This paper surveys the contributions of five mathematicians --- Eugenio Beltrami (1835--1899), Camille Jordan (1838--1921), James Joseph Sylvester (1814--1897), Erhard Schmidt (1876--1959), and Hermann Weyl (1885--1955) --- who were responsible for establishing the existence of the singular value decomposition and developing its theory.

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Abstract. In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the S-lemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry. Key words. S-lemma, S-procedure, control theory, nonconvex theorem of alternatives, numerical range, relaxation theory, semidefinite optimization, generalized convexities

In this paper we study limits of generalized state space systems under high gain feedback modulo system equivalence. Different group actions on the space of system pencils are considered and related to the action of pencil equivalence. A recent result on the orbit closure problem for pencils is applied to obtain necessary conditions for a system to be a limit of a given system under high gain feedback. These conditions are shown to be sufficient for arbitrary state space systems. The result is used to investigate a high gain version of Rosenbrock's problem: invariant factor assignment in the limit via high gain state feedback. 1 Introduction In the sixties the theory of matrix pencils [24], [14], [4] created by Weierstrass (1867) and Kronecker (1890) was the main mathematical source of inspiration for the emerging structure theory of linear state space systems, see [21], [13]. In the late seventies and eighties, Rosenbrock's description of linear systems by polynomial system matrices ...

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The main theorem of this paper complements the tame-wild dichotomy for commutative Noetherian rings, obtained by Klingler and Levy [14]–[16]. They gave a complete classification of all finitely generated modules over Dedekind-like rings (cf. Definition 1.1) and showed that, over any ring that is not a homomorphic image of a Dedekind-like ring, the category

. The problem of determining when two central extension groups E 1 ,E 2 of Hn in 0 \Gamma! IR \Gamma! E 1 ! Hn ! 1 and 0 ! IR ! E 2 ! Hn \Gamma! 1 are isomorphic as Lie groups is reduced to the classical problem of the equivalence of pencils of antisymmetric matrices. A technique of cohomologically trivializing trilinear functions from IR 2 to IR is used to explicitly calculate 2-cocycles on H 1 which are not cohomologous to bilinear ones, thus providing a counterexample to a recent paper of Moskowitz, although for the higher groups Hn , n 2, all IR-valued 2-cocycles are cohomologous to bilinear ones. Finally, a generalization, from semidirect products to arbitrary extensions, of a set of cocycle equations of Mackey and Tahara is derived and solved in the particular case of the Heisenberg group Hn . Consequences for the projective representation theory of Hn are given. I. Introduction In this paper the following question is completely answered for n = 1 in theorem 8.3: When are two...

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