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Vladimír Souček

Bio: Vladimír Souček is an academic researcher from Charles University in Prague. The author has contributed to research in topics: Clifford analysis & Invariant (mathematics). The author has an hindex of 26, co-authored 137 publications receiving 3285 citations. Previous affiliations of Vladimír Souček include Czechoslovak Academy of Sciences & University of York.


Papers
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Book
30 Apr 1992
TL;DR: Clifford algebras over lower dimensional Euclidean spaces and spinor spaces have been studied in this paper, where the Penrose transform has been applied to the Clifford analysis.
Abstract: Clifford algebras over lower dimensional Euclidean spaces Clifford algebras and spinor spaces monogenic functions special functions and methods monogenic differential forms and residues Clifford analysis and the Penrose transform.

303 citations

Journal ArticleDOI
TL;DR: The Bernstein-Gelfand Gelfand sequences as mentioned in this paper extend the complexes of homogeneous vector bundles to curved Cartan geometries. But they do not extend the Bernstein sequences to Cartan geometry.
Abstract: The Bernstein-Gelfand-Gelfand sequences extend the complexes of homogeneous vector bundles to curved Cartan geometries

248 citations

Journal ArticleDOI
TL;DR: In this paper, the Hermitean Dirac operators are shown to arise as generalized gradients when projecting the gradient on the invariant subspaces mentioned, which actually implies their invariance under the action of Spin J (2n;\({\mathbb{R}}\).
Abstract: Hermitean Clifford analysis focusses on h–monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Monogenicity is expressed here by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a Clifford realisation of the unitary group. In this contribution we present a deeper insight in the transition from the orthogonal setting to the Hermitean one. Starting from the orthogonal Clifford setting, by simply introducing a so-called complex structure J ∈ SO(2n;\({\mathbb{R}}\)), the fundamental elements of the Hermitean setting arise in a quite natural way. Indeed, the corresponding projection operators 1/2 (1 ± iJ) project the initial basis (eα, α = 1, . . . , 2n) onto the Witt basis and moreover give rise to a direct sum decomposition of Open image in new window into two components, where the SO(2n;\({\mathbb{R}}\))-elements leaving those two subspaces invariant, commute with the complex structure J. They generate a subgroup which is doubly covered by a subgroup of Spin(2n;\({\mathbb{R}}\)), denoted Spin J (2n;\({\mathbb{R}}\)), being isomorphic with the unitary group U(n;\({\mathbb{C}}\)). Finally the two Hermitean Dirac operators are shown to originate as generalized gradients when projecting the gradient on the invariant subspaces mentioned, which actually implies their invariance under the action of Spin J (2n;\({\mathbb{R}}\)). The eventual goal is to extend the complex structure J to the whole Clifford algebra Open image in new window , in order to conceptually unravel the true meaning of Hermitean monogenicity and its connections to orthogonal monogenicity.

103 citations

Journal ArticleDOI
TL;DR: In this paper, the Hermitean Dirac operators are shown to originate as generalized gradients when projecting the gradient on invariant subspaces, which are invariant under the action of a Clifford realization of the unitary group.
Abstract: Hermitean Clifford analysis focuses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space, where h-monogenicity is expressed by means of two complex and mutually adjoint Dirac operators, which are invariant under the action of a Clifford realization of the unitary group. In part 1 of the article the fundamental elements of the Hermitean setting have been introduced in a natural way, i.e., by introducing a complex structure on the underlying vector space, eventually extended to the whole complex Clifford algebra . The two Hermitean Dirac operators are then shown to originate as generalized gradients when projecting the gradient on invariant subspaces. In this part of the article, the aim is to further unravel the conceptual meaning of h-monogenicity, by studying possible splittings of the corresponding first-order system into independent parts without changing the properties of the solutions. In this way further connections with holomorphic functions of several c...

88 citations


Cited by
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Journal ArticleDOI
TL;DR: Finite element exterior calculus as mentioned in this paper is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations, which brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological and algebraic structures which underlie well-posedness of the PDE problem being solved.
Abstract: Finite element exterior calculus is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are re- vealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Lapla- cian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.

1,044 citations

Book
01 Jan 2001
TL;DR: The Poincare, Cousin, Runge, and Runge problems as discussed by the authors have been studied in the context of analytic spaces and pseudoconcave spaces, as well as holomorphic functions and analytic sets.
Abstract: Fundamental theory: Holomorphic functions and domains of holomorphy Implicit functions and analytic sets The Poincare, Cousin, and Runge problems Pseudoconvex domains and pseudoconcave sets Holomorphic mappings Theory of analytic spaces: Ramified domains Analytic sets and holomorphic functions Analytic spaces Normal pseudoconvex spaces Bibliography Index.

493 citations

Book
30 Nov 1995
TL;DR: The limit theorem for the Riemann Zeta-function in the complex plane was proved for Dirichlet polynomials with multiplicative coefficients in this paper, as well as the limit theorem in the space of analytic functions.
Abstract: Preface 1 Elements of the probability theory 2 Dirichlet series and Dirichlet polynomials 3 Limit theorems for the modulus of the Riemann Zeta-function 4 Limit theorems for the Riemann Zeta-function on the complex plane 5 Limit theorems for the Riemann Zeta-function in the space of analytic functions 6 Universality theorem for the Riemann Zeta-function 7 Limit theorem for the Riemann Zeta-function in the space of continuous functions 8 Limit theorems for Dirichlet L-functions 9 Limit theorem for the Dirichlet series with multiplicative coefficients References Notation Subject index

325 citations