Thesis (M.Sc.)--University of Melbourne, [Dept. of Mathematics and Statistics], 2009. MSc (by Research) / Typescript. Includes bibliographical references.

Thesis (Ph. D.)--West Virginia University, 2000. / Title from document title page. Document formatted into pages; contains iii, 56 p. Includes abstract. Includes bibliographical references (p. 54-56).

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In this paper, we use the techniques of plethystic substitution to reformulate the difference raising operators presented by Di Francesco and Kedem. A connection between these operators and Shimozono and Zabrocki's parabolic Jing operators is presented. In particular, we find that these operators are a renormalization of a particular case of the parabolic Jing operators. / Master of Science / In response to an open problem in Physics, an idea is presented by Di Francesco and Kedem in [1]. A connection between this idea and a Math idea presented by Shimozono and Zabrocki in [9] is presented. It is common that unknown overlap exists when authors from different fields work on similar problems. This connection is seen once the techniques used by Di Francesco and Kedem are interpreted in the language used by Shimozono and Zabrocki. In particular, we find that the idea in [1] is a specialization of that in [9].

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 1999. / Includes bibliographical references. Also available on the Internet.

The ring of symmetric functions Λ, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the symmetric group. One may define a coproduct on Λ by the plethystic addition on alphabets. In this way the ring of symmetric functions becomes a Hopf algebra. The Littlewood–Richardson numbers may be viewed as the structure constants for the co-product in the Schur basis. The first part of this thesis, inspired by the umbral calculus of Gian-Carlo Rota, is a study of the co-algebra maps of Λ. The Macdonald polynomials are a somewhat mysterious qt-deformation of the Schur functions. The second part of this thesis contains a proof a generating function identity for the Macdonald polynomials which was originally conjectured by Kawanaka.

Thesis (Ph. D.)--University of Wisconsin, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

The original Macdonald polynomials P_μ form a basis for the vector space of symmetric functions which specializes to several of the common bases such as the monomial, Schur, and elementary bases. There are a number of different types of Macdonald polynomials obtained from the original P_μ through a combination of algebraic and plethystic transformations one of which is the modified Macdonald polynomial H̃_μ. In this dissertation, we study a certain specialization F̃_μ(q,t) which is the coefficient of x₁x₂…x_N in H̃_μ and also the Hilbert series of the Garsia-Haiman module M_μ. Haglund found a combinatorial formula expressing F̃_μ as a sum of n! objects weighted by two statistics. Using this formula we prove a q,t-analogue of the hook-length formula for hook shapes. We establish several new combinatorial operations on the fillings which generate F̃_μ. These operations are used to prove a series of recursions and divisibility properties for F̃_μ. / Ph. D.

In this thesis we study relativistic models of gravitating uids with heat ow and electric charge. Firstly, we derive the model of a charged shear-free spherically symmetric cosmological model with heat ow. The solution of the Einstein-Maxwell equations of the system is governed by the pressure isotropy condition. This condition is a highly nonlinear partial di erential equation. We analyse this master equation using Lie's group theoretic approach. The Lie symmetry generators that leave the equation invariant are found. We provide exact solutions to the gravitational potentials using the rst symmetry admitted by the equation. Our new exact solutions contain the earlier results of Msomi et al (2011) without charge. Using the second symmetry we are able to reduce the order of the master equation to a rst order highly nonlinear di erential equation. Secondly, we study a shear-free spherically symmetric cosmological model with heat ow in higher dimensions. We establish the Einstein eld equations and nd the governing pressure isotropy condition. We use an algorithm due to Deng (1989) to provide several new classes of solutions to the model. The four-dimensional case is contained in our general result. Solutions due to Bergmann (1981), Maiti (1982), Modak (1984) and Sanyal and Ray (1984) for the four-dimensional case are regained. We also establish a new class of solutions that contains the results of Deng (1989) from four dimensions. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2011.

In this dissertation we discuss three problems. We first show the classical q-Stirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. We extend this enumerative result via a decomposition of a new poset which we call the Stirling poset of the second kind. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done. We also give a bijective argument showing the (q, t)-Stirling numbers of the first and second kind are orthogonal. In the second part we give combinatorial proofs of q-Stirling identities via restricted growth words. This includes new proofs of the generating function of q-Stirling numbers of the second kind, the q-Vandermonde convolution for Stirling numbers and the q-Frobenius identity. A poset theoretic proof of Carlitz’s identity is also included. In the last part we discuss a new expression for q-binomial coefficients based on the weighting of certain 01-permutations via a new bistatistic related to the major index. We also show that the bistatistics between the inversion number and major index are equidistributed. We generalize this idea to q-multinomial coefficients evaluated at negative q values. An instance of the cyclic sieving phenomenon related to flags of unitary spaces is also studied.