Let G be a finite group and X a subset of G. The commuting graph C(G,X) is the graph whose vertex set is X with two distinct elements of X joined by an edge whenever they commute in the group G. This thesis studies the structure of commuting graphs C(G,X) when G is either a symmetric group Sym(n) or a sporadic group McL, and X a conjugacy class for elements of order three. We describe how this graph can be useful in understanding various aspects of the structure of the group with a particular emphasis on the connectivity of the graph, the properties of the discs around some fixed vertex and the diameter of the graph.

Representation theory is the study of abstract algebraic structures by representing their elements as linear transformations or matrices. It provides a bridge between the abstract symbolic mathematics and its explicit applications in nearly every branch of mathematics. Combinatorial representation theory aims to use combinatorial objects to model representations, thus answering questions in this field combinatorially. Combinatorial objects are used to help describe, count and generate representations. This has led to a rich symbiotic relationship where combinatorics has helped answer algebraic questions and algebraic techniques have helped answer combinatorial questions. In this thesis we discuss the representation theory of the symmetric group and the general linear group. The theory of these two families of groups is often considered the corner stone of combinatorial representation theory. Results and techniques arising from the study of these groups have been successfully generalized to a very wide class of groups. An overview of some of the generalizations can be found in [BR99]. There are also many avenues for further generalizations which are currently being explored. The constructions of the Specht and Schur modules that we discuss here use the concept of Young tableaux. Young tableaux are combinatorial objects that were introduced by the Reverend Alfred Young, a mathematician at Cambridge University, in 1901. In 1903, Georg Frobenius applied them to the study of the symmetric group. Since then, they have been found to play an important role in the study of symmetric functions, representation theory of the symmetric and complex general linear groups and Schubert calculus of Grassmannians. Applications of Young tableaux to other branches of mathematics are still being discovered. When drawing and labelling Young tableaux there are a few conflicting conventions in the literature, throughout this thesis we shall be following the English notation. In chapter 1 we shall make a few definitions and state some results which will be used in this thesis. In chapter 2 we discuss the representations of the symmetric group. In this chapter we define the Specht modules and prove that they describe all the irreducible representations of Sn. We conclude with a discussion about the ring of Sn representations which is used to prove some identities of Specht modules. In chapter 3 we discuss the representations of the general linear group. In this chapter we define the Schur modules and prove that they describe all the irreducible rational representations of GLmC. We also show that the set of tableaux forms an indexing set for a basis of the Schur modules. In chapter 4 we describe a relation between the Specht and Schur modules. This is a corollary to the more general Schur-Weyl duality, an overview of which can be found in [BR99]. The appendix contains the code and screen-shots of two computer programs that were written as part of this thesis. The programs have been written in C++ and the data structures have been implemented using the Standard Template Library. The first program gives us information about the representations of Sn for a given n. For a user defined n it will list all the Specht modules corresponding to that n, their dimensions and the standard tableaux corresponding to their basis elements. The second program gives information about a certain representation of GLmC. For a user defined m and λ it gives the dimension and the semistandard tableaux corresponding to the basis elements of the Schur module Eλ .

The goal of this thesis is to first give an overview of the modern approach, using the paper of A. Vershik and A. Okounkov, to inductively parametrizing all irreducible representations of the symmetric groups. This theory is then used to answer questions concerning to central projections in the group algebra. We index units first by partitions, and then by so called standard tableaux. We also present a new result and discuss future research exploring the connections between this theory and Quantum Information.

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The theory of infinite multiplication has been studied in the case of the Hawaiian earring group, and has been seen to simplify the description of that group. In this paper we try to extend the theory of infinite multiplication to other groups and give a few examples of how this can be done. In particular, we discuss the theory as applied to symmetric groups and braid groups. We also give an equivalent definition to K. Eda's infinitary product as the fundamental group of a modified wedge product.

The total character of a finite group G is the sum of the irreducible characters of G. When the total character of a finite group can be written as a monic polynomial with integer coefficients in an irreducible character of G, we say that G is a total character group. In this thesis we examine the total character of the dicyclic group of order 4n, the non-abelian groups of order p^3, and the symmetric group on n elements for all n ≥ 1. The dicyclic group of order 4n is a total character group precisely when n is congruent to 2 or 3 mod 4, and the associated polynomial is a sum of Chebyshev polynomials of the second kind. The irreducible characters paired with these polynomials are exactly the faithful characters of the dicyclic group. In contrast, the non-abelian groups of order p^3 and the symmetric group on n elements with n ≥ 4 are not total character groups. Finally, we examine the special case when G is a total character group and the polynomial is of degree 2. In this case, we say that G is a quadratic total character group. We classify groups which are both quadratic total character groups and p-groups.

<p> This study investigates the flow patterns and wall shear rate distributions downstream from a series of three glass model symmetric bifurcations, typical of the blood vessels in man. The models have a single included angle of 75° and total output to input flow area ratios of 0.75, 1.02 and 1.29, covering the physiological range. The Reynolds numbers studied (based on parent tube) were 400, 800 and 1200 in steady flow.</p> <p> Local fluid velocities were obtained at a number of axial positions along the bifurcation daughter tube via a neutrally buoyant tracer particle technique utilizing cine photography. This provided sufficient information to determine the three velocity components for each particle. The tangential and radial components were in general less than 6% of the mean axial velocity. In the case of the axial components, an analytical representation of the velocity in polar coordinates was obtained. This analytical function permits evaluation of wall shear rate distribution.</p> <p> The velocity pro£iles were found to be symmetric with respect to the plane of the bifurcation. At two diameters downstream from the carina the velocity profiles in the plane of the bifurcation showed a high peak near the inside wall of the branch. With distance downstream the peak was convected tangentially evening out the profile towards an axially symmetric mountain plateau with a dished top.</p> <p> Wall shear rate as a function of θ at constant axial position was represented by displaced cosine function. The highest shear rates always occurred on the inside wall of the daughter tube and the lowest on the outside wall. In general, the largest deviation from developed shear rates occurred close to the carina.</p> <p> The largest positive deviation in wall shear rate from developed values was found in the small area ratio bifurcation and the lowest wall shear rate value was found in the large area ratio bifurcation (a = 1.29) indicating possible flow separation near the carina. The biological implications of the shear rate information generated are discussed.</p> / Thesis / Master of Engineering (MEngr)

<p> An attempt was made to measure the yields of the stable cadmium isotopes produced in the thermal-neutron fission of 233U and 235U using a solid-source mass spectrometer. The results for 235U fission indicate that there is structure in the mass-yield curve for the region studied which takes the form of a depression around masses 112-114. The origin of this structure is discussed in terms of the various mechanisms which have been proposed to explain the nature of the mass distribution in the symmetric region.</p> <p> The 233U study was unsuccessful because of the experimental difficulties encountered, primarily the interference from terrestrial cadmium and the low recovery of the fission products. </p> / Thesis / Master of Science (MSc)

A graph in which every edge is labeled positive or negative is called a signed graph. We determine the number of ways to sign the edges of the McGee graph with exactly two negative edges up to switching isomorphism. We characterize signed graphs that are both sign-symmetric and have a frustration index of 1. We prove some results about which signed graphs on complete multipartite graphs have frustration indices 2 and 3. In the final part, we derive the relationship between the frustration index and the number of parts in a sign-symmetric signed graph on complete multipartite graphs.