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In Pontrjagin's theory of duality for compact abelian groups, the following theorem is known : Let G be a compact abelian group, G* the dual group. Then the topological dimension of G, in the sense of Lebesgue, is equal to the rank of discrete abelian group G*. It was Prof. T. Tannaka who has called my attention to the lack of corresponding theorem in non-commutative case. I intend to give, in this note, a theorem of this kind in the following form: THEOREM A. Let G be an arbitrary compact group, G the aggregate of continuous finite dimensional representstions of G, C[ G ] the algebra over the complex numbers C generated by the coefficients of representations in G, i.e., the "representative ring" of G in the sense of C. Chevalley^. Then the topological dimension of G, in the sense of Lebesgue, is equal to the transcendental degree of C[G] over C.

A mathematical analysis is given of invariant tridiagonal multiplication operators.

Apparent ambiguities of definitions of masses and lifetimes of unstable particles that depend either on the introduction of unperturbed Hamiltonians and their eigenstates, or on an assumed correspondence between resonances and elementary fields, are noted. TheS-matrix definition is unambiguous; the positions of poles in the first unphysical sheets are given by the zeros of the Fredholm denominator function, which is a function only of an appropriate center-of-mass energy. The mass and lifetime of a particle are consequently independent of the variables of the scattering process or of the particular process to which the particle contributes. The invariance of the Fredholm denominator under charge conjugation, which is a consequence of CPT invariance, ensures the equality of masses and lifetimes of relatively conjugate antiparticles. Unstable particles are closely akin to stable ones; by the factorization of the residues of unstable-particle poles, unstable-particle scattering functions quite analogous to ordinary scattering functions can be unambiguously defined. Like ordinary scattering functions they are defined only on the mass shell, the fixed masses of the unstable particles being well-defined complex numbers. The needed factorizability of the residue is an immediate consequence of Fredholm’s second theorem. The continuation, by means of unitarity, through the multiparticle physical cuts onto unphysical sheets is discussed.

In this paper, the author establishes an analogue of an integral representation for positive definite hermitian matrices. This result is then used to derive a number of known inequalities and to obtain various generalizations.

The Mathieu groups are the finite simple groups M11, M12, M22, M23, M24 given originally as permutation groups on respectively 11, 12, 22, 23, 24 symbols. Their definition can best be found in the work of Witt [1]. Using a concept from Lie group theory we can describe the Schur multiplier of a group as the center of a “simply-connected” covering of that group. A precise definition will be given later. We also mention that the Schur multiplier of a group is the second cohomology group of that group acting trivially on the complex roots of unity. The purpose of this paper is to determine the Schur multipliers of the five Mathieu groups.

Let us begin by listing some notations. We shall denote by K the field of complex numbers, by K[x] = K [ x 1 . . . . . x ~] a polynemial ring over ~K in n variables, and by K n the n-dimensional vector space over /~ . The complex, ,qonjugation in K, and its natural extensions to K[x] and K ~, wilt~be indicated by the ,superscript ~ over the respective elements. Le t 7 = (71 . . . . , 7~)be an element of K~., I t is called real if ~ = 7, tha t is, if 71 . . . . . 7 = are all real. The,norm,l] ? H of 7 is defined as the:non-negative number satisfying n II

of proof bygeneralized successive approximation methods carry over readily tocomplexBanachspacesthusfurnishingexistenceanduniqueness theoremsfor analytic solutions of generalized Pfaffians in complex Banach spaces.2Third Example: Let x be a (complex) numerical variable so that N1is the (complex) Banach space of complex numbers. In this case thepartial differential equation is