On Attractors for Multivalued Semigroups Defined by Generalized Semiflows

In this paper we extend results from Semigroup Theory on existence and characterization of attractors in order to include multivalued semigroups T(t) defined by generalized semiflows . In particular we show that, if is continuous, possesses a Lyapunov function, and has a global attractor which is maximal compact invariant, then  =  W ^u (Z()), where Z() is the stationary solutions set and W ^u (Z()) is the unstable set of Z(). We introduce the -attractor concept which does not enjoy any uniformity on time of attraction and we prove, under suitable conditions, that the global -attractor is the set of asymptotic states described by Z(). Jacson Simsen is supported by CAPES-Brazil.     

Applications of the Category of Linear Complexes of Tilting Modules Associated with the Category

We use the category of linear complexes of tilting modules for the BGG category \mathfrakg\mathfrak{g}, to reprove in purely algebraic way several known results about obtained earlier by different authors using geometric methods. We also obtain several new results about the parabolic category .     

Universal Coefficient Theorem in Triangulated Categories

We consider a homology theory on a triangulated category with values in an abelian category. If the functor h reflects isomorphisms, is full and is such that for any object x in there is an object X in with an isomorphism between h(X) and x, we prove that is a hereditary abelian category, all idempotents in split and the kernel of h is a square zero ideal which as a bifunctor on is isomorphic to. The second author is a researcher from CONICET, Argentina.     

Depth Generated Simple Lie Algebras

A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra . This determines a depth functor from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras are simple for all i ∈ ω. In particular, the (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras. Presented by Don Passman.     

Dynamic block GMRES: an iterative method for block linear systems

We present variants of the block-GMRES() algorithms due to Vital and the block-LGMRES(,) by Baker, Dennis and Jessup, obtained with replacing the standard QR factorization by a rank-revealing QR factorization in the Arnoldi process. The resulting algorithm allows for dynamic block deflation whenever there is a linear dependency between the Krylov vectors or the convergence of a right-hand-side occurs. implementations of the algorithms were tested on a number of test matrices and the results show that in some cases a substantial reduction of the execution time is obtained. Also a parallel implementation of our variant of the block-GMRES() algorithm, using and was tested on parallel computer, showing good parallel efficiency. This work was carried out while the author was at IM/UFRGS.     

Stability results for scattered‐data interpolation on Euclidean spheres

Let denote the unit sphere in and the geodesic distance in . A spherical‐basis function approximant is a function of the form , where are real constants, is a fixed function, and is a set of distinct points in . It is known that if is a strictly positive definite function in , then the interpolation matrix is positive definite, hence invertible, for every choice of distinct points and every positive integer M. The paper studies a salient subclass of such functions , and provides stability estimates for the associated interpolation matrices. This revised version was published online in June 2006 with corrections to the Cover Date.     

Symmetric inverse topological semigroups of finite rank ≤ <Emphasis Type="Italic">n</Emphasis>

We establish topological properties of the symmetric inverse topological semigroup of finite transformations of the rank ≤ n. We show that the topological inverse semigroup is algebraically h -closed in the class of topological inverse semigroups. Also we prove that a topological semigroup S with countably compact square S×S does not contain the semigroup for infinite cardinal λ and show that the Bohr compactification of an infinite topological symmetric inverse semigroup of finite transformations of the rank ≤ n is the trivial semigroup.     

The Representations of Cyclotomic BMW Algebras,II

In this paper, we give a recursive formula to compute the Gram determinant associated to each cell module of the cyclotomic BMW algebras over an integral domain. As a by-product, we determine explicitly when is semisimple over a field. This generalizes our previous result on Birman-Murakami-Wenzl algebras in Rui and Si (J Reine Angew Math 631:153–180, 2009).     

The best m-term approximation and greedy algorithms

Two theorems on nonlinear ‐term approximation in , are proved in this paper. The first one (theorem 2.1) says that if a basis is ‐equivalent to the Haar basis then a near best >‐term approximation to any can be realized by the following simple greedy type algorithm. Take the expansion and form a sum of terms with the largest out of this expansion. The second one (theorem 3.3) states that nonlinear ‐term approximations with regard to two dictionaries: the Haar basis and the set of all characteristic functions of intervals are equivalent in a very strong sense. This revised version was published online in June 2006 with corrections to the Cover Date.     

Equivalence of $n$-point Gauss–Chebyshev rule and $4n$-point midpoint rule in computing the period of a Lotka–Volterra system

Two integrals (3.6), (4.7) for the period of a periodic solution of the Lotka–Volterra system are presented in terms of two inverse functions of restricted on , , respectively. In computing this period numerically, the integral (3.6), which possesses a weak singularity of the square root type at each endpoint of the integration, is an excellent example of using the Gauss–Chebyshev integration rule of the first kind; while the integral (4.7), which is an integral of a smooth periodic function over its period , is an excellent example of using the midpoint rule, but not the trapezoidal rule, suggested by Waldvogel [39, 40], due to a removable singularity of the integrand at , , , , and , respectively. This paper shows, in computing the period of a periodic solution of the Lotka–Volterra system, the -point Gauss–Chebyshev integration rule of the first kind applied to the integral (3.6) becomes the -point midpoint rule to the integral (4.7). Dedicated to R. Bruce Kellogg on the occasion of his 75th birthday.     

Frames in spaces with finite rate of innovation

Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space modeling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wide-band communication. In particular, the space is generated by a family of well-localized molecules of similar size located on a relatively separated set using coefficients, and hence is locally finitely generated. Moreover that space includes finitely generated shift-invariant spaces, spaces of non-uniform splines, and the twisted shift-invariant space in Gabor (Wilson) system as its special cases. Use the well-localization property of the generator , we show that if the generator is a frame for the space and has polynomial (sub-exponential) decay, then its canonical dual (tight) frame has the same polynomial (sub-exponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator for the space with , and of the polynomial (sub-exponential) decay property of the mask associated with a refinable function that has polynomial (sub-exponential) decay.      

Gap, Excess and Bornological Convergence

Let be the nonempty subsets of a metric space 〈X, d〉. Some classical convergences in - such as convergence in Hausdorff distance, Attouch-Wets convergence and Wijsman convergence - have been shown to be compatible with the weak topology on induced by all gap and excess functionals with fixed left argument ranging in some bornology. Here we consider an arbitrary ideal of subsets of X and compare the gap and excess topology so generated with the corresponding convergence defined in terms of truncations by elements of the ideal. Dedicated to the memory of Flora Daniel.     

Holonomic -modules and positive characteristic

We discuss a hypothetical correspondence between holonomic -modules on an algebraic variety X defined over a field of zero characteristic, and certain families of Lagrangian subvarieties in the cotangent bundle to X. The correspondence is based on the reduction to positive characteristic. This article is based on the 5th Takagi Lectures that the author delivered at the University of Tokyo on October 4 and 5, 2008.     

Non-linear sampling recovery based on quasi-interpolant wavelet representations

We investigate a problem of approximate non-linear sampling recovery of functions on the interval expressing the adaptive choice of n sampled values of a function to be recovered, and of n terms from a given family of functions Φ. More precisely, for each function f on , we choose a sequence of n points in , a sequence of n functions defined on and a sequence of n functions from a given family Φ. By this choice we define a (non-linear) sampling recovery method so that f is approximately recovered from the n sampled values f(ξ ¹), f(ξ ²),..., f(ξ ⁿ ), by the n-term linear combination

Convexities of partial monounary algebras

In the present paper we prove that the collection of all convexities of partial monounary algebras is finite; namely, it has exactly 23 elements. Further, we show that for each element there exists a subset of such that is generated by and card . This work was supported by the Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by Grant VEGA 1/3003/06.     

A rank six geometry related to the McLaughlin sporadic simple group

In the literature, there are but a few incidence geometries on which the McLaughlin sporadic group acts as a flag-transitive automorphism group. Their highest rank is four. In the present paper, we construct a geometry of rank six on which acts flag-transitively and which has the following diagram.      

Note on the Construction of Free Monoids

We construct free monoids in a monoidal category with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains. In particular this will be the case if tensoring preserves sifted colimits.     

Gabor frames by sampling and periodization

By sampling the window of a Gabor frame for belonging to Feichtinger’s algebra, , one obtains a Gabor frame for . In this article we present a survey of results by R. Orr and A.J.E.M. Janssen and extend their ideas to cover interrelations among Gabor frames for the four spaces , , and . Some new results about general dual windows with respect to sampling and periodization are presented as well. This theory is used to show a new result of the Kaiblinger type to construct an approximation to the canonical dual window of a Gabor frame for .      

Quasi-equational bases for graphs of semigroups,monoids and groups

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. Let denote by the class of all graphs of algebras from a class . We prove that if is a class of semigroups possessing a nontrivial member with a neutral element, then does not have finite quasi-equational basis. We deduce that, for a class of monoids or groups with a nontrivial member, also does not have finite quasi-equational basis.     

Solution of elementary equations in the Minkowski geometric algebra of complex sets

The solution of elementary equations in the Minkowski geometric algebra of complex sets is addressed. For given circular disks and with radii a and b, a solution of the linear equation in an unknown set exists if and only if ab. When it exists, the solution is generically the region bounded by the inner loop of a Cartesian oval (which may specialize to a limaçon of Pascal, an ellipse, a line segment, or a single point in certain degenerate cases). Furthermore, when a<b<1, the solution of the nonlinear monomial equation is shown to be the region that is bounded by a single loop of a generalized form of the ovals of Cassini. The latter result is obtained by considering the nth Minkowski root of the region bounded by the inner loop of a Cartesian oval. Preliminary consideration is also given to the problems of solving univariate polynomial equations and multivariate linear equations with complex disk coefficients.