This thesis is concerned with the Galois groups of the root fields of the equations x[superscript]P - a = 0, (x[superscript]p - a)•(x[superscript]q - b) = 0 and (x[superscript]q - b) [superscript]p - a = 0, where p and q are distinct primes, and a and b are rationals. The correspondence of subflelds and subgroups is studied for each of the three cases. The field [formula omitted] formed by adjoining to the rational field F the elements [formula omitted and ⍺, a primitive pth root of unity, is shown to be the root field of x[superscript]p - a = 0, normal over F of degree p(p-l). The Galois group of [formula omitted] over F Is found to be the metacyclic group constructed from generators s and t subject to relations s[superscript]p = 1, t[superscript]p ⁻¹ = 1 and st = ts[superscript]r, where r is a primitive root modulo p, and where s is the automorphism which maps [formula omitted] onto [formula omitted] while t is the automorphism which maps ⍺. onto ⍺ [superscript]r. Various subgroups and corresponding subflelds are studied and nine theorems proven on their correspondences, illustrated with a partial lattice diagram. The field [formula omitted]where β is a primitive qth root of unity is shown to be the root field of (x[superscript]p - a)-(x[superscript]q - b) = 0 and the Galois group is proven to be the direct product of two of the type for the field [formula omitted]. The field [formula omitted] for i = 1, 2, 3 ... p, which is the root field of the equation (x[superscript]q - b) [superscript]p - a = 0 is studied and shown to have degree pq[superscript]p-(p-1)•(q-1). The Galois group is found to be generated by four independent generators: s, t, u, v subject to eleven defining relations. Here the elements s, t, u, v are the automorphisms which respectively map [formula omitted] onto [formular omitted], ⍺ onto ⍺[superscript r, [formula omitted] onto [formula omitted] β onto β [superscript]where w is a primitive root modulo q. A partial lattice diagram illustrates the correspondence of subgroups and subflelds. The thesis was carried out under the supervision of Dr. D. C. Murdoch. / Science, Faculty of / Mathematics, Department of / Graduate

Theorem : Let X be simply connected . H[sub q](X) be finitely generated for each q. π[sub q](X) be finite for each q < n. n>> 1. Then , H[sub q] , π[sub q](X) --> H[sub q](X) has finite kernel for q < 2n has finite cokernel for q < 2n+l ker h[sub 2n+1] Q = ker u where , u is the cup product or the square free cup product on R[sup n+1](x) depending on whether n+1 is even or odd , respectively . ( R[sup N+1](X) is a quotient group of H[sub Q][sup n+1](x) to be defined in this thesis ) / Science, Faculty of / Mathematics, Department of / Graduate

While there is considerable literature about algebras satisfying a polynomial identity, there are only scant results about varieties of algebras. For such an algebra we can introduce the notions of bimodule, birepresentation and universal enveloping algebra as an extension of the notions of module and representation for associative algebras. Moreover, it is possible to define injective hulls for these restricted representations. We derive a rather concrete structure theorem of I-bimodules M for a finite dimensional algebra in a certain variety by studying a universal enveloping algebra and injective hulls. / Science, Faculty of / Mathematics, Department of / Graduate

Definition: Let k be an algebraically closed field. An algebraic monoid is a triple (E,m,l) such that E is an algebraic variety defined over k, m : ExE → E is an associative morphism and 1 € E is a two—sided unit for m. The object of this thesis is to expose several fundamental topics in the theory of algebraic monoids. My results may be divided into three types; general theory of irreducible affine monoids, structure and classification of semi—simple rank one reductive monoids, and theory of general monoid varieties (not necessarily affine). I General Theory of Affine Monoids II Reductive Monoids of Semi-simple Rank One III General Monoid Varieties [Please see document for entire abstract] / Science, Faculty of / Mathematics, Department of / Graduate

A set-theoretical point of view to study algebraic numbers has been introduced. We extend a result of Navarro-Bermudez concerning shift invariant measures in the Cantor space which are topologically equivalent to shift invariant measures which correspond to some algebraic integers. It is known that any transcendental numbers and rational numbers in the unit interval are not binomial. We proved that there are algebraic numbers of degree greater than two so that they are binomial numbers. Algebraic integers of degree 2 are proved not to be binomial numbers. A few compositive relations having to do with algebraic numbers on the unit interval have been studied; for instance, rationally related, integrally related, binomially related, B1-related relations. A formula between binomial numbers and binomial coefficients has been stated. A generalized algebraic equation related to topologically equivalent measures has also been stated.

Through the definition of a new insertion algorithm this paper seeks to provide an alternative to the existing bijections between permutations and certain kinds of tableaux. We will define two versions of each algorithm covered, both the existing ones and the novel one. These different constructions will include one using a lot of small intermediate steps and one which directly creates the tableaux from the permutation. After showing that these are equivalent, we will briefly discuss the results of pattern avoidance on tableau shape. / Doctor of Philosophy / Building up tableaux from permutations can be a helpful way to get information about that permutation without having to check by hand. Different methods of building tableaux will tell us different types of information about the permutation. For that reason, we are defining a new method of building tableaux so that we can extract useful information from the permutations used.

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Undergraduate honors paper--Mount Holyoke College, 2006. Dept. of Mathematics. / Includes bibliographical references (leaves 66-67).

The creation of smooth interpolating curves and surfaces is an important aspect of computer graphics. Trigonometric interpolation in the form of the Fourier transform has been a popular technique. For computer graphics, simpler curves and surfaces like the B´ezier curve and B-spline curve have been more popular due to the computational efficiency. Fitting B-spline or B´ezier curves or surfaces to unorganised data points has been more challenging since these curves are not naturally interpolating. Normally a system of equations needs to be solved to obtain the curves or surfaces with the added problem of identifying data points to form piecewise continuous surfaces. We solve the problem of periodic interpolating curves and surfaces using harmonic interpolation [73]. We extend harmonic interpolation to handle an even number of data points. We then show how harmonic interpolation can be applied using geometry images [29] to create smooth interpolating surfaces. We introduce algorithms to manipulate the amount of interpolated points, and the location of the interpolated points. Finally, we show how a smooth interpolating surface created by harmonic interpolation can be converted to a series of B´ezier surfaces. The combination of techniques allows us to quickly create a smooth interpolating surface from a set of unorganised points that have a known spherical structure. Keywords: Interpolation, harmonic interpolation, trigonometric interpolation, B´ezier curves surface fitting, tensor product surfaces. / Prof. W.F. Steeb

Thesis (MA. (Mathematics)) -- University of Limpopo, 2012 / Topological structures called Yosida frames and related algebraic frames are studied in the realm of Pointfree Topology. It is shown that in algebraic frames regular elements are those for which compact elements are rather below the regular elements, and algebraic frames are regular if and only if every compact element is rather below itself if and only if the frame has the Finite Intersection Property (FIP) and each prime element is minimal. We also show that Yosida frames are those algebraic frames with the Finite Intersection Property and are finitely subfit; that these frames are also those semi-simple algebraic frames with FIP and a disjointification where dim (L)≤ 1; and we prove that in an algebraic frame with FIP, it holds that dom (L) = dim (L). In relation to normality in Yosida frames, we show that in a coherent normal Yosida frame L, the frame is subfit if and only if it is regular if and only if it is zero- dimensional if and only if every compact element is complemented.