(PHD, 1971)

Abstract:

We use a nonstandard model of analysis to study two main topics in complex analysis.

UNIFORM CONTINUITY AND RATES OF GROWTH OF MEROMORPHIC FUNCTIONS
is a unified nonstandard approach to several
theories; the Julia-Milloux theorem and Julia exceptional functions,
Yosida’s class (A), normal meromorphic functions, and Gavrilov’s
W_{p} classes. All of these theories are reduced to the study of uniform
continuity in an appropriate metric by means of S-continuity in the
nonstandard model (which was introduced by A. Robinson).

The connection with the classical Picard theorem is made through a generalization of a result of A. Robinson on S-continuous *-holomorphic functions.

S-continuity offers considerable simplifications over the standard sequential approach and permits a new characterization of these growth requirements.

BOUNDED ANALYTIC FUNCTIONS AS THE DUAL OF A
BANACH SPACE is a nonstandard approach to the pre-dual Banach
space for H^{∞}(D) which was introduced by Rubel and Shields.

A new characterization of the pre-dual by means of the nonstandard hull of a space of contour integrals infinitesimally near the boundary of an arbitrary region is given.

A new characterization of the strict topology is given in terms of the infinitesimal relation: "h b k provided ||h-k|| is finite and h(z) ≈ k(z) for z∈(*D)".

A new proof of the noncoincidence of the strict and Mackey topologies is given in the case of a smooth finitely connected region. The idea of the proof is that the infinitesimal relation: “h γ k provided ||h-k|| is finite and h(z) ≈ k(z) on nearly all of the boundary”, gives rise to a compatible topology finer than the strict topology.

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(PHD, 1969)

Abstract:

If R is a ring with identity, let N(R) denote the Jacobson radical of R. R is local if R/N(R) is an artinian simple ring and ∩N(R)^{i} = 0. It is known that if R is complete in the N(R)-adic topology then R is equal to (B)_{n}, the full n by n matrix ring over B where E/N(E) is a division ring. The main results of the thesis deal with the structure of such rings B. In fact we have the following.

If B is a complete local algebra over F where B/N(B) is a finite dimensional normal extension of F and N(B) is finitely generated as a left ideal by k elements, then there exist automorphisms g_{i},…,g_{k} of B/N(B) over F such that B is a homomorphic image of B/N[[x_{1},…,x_{k};g_{1},…,g_{k}]] the power series ring over B/N(B) in noncommuting indeterminates x_{i}, where x_{i}b = g_{i}(b)x_{i} for all b ϵ B/N.

Another theorem generalizes this result to complete local rings which have suitable commutative subrings. As a corollary of this we have the following. Let B be a complete local ring with B/N(B) a finite field. If N(B) is finitely generated as a left ideal by k elements then there exist automorphisms g_{1},…,g_{k} of a v-ring V such that B is a homomorphic image of V [[x_{1},…,x_{k};g_{1},…,g_{k}]].

In both these results it is essential to know the structure of N(B) as a two sided module over a suitable subring of B.

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(PHD, 1960)

Abstract: NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document. Let […] be a set of finite groups and define […] to be the intersection of all sets of groups which contain […] and are closed under the operations of subgroup, factor group and direct product. The equivalence relation defined by […] if […] = […] is studied and it is shown that if Qn and Dn are the generalized quaternion group of order 2n and the dihedral group of order 2n then […] = […]. A group G is called decomposable if […] with […] the set of proper subgroups and factor groups of G. It is shown that if G is decomposable then G must contain a proper subgroup or factor group whose class is the same as the class of G and one whose derived length is the same as the derived length of G. The set of indecomposable p-groups of class two are characterized and for […] their defining relations are compiled. It is also shown that if the exponent of G is p and the class of G is greater than two then G is decomposable if G/Z(G) is a direct product. Finally the equivalence relation given above is modified and its connection with the isoclinism relation of P. Hall is investigated. It is shown that for a certain class of p-groups this relation is equivalent to isoclinism

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