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.

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Finite element exterior calculus, homological techniques, and applications

Groupes et algébres de lie

Introduction to lie algebras and representation theory

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.

Function theory in several complex variables

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

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Limit Theorems for the Riemann Zeta-Function

Euclidean Clifford analysis is a higher dimensional function theory studying so--called monogenic functions, i.e. null solutions of the rotation invariant, vector valued, first order Dirac operator D. In the more recent branch Hermitean Clifford analysis, this rotational invariance has been broken by introducing a complex structure J on Euclidean space and a corresponding second Dirac operator D_J, leading to the system of equations D f = 0 = D_J f expressing so-called Hermitean monogenicity. The invariance of this system is reduced to the unitary group U(n). In this paper we decompose the spaces of homogeneous monogenic polynomials into U(n)-irrucibles involving homogeneous Hermitean monogenic polynomials and we carry out a dimensional analysis of those spaces. Meanwhile an overview is given of so-called Fischer decompositions in Euclidean and Hermitean Clifford analysis.

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Fischer decompositions in Euclidean and Hermitean Clifford analysis

In this note, we describe the Gel’fand‐Tsetlin procedure for the construction of an orthogonal basis in spaces of Hermitean monogenic polynomials of a fixed bidegree. The algorithm is based on the Cauchy‐Kovalevskaya extension theorem and the Fischer decomposition in Hermitean Clifford analysis.

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Gel'fand-Tsetlin procedure for the construction of orthogonal bases in Hermitean Clifford analysis

In [J. Bures, R. Lavicka, V. Soucek, Elements of quaternionic analysis and Radon transform, Textos de Matematica 42, Departamento de Matematica, Universidade de Coimbra, 2009], the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on R^(n+1) using the Radon and dual Radon transforms. The main aim of this paper is to further develop this approach. In fact, the Radon transform for functions with values in the Clifford algebra R_n is mapping solutions of the generalized Cauchy-Riemann equation, i.e., monogenic functions, to a parametric family of holomorphic functions with values in R_n and, analogously, the dual Radon transform is mapping parametric families of holomorphic functions as above to monogenic functions. The parametric families of holomorphic functions considered in the paper can be viewed as a generalization of the so-called slice monogenic functions. An important part of the problem solved in the paper is to find a suitable definition of the function spaces serving as the domain and the target of both integral transforms.

The Radon transform between monogenic and generalized slice monogenic functions

In the paper the complex-quaternionic analysis and its spinor version is used for the study of integral formulas for spin ½ massless fields. The basic corrspondence between the complexified Fueter equation and the massless field equation (spin½) is described first, together with the corresponding Cauchy integral formulas. It is shown then how the complexified Cauchy integral formula can be used to give the connection between elliptic type (boundary value type) integral formulas on Euclidean spacetime and Kirchhoff type (initial value type) integral formulas on Minkowski space (for spin ½ massless fields). The explicit formulas showing such connection with the integral formula described by Penrose are given.

Complex-quaternionic analysis applied to spin–½ massless fields

https://biblio.ugent.be/publication/1938058/file/1938059.pdf

Vladimír Souček

Papers

Fischer decompositions in Euclidean and Hermitean Clifford analysis

Gel'fand-Tsetlin procedure for the construction of orthogonal bases in Hermitean Clifford analysis

The Radon transform between monogenic and generalized slice monogenic functions

Complex-quaternionic analysis applied to spin–½ massless fields

The Cauchy–Kovalevskaya extension theorem in Hermitian Clifford analysis