In this treatise the convergence of iterative algorithms for the solution of non-linear operator equations in complex linear topological spaces are studied from the point of view of fixed-point theorems in such spaces... It was felt that the concept of the Gâteaux differential is a more natural one to use in connection with linear topological spaces. The beauty of the developed technique we mentioned earlier is essentially due to the fact that we are considering spaces over the complex number field. The resulting convergence theorems have also the added advantage of imposing no conditions on the second or higher order differentials of the operator T, as would be the case in an obvious extension ( which was not written down) of Kantorovich's work to such real linear topological spaces. [...]

[...]In the chapters which follow we study a general class of discrete distributions which we call Laurent Series Distributions (LSD), where the random variables can assume positive as well as negative integral values. [...]

In [1], Birkhoff and Pierce introduced the concept of f-ring. In (7], [9], and [10] it was shown that certain Archimedian f-rings can be represented as rings of continuous extended real valued functions on various topological spaces. It was noted that the m-topology on rings of continuous real valued functions (see problem 2N of [6]) could be generalized to arbitrary f-rings provided they were convex in the sense of [5]. It was also shown that every f-ring can be embedded in a smallest convex f-ring. [...]

In this thesis we study the differential equation [...equation...] which includes the Riccati equation, Abel's equation, Bernoulli's equation, and the linear equation as special cases. [...]

The problem was solved by Artin in 1927, [2], using the typical non-constructive machinery of modern algebra, and in 1955 Artin asked Kreisel if the proof could be made constructive. Kreisel asserted that this was possible, and published indications of his method in two places, [3] and [4]. Although [3] gives more detail than, [4], both papers are rather cryptic, neither giving a complete connected account: moreover, for the logical part of the argument, Kreisel uses the Hilbert-Bernays e-theorems. [...]

The basic aim of this thesis is to extend the definition of an n-dimensional vector measure so as to allow it to assume infinite values. In the 1-dimensional case, when one extends the notion of a non-negative measure to that of a signed measure which may assume negative values, it is necessary to assume that the signed measure takes on at most one of the values (+ oo) or(- oe). In a similar fashion it is shown that in arder to successfully extend the definition of a finite-valued n-dimensional measure, it is necessary to suppose that the extended measure assumes at most one infinite value. [...]

Let G be an abstract group generated by a finite set of generators g1, g2, ..., gr and defined by a finite set of relations ... [ equation ] ... 1.2 Method. The method of coset enumeration is as follows. G is defined as above by the r generators gl, ..., gr and the k relations (1.1). [...]

After the new impulse given to the theory of algebraic equations by the discoveries of Lagrange and Vandermonde in 1770, Ruffini tried to solve the problem where Lagrange had left it, i.e., proved the impossibility of solving by radicals the general equation of the fifth degree. His proof still remains unclear but nevertheless is very similar to the proof obtained by Abel later. Looking for new types of equations solvable by radicals, the latter reached the conception of "abelian" extensions and showed the solvability by radicals in this case. He defined the notion of irreducible polynomials over a given field. After him, Galois defined what was to be called the Galois group of a polynomial and showed that a polynomial is solvable by radicals if its Galois group is solvable. [...]

The purpose of this paper is two fold. Firstly, to introduce and study the Silov Boundary of a Banach algebra of continuous complex valued functions defined on a compact Hausdorff space X. Secondly, to apply the definition of Silov Boundary to more general families of functions, namely, to linear spaces and semi-groups of functions. It will be seen that in these latter cases, existence of the Silov Boundary depends on certain restricting assumptions about the linear space or semi-group. [...]

It was first pointed out by Doob and Kakutani the connection between classical potential theory and Brownian motion. In [10] one finds that if P(t,x,A) is the probability transition function, i.e. P(t,x,A) = probability that a particle moves from the point x to the Borel subset A of a set I in time t, then the potential kernal, K(x,A), is defined as follows [...]