Education Commons^™

Compositions, Partitions, And Fibonacci Numbers, Andrew Sills

Department of Mathematical Sciences Faculty Publications

A bijective proof is given for the following theorem: the number of compositions of n into odd parts equals the number of compositions of n+ 1 into parts greater than one. Some commentary about the history of partitions and compositions is provided.

Multiplicative Noise For Masking Numerical Microdata Data With Constraints, Anna Oganian

Department of Mathematical Sciences Faculty Publications

Before releasing databases which contain sensitive information about individuals, statistical agencies have to apply Statistical Disclosure Limitation (SDL) methods to such data. The goal of these methods is to minimize the risk of disclosure of the confidential information and at the same time provide legitimate data users with accurate information about the population of interest. SDL methods applicable to the microdata (i. e. collection of individual records) are often called masking methods. In this paper, several multiplicative noise masking schemes are presented. These schemes are designed to preserve positivity and inequality constraints in the data together with the vector of …

Weak Type Inequalities For Maximal Operators Associated To Double Ergodic Sums, Paul Hagelstein, Alexander M. Stokolos

Department of Mathematical Sciences Faculty Publications

Given an approach region Γ ∈ Z₊² and a pair U, V of commuting nonperiodic measure preserving transformations on a probability space (Ω, Σ, μ), it is shown that either the associated multiparameter ergodic averages of any function in L¹(Ω) converge a.e. or that, given a positive increasing function ϕ on [0,∞) that is o(log x) as x → ∞, there exists a function g ∈ Lϕ(L)(Ω) whose associated multiparameter ergodic averages fail to converge a.e.

Spectral Asymptotics Of Laplacians Associated To One-Dimensional Iterated Function Systems With Overlaps, Sze-Man Ngai

Department of Mathematical Sciences Faculty Publications

We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

Polynomial Generalizations Of Two-Variable Ramanujan Type Identities, James Mclaughlin, Andrew Sills

Department of Mathematical Sciences Faculty Publications

We provide finite analogs of a pair of two-variable q-series identities from Ramanujan's lost notebook and a companion identity.

Independence Number And Disjoint Theta Graphs, Shinya Fujita, Colton Magnant

Department of Mathematical Sciences Faculty Publications

The goal of this paper is to find vertex disjoint even cycles in graphs. For this purpose, define a θ-graph to be a pair of vertices u,v with three internally disjoint paths joining u to v. Given an independence number α and a fixed integer k, the results contained in this paper provide sharp bounds on the order f(k,α) of a graph with independence number α(G)≤α which contains no k disjoint θ-graphs. Since every θ-graph contains an even cycle, these results provide k disjoint even cycles in graphs of order at least f(k,α)+1. We also discuss the relationship between this …

Unified Analysis Of Kernel-Based Interior-Point Methods For P *(Κ)-Lcp, Goran Lesaja, C. Roos

Department of Mathematical Sciences Faculty Publications

We present an interior-point method for the P∗(κ)-linear complementarity problem (LCP) that is based on barrier functions which are defined by a large class of univariate functions called eligible kernel functions. This class is fairly general and includes the classical logarithmic function and the self-regular functions, as well as many non-self-regular functions as special cases. We provide a unified analysis of the method and give a general scheme on how to calculate the iteration bounds for the entire class. We also calculate the iteration bounds of both long-step and short-step versions of the method for several …

On Weighted Distributions And Mean Advantage Over Inferiors Functions, Broderick O. Oluyede, Norou Diawara

Department of Mathematical Sciences Faculty Publications

In this note, some fundamental results including relationship be-tween weighted distribution functions and mean advantage over inferi-ors functions are established. Ordering of reliability and/or distribution functions via mean advantage over inferiors functions and related func-tions for parent and weighted reliability functions are presented. Some applications and examples are given.

Long Path Lemma Concerning Connectivity And Independence Number, Shinya Fujita, Alexander Halperin, Colton Magnant

Department of Mathematical Sciences Faculty Publications

We show that, in a k-connected graph G of order n with α(G)=α, between any pair of vertices, there exists a path P joining them with

|P|≥min{n,(k−1)(n−k)/α +k}.

This implies that, for any edge e∈E(G), there is a cycle containing e of length at least

min{n,(k−1)(n−k)/α +k}.

Moreover, we generalize our result as follows: for any choice S of s≤k vertices in G, there exists a tree T whose set of leaves is S with

|T|≥min{n,(k−s+1)(n−k)/α +k}.

Kernel-Based Interior-Point Methods For Cartesian P*(Κ)-Linear Complementarity Problems Over Symmetric Cones, Goran Lesaja

Department of Mathematical Sciences Faculty Publications

We present an interior point method for Cartesian P*(k)-Linear Complementarity Problems over Symmetric Cones (SCLCPs). The Cartesian P*(k)-SCLCPs have been recently introduced as the generalization of the more commonly known and more widely used monotone SCLCPs. The IPM is based on the barrier functions that are defined by a large class of univariate functions called eligible kernel function which have recently been successfully used to design new IPMs for various optimization problems. Eligible barrier (kernel) functions are used in calculating the Nesterov-Todd search directions and the default step-size which leads to a very good complexity results for the method. For …

Meander Graphs And Frobenius Seaweed Lie Algebras, Colton Magnant, Vincent E. Coll, Anthony Giaquinto

Department of Mathematical Sciences Faculty Publications

The index of a seaweed Lie algebra can be computed from its associated meander graph. We examine this graph in several ways with a goal of determining families of Frobenius (index zero) seaweed algebras. Our analysis gives two new families of Frobenius seaweed algebras as well as elementary proofs of known families of such Lie algebras.

On An Identity Of Gessel And Stanton And The New Little Göllnitz Identities, Carla D. Savage, Andrew Sills

Department of Mathematical Sciences Faculty Publications

We show that an identity of Gessel and Stanton [I. Gessel, D. Stanton, Applications of q-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983) 197, Eq. (7.24)] can be viewed as a symmetric version of a recent analytic variation of the little Göllnitz identities. This is significant, since the Göllnitz–Gordon identities are considered the usual symmetric counterpart to little Göllnitz theorems. Is it possible, then, that the Gessel–Stanton identity is part of an infinite family of identities like those of Göllnitz–Gordon?

Toward this end, we derive partners and generalizations of the Gessel–Stanton identity. We show that …

Hybrid Proofs Of The Q-Binomial Theorem And Other Identities, Dennis Eichhorn, James Mclaughlin, Andrew Sills

Department of Mathematical Sciences Faculty Publications

We give "hybrid" proofs of the q-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version.

We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities due to Ramanujan.

Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities.

Core Compactness And Diagonality In Spaces Of Open Sets, Francis Jordan, Frédéric D. Mynard

Department of Mathematical Sciences Faculty Publications

We investigate when the space OX of open subsets of a topological space X endowed with the Scott topology is core compact. Such conditions turn out to be related to infraconsonance of X, which in turn is characterized in terms of coincidence of the Scott topology of OX × OX with the product of the Scott topologies of OX at (X,X). On the other hand, we characterize diagonality of OX endowed with the Scott convergence and show that this space can be diagonal without being pretopological. New examples are provided to clarify the relationship between pretopologicity, topologicity and diagonality of …