Invariants and spaces of zero solutions of linear partial differential operators

It is shown that for open convex , d > 1 and a nontrivial polynomial P the space does not have property . If P is elliptic or homogeneous, then this holds for every open Ω. For even cannot occur and if it occurs for some Ω, then P must be hypoelliptic. Received: 18 July 2005     

{\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pairs}}

Let R be a unital associative ring and two classes of left R-modules. In this paper we introduce the notion of a In analogy to classical cotorsion pairs as defined by Salce [10], a pair of subclasses and is called a if it is maximal with respect to the classes and the condition for all and Basic properties of are stated and several examples in the category of abelian groups are studied. Received: 17 March 2005     

Line bundles over a moduli space of logarithmic connections on a Riemann surface

We consider logarithmic connections, on rank n and degree d vector bundles over a compact Riemann surface X, singular over a fixed point x₀ ∈ X with residue in the center of the integers n and d are assumed to be mutually coprime. A necessary and sufficient condition is given for a vector bundle to admit such a logarithmic connection. We also compute the Picard group of the moduli space of all such logarithmic connections. Let denote the moduli space of all such logarithmic connections, with the underlying vector bundle being of fixed determinant L, and inducing a fixed logarithmic connection on the determinant line L. Let be the Zariski open dense subset parametrizing all connections such that the underlying vector bundle is stable. The space of all global sections of certain line bundles on are computed. In particular, there are no nonconstant algebraic functions on Therefore, there are no nonconstant algebraic functions on although is biholomorphic to a representation space which admits nonconstant algebraic functions. The moduli space admits a natural compactification by a smooth divisor. We investigate numerically effectiveness of this divisor at infinity. It turns out that the divisor is not numerically effective in general. Received: March 2004 Revision: May 2004 Accepted: May 2004     

Sobolev spaces with only trivial isometries, II

In this article, we obtain a canonical form for surjective linear isometries provided U is an open, bounded, connected, domain with Lipschitz boundary, and . We will show there exists |c| = 1 and mapping τ that is a composition of a translation and a sign-changing permutation of coordinates such that Tf = cf(τ). As a corollary, if , all surjective isometries have this trivial form by the Sobolev Imbedding Theorem.      

Exponential Dichotomy on the Real Line and Admissibility of Function Spaces

The purpose of this paper is to give characterizations for uniform exponential dichotomy of evolution families on the real line. We consider a general class of Banach function spaces denoted and we prove that if with and the pair is admissible for an evolution family then is uniformly exponentially dichotomic. By an example we show that the admissibility of the pair for an evolution family is not a sufficient condition for uniform exponential dichotomy. As applications, we deduce necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pairs and with      

Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let denote a Bose-Mesner algebra on a finite nonempty set X. Fix p X, and let and denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of with respect to p. By a hyper-duality of , we mean an automorphism of such that for all ; and is a duality of . is said to be hyper-self-dual whenever there exists a hyper-duality of . We say that is strongly hyper-self-dual whenever there exists a hyper-duality of which can be expressed as conjugation by an invertible element of . We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.     

Strong convergence of viscosity approximation methods with strong pseudocontraction for Lipschitz pseudocontractive mappings

In this paper, for a Lipschitz pseudocontractive mapping T, we study the strong convergence of iterative schemes generated by

Sampling Theory Associated with a Symmetric Operator with Compact Resolvent and De Branges Spaces

Let be a symmetric operator with compact resolvent defined in a Hilbert space For any fixed we consider an entire function K_a which involves the resolvent of Associated with K_a we obtain, by duality in a Hilbert space of entire functions which becomes a De Branges space of entire functions. This property provides a characterization of regardless of the anti-linear mapping which has as its range space. There exists also a sampling formula allowing to recover any function in from its samples at the sequence of eigenvalues of This work has been supported by the grant BFM2003–01034 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología.     

On the nuclearity of integral operators

Let X be a nonempty measurable subset of and consider the restriction of the usual Lebesgue measure σ of to X. Under the assumption that the intersection of X with every open ball of has positive measure, we find necessary and sufficient conditions on a L²(X)-positive definite kernel in order that the associated integral operator be nuclear. Taken nuclearity for granted, formulas for the trace of the operator are derived. Some of the results are re-analyzed when K is just an element of .      

A family of transitive modular Lie superalgebras with depth one

The embedding theorem is established for Z-graded transitive modular Lie superalgebras g=(?)satisfying the conditions: (i)g0(?)(g-1)and g0-module g-1 is isomorphic to the natural(?)(g-1)-module; (ii)dim g1=2/3n(2n~2 1),where n=1/2dim g-1. In particular,it is proved that the finite-dimensional simple modular Lie superalgebras satisfying the conditions above are isomorphic to the odd Hamiltonian superalgebras.The restricted Lie superalgebras are also considered.     

A class number formula for the p-cyclotomic field

Let p be an odd prime number and . Let be the classical Stickelberger ideal of the group ring . Iwasawa [6] proved that the index equals the relative class number of . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal of , and studied some of its properties. In this note, we prove that when mod 4 and [G : H] = 2, the index equals the quotient . Received: 13 January 2006     

Equivalence Bimodule between Spherical Noncommutative Tori

Let n _i ,m _j > 1. In [5], the sperical noncommutative torus was defined by twisting in by a totally skew multiplier p on for T ^pd a pd-homogeneous C*-algebra over . It is shown that is strongly Morita equivalent to . This work is supported by Grant No. 1999-2-102-001-3 from the interdisciplinary research program year of the KOSEF     

Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrödinger Equation

In the first part of these notes, we deal with first order Hamiltonian systems in the form where the phase space X may be infinite dimensional so as to accommodate some partial differential equations. The Hamiltonian is required to be invariant with respect to the action of a group of isometries where is skew-symmetric and JA  = AJ. A standing wave is a solution having the form for some and such that . Given a solution of this type, it is natural to investigate its stability with respect to perturbations of the initial condition. In this context, the appropriate notion of stability is orbital stability in the usual sense for a dynamical system. We present some of the important criteria for establishing orbital stability of standing waves. In the second part we consider the nonlinear Schr?dinger equation which provides an interesting example of this situation where standing waves appear as time-harmonic solutions. We show how the general theory applies to this case and review what is known about stability. Received: January 2008     

Passive Systems with a Normal Main Operator and Quasi-selfadjoint Systems

Passive systems with and as an input and output space and as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system with is said to be quasi-selfadjoint if ran . The subclass of the Schur class is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass and the Q-function of T is given. Received: December 16, 2007., Accepted: March 4, 2008.     

The Torsion Theory Generated by <Emphasis Type="Italic">M</Emphasis>-Small Modules

Let M be a right R-module, the class of all M-small modules, and P a projective cover of M in [M]. We consider the torsion theories = ( ), = ( ), and = ( ) in [M], where is the torsion theory generated by is the torsion theory cogenerated by , and is the dual Lambek torsion theory. We study some conditions for to be cohereditary, stable, or split, and prove that Rej(M, ) = M = (= = ) = Gen_M(P) .2000 Mathematics Subject Classification: 16S90     

Semilinear representations of PGL

Let L be the function field of a projective space over an algebraically closed field k of characteristic zero, and H be the group of projective transformations. An H-sheaf on is a collection of isomorphisms for each g ∈ H satisfying the chain rule. We construct, for any n > 1, a fully faithful functor from the category of finite-dimensional L-semilinear representations of H extendable to the semigroup End(L/k) to the category of coherent H-sheaves on The paper is motivated by a study of admissible representations of the automorphism group G of an algebraically closed extension of k of countable transcendence degree undertaken in [4]. The semigroup End(L/k) is considered as a subquotient of G, hence the condition on extendability. In the appendix it is shown that, if is either H, or a bigger subgroup in the Cremona group (generated by H and a certain pair of involutions), then any semilinear of degree one is an integral L-tensor power of It is also shown that this bigger subgroup has no non-trivial representations of finite degree if n > 1.     

Entropy and Decay of Correlations for Real Analytic Semi-Flows

We consider real analytic suspension semi-flows over uniformly expanding real-analytic map of the interval. We show that for any -invariant equilibrium measure related to an analytic potential g, there exists a Banach space of test functions such that for generic observables in , the corresponding correlation functions cannot decay faster than , where h_g is the measure theoretic entropy of . This statement implies the existence of essential spectrum for the Perron-Frobenius operator associated to the semi-flow, when acting on any reasonable Banach space. Submitted: September 16, 2008. Accepted: March 30, 2009.     

New results on the peak algebra

The peak algebra is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of [n−1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of . We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements for the peak algebra. We use these bases to describe the Jacobson radical of and to characterize the elements of in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals of , j = 0,..., , such that is the linear span of sums of permutations with a common set of interior peaks and is the peak algebra. We extend the above results to , generalizing results of Schocker (the case j = 0). Aguiar supported in part by NSF grant DMS-0302423 Orellana supported in part by the Wilson Foundation     

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.