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.

/pdf/finite-element-exterior-calculus-homological-techniques-and-4129s2mxlu.pdf

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

/pdf/limit-theorems-for-the-riemann-zeta-function-s804sbo6qf.pdf

Limit Theorems for the Riemann Zeta-Function

This paper is a continuation of the paper [De Bie H., Orsted B., Somberg P., Soucek V., Trans. Amer. Math. Soc. 364 (2012), 3875-3902], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it gives extensions of many results obtained in [Ben Said S., Kobayashi T., Orsted B., Compos. Math. 148 (2012), 1265-1336]. We establish the analogues of Bochner's formula and the Heisenberg uncertainty relation in the framework of the (holomorphic) Hermite semigroup, and also give a detailed analytic treatment of the series expansion of the associated integral transform.

/pdf/the-clifford-deformation-of-the-hermite-semigroup-22wcfyn97k.pdf

The Clifford Deformation of the Hermite Semigroup

Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras

An explicit algorithmic construction is given for orthogonal bases for spaces of homogeneous polynomials, in the context of Hermitean Clifford analysis, which is a higher dimensional function theory centred around the simultaneous null solutions of two Hermitean conjugate complex Dirac operators.

/pdf/gelfand-tsetlin-bases-of-orthogonal-polynomials-in-hermitean-c8v7itbrj6.pdf

Gelfand-Tsetlin Bases of Orthogonal Polynomials in Hermitean Clifford Analysis

The homological version of the Cauchy integral formula is formulated in the paper for solutions of corresponding equations in complexified hypercomplex analysis. Many different cases are treated in unified manner, including some higher order operators. The notion of index of n-cycles is defined in this complexified situation and its properties are studied.

Generalized hypercomplex analysis and its integral formulas

In this note we describe explicitly irreducible decompositions of kernels of the Hermitean Dirac Operators. In [6], it is shown that these decompositions are essential for a construction of orthogonal (or even Gelfand‐Tsetlin) bases of homogeneous Hermitean monogenic polynomials.

/pdf/fischer-decompositions-of-kernels-of-hermitean-dirac-1kz6rx8ff4.pdf

Vladimír Souček

Papers

The Clifford Deformation of the Hermite Semigroup

Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras

Gelfand-Tsetlin Bases of Orthogonal Polynomials in Hermitean Clifford Analysis

Generalized hypercomplex analysis and its integral formulas

Fischer Decompositions of Kernels of Hermitean Dirac Operators