On polynomial representations of strange Lie superalgebras of <Emphasis Type="Italic">Q</Emphasis>-type

In this paper, both canonical and noncanonical polynomial representations of Lie superalgebara of Q-type are investigated. It turns out that not all these representations are completely reducible. Moreover, the representation spaces has only two proper submodules when it is completely reducible, and has a unique composition series when it is not completely reducible.     

基于正交规范化的小波的有限差分表示

本文研究了在时域内小波的一种表达形式.利用正交规范化,获得了小波的有限差分表示.不仅该形式构造了任意次B样条正交小波.而且在时域中用来直接获得小波滤波器是有效的.     

Analysis on the minimal representation of O(p,q) I. Realization via conformal geometry

This is the first in a series of papers devoted to an analogue of the metaplectic representation, namely the minimal unitary representation of an indefinite orthogonal group; this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of Kirillov–Kostant. We begin by applying methods from conformal geometry of pseudo-Riemannian manifolds to a general construction of an infinite-dimensional representation of the conformal group on the solution space of the Yamabe equation. By functoriality of the constructions, we obtain different models of the unitary representation, as well as giving new proofs of unitarity and irreducibility. The results in this paper play a basic role in the subsequent papers, where we give explicit branching formulae, and prove unitarization in the various models.     

The explicit chaotic representation of the powers of increments of Lévy processes

This paper presents a computationally explicit formula of the chaotic representation property (CRP) for the powers of increments of a Lévy process. The formula can be used to obtain the integrands of the CRP in terms of orthogonal compensated power jump processes and the CRP in terms of Poisson random measures. Simulation results demonstrate that the performance of the representation is satisfactory. The CRP of a number of financial derivatives can be found by expressing them in terms of the powers of increments of the underlying Lévy process using Taylor's expansion.     

Representations of a p-Elementary Form by a Genus

We study the branching of representations of a p-elementary quadratic form by a genus of positive definite locally p-two-dimensional forms. A primitive representation of a p-elementary form is decomposed into a direct sum of minimal indecomposable representations; the latter representations are found in an explicit form. For the case of branching, we find local multipliers of the weight of representations of a form by a genus. As an application, we calculate the number of embeddings into the classical root lattices. The method of orthogonal complement is applied in constructing new genera of quadratic forms. Bibliography: 9 titles.     

Circular β ensembles,CMV representation,characteristic polynomials

In this note we first briefly review some recent progress in the study of the circular β ensemble on the unit circle, where β > 0 is a model parameter. In the special cases β = 1,2 and 4, this ensemble describes the joint probability density of eigenvalues of random orthogonal, unitary and sympletic matrices, respectively. For general β, Killip and Nenciu discovered a five-diagonal sparse matrix model, the CMV representation. This representation is new even in the case β = 2; and it has become a powerful tool for studying the circular β ensemble. We then give an elegant derivation for the moment identities of characteristic polynomials via the link with orthogonal polynomials on the unit circle. This work was supported by National Natural Science Foundation of China (Grant No. 10671176)     

A Reflection Principle and an Orthogonal Decomposition Concernig Hypoelliptic Equations

A jump relation for a boundary integral representation of solutions of hypoelliptic equations is described by a reflection principle. An orthogonal decomposition of L² can be proved by the jump relation. In the orthogonal complement of the space of regular functions, i.e. the space of solutions of the homogeneous equation, the inhomogeneous adjoint equation has a solution with homogeneous boundary values. As a conclusion, one obtains Sobolev's regularity theorem. Furthermore it will be proved that the existence of the orthogonal decomposition and Sobolev's regularity theorem are equivalent. Theorems of Runge's type will be proved in order to determine countable dense subsets of the space of regular functions.     

Geometric aspects of frame representations of abelian groups

We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.

     

On Local Coefficients for Non-generic Representations of Some Classical Groups

This paper is concerned with representations of split orthogonal and quasi-split unitary groups over a nonarchimedean local field which are not generic, but which support a unique model of a different kind, the generalized Bessel model. The properties of the Bessel models under induction are studied, and an analogue of Rodier's theorem concerning the induction of Whittaker models is proved for Bessel models which are minimal in a suitable sense. The holomorphicity in the induction parameter of the Bessel functional is established. Local coefficients are defined for each irreducible supercuspidal representation which carries a Bessel functional and also for a certain component of each representation parabolically induced from such a supercuspidal. The local coefficients are related to the Plancherel measures, and their zeroes are shown to be among the poles of the standard intertwining operators.     

A note on generating finer‐grain parallelism in a representation tree

The representation tree lies at the heart of the algorithm of Multiple Relatively Robust Representations for computing orthogonal eigenvectors of a symmetric tridiagonal matrix without Gram–Schmidt. A representation tree describes the incremental shift relations between relatively robust representations of eigenvalue clusters of an unreduced tridiagonal matrix, which are needed to strongly separate close eigenvalues in the relative sense. At the bottom of the representation tree, each leaf defines a relatively isolated eigenvalue to high relative accuracy. The shape of the representation tree plays a pivotal role for complexity and available parallelism: a deeper tree consisting of multiple levels of nodes involves tasks associated to more work (i.e., eigenvalue refinement to resolve eigenvalue clusters) and less parallelism (i.e., a longer critical path as well as potential data movement and synchronization). An embarrassingly parallel, ideal tree on the other hand consists of a root and leaves only. As highly parallel hybrid graphics processing unit/multicore platforms with large memory now become available as commodity platforms, exploiting parallelism in traditional algorithms becomes key to modernizing the components of standard software libraries such as LAPACK. This paper focuses on LAPACK's Multiple Relatively Robust Representations algorithm and investigates the critical case where a representation tree contains a long sequential chain of large (fat) nodes that hamper parallelism. This key problem needs to be addressed as it concerns all sorts of computing environments, distributed computing, symmetric multiprocessor, as well as hybrid graphics processing unit/multicore architectures. We present an improved representation tree that often offers a significantly shorter critical path and finer computational granularity of smaller tasks that are easier to schedule. In a study of selected synthetic and application matrices, we show that an average 75% reduction in the length of the critical path and 82% reduction in task granularity can be achieved. Copyright © 2011 John Wiley & Sons, Ltd.     

Lie algebroid modules and representations up to homotopy

We establish a relationship between two different generalizations of Lie algebroid representations: representation up to homotopy and Vaĭntrob’s Lie algebroid modules. Specifically, we show that there is a noncanonical way to obtain a representation up to homotopy from a given Lie algebroid module, and that any two representations up to homotopy obtained in this way are equivalent in a natural sense. We therefore obtain a one-to-one correspondence, up to equivalence.     

Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras

The non-commutative analytic Toeplitz algebra is the WOT-closed algebra generated by the left regular representation of the free semigroup onn generators. We obtain a distance formula to an arbitrary WOT-closed right ideal and thereby show that the quotient is completely isometrically isomorphic to the compression of the algebra to the orthogonal complement of the range of the ideal. This is used to obtain Nevanlinna-Pick type interpolation theoremsFirst author partially supported by an NSERC grant and a Killam Research Fellowship.Second author partially supported by an NSF grant.     

Ramification of a finite flat group scheme over a local field

In this paper, we analyze ramification in the sense of Abbes-Saito of a finite flat group scheme over the ring of integers of a complete discrete valuation field of mixed characteristic (0,p). We deduce that its Galois representation depends only on its reduction modulo explicitly computed p-power. We also give a new proof of a theorem of Fontaine on ramification of a finite flat Galois representation, and extend it to the case where the residue field may be imperfect.     

On Polynomial Representations of Strange Lie Superalgebras of Q-Type

In this paper,both canonical and noncanonical polynomial representations of Lie superalgebara of Q-type are investigated.It turns out that not all these representations are completely reducible.Moreover,the representation spaces has only two proper submodules when it is completely reducible,and has a unique composition series when it is not completely reducible.     

Complete reducibility of gyrogroup representations

Abstract

In this article, we show that any finite gyrogroup can be represented on a space of complex-valued functions. In particular, we prove that any linear representation of a finite gyrogroup on a finite-dimensional complex inner product space is unitary and hence is completely reducible using strong connections between linear actions of groups and gyrogroups. Also, we provide an example of a unitary representation of an arbitrary finite gyrogroup, which resembles the group-theoretic left regular representation.     

The pascal automorphism has a continuous spectrum

In this paper we describe the Pascal automorphism and present a sketch of the proof that its spectrum is continuous on the orthogonal complement of the constants.     

Probability of a fuzzy event: An axiomatic approach

Axioms are proposed that could justify the natural definition of the probability of a fuzzy event initially given by Zadeh. They are based (1) on the postulate that the sum of the conditional probability of a fuzzy event and of its complement given any fuzzy event adds to one or (2) on soft independence for orthogonal sets with independent constitutive elements. A general postulate is also required concerning the complement of a fuzzy set. The classical definition of the operator representing the complement can also be deduced.     

A matrix description of a wild category

A bocsA of representation wild type over an algebraically closed fieldk and its representation categoryR A are concretely described by using matrix method. It is proved thatA has almost split sequences and strong homogeneous property in the case chk=0. Thus there exists neither proper projective nor proper injective object inR A. And for each indecomposable object M inR A, there exists a proper almost split sequence, which is given explicitly. Project supported by the National Natural Science Foundation of China(Grant Nos. 19331013; 19671013).     

Unitary representations and modular actions

We call a measure-preserving action of a countable discrete group on a standard probability space tempered if the associated Koopman representation restricted to the orthogonal complement to the constant functions is weakly contained in the regular representation. Extending a result of Hjorth, we show that every tempered action is antimodular, i.e., in a precise sense “orthogonal” to any Borel action of a countable group by automorphisms on a countable rooted tree. We also study tempered actions of countable groups by automorphisms on compact metrizable groups, where it turns out that this notion has several ergodic theoretic reformulations and fits naturally in a hierarchy of strong ergodicity properties strictly between ergodicity and strong mixing. Bibliography:s 25 titles. Dedicated to Professor Anatoly Vershik on the occasion of his 70th birthday Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 97–144.