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Automorphism Groups of Strong Bruhat Orders of Coxeter Groups
In this dissertation, we describe the automorphism groups for the strong Bruhat orders A_n-1, B_n, and D_n. In particular, the automorphism group of A_n-1 for n ≥ 3 is isomorphic to the dihedral group of order eight, D_4; the automorphism group of B_n for n ≥ 3 is isomorphic to C_2 x C_2 where C_2 is the cyclic group of order two; the automorphism group of D_n for n > 5 and n even is isomorphic to C_2 x C_2 x C_2; and the automorphism group of D_n for n ≥ 5 and n odd is isomorphic to the dihedral group D_4.
Conway's Link Polynomial: a Generalization of the Classic Alexander's Knot Polynomial
The problem under consideration is that of determining a simple and effective invariant of knots. To this end, the Conway polynomial is defined as a generalization of Alexander's original knot polynomial. It is noted, however, that the Conway polynomial is not a complete invariant. If two knots are equivalent, as defined in this investigation, then they receive identical polynomials. Yet, if two knots have identical polynomials, no information about their equivalence may be obtained. To define the Conway polynomial, the Axioms for Computation are given and many examples of their use are included. A major result of this investigation is the proof of topological invariance of these polynomials and the proof that the axioms are sufficient for the calculation of the knot polynomial for any given knot or link.
Dimension Theory
This paper contains a discussion of topological dimension theory. Original proofs of theorems, as well as a presentation of theorems and proofs selected from Ryszard Engelking's Dimension Theory are contained within the body of this endeavor. Preliminary notation is introduced in Chapter I. Chapter II consists of the definition of and theorems relating to the small inductive dimension function Ind. Large inductive dimension is investigated in Chapter III. Chapter IV comprises the definition of covering dimension and theorems discussing the equivalence of the different dimension functions in certain topological settings. Arguments pertaining to the dimension o f Jn are also contained in Chapter IV.
Some Properties of Noetherian Rings
This paper is an investigation of several basic properties of noetherian rings. Chapter I gives a brief introduction, statements of definitions, and statements of theorems without proof. Some of the main results in the study of noetherian rings are proved in Chapter II. These results include proofs of the equivalence of the maximal condition, the ascending chain condition, and that every ideal is finitely generated. Some other results are that if a ring R is noetherian, then R[x] is noetherian, and that if every prime ideal of a ring R is finitely generated, then R is noetherian.
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