations we consider is that both the corresponding self-adjoint generators and their domains may depend explicitly on time, whereas the associated quadratic form domains may not. Furthermore the evolution operators we are interested in satisfy the equations in a weak sense. Under such conditions the approximation formulae we prove for U(t; s) involve weak operator limits of products of suitable approximating functions taking values in L(H), the algebra of all linear bounded operators on H. Our results may be relevant to the numerical analysis of U(t; s) and we illustrate them by considering two evolution problems in quantum mechanics. 1 Introduction and Outline Let H be an arbitrary complex Hilbert space and let L(H) be the algebra of all bounded linear operators de…ned on H. Our purpose in this article is to prove approximation formulae for the solutions to initial-value problems of the form i du(t) dt = H(t)u(t); 0 s < t T; u(s) = v; (1) where the H(t)’s are given self-adjoint and positive operators in H, with T 2 (0;+1) arbitrary. More speci…cally, assuming there exists a unitary evolution system UH(t; s)s;t2[0;T ] on H that solves (1) in a suitably weak sense, we display a large one-parameter family of functions Ft : R 7! L(H) such that formulae of the form UH(t; s) = lim n!+1 0 Y =n 1 Fs+ n (t s) t s n (2)