Numerical Properties of Shifted Tridiagonal LU Factorizations

In this paper, we consider shifted tridiagonal matrices. We prove that the standard algorithm to compute the LU factorization in this situation is mixed forward-backward stable and, therefore, componentwise forward stable. Moreover, we give a formula to compute the corresponding condition number in O(n) flops. This research has been partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain through grants BFM2003-06335-C03-02 and MTM2006-06671 as well as by the Postdoctoral Fellowship EX2004-0658 provided by Ministerio de Educación y Ciencia of Spain.     

On a question of chambers and makan

This work has been supported by Proyecto PB 90-0414-C03-01 of DGICYT, Ministerio de Educación Ciencia of Spain.     

Optimality Conditions for Metrically Consistent Approximate Solutions in Vector Optimization

In this paper, approximate solutions of vector optimization problems are analyzed via a metrically consistent ε-efficient concept. Several properties of the ε-efficient set are studied. By scalarization, necessary and sufficient conditions for approximate solutions of convex and nonconvex vector optimization problems are provided; a characterization is obtained via generalized Chebyshev norms, attaining the same precision in the vector problem as in the scalarization. This research was partially supported by the Ministerio de Educación y Ciencia (Spain), Project MTM2006-02629 and by the Consejería de Educación de la Junta de Castilla y León (Spain), Project VA027B06. The authors are grateful to the anonymous referees for helpful comments and suggestions.     

Evolution families and the Loewner equation II: complex hyperbolic manifolds

We prove that evolution families on complex complete hyperbolic manifolds are in one to one correspondence with certain semicomplete non-autonomous holomorphic vector fields, providing the solution to a very general Loewner type differential equation on manifolds. M. D. Contreras and S. Díaz-Madrigal were partially supported by the Ministerio de Ciencia e Innovación and the European Union (FEDER), project MTM2006-14449-C02-01, by La Consejería de Educación y Ciencia de la Junta de Andalucía, and by the European Science Foundation Research Networking Programme HCAA.     

Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations

We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case, we consider in this paper nonincreasing mappings as well as non monotone mappings. We also present some applications to first–order ordinary differential equations with periodic boundary conditions, proving the existence of a unique solution admitting the existence of a lower solution. Research partially supported by Ministerio de Educación y Ciencia and FEDER, Project MTM2004-06652-C03-01, and by Xunta de Galicia and FEDER, Projects PGIDIT02PXIC20703PN and PGIDIT05PXIC20702PN     

Rational quadrature formulae on the unit circle with arbitrary poles

Interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered. The poles are prescribed under the only restriction of not lying on the unit circle. A computable upper bound of the error is obtained which is valid for any choice of poles, arbitrary weight functions and any degree of exactness provided that the integrand is analytic on a neighborhood of the unit circle. A number of numerical examples are given which show the advantages of using such rules as well as the sharpness of the error bound. Also, a comparison is made with other error bounds appearing in the literature. The work of the first author was supported by the Dirección General de Investigación, Ministerio de Educación y Ciencia, under grants MTM2006-13000-C03-02 and MTM2006-07186 and by UPM and Comunidad de Madrid under grant CCG06-UPM/MTM-539. The work of the second author was partially supported by the Dirección General de Investigación, Ministerio de Educación y Ciencia, under grant MTM2005-08571.     

A multi-local optimization algorithm

The development of efficient algorithms that provide all the local minima of a function is crucial to solve certain subproblems in many optimization methods. A “multi-local” optimization procedure using inexact line searches is presented, and numerical experiments are also reported. An application of the method to a semi-infinite programming procedure is included. This work was partially supported by Ministerio de Educación y Ciencia, Spain, DGICYT grant PB93-0703. Author (*) was supported by the Consellería d'Educació i Ciència of the Generalitat Valenciana.     

Real characters and degrees

Several classical theorems on character degrees are revisited from the point of view of the real characters. The research of the first two authors is partially supported by the Spanish Ministerio de Educación y Ciencia (MTM2004-06067-C02-01). The third author gratefully acknowledges the support of the NSF (grant DMS-0600967)     

Characterizing efficiency without linear structure: a unified approach

In this paper, we study a general optimization problem without linear structure under a reflexive and transitive relation on a nonempty set E, and characterize the existence of efficient points and the domination property for a subset of E through a generalization of the order-completeness condition introduced earlier. Afterwards, we study the abstract optimization problem by using generalized continuity concepts and establish various existence results. As an application, we extend and improve several existence results given in the literature for an optimization problem involving set-valued maps under vector and set criteria. The work of the first author was partially supported by CONICYT-Chile through FONDECYT 107-0689 and FONDAP-Matemáticas Aplicadas II; whereas that of the second and third author were supported in part by Ministerio de Educación y Ciencia (Spain), project MTM2006-02629 and by Junta de Castilla y León (Spain) Project VA027B06.     

A counterexample on non-archimedean regularity

A non-regular inductive sequence of non-archimedean reflexive Fréchet spaces is constructed. On the other hand, it is proved that every inductive sequence of reflexive Banach spaces over a spherically complete field is regular. Also, some applications are given. Research partially supported by Ministerio de Educación y Ciencia, MTM2006-14786. Authors’ addresses: N. De Grande-De Kimpe, Groene Laan 36 (302) B 2830, Willebroek, Belgium; C. Perez-Garcia, Department of Mathematics, Facultad de Ciencias, Universidad de Cantabria, Avda. de los Castros s/n, 39071 Santander, Spain     

An extension to rapid transit network design problem

The rapid transit network design problem consists of the location of train alignments and stations, in a context where the demand makes its own decisions about the mode and route. The originality of this study is to incorporate in the model the line locations constraints with a bounded but variable number of lines, and lines with no predetermined origins and destinations. The computational experiments show the necessity of this extension to solve large networks, principally because of its computational advantage. The project has been supported by Ministerio de Educación y Ciencia (Spain) under project TRA-2005-09068-C03-01/MODAL, and by Ministerio de Fomento (Spain) under project 2005/70029/T05.     

Gorenstein quivers

In this paper we characterize when the path ring associated to a quiver is Gorenstein (in the sense of Iwanaga [9]). Then, by using the notion of a Gorenstein category (cf. [2]), we extend the classes of quivers whose corresponding category of representations has finite Gorenstein global dimension. This extension includes non-noetherian quivers. E. E., S.E., and J.R.G.R., partially supported by the DGI MTM2005-03227. Estrada’s work was supported by a MEC/Fulbright grant from the Spanish Secretaría de Estado de Universidades e Investigación del Ministerio de Educación y Ciencia. Received: 28 February 2006     

Local embeddings of some families of subgroups of finite groups

In this paper we study the influence of the partial cover and avoidance property on the subgroups of some relevant families of subgroups in a finite group.     

Semigroups of Composition Operators in BMOA and the Extension of a Theorem of Sarason

In this paper we deal with the maximal subspace in BMOA where a general semigroup of analytic functions on the unit disk generates a strongly continuous semigroup of composition operators. Particular cases of this question are related to a well-known theorem of Sarason about VMOA. Our results describe analytically that maximal subspace and provide a condition which is sufficient for the maximal subspace to be exactly VMOA. A related necessary condition is also proved in the case when the semigroup has an inner Denjoy-Wolff point. As a byproduct we provide a generalization of the theorem of Sarason. This research has been partially supported by the Ministerio de Educación y Ciencia projects n. MTM2006-14449-C02-01 and MTM2005-08350-C03-03 and by La Consejería de Educación y Ciencia de la Junta de Andalucía.     

A Set-Valued Analysis Approach to Second Order Differentiation of Nonsmooth Functions

A theory of second order differentiability for real functions with calm Clarke gradient at points of strict differentiability is presented from a set-valued analysis perspective by using the theory of graphical derivatives of set-valued maps. Connections with other second order theories and applications in optimization are given. This work is partially supported by Ministerio de Educación y Ciencia (Spain), project MTM2006-02629 and IngenioMathematica (i-MATH) CSD2006-00032 (Consolider-Ingenio 2010).     

Groups with three real valued irreducible characters

We obtain strong restrictions on the structure of a Sylow 2-subgroup of a group with at most three real valued irreducible characters. This extends results of Iwasaki, who studied groups with at most two real valued irreducible characters. This research was partially supported by the Spanish Ministerio de Educación y Ciencia, MTM2004-06067-C02-01 and MTM2004-04665, and the FEDER. The first author was also supported by the Programa Ramón y Cajal and the Generalitat Valenciana.     

New Constraint Qualification and Conjugate Duality for Composed Convex Optimization Problems

The approach to optimal control problems based on a purely variational reformulation may lead to new existence results by using fine, general existence theorems for variational problems without convexity assumptions. We illustrate this perspective here for autonomous one-dimensional problems and defer the study of more complex situations to later work. This research was supported by Project MTM2004-07114 from Ministerio de Educación y Ciencia (Spain) and by Grant PAI05-029 from JCCM (Castilla-La Mancha).     

On the Use of Cellular Automata in Symmetric Cryptography

In this work, pseudorandom sequence generators based on finite fields have been analyzed from the point of view of their cryptographic application. In fact, a class of nonlinear sequence generators has been modelled in terms of linear cellular automata. The algorithm that converts the given generator into a linear model based on automata is very simple and is based on the concatenation of a basic structure. Once the generator has been linearized, a cryptanalytic attack that exploits the weaknesses of such a model has been developed. Linear cellular structures easily model sequence generators with application in stream cipher cryptography.Work supported by Ministerio de Educación y Ciencia (Spain), Projects SEG2004-02418 and SEG2004-04352-C04-03.     

New Second-Order Directional Derivative and Optimality Conditions in Scalar and Vector Optimization

We present a new second-order directional derivative and study its properties. Using this derivative and the parabolic second-order derivative, we establish second-order necessary and sufficient optimality conditions for a general scalar optimization problem by means of the asymptotic and parabolic second-order tangent sets to the feasible set. For the sufficient conditions, the initial space must be finite dimensional. Then, these conditions are applied to a general vector optimization problem obtaining second-order optimality conditions that generalize the differentiable case. For this aim, we introduce a scalarization, and the relationships between the different types of solutions to the vector optimization problem and the scalarized problem are studied. This research was partially supported by the Ministerio de Educación y Ciencia (Spain), under projects MTM2006-02629 and Ingenio Mathematica (i-MATH) CSD2006-00032 (Consolider-Ingenio 2010), and by the Consejería de Educación de la Junta de Castilla y León (Spain), Project VA027B06. The authors are grateful to the anonymous referees for valuable comments and suggestions.     

Infinitesimal generators associated with semigroups of linear fractional maps

We characterize the infinitesimal generator of a semigroup of linear fractional self-maps of the unit ball in ℂⁿ, n ≥ 1. For the case n = 1, we also completely describe the associated Koenigs function and solve the embedding problem from a dynamical point of view, proving (among other things) that a generic semigroup of holomorphic self-maps of the unit disc is a semigroup of linear fractional maps if and only if it contains a linear fractional map for some positive time. Partially supported by the Ministerio de Ciencia y Tecnología and the European Union (FEDER) project BFM2003-07294-C02-02 and by La Consejería de Educación y Ciencia de la Junta de Andalucía.