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.     

Deformation Techniques for Sparse Systems

We exhibit a probabilistic symbolic algorithm for solving zero-dimensional sparse systems. Our algorithm combines a symbolic homotopy procedure, based on a flat deformation of a certain morphism of affine varieties, with the polyhedral deformation of Huber and Sturmfels. The complexity of our algorithm is cubic in the size of the combinatorial structure of the input system. This size is mainly represented by the cardinality and mixed volume of Newton polytopes of the input polynomials and an arithmetic analogue of the mixed volume associated to the deformations under consideration. Research was partially supported by the following grants: UBACyT X112 (2004–2007), UBACyT X847 (2006–2009), PIP CONICET 2461, PIP CONICET 5852/05, ANPCyT PICT 2005 17-33018, UNGS 30/3005, MTM2004-01167 (2004–2007), MTM2007-62799 and CIC 2007–2008.     

Sobolev Spaces with Respect to Measures in Curves and Zeros of Sobolev Orthogonal Polynomials

In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial results about these Sobolev spaces with respect to measures; in particular, we prove a main result in the theory: they are Banach spaces. J.M. Rodriguez supported in part by three grants from M.E.C. (MTM 2006-13000-C03-02, MTM 2006-11976 and MTM 2007-30904-E), Spain, and by a grant from U.C.III M./C.A.M. (CCG07-UC3M/ESP-3339), Spain. J.M. Sigarreta supported in part by a grant from M.E.C. (MTM 2006-13000-C03-02), Spain, and by a grant from U.C.III M./C.A.M. (CCG07-UC3M/ESP-3339), Spain.     

A maschke type theorem for weak Hopf algebras

In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke＇s Theorem for （H, B）-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.     

Partially ordered cooperative games: extended core and Shapley value

In this paper we analyze cooperative games whose characteristic function takes values in a partially ordered linear space. Thus, the classical solution concepts in cooperative game theory have to be revisited and redefined: the core concept, Shapley–Bondareva theorem and the Shapley value are extended for this class of games. The classes of standard, vector-valued and stochastic cooperative games among others are particular cases of this general theory. The research of the authors is partially supported by Spanish DGICYT grant numbers MTM2004-0909, HA2003-0121, HI2003-0189, MTM2007-67433-C02-01, P06-FQM-01366.     

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.     

Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

This paper studies the construction of geometric integrators for nonholonomic systems. We develop a formalism for nonholonomic discrete Euler–Lagrange equations in a setting that permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie groupoid. We also discuss the existence of nonholonomic evolution operators in terms of the discrete nonholonomic Legendre transformations and in terms of adequate decompositions of the prolongation of the Lie groupoid. The characterization of the reversibility of the evolution operator and the discrete nonholonomic momentum equation are also considered. Finally, we illustrate with several classical examples the wide range of application of the theory (the discrete nonholonomic constrained particle, the Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on a rotating table and the two wheeled planar mobile robot). This work was partially supported by MEC (Spain) Grants MTM 2006-03322, MTM 2007-62478, MTM 2006-10531, project “Ingenio Mathematica” (i-MATH) No. CSD 2006-00032 (Consolider-Ingenio 2010) and S-0505/ESP/0158 of the CAM.     

High order iterative schemes for quadratic equations

High order iterative methods for quadratic equations are studied. A bi-parametric family that includes some well known iterative schemes is introduced. A unified semilocal convergence theorem is presented. The implementation and some applications to partial differential equations are finally discussed. Research supported in part by the Spanish grant MTM2007-62945.     

Dynamics of the differentiation operator on weighted spaces of entire functions

The continuity of the differentiation operator on weighted Banach spaces of entire functions with sup-norm has been characterized recently by Harutyunyan and Lusky. We give necessary and sufficient conditions to ensure that the differentiation operator on these weighted Banach spaces of entire functions is hypercyclic or chaotic, when it is continuous. This research was partially supported by MEC and FEDER Project MTM2007-62643.     

A method for the resolution of the Jacobi equation <Emphasis Type="Italic">Y</Emphasis>″ + <Emphasis Type="Italic">RY</Emphasis> = 0 on the manifold <Emphasis Type="Italic">Sp</Emphasis>(2)/<Emphasis Type="Italic">SU</Emphasis>(2)

In this paper a method for the resolution of the differential equation of the Jacobi vector fields in the manifold V ₁ = Sp(2)/SU(2) is exposed. These results are applied to determine areas and volumes of geodesic spheres and balls. Work partially supported by DGI (Spain) and FEDER Projects MTM 2004-06015-C02-01 and MTM 2007-65852 (first author) and by Research Project PGIDIT05PXIB16601PR (second author). Authors’ addresses: A. M. Naveira, Departamento de Geometría y Topología. Facultad de Matemáticas, Avda. Andrés Estellés, N1, 46100 – Burjassot, Valencia, Spain; A. D. Tarrío Tobar, E. U. Arquitectura Técnica, Campus A Zapateira. Universidad de A Coru?a, 15192 – A Coru?a, Spain     

Controllability of Semilinear Differential Systems with Nonlocal Initial Conditions in Banach Spaces

In this note, we establish a sufficient condition for the controllability of a first-order semilinear differential system with nonlocal initial conditions in Banach spaces. The approach used is the Sadovskii fixed-point theorem combined with operator semigroups. Particularly, the compactness of the operator semigroups is not needed in this article. Y.K. Chang was supported by Tianyuan Youth Fund of Mathematics in China (10826063), NNSF of China (10801065), the Scientific Research Fund of Gansu Provincial Education Department (20868), and Qing Lan Talent Engineering Funds (QL-05-16A) of Lanzhou Jiaotong University. J.J. Nieto was supported by Ministerio de Educacion y Ciencia-FEDER Project MTM2007-61724, and by Xunta de Galicia-FEDER Project PGIDIT05PXIC20702PN.     

Quasiconvexification of sets in optimal design

By a suitable reformulation of a typical optimal design problem in conductivity, we treat the issue of determining the quasiconvexification (to fixed volume fraction) of certain sets of matrices that are union of two manifolds. By means of a helpful lemma, we are able to explicitly compute such hulls in a rather straightforward fashion without the need of seeking laminates by hand, and hence avoiding some tedious computations. We examine the linear case, both elliptic and hyperbolic, in 2 and higher dimension (by looking at a div–curl situation), as well as an interesting non-linear example that provides some clue as to what kind of characterization one may hope for in non-linear situations. This work was supported by the research projects MTM2007-62945 of the MCyT (Spain) and PCI08-0084-0424 of the JCCM (Castilla-La Mancha).     

Stability of Indices in the KKT Conditions and Metric Regularity in Convex Semi-Infinite Optimization

This paper deals with a parametric family of convex semi-infinite optimization problems for which linear perturbations of the objective function and continuous perturbations of the right-hand side of the constraint system are allowed. In this context, Cánovas et al. (SIAM J. Optim. 18:717–732, [2007]) introduced a sufficient condition (called ENC in the present paper) for the strong Lipschitz stability of the optimal set mapping. Now, we show that ENC also entails high stability for the minimal subsets of indices involved in the KKT conditions, yielding a nice behavior not only for the optimal set mapping, but also for its inverse. Roughly speaking, points near optimal solutions are optimal for proximal parameters. In particular, this fact leads us to a remarkable simplification of a certain expression for the (metric) regularity modulus given in Cánovas et al. (J. Glob. Optim. 41:1–13, [2008]) (and based on Ioffe (Usp. Mat. Nauk 55(3):103–162, [2000]; Control Cybern. 32:543–554, [2003])), which provides a key step in further research oriented to find more computable expressions of this regularity modulus. This research was partially supported by Grants MTM2005-08572-C03 (01-02) and MTM2006-27491-E (MEC, Spain, and FEDER, E.U.), ACOMP06/117-203 and ACOMP/2007/247-292 (Generalitat Valenciana, Spain), and CIO (UMH, Spain).     

On Impulsive Hyperbolic Differential Inclusions with?Nonlocal Initial Conditions

This paper is focused mainly upon existence of solutions for a second-order impulsive hyperbolic differential inclusions with nonlocal initial conditions. By using some well-known fixed-point theorems, existence theorems are established when the multivalued map has convex or nonconvex values. As applications of these main theorems, some consequences are given for the sublinear growth cases. Y.-K. Chang was supported by “Qing Lan” Talent Engineering Funds (QL-05-16A), by Lanzhou Jiaotong University, the Scientific Research Fund of Gansu Provincial Education Department (0704-14). J.J. Nieto was supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, Project PGIDIT05PXIC20702PN.     

Robust estimation and classification for functional data via projection-based depth notions

Five notions of data depth are considered. They are mostly designed for functional data but they can be also adapted to the standard multivariate case. The performance of these depth notions, when used as auxiliary tools in estimation and classification, is checked through a Monte Carlo study. Research partially supported by Spanish grants MTM2004-00098 (A. Cuevas and R. Fraiman) and MTM2005-00820 (M. Febrero).     

Gromov hyperbolicity of Denjoy Domains

In this paper we characterize the Gromov hyperbolicity of the double of a metric space. This result allows to give a characterization of the hyperbolic Denjoy domains, in terms of the distance to of the points in some geodesics. In the particular case of trains (a kind of Riemann surfaces which includes the flute surfaces), we obtain more explicit criteria which depend just on the lengths of what we have called fundamental geodesics. Research partially supported by three grants from M.E.C. (MTM 2006-11976, MTM 2006-13000-C03-02 and MTM 2004-21420-E), Spain.     

Complexity of Bezout’s Theorem VII: Distance Estimates in the Condition Metric

We study geometric properties of the solution variety for the problem of approximating solutions of systems of polynomial equations. We prove that given the two pairs (f _i ,ζ _i ), i=1,2, there exist a short path joining them such that the complexity of following the path is bounded by the logarithm of the condition number of the problems. Research was partially supported by MTM2007-62799 and by an NSERC discovery grant.     

Strong factorization of operators on spaces of vector measure integrable functions and unconditional convergence of series

In this paper, we characterize the space of multiplication operators from an L ^p -space into a space L ¹(m) of integrable functions with respect to a vector measure m, as the subspace defined by the functions that have finite p-semivariation. We prove several results concerning the Banach lattice structure of such spaces. We obtain positive results—for instance, they are always complete, and we provide counterexamples to prove that other properties are not satisfied—for example, simple functions are not in general dense. We study the operators that factorize through , and we prove an optimal domain theorem for such operators. We use our characterization to generalize the Bennet–Maurey–Nahoum Theorem on decomposition of functions that define an unconditionally convergent series in L ¹[0,1] to the case of 2-concave Banach function spaces. The research was partially supported by Proyecto MTM2005-08350-c03-03 and MTN2004-21420-E (J. M. Calabuig). Proyecto CONACyT 42227 (F. Galaz-Fontes). Proyecto MTM2006-11690-c02-01 and Feder (E. Jiménez-Fernández and E. A. Sánchez Pérez).     

The Diagonal Functors

We obtain new categorical proofs that generalize the diagonal principles introduced in Castillo and Moreno (Israel J. Math. 140:253–270, 2004) to study the automorphic and partially automorphic character of Banach spaces. We then introduce and study the automorphy index for a Banach space, showing that while . This research has been supported in part by project MTM2004-02635.     

Smash Products for Secondary Homotopy Groups

We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered. The second author was partially supported by the Spanish Ministry of Education and Science under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research contract.