Exact Solution of the Discrete (1+1)-dimensional RSOS Model with Field and Surface Interactions

We present the solution of a linear Restricted Solid--on--Solid (RSOS) model in a field. Aside from the origins of this model in the context of describing the phase boundary in a magnet, interest also comes from more recent work on the steady state of non-equilibrium models of molecular motors. While similar to a previously solved (non-restricted) SOS model in its physical behaviour, mathematically the solution is more complex. Involving basic hypergeometric functions ${}_3\phi_2$, it introduces a new form of solution to the lexicon of directed lattice path generating functions.Comment: 10 pages, 2 figure

Generating sequences from the sums of binomial coefficients in a residue class modulo q.

For non-negative integers r we examine four families of alternating and non-alternating sign closed form binomial sums, Fs;ab(r; t; q), in a generalised congruence modulo q. We explore sums of squares and divisibility properties such as those determined by Weisman (and Fleck). Extending r to all integers we express the sequences in terms of closed form roots of unity and subsequently cosines. By a renumbering of these sequences we build eight new \diagonalised" sequences, Ls;abc(r; t; q), and construct equivalent closed forms and sums of squares relations. We modify Fibonacci type polynomials to construct order m recurrence polynomials that satisfy these diagonalised sequences. These recurrence polynomial sequences are shown to satisfy second order differential equations and exhibit orthogonal relations. From these latter relations we establish three term recurrence relations both between and within sequences. By the application of the reciprocal recurrence polynomial and hypergeometric functions, generating functions for these renumbered sequences are determined. Then employing these latter functions, we establish theorems that enable us to express each of the new sequences in terms of a Minor Corner Layered (MCL) determinant. When r is a negative integer and q = 2m+b is unspecified, the MCL determinants produce sequences of polynomials in m. For particular sequences we truncate these polynomials to contain only the leading coefficient and find that the truncated polynomial is equal to that of a Dirichlet series of the form zeta, lambda, beta or eta. From this relationship, recurrence polynomials for these latter functions are established Finally we develop a congruence for the denominator of the uncancelled modified Bernoulli numbers of the first kind, Bn=n!, and consequently a similar congruence for the zeta function at positive even valued integers. Furthermore we determine that these congruences obey the Fleck congruence

Mock Modular Forms and Class Numbers of quadratic forms

This thesis deals with the question for possible recurrence relations among Fourier coefficients of a certain class of mock modular forms. The most prominent examples of such Fourier coefficients are the Hurwitz class numbers of binary quadratic forms, which satisfy many well-known recurrence relations. As examples one should mention the Kronecker-Hurwitz class number relations and the famous Eichler-Selberg trace formula for Hecke operators on spaces of cusp forms. In 1975, H. Cohen conjectured an infinite family of such class number relations which are intimately related to the aforementioned Eichler-Selberg trace formula. In this thesis, I prove Cohen's conjecture and other similar class number formulas using important results from the theory of mock modular forms. By applying a different method I prove at the end that such recurrence relations are a quite general phenomenon for Fourier coefficients of mock theta functions and mock modular forms of weight $\tfrac 32$. As special cases, one gets an alternative proof for the aforementioned class number relations

From abstract programs to precise asymptotic closed-form bounds

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Informática, Departamento de Sistemas Informáticos y Computación, leída el 29-05-2014.Depto. de Sistemas Informáticos y ComputaciónFac. de InformáticaTRUEunpu

STK /WST 795 Research Reports

These documents contain the honours research reports for each year for the Department of Statistics.Honours Research Reports - University of Pretoria 20XXStatisticsBSs (Hons) Mathematical Statistics, BCom (Hons) Statistics, BCom (Hons) Mathematical StatisticsUnrestricte