A well-known theorem of Zygmund (6) states that if n 1 < n 2 <. .. is a sequence of integers satisfying a (1) n~ +i/n~ > l+c (c > 0), k=1 converges for at least one x ; in fact the set of x for which (2) converges is of power c in any interval. Paley and Mary Weiss (5) extended this theorem for power series, i .e. (3) Y a i.znk k=1 converges for at least one z with I z I = 1 ; in fact the set o...

In a recent paper (Square roots in locally euclidean groups, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 446-449), A. M. Gleason proved that, in a locally euclidean group G which has no small subgroups, there exist neighborhoods M and N of the neutral element e such that every element in M has a unique square root in N. The author clearly considered this result to be a step towards proving that, ...

In this paper, we present a formula for computing the Tutte polynomial of the signed graph formed from a labeled graph by edge replacements in terms of the chain polynomial of the labeled graph. Then we define a family of ‘ring of tangles’ links and consider zeros of their Jones polynomials. By applying the formula obtained, Beraha-Kahane-Weiss’s theorem and Sokal’s lemma, we prove that zeros o...

Gleason-type theorems for quantum theory allow one to recover the state space by assuming that (i) states consistently assign probabilities measurement outcomes and (ii) there is a unique every such assignment. We identify class of general probabilistic theories which also admit theorems. It contains satisfying no-restriction hypothesis as well others can simulate an unrestricted arbitrarily wh...

The Gleason-Pierce-Ward theorem gives constraints on the divisor and field size of a linear divisible code over a finite field whose dimension is half of the code length. This result is a departure point for the study of self-dual codes. In recent years, additive codes have been studied intensively because of their use in additive quantum codes. In this work, we generalize the GleasonPierce-War...