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2008

Education

Department of Mathematical Sciences Faculty Publications

Articles 1 - 8 of 8

Full-Text Articles in Physical Sciences and Mathematics

Ramanujan–Slater Type Identities Related To The Moduli 18 And 24, James Mclaughlin, Andrew Sills Aug 2008

Ramanujan–Slater Type Identities Related To The Moduli 18 And 24, James Mclaughlin, Andrew Sills

Department of Mathematical Sciences Faculty Publications

We present several new families of Rogers–Ramanujan type identities related to the moduli 18 and 24. A few of the identities were found by either Ramanujan, Slater, or Dyson, but most are believed to be new. For one of these families, we discuss possible connections with Lie algebras. We also present two families of related false theta function identities.

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Rogers-Ramanujan-Slater Type Identities, James Mclaughlin, Andrew Sills, Peter Zimmer May 2008

Rogers-Ramanujan-Slater Type Identities, James Mclaughlin, Andrew Sills, Peter Zimmer

Department of Mathematical Sciences Faculty Publications

In this survey article, we present an expanded version of Lucy Slater's famous list of identities of the Rogers-Ramanujan type, including identities of similar type, which were discovered after the publication of Slater's papers, and older identities (such as those in Ramanujan's lost notebook) which were not included in Slater's papers. We attempt to supply the earliest known reference for each identity. Also included are identities of false theta functions, along with their relationship to Rogers-Ramanujan type identities. We also describe several ways in which pairs/larger sets of identities may be related, as well as dependence relationships between identities.

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Disturbing The Q-Dyson Conjecture, Andrew Sills Jan 2008

Disturbing The Q-Dyson Conjecture, Andrew Sills

Department of Mathematical Sciences Faculty Publications

I discuss the computational methods behind the formulation of some conjectures related to variants on Andrews’ q-Dyson conjecture.

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Harmonic Analysis Related To Schrödinger Operators, Gestur Olafsson, Shijun Zheng Jan 2008

Harmonic Analysis Related To Schrödinger Operators, Gestur Olafsson, Shijun Zheng

Department of Mathematical Sciences Faculty Publications

In this article, we give an overview of some recent developments in Littlewood-Paley theory for Schrödinger operators. We extend the Littlewood-Paley theory for special potentials considered in our previous work [J. Fourier Anal. Appl. 12 (2006), no. 6, 653–674; MR2275390]. We elaborate our approach by considering a potential in C∞0 or the Schwartz class in one dimension. In particular, the low energy estimates are treated by establishing some new and refined asymptotics for the eigenfunctions and their Fourier transforms. We give a maximal function characterization of the Besov spaces and Triebel-Lizorkin spaces associated with H. Then we …

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A Note On Extreme Bernoulli And Dependent Families Of Bivariate Distributions, Broderick O. Oluyede, Marvis Pararai Jan 2008

A Note On Extreme Bernoulli And Dependent Families Of Bivariate Distributions, Broderick O. Oluyede, Marvis Pararai

Department of Mathematical Sciences Faculty Publications

The objective and purpose of this note is to generate bivariate distributions via extreme Bernoulli distributions and obtain results on positively and negatively dependent families of bivariate binomial distributions generated by extreme Bernoulli distributions. Some distributional properties and results are presented. The factorial moment generating functions, correlation functions, conditional distributions and the regression functions are given.

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A Partition Bijection Related To The Rogers-Selberg Identities And Gordon's Theorem, Andrew Sills Jan 2008

A Partition Bijection Related To The Rogers-Selberg Identities And Gordon's Theorem, Andrew Sills

Department of Mathematical Sciences Faculty Publications

We provide a bijective map from the partitions enumerated by the series side of the Rogers–Selberg mod 7 identities onto partitions associated with a special case of Basil Gordon's combinatorial generalization of the Rogers–Ramanujan identities. The implications of applying the same map to a special case of David Bressoud's even modulus analog of Gordon's theorem are also explored.

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On The Ordinary And Signed Göllnitz-Gordon Partitions, Andrew Sills Jan 2008

On The Ordinary And Signed Göllnitz-Gordon Partitions, Andrew Sills

Department of Mathematical Sciences Faculty Publications

A partition of an integer n is a representation of n as an unordered sum of positive integers. In a recent paper [1], Andrews introduced the notion of a "signed partition," that is, a representation of a positive integer as an unordered sum of integers, some possibly negative.

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Inequalities And Exponential Approximations For Residual Life Reliability Functions, Broderick O. Oluyede, Marvis Pararai Jan 2008

Inequalities And Exponential Approximations For Residual Life Reliability Functions, Broderick O. Oluyede, Marvis Pararai

Department of Mathematical Sciences Faculty Publications

Given that a unit is of age t, the remaining life after time t is random. The expected value of this random residual life is called the mean residual life at time t. Specifically, if T is the life of a component with distribution function F, then δF (t) = E(T −t|T > t) is called the mean residual life function (MRLF). It is well known that the class of distributions with decreasing mean residual life (DMR) contains the class of distributions with increasing hazard rate (IHR). In this note, …

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