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

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 \(\mathfrak {sl}(2,\mathbb {C}) \oplus \mathfrak {sl}(2,\mathbb {C}) = \mathfrak {so}(4,\mathbb {C})\).

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Fischer decomposition in symplectic harmonic analysis

In this note, we describe quite explicitly the Howe duality for Hodge systems and connect it with the well-known facts of harmonic analysis and Clifford analysis. In Section 2, we recall briefly the Fisher decomposition and the Howe duality for harmonic analysis. In Section 3, the well-known fact that Clifford analysis is a real refinement of harmonic analysis is illustrated by the Fisher decomposition and the Howe duality for the space of spinor-valued polynomials in the Euclidean space under the so-called L-action. On the other hand, for Clifford algebra valued polynomials, we can consider another action, called in Clifford analysis the H-action. In the last section, we recall the Fisher decomposition for the H-action obtained recently. As in Clifford analysis the prominent role plays the Dirac equation in this case the basic set of equations is formed by the Hodge system. Moreover, analysis of Hodge systems can be viewed even as a refinement of Clifford analysis. In this note, we describe the Howe duality for the H-action. In particular, in Proposition 1, we recognize the Howe dual partner of the orthogonal group O(m) in this case as the Lie superalgebra sl(2 1). Furthermore, Theorem 2 gives the corresponding multiplicity free decomposition with an explicit description of irreducible pieces.

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The howe duality for hodge systems

The Joseph ideal in the universal enveloping algebra U(so(m)) is the annihilator ideal of the so(m)-representation on the harmonic functions on R^m-2. The Joseph ideal for sp(2n) is the annihilator ideal of the Segal–Shale–Weil (metaplectic) representation. Both ideals can be constructed in a unified way from a quadratic relation in the tensor algebra of so(m) or sp(2n). In this paper, we construct two analogous ideals for the orthosymplectic Lie super-algebra sop(m|2n) and prove that they have unique characterizations that naturally extend the classical case. Then we show that these two ideals are the annihilator ideals of, respectively, the representation on the spherical harmonics and a generalization of the metaplectic representation to spo(2n|m). This proves that these ideals are reasonable candidates to establish the theory of Joseph-like ideals for Lie super-algebras. We also discuss the relation between the Joseph ideal of osp(m|2n) and the algebra of symmetries of the super-conformal Laplace operator, regarded as an intertwining operator between principal series representations for sop(m|2n) . As a side result, we obtain the proof of a conjecture of Eastwood about the Cartan product of irreducible representations of semisimple Lie algebras made in [10].

https://biblio.ugent.be/publication/3199502/file/3199512.pdf

Joseph Ideals and Harmonic Analysis for

Relative BGG sequences: I. Algebra

This is the second in a series of articles on a natural modification of the normal tractor connection on parabolic geometries, which naturally prolongs an underlying overdetermined system of invariant differential equations. We give a short review of the general procedure developed in Hammerl et al. (preprint) and then compute the prolongation covariant derivatives for a number of interesting examples in projective, conformal, and Grassmannian geometries.

Vladimír Souček

Papers

Fischer decomposition in symplectic harmonic analysis

The howe duality for hodge systems

Joseph Ideals and Harmonic Analysis for

Relative BGG sequences: I. Algebra

Invariant prolongation of overdetermined PDEs in projective, conformal, and Grassmannian geometry