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The partition functions of the affine Pasquier models on the cylinder are calculated in the continuum limit. The partition functions of the models based upon the Â(_n) cycle graphs are first found from the appropriate Coulomb-gas equivalence. Their relationship with the D(_n) and Ề(_6,7,8) models is established by constructing an affine analogue to the classical intertwiners using a Temperley-Lieb algebraic equivalence. From this relationship, each of the partition functions is constructed. We write our results in terms of 'generating polynomials' establishing explicitly the precise operator content of the conformally invariant continuum field theories. A numerical study is undertaken to establish the validity of the partition functions as calculated. We conclude that the partition functions calculated are correct. The partition functions are further studied and the connection with the McKay correspondence established. We establish a simple form for the partition functions in terms of degenerate c = 1 Virasoro characters and Chebychev polynomials of the second kind. From this, we establish the role within the partition functions played by the affine Coxeter element, a particular member of the Weyl group associated with the defining graph of the model. Some of the resulting consequences of this role are explored.

A research report submitted to the Faculty of Science, University of the Witwatersrand, in partial fulfillment of the degree of Master of Science. / The Toeplitz form is a most useful and important teo! in many areas of applied. mathematics today including signal processing, time-series analysis and prediction theory. It is even used in quantum mechanics in Ising model correlation functions. (Abbreviation abstract) / AC 2018

This project presents a modified method of numerical integration for a “well behaved� function over the finite interval [-1,1]. Similar to the Clenshaw-Curtis quadrature rule, this new algorithm relies on expressing the integrand as an expansion of Chebyshev polynomials of the second kind. The truncated series is integrated term-by-term yielding an approximation for the integral of which we wish to compute. The modified method is then contrasted with its predecessor Clenshaw-Curtis, as well as the classical method of Gauss-Legendre in terms of convergence behavior, error analysis and computational efficiency. Lastly, illustrative examples are shown which demonstrate the dependence that the convergence has on the given function to be integrated.

The classical moment problem concerns distribution functions on the real line. The central feature is the connection between distribution functions and the moment sequences which they generate via a Stieltjes integral. The solution of the classical moment problem leads to the well known theorem of Favard which connects orthogonal polynomial sequences with distribution functions on the real line. Orthogonal polynomials in their turn arise in the computation of measures via continued fractions and the Nevanlinna parametrisation. In this dissertation classical orthogonal polynomials are investigated rst and their connection with hypergeometric series is exhibited. Results from the moment problem allow the study of a more general class of orthogonal polynomials. q-Hypergeometric series are presented in analogy with the ordinary hypergeometric series and some results on q-Laguerre polynomials are given. Finally recent research will be discussed.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. April 2013 / We study the action of the dilatation operator on restricted Schur polynomials labeled by Young diagrams with p long columns or p long rows. A new version of Schur-Weyl duality provides a powerful approach to the computation and manipulation of the symmetric group operators appearing in the restricted Schur polynomials. Using this new technology, we are able to evaluate the action of the one loop dilatation operator. The result has a direct and natural connection to the Gauss Law constraint for branes with a compact world volume. We find considerable evidence that the dilatation operator reduces to a decoupled set of harmonic oscillators. This strongly suggests that integrability in N = 4 super Yang-Mills theory is not just a feature of the planar limit, but extends to other large N but non-planar limits.

Let D be an integral domain and f a polynomial that is integer-valued on D. We prove that Int(f(D);D) has the Skolem Property and give a description of its spectrum. For certain discrete valuation domains we give a basis for the ring of integer-valued even polynomials. For these discrete valuation domains, we also give a series expansion of continuous integer-valued functions. / by Mario V. Marshall. / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.

by Wang Jun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 82-86). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- The polynomial zero-one programming problem --- p.2 / Chapter 1.3 --- Motivation --- p.3 / Chapter 1.4 --- Thesis outline --- p.4 / Chapter 2 --- Literature Survey --- p.6 / Chapter 2.1 --- Lawler and Bell's method --- p.7 / Chapter 2.2 --- The covering relaxation algorithm for polynomial zero-one pro- gramming --- p.8 / Chapter 2.3 --- The method of reducing polynomial integer problems to linear zero- one problems --- p.9 / Chapter 2.4 --- Pseudo-boolean programming --- p.11 / Chapter 2.5 --- The Balasian-based algorithm for polynomial zero-one programming --- p.12 / Chapter 2.6 --- The hybrid algorithm for polynomial zero-one programming --- p.12 / Chapter 3 --- The Balasian-based Algorithm --- p.14 / Chapter 3.1 --- The additive algorithm for linear zero-one programming --- p.15 / Chapter 3.2 --- Some notations and definitions referred to the Balasian-based al- gorithm --- p.17 / Chapter 3.3 --- Identification of all the feasible solutions to the master problem --- p.18 / Chapter 3.4 --- Consistency check of the feasible partial solutions --- p.19 / Chapter 4 --- The p-norm Surrogate Constraint Method --- p.21 / Chapter 4.1 --- Introduction --- p.21 / Chapter 4.2 --- Numerical example --- p.23 / Chapter 5 --- The P-norm Surrogate-constraint Algorithm --- p.26 / Chapter 5.1 --- Main ideas --- p.26 / Chapter 5.2 --- The standard form of the polynomial zero-one programming problem --- p.27 / Chapter 5.3 --- Definitions and notations --- p.29 / Chapter 5.3.1 --- Partial solution in x --- p.29 / Chapter 5.3.2 --- Free term --- p.29 / Chapter 5.3.3 --- Completion --- p.29 / Chapter 5.3.4 --- Feasible partial solution --- p.30 / Chapter 5.3.5 --- Consistent partial solution --- p.30 / Chapter 5.3.6 --- Partial solution in y --- p.30 / Chapter 5.3.7 --- Free variable --- p.31 / Chapter 5.3.8 --- Augmented solution in x --- p.31 / Chapter 5.4 --- Solution concepts --- p.33 / Chapter 5.4.1 --- Fathoming --- p.33 / Chapter 5.4.2 --- Backtracks --- p.41 / Chapter 5.4.3 --- Determination of the optimal solution in y --- p.42 / Chapter 5.5 --- Solution algorithm --- p.42 / Chapter 6 --- Numerical Examples --- p.46 / Chapter 6.1 --- Solution process by the new algorithm --- p.46 / Chapter 6.1.1 --- Example 5 --- p.46 / Chapter 6.1.2 --- Example 6 --- p.57 / Chapter 6.2 --- Solution process by the Balasian-based algorithm --- p.61 / Chapter 6.3 --- Comparison between the p-norm surrogate constraint algorithm and the Balasian-based algorithm --- p.71 / Chapter 7 --- Application to the Set Covering Problem --- p.74 / Chapter 7.1 --- The set covering problem --- p.74 / Chapter 7.2 --- Solving the set covering problem by using the new algorithm . .。 --- p.75 / Chapter 8 --- Conclusions and Future Work --- p.80 / Bibliography --- p.82

Complex dynamics is the study of iteration of functions which map the complex plane onto itself. In general, their dynamics are quite complicated and hard to explain but for some simple classes of functions many interesting results can be proved. For example, one often studies the class of rational functions (i.e. quotients of polynomials) or, even more specifically, polynomials. Each such function f partitions the extended complex plane C into two regions, one where iteration of the function is chaotic and one where it is not. The nonchaotic region, called the Fatou Set, is the set of all points z such that, under iteration by f, the point z and all its neighbors do approximately the same thing. The remainder of the complex plane is called the Julia set and consists of those points which do not behave like all closely neighboring points. The Julia set of a polynomial typically has a complicated, self similar structure. Many questions can be asked about this structure. The one that we seek to investigate is the notion of the dimension of the Julia set. While the dimension of a line segment, disc, or cube is familiar, there are sets for which no integer dimension seems reasonable. The notion of Hausdorff dimension gives a reasonable way of assigning appropriate non-integer dimensions to such sets. Our goal is to investigate the behavior of the Hausdorff dimension of the Julia sets of a certain simple class of polynomials, namely fd,c(z) = zd + c. In particular, we seek to determine for what values of c and d the Hausdorff dimension of the Julia set varies continuously with c. Roughly speaking, given a fixed integer d > 1 and some complex c, do nearby values of c have Julia sets with Hausdorff dimension relatively close to each other? We find that for most values of c, the Hausdorff dimension of the Julia set does indeed vary continuously with c. However, we shall also construct an infinite set of discontinuities for each d. Our results are summarized in Theorem 10, Chapter 2. In Chapter 1 we state and briefly explain the terminology and definitions we use for the remainder of the paper. In Chapter 2 we will state the main theorems we prove later and deduce from them the desired continuity properties. In Chapters 3 we prove the major results of this paper.

Thesis (Ph.D.)--McMaster University, 2004. / Supervisor: N. Balakrishnan. Includes bibliographical references (p. 98-101).