Ph.D. / After giving basic definitions concerning additive hereditary properties of graphs, this document is divided into three main sections, concerning minimal reducible bounds, minimal forbidden subgraphs and prime ideals. We prove that every irreducible property has at least one minimal reducible bound, and that if an irreducible property P is contained in a reducible property R, then there is a minimal reducible bound for P contained in R. In addition we show that every reducible property serves as a minimal reducible bound for some irreducible property. Several applications of these results are given in the case of hom-properties, mainly to show the existence of minimal reducible bounds of certain types. We prove that every degenerate property has a minimal reducible bound where one factor is the class of edgeless graphs and provide evidence that this may also be true for an arbitrary irreducible property. We also consider edge partitions and we show that the results for minimal decomposable bounds are similar to those for minimal reducible bounds. The second set of results deals with minimal forbidden subgraphs of graph properties. We show that every reducible property has infinitely many minimal forbidden subgraphs since the set of all the cyclic blocks making up these graphs is infinite. Finally we consider the lattice of all additive hereditary properies and study the prime ideals in this lattice. We give an example of a prime ideal that is not co-principal and give some requirements that non co-principal prime ideals should satisfy. 'vVe prove that the set of properties with bounded chromatic number is not a prime ideal by showing that a property P with bounded chromatic number can be written as the intersection of two properties with unbounded chromatic number if and only if P contains all trees.

Three concepts of ordinal numbers are examined with a view to their intuitiveriess and existence in two principle systems of axiomatic set theory. The first is based on equivalence classes of the similarity relation between well-ordered sets. Two alternatives are suggested in later chapters for overcoming the problems arizing from this definition. Next, ordinal numbers are defined as certain representatives of these equivalence classes,, and one of several such possible definitions is taken for proving the fundamental properties of these ordinals. Finally, a generalization of Peano's axioms provides us with a method of defining ordinal numbers which are the ultimate result of abstractions. / Science, Faculty of / Mathematics, Department of / Graduate

Magnetic space groups (MSGs) were first introduced (under a different name) by Heesch more than 30 years ago, and a list of all of them was published by Belov, Neronova and Srairnova in 1955. However, no mathematically rigorous derivation of MSGs can be found in the existing literature, although an outline of a method for obtaining a large class of MSGs was published by Zaraorzaev in 1957. In this thesis a systematic rigorous method for constructing MSGs is described in detail, and a proof that the method in fact gives all the MSGs is presented. The method also leads in a natural way to a classification of MSGs which is useful for a systematic study of the arrangements of spins in ferromagnetic, ferrimagnetic and antiferromagnetic crystals. The first and the last chapter of the thesis deal with the physical aspects of the problem, the remaining chapters with purely mathematical aspects of it. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

The use of spline basis functions in solving least squares approximation problems is investigated. The question as to which are appropriate basis functions to use is discussed along with the reasons why the final choice was made. The Householder transformation method for solving the fixed knot spline approximation problem is examined. Descriptions of both an automatic procedure using function minimization and an interactive procedure using a graphics terminal for solving the variable knot spline approximation problem are given. In conclusion, numerical results using the interactive system are presented and analyzed. / Science, Faculty of / Computer Science, Department of / Graduate

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<p>The classical Main Conjecture (MC) in Iwasawa Theory relates values of p-adic L-function associated to 1-dimensional Artin characters over a totally real number field F to values of characteristic polynomials attached to certain Iwasawa modules. Wiles [47] proved the MC for odd primes p over arbitrary totally real base fields F and for the prime 2 over abelian totally real fields F.</p> <p>An equivariant version of the MC, which combines the information for all characters of the Galois group of a relative abelian extension E/F of number fields with F totally real, was formulated and proven for odd primes p by Ritter and Weiss in [33] under the assumption that the corresponding Iwasawa module is finitely generated over ℤ_p ("µ=0"). This assumption is satisfied for abelian fields and conjectured to be true in general.</p> <p>In this thesis we formulate an Equivariant Main Conjecture (EMC) for all prime numbers p, which coincides with the version of Ritter and Weiss for odd p, and we provide a unified proof of the EMC for all primes p under the assumptions µ=0 and the validity of the 2-adic MC. The proof combines the approach of Ritter and Weiss with ideas and techniques used recently by Greither and Popescu [13] to give a proof of a slightly different formulation of an EMC under the same assumptions (p odd and µ=0) as in [33].</p> <p>As an application of the EMC we prove the Coates-Sinnott Conjecture, again assuming µ=0. We also show that the p-adic version of the Coates-Sinnott Conjecture holds without the assumption µ=0 for abelian Galois extensions E/F of degree prime to p. These generalize previous results for odd primes due to Nguyen Quang Do in [27], Greither-Popescu [13], and Popescu in [30].</p> / Doctor of Philosophy (PhD)

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by Li Jian Lin. / "November 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 85-90) / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.