Abstract: The law of the iterated logarithm provides a family of bounds all of the same order such that with probability one only finitely many partial sums of a sequence of independent and identically distributed random variables exceed some members of the family, while for others infinitely many do so. In the former case, the total number of such excesses has therefore a proper probability distribution, and it is shown here that whenever the law applies this distribution possesses no moments of positive order. This result further elucidates the celebrated precision of this law of probability concerning fluctuations of sums of random variables.

Abstract: This paper deals with the Dirichlet problem for the hyperbolic differential equation $${U_{xy}} = 0$$ (1) i. e. with the problem of determining a solution u of (1) from given values on a closed curve C.1 Simple examples show that the Dirichlet problem for a hyperbolic equation has a completely different character from that of the corresponding problem for an elliptic equation such as the potential equation $${U_{xx}} + {U_{yy}} = 0$$ (2)

Abstract: The purpose of this paper is the derivation of a classification of the representations of finite groups with special reference to groups which satisfy the following two conditions: a. Every element is equivalent to its reciprocal, i. e., all classes are ambivalent. b. The Kronecker (or “direct”) product of any two irreducible representations of the group contains no representation more than once.