The extended Freudenthal Magic Square and Jordan algebras

The Lie superalgebras in the extended Freudenthal Magic Square in characteristic 3 are shown to be related to some known simple Lie superalgebras, specific to this characteristic, constructed in terms of orthogonal and symplectic triple systems, which are defined in terms of central simple degree three Jordan algebras. I. Cunha was supported by CMUC, Department of Mathematics, University of Coimbra. A. Elduque was supported by the Spanish Ministerio de Educación y Ciencia and FEDER (MTM 2004-081159-C04-02) and by the Diputación General de Aragón (Grupo de Investigación de álgebra).     

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.     

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.     

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.     

Local Rings of Rings of Quotients

The aim of this paper is to characterize those elements in a semiprime ring R for which taking local rings at elements and rings of quotients are commuting operations. If Q denotes the maximal ring of left quotients of R, then this happens precisely for those elements if R which are von Neumann regular in Q. An intrinsic characterization of such elements is given. We derive as a consequence that the maximal left quotient ring of a prime ring with a nonzero PI-element is primitive and has nonzero socle. If we change Q to the Martindale symmetric ring of quotients, or to the maximal symmetric ring of quotients of R, we obtain similar results: an element a in R is von Neumann regular if and only if the ring of quotients of the local ring of R at a is isomorphic to the local ring of Q at a. Partially supported by the Ministerio de Educación y Ciencia and Fondos Feder, jointly, trough projects MTM2004-03845, MTM2007-61978 and MTM2004-06580-C02-02, MTM2007-60333, by the Junta de Andalucía, FQM-264, FQM336 and FQM02467 and by the Plan de Investigación del Principado de Asturias FICYT-IB05-017.     

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)     

All periodic minimizers are unstable

We consider the periodic action functional associated to some Lagrangian verifying the Legendre convexity condition and show that all minimizers are unstable. Supported by D.G.I. MTM2005-03483, Ministerio de Educación y Ciencia, Spain.     

The isodiametric problem for lattice-point-free convex sets

The isodiametric problem in the Euclidean plane is solved for lattice-point-free convex sets: we characterize the planar convex sets containing no points of the rectangular lattice in their interior, which have maximum area, for each given value of the diameter. Then we use this result for giving also an answer to the isodiametric problem in the case of an arbitrary lattice. The first author is supported by Dirección General de Investigación (MEC) MTM2004-04934-C04-02 and by Fundación Séneca (C.A.R.M.) 00625/PI/04.     

Holomorphic Operators Generating Dense Images

The existence of infinite dimensional closed linear spaces of holomorphic functions f on a domain G in the complex plane such that Tf has dense images on certain subsets of G, where T is a continuous linear operator, is analyzed. Necessary and sufficient conditions for T to have the latter property are provided and applied to obtain a number of concrete examples: infinite order differential operators, composition operators and multiplication operators, among others. This work was supported in part by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 and by MEC DGES Grants MTM2006-13997-C02-01 and MTM2004-21420-E.     

Cyclicity of coefficient multipliers: Linear structure

We characterize various kinds of cyclicity of sequences of coefficient multipliers, which are operators defined on spaces of holomorphic functions. In the case of a single coefficient multiplier we characterize its cyclicity, which contrasts with the fact that such operators are never supercyclic. Moreover, it is proved that for each cyclic function there is a dense part of the linear span of its orbit all of whose vectors are cyclic. The authors have been partially supported by MCYT Grants BFM2003-03893-C02-01, MTM2004-21420-E and Plan Andaluz de Investigación Junta de Andalucía FQM-127.     

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.     

Interpolation of Ideal Measures by Abstract K and J Spaces

We work with the abstract K and J interpolation method generated by a sequence lattice Г. We investigate the deviation of an interpolated operator from a given operator ideal by establishing formulae for the ideal measure of the interpolated operator in terms of the ideal measures of restrictions of the operator. Formulae are given in terms of the norms of the shift operators on Г.     

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 new coupling of mixed finite element and boundary element methods for an exterior Helmholtz problem in the plane

This paper deals with the scattering of time harmonic electromagnetic waves by an infinitely long cylinder containing a non-homogeneous conducting medium. More precisely, we study the transverse magnetic field that solves an interface problem holding between the cross section of the cylinder and the exterior two-dimensional free space. We apply a dual-mixed variational formulation in the obstacle coupled with a boundary integral equation method in the unbounded homogeneous space. A Fredholm alternative is utilized to prove that this continuous formulation is well posed. We define the corresponding discrete scheme by using the lowest order rotated Raviart-Thomas finite elements for the magnetic field and spectral elements for the boundary unknown. Then, we show that the resulting Galerkin scheme is uniquely solvable and convergent, and prove optimal error estimates. Finally, we illustrate our analysis with some results from computational experiments. This research was partially supported by CONICYT-Chile through the FONDAP Program in Applied Mathematics, by the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program, and by the Ministerio de Educación y Ciencia of Spain, through the project No. MTM2004-05417.     

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.     

A high order uniformly convergent alternating direction scheme for time dependent reaction–diffusion singularly perturbed problems

In this work we design and analyze an efficient numerical method to solve two dimensional initial-boundary value reaction–diffusion problems, for which the diffusion parameter can be very small with respect to the reaction term. The method is defined by combining the Peaceman and Rachford alternating direction method to discretize in time, together with a HODIE finite difference scheme constructed on a tailored mesh. We prove that the resulting scheme is ε-uniformly convergent of second order in time and of third order in spatial variables. Some numerical examples illustrate the efficiency of the method and the orders of uniform convergence proved theoretically. We also show that it is easy to avoid the well-known order reduction phenomenon, which is usually produced in the time integration process when the boundary conditions are time dependent. This research has been partially supported by the project MEC/FEDER MTM2004-01905 and the Diputación General de Aragón.     

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).     

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant Members of VBAC (Vector Bundles on Algebraic Curves). Second and Third authors partially supported by Ministerio de Educación y Ciencia and Conselho de Reitores das Universidades Portuguesas through Acción Integrada Hispano-Lusa HP2002-0017 (Spain)/E–30/03 (Portugal). First and Second authors partially supported by Ministerio de Educación y Ciencia (Spain) through Project MTM2004-07090-C03-01. Third author partially supported by the Centro de Matemática da Universidade do Porto and the project POCTI/MAT/58549/2004, financed by FCT (Portugal) through the programmes POCTI and POSI of the QCA III (2000–2006) with European Community (FEDER) and national funds. The second author visited the IHES with the partial support of the European Commission through its 6th Framework Programme “Structuring the European Research Area” and the Contract No. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action     

Wall Rational Functions and Khrushchev’s Formula for Orthogonal Rational Functions

The Nevalinna–Pick algorithm yields a continued fraction expansion of every Schur function, whose approximants are identified. These approximants are quotients of rational functions which can be understood as the rational analogs of the Wall polynomials. The properties of these Wall rational functions and the corresponding approximants permit us to obtain a Khrushchev’s formula for orthogonal rational functions. An introduction to the convergence of the Wall approximants in the indeterminate case is presented. This work was partially realized during two stays of the second author at the Norwegian University of Science and Technology (NTNU) financed respectively by Secretaría de Estado de Universidades e Investigación from the Ministry of Education and Science of Spain and by the Department of Mathematical Sciences of NTNU. The work of the second author was also partially supported by the Spanish grants from the Ministry of Education and Science, project code MTM2005-08648-C02-01, and the Ministry of Science and Innovation, project code MTM2008-06689-C02-01, and by Project E-64 of Diputación General de Aragón (Spain).     

Strategic absentmindedness in finitely repeated games

In this paper we consider finitely repeated games in which players can unilaterally commit to behave in an absentminded way in some stages of the repeated game. We prove that the standard conditions for folk theorems can be substantially relaxed when players are able to make this kind of compromises, both in the Nash and in the subgame perfect case. We also analyze the relation of our model with the repeated games with unilateral commitments studied, for instance, in García-Jurado et al. (Int. Game Theory Rev. 2:129–139, 2000). Authors acknowledge the financial support of Ministerio de Educaci ón y Ciencia, FEDER and Fundación Séneca de la Región de Murcia through projects SEJ2005-07637-C02-02, ECO2008-03484-C02-02, MTM2005-09184-C02-02, MTM2008-06778-C02-01 and 08716/PI/08.