Mathematics Commons^™

A Combinatorial Approach To Hyperharmonic Numbers, Arthur T. Benjamin, David Gaebler '04, Robert Gaebler '04

All HMC Faculty Publications and Research

Hyperharmonic numbers arise by taking repeated partial sums of harmonic numbers. These numbers can be expressed in terms of r-Stirling numbers, leading to combinatorial interpretations of many interesting identities.

A Probabilistic View Of Certain Weighted Fibonacci Sums, Arthur T. Benjamin, Judson D. Neer, Daniel T. Otero, James A. Sellers

All HMC Faculty Publications and Research

In this article, we pursue the reverse strategy of using probability to derive a_n and develop an exponential generating function for a_n in Section 3. In Section 4, we present a method for finding an exact, non-recursive, formula for a_n.

The Fibonacci Numbers -- Exposed More Discretely, Arthur T. Benjamin, Jennifer J. Quinn

All HMC Faculty Publications and Research

No abstract provided in this article.

Balanced Configurations Of Lattice Vectors And Gkz-Rational Toric Fourfolds In P^6, Eduardo Cattani, Alicia Dickenstein

Eduardo Cattani

We introduce a notion of balanced configurations of vectors. This is motivated by the study of rational A-hypergeometric functions in the sense of Gelfand, Kapranov and Zelevinsky. We classify balanced configurations of seven plane vectors up to GL(2,R)-equivalence and deduce that the only gkz-rational toric four-folds in P6 are those varieties associated with an essential Cayley configuration. We show that in this case, all rational A-hypergeometric functions may be described in terms of toric residues. This follows from studying a suitable hyperplane arrangement.

Vertex-Magic Labeling Of Trees And Forests, I. D. Gray, J. Macdougall, John P. Mcsorley, Walter D. Wallis

Articles and Preprints

A vertex-magic total labeling of a graph G(V,E) is a one-to-one map λ from E ∪ V onto the integers {1, 2, . . . , |E| + |V|} such that

λ(x) + Σ λ(xy) where the sum is over all vertices y adjacent to x, is a constant, independent of the choice of vertex x. In this paper we examine the existence of vertex-magic total labelings of trees and forests. The situation is quite different from the conjectured behavior of edge-magic total labelings …

On The Divergence In The General Sense Of Q-Continued Fractions On The Unit Circle, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

We show, for each q-continued fraction G(q) in a certain class of continued fractions, that there is an uncountable set of points on the unit circle at which G(q) diverges in the general sense. This class includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fraction. We discuss the implications of our theorems for the general convergence of other q-continued fractions, for example the G¨ollnitz-Gordon continued fraction, on the unit circle.

Multi-Variable Polynomial Solutions To Pell's Equation And Fundamental Units In Real Quadratic Fields, James Mclaughlin

Mathematics Faculty Publications

Solving Pell’s equation is of relevance in finding fundamental units in real quadratic fields and for this reason polynomial solutions are of interest in that they can supply the fundamental units in infinite families of such fields. In this paper an algorithm is described which allows one to construct, for each positive integer n, a finite collection, {Fi}, of multi-variable polynomials (with integral coefficients), each satisfying a multi-variable polynomial Pell’s equation C 2 i − FiH 2 i = (−1)n−1 , where Ci and Hi are multi-variable polynomials with integral coefficients. Each positive integer whose square-root has a regular continued …

Polynomial Solutions To Pell's Equation And Fundamental Units In Real Quadratic Fields, James Mclaughlin

Mathematics Faculty Publications

Finding polynomial solutions to Pell’s equation is of interest as such solutions sometimes allow the fundamental units to be determined in an infinite class of real quadratic fields. In this paper, for each triple of positive integers (c, h, f) satisfying c 2 − f h2 = 1, where (c, h) are the smallest pair of integers satisfying this equation, several sets of polynomials (c(t), h(t), f(t)) which satisfy c(t) 2 − f(t) h(t) 2 = 1 and (c(0), h(0), f(0)) = (c, h, f) are derived. Moreover, it is shown that the pair (c(t), h(t)) constitute the fundamental polynomial …

Directed Acyclic Graphs, Stephen B. Maurer , '67

Mathematics & Statistics Faculty Works

No abstract provided.

Notes On Interpolation In The Generalized Schur Class. Ii. Nudelman's Problem, Daniel Alpay, T. Constantinescu, A. Dijksma, J. Rovnyak, A. Dijksma

Mathematics, Physics, and Computer Science Faculty Articles and Research

An indefinite generalization of Nudel′man’s problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data. Besides results on existence criteria for Pick-Nevanlinna and Carath´eodory-Fej´er interpolation, the method yields new results on generalized interpolation in the sense of Sarason and boundary interpolation, including properties of the finite Hilbert transform relative to weights. The main theorem appeals to the Ball and Helton almost-commutant lifting theorem to provide criteria for the existence of a solution to Nudel′man’s problem.

Working Across Cultures, John Hooker

John Hooker

No abstract provided.

A Family Of Isomorphic Fusion Algebras Of Twisted Quantum Doubles Of Finite Groups, Christopher Goff

Christopher Goff