共查询到20条相似文献,搜索用时 78 毫秒
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GuangBin Ren 《中国科学 数学(英文版)》2010,53(12):3153-3162
In the framework of superspace in Clifford analysis for the Dunkl version, the Fischer decomposition is established for solutions of the Dunkl super Dirac operators. The result is general without restrictions on multiplicity functions or on super dimensions. The Fischer decomposition provides a module for the Howe dual pair G × osp(1|2) on the space of spinor valued polynomials with G the Coxeter group, while the generators of the Lie superspace reveal the naturality of the Fischer decomposition. 相似文献
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The generalization of Berezin's Grassmann algebra integral to a Clifford algebra is shown to be translation-invariant in a certain sense. This enables the construction of analogs of twisted convolutions of Grassmann algebra elements and of the Fourier-Weyl transformation, which is an isomorphism from a Clifford algebra to the Grassmann algebra over the dual space, equipped with a twisted convolution product. As an application a noncommutative central limit theorem for states of a Clifford algebra is proved. 相似文献
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Summary This paper presents a theory of Clifford algebra-valued martingales on a -finte measure space, with respect to a pseudoaccretive weight. A novel dual pair system of Haar functions associated with the Clifford martingale is constructed, and Littlewood-Paley estimates are established. The dual pair system of Clifford Haar functions is used to give a new proof of the boundedness of the Cauchy principal value integral on Lipschitz surfaces, and of the Clifford-valued T(b) theorem.Research supported by the Australian Research Council.Research supported by the National Science Foundation of China.Research carried out as a National Research Fellow. 相似文献
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I. A. Borovikov 《Proceedings of the Steklov Institute of Mathematics》2016,293(1):99-106
The central focus in the paper is on studying the relation between real dual spaces and dual Clifford modules. Extension problems for linear functionals on Clifford modules are addressed; in particular, an analog of the Hahn–Banach theorem is established. 相似文献
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谢永红 《数学年刊A辑(中文版)》2014,35(2):235-246
研究了取值于实Clifford代数空间Cl_(n+1,0)(R)中对偶的k-hypergenic函数.首先,给出了对偶的k-hypergenic函数的一些等价条件,其中包括广义的Cauchy-Riemann方程.其次,给出了对偶的hypergenic函数的Cauchy积分公式,并且应用其证明了(1-n)-hypergenic函数的Cauchy积分公式.最后,证明了对偶的hypergenic函数的Cauchy积分公式右端的积分是U\Ω_2中对偶的hypergenic函数. 相似文献
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Ekaterina Pervova 《Linear and Multilinear Algebra》2013,61(9):1785-1828
AbstractWe consider the diffeological version of the Clifford algebra of a diffeological finite dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). After considering the natural diffeology of the Clifford algebra, and considering which of its standard properties re-appear in the diffeological context (most of them), we turn to our main interest, which is constructing the pseudo-bundles of Clifford algebras associated to a given (finite dimensional) diffeological vector pseudo-bundle, and those of the usual Clifford modules (the exterior algebras). The substantial difference that emerges with respect to the standard context, and paves the way to various questions that do not have standard analogues, stems from the fact that the notion of a diffeological pseudo-bundle is very different from the usual bundle, and this under two main respects: it may have fibres of different dimensions, and even if it does not, its total and base spaces frequently are not smooth, or even topological, manifolds. 相似文献
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Tim Römer 《manuscripta mathematica》2001,104(1):39-48
In this paper we consider the category of squarefree modules over the polynomial ring and an exact duality functor, which
is an extension of the Alexander dual of a simplicial complex. We give a relationship between the squarefree components of
local cohomology groups of a squarefree module and the Tor groups of its dual. With this result it is shown that a squarefree
module is sequentially Cohen–Macaulay if and only if the dual is componentwise linear.
Received: 7 June 1999 / Revised version: 6 September 2000 相似文献
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F. Baaske S. Bernstein F. Sommen 《Journal of Difference Equations and Applications》2013,19(2):271-295
The main purpose of this paper was to study solutions of the heat equation in the setting of discrete Clifford analysis. More precisely we consider this equation with discrete space and continuous time. Thereby we focus on representations of solutions by means of dual Taylor series expansions. Furthermore we develop a discrete convolution theory, apply it to the inhomogeneous heat equation and construct solutions for the related Cauchy problem by means of heat polynomials. 相似文献
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This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated
extended algebras, and to explore these concepts also from the counterspace viewpoint. The presented formalism explains how
the concept of chirality stems from the bracket, as defined by Rota et all [1]. The exterior (regressive) algebra is shown
to share the exterior (progressive) algebra in the direct sum of chiral and achiral subspaces. The duality between scalars
and volume elements, respectively under the progressive and the regressive products is shown to have chirality, in the case
when the dimension n of the Peano space is even. In other words, the counterspace volume element is shown to be a scalar or a pseudoscalar, depending
on the dimension of the vector space to be respectively odd or even. The de Rham cochain associated with the differential
operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove
that the exterior algebra over the space and the exterior algebra constructed on the counterspace are only pseudoduals each other, if we introduce chirality. The extended Clifford algebra is introduced in the light of the periodicity theorem
of Clifford algebras context, wherein the Clifford and extended Clifford algebras
can be embedded in
which is shown to be exactly the extended Clifford algebra. We present the essential character of the Rota’s bracket, relating
it to the formalism exposed by Conradt [25], introducing the regressive product and subsequently the counterspace. Clifford
algebras are constructed over the counterspace, and the duality between progressive and regressive products is presented using
the dual Hodge star operator. The differential and codifferential operators are also defined for the extended exterior algebras
from the regressive product viewpoint, and it is shown they uniquely tumble right out progressive and regressive exterior
products of 1-forms.
R. da Rocha is supported by CAPES 相似文献
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Jawad Abuhlail 《Topology and its Applications》2011,158(3):457-467
We introduce a dual Zariski topology on the spectrum of fully coprime R-submodules of a given duo module M over an associative (not necessarily commutative) ring R. This topology is defined in a way dual to that of defining the Zariski topology on the prime spectrum of R. We investigate this topology and clarify the interplay between the properties of this space and the algebraic properties of the module under consideration. 相似文献
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首先给出了复Clifford分析中的复k-超单演函数的定义,进一步得到了复Clifford分析中的复k-超单演函数的一些等价条件,从而使复Clifford分析中的复k-超单演函数与其满足的方程之间建立了联系. 相似文献
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Prolate spheroidal wave functions (PSWFs) possess many remarkable properties. They are orthogonal basis of both square integrable space of finite interval and the Paley–Wiener space of bandlimited functions on the real line. No other system of classical orthogonal functions is known to obey this unique property. This raises the question of whether they possess these properties in Clifford analysis. The aim of the article is to answer this question and extend the results to more flexible integral transforms, such as offset linear canonical transform. We also illustrate how to use the generalized Clifford PSWFs (for offset Clifford linear canonical transform) we derive to analyze the energy preservation problems. Clifford PSWFs is new in literature and has some consequences that are now under investigation. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
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LetR be a commutative ring,I an invertibleR-module, and consider quadratic spaces with values inI. The Clifford algebra of such a quadratic space is an algebra over the generalized Rees ring associated toI. We discuss the relation between the Witt module of quadratic spaces with values inI and the graded Witt ring and the graded Brauer-Wall group of the generalized Rees ring. This leads to the introduction of three distinguished subgroups of the graded Brauer-Wall group of the generalized Rees ring. The image of the Clifford functor is a subgroup of one of these three subgroups (the type 1 subgroup). 相似文献
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Lin ZHU Xiu Hong FENG Yan Lin YU 《数学学报(英文版)》2007,23(9):1629-1638
In this paper, we give identifications of bispinor space with Grassmann algebra, and with Clifford algebra. The multiplication in Clifford algebra provides an action on them. Lastly we have researched on the geometry of bispinor space, and define Dirac operators to get a Pythagoras equality. 相似文献
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The design of linear algebra and geometry 总被引:2,自引:0,他引:2
David Hestenes 《Acta Appl Math》1991,23(1):65-93
Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra into the foundations of linear algebra. There is a natural extension of linear transformations on a vector space to the associated Clifford algebra with a simple projective interpretation. This opens up new possibilities for coordinate-free computations in linear algebra. For example, the Jordan form for a linear transformation is shown to be equivalent to a canonical factorization of the unit pseudoscalar. This approach also reveals deep relations between the structure of the linear geometries, from projective to metrical, and the structure of Clifford algebras. This is apparent in a new relation between additive and multiplicative forms for intervals in the cross-ratio. Also, various factorizations of Clifford algebras into Clifford algebras of lower dimension are shown to have projective interpretations.As an important application with many uses in physics as well as in mathematics, the various representations of the conformal group in Clifford algebra are worked out in great detail. A new primitive generator of the conformal group is identified. 相似文献
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Structures and objects used in Hamiltonian secondary quantization are discussed. By the secondary quantization of a Hamiltonian system ?, we mean the Schrödinger quantization of another Hamiltonian system ?1 for which the Hamiltonian equation is the Schrödinger one obtained by the quantization of the original Hamiltonian system ?. The phase space of ?1 is the realification ?R of the complex Hilbert space ? of the quantum analogue of ? equipped with the natural symplectic structure. The role of a configuration space is played by the maximal real subspace of ?. 相似文献
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F. Brackx H. De Schepper D. Eelbode R. Lávička V. Souček 《Annals of Global Analysis and Geometry》2014,46(4):409-430
In the framework of quaternionic Clifford analysis in Euclidean space \(\mathbb {R}^{4p}\) , which constitutes a refinement of Euclidean and Hermitian Clifford analysis, the Fischer decomposition of the space of complex valued polynomials is obtained in terms of spaces of so-called (adjoint) symplectic spherical harmonics, which are irreducible modules for the symplectic group Sp \((p)\) . Its Howe dual partner is determined to be \(\mathfrak {sl}(2,\mathbb {C}) \oplus \mathfrak {sl}(2,\mathbb {C}) = \mathfrak {so}(4,\mathbb {C})\) . 相似文献