Informatics Publishing Limited

M. Venkataraman ¹

Affiliations
1 University of Madras in Madurai, Madurai-2, IN

Abstract

The aim of this paper is to find (cf. Theorems 4.2; 4.3) a canonical form, and a complete set of invariants, for self-adjoint transformations in a Hilbert space-up to unitary equivalence. While the basic mode of approach is essentially the same as in the excellent exposition in Halmos [1], the final result will be seen to be very much simpler. We do not have to work with the fairly involved concepts of rows, columns, etc.

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M. Venkataraman ¹

Affiliations
1 Annamalai University, IN

Abstract

DeLongchamps⁽¹⁾ (1877), and Pesc⁽²⁾(1891) proved by elementary methods⁽³⁾ the following chain of theorems:

Theorem. Given n lines in a plane, the centre-circles of sets of (n-I) of these will be concurrent, also the centres of these will lie on a circle-the centre-circle of the n lines. The centre-circle of thret lines is the circumcircle of the triangle formed by them.

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M. Venkataraman ¹

Affiliations
1 Annamalai University, IN

Abstract

A Chain theorem in geometry is a sequence of theorems, each affirming the existence of a geometrical construct derived from a number of primary elements. The construct at any stage is used to obtain that at the next stage, the resulting configuration growing with the number of initial primary elements. An important type of these chains is the Cox-chain and the chain associated with the names of Miquel, Clifford, Cantor and Grace.

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