Discrete Mathematics and Combinatorics Commons^™

Counting On Determinants, Arthur T. Benjamin, Naiomi T. Cameron

All HMC Faculty Publications and Research

No abstract provided in this article.

Lower Bounds For Simplicial Covers And Triangulations Of Cubes, Adam Bliss '03, Francis E. Su

All HMC Faculty Publications and Research

We show that the size of a minimal simplicial cover of a polytope P is a lower bound for the size of a minimal triangulation of P, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and we improve lower bounds for covers and triangulations in dimensions 4 through at least 12 (and possibly more dimensions as well). Important ingredients are an analysis of the number of exterior faces that a simplex in the cube can have of a specified dimension and volume, and a characterization of corner simplices in terms of their …

Double Arrays, Triple Arrays And Balanced Grids, John P. Mcsorley, Nicholas C. Phillips, Walter D. Wallis, Joseph L. Yucas

Articles and Preprints

Triple arrays are a class of designs introduced by Agrawal in 1966 for two-way elimination of heterogeneity in experiments. In this paper we investigate their existence and their connection to other classes of designs, including balanced incomplete block designs and balanced grids.

The Terwilliger Algebra Of An Almost-Bipartite P- And Q-Polynomial Association Scheme, John S. Caughman Iv, Mark S. Maclean, Paul M. Terwilliger

Mathematics and Statistics Faculty Publications and Presentations

Let Y denote a D-class symmetric association scheme with D≥3, and suppose Y is almost-bipartite P- and Q-polynomial. Let x denote a vertex of Y and let T=T(x) denote the corresponding Terwilliger algebra. We prove that any irreducible T-module W is both thin and dual thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T-module is determined by two parameters, the dual endpoint and diameter of W. We find a recurrence which gives the multiplicities with which the …

Combinatorial Proofs Of Fermat's, Lucas's, And Wilson's Theorems, Peter G. Anderson, Arthur T. Benjamin, Jeremy A. Rouse

All HMC Faculty Publications and Research

No abstract provided in this article.

Convergence Analysis Of Mcmc Method In The Study Of Genetic Linkage With Missing Data, Diana Fisher

Theses, Dissertations and Capstones

Computational infeasibility of exact methods for solving genetic linkage analysis problems has led to the development of a new collection of stochastic methods, all of which require the use of Markov chains. The purpose of this work is to investigate the complexities of missing data in pedigree analysis using the Monte Carlo Markov Chain (MCMC) method as compared to the exact results. Also, we attempt to determine an association between missing data in a familial pedigree and the convergence to stationarity of a descent graph Markov chain implemented in the stochastic method for parametric linkage analysis.

In particular, we will …

Reduction Of The Gibbs Phenomenon Via Interpolation Using Chebyshev Polynomials, Filtering And Chebyshev-Pade' Approximations, Rob-Roy L. Mace

Theses, Dissertations and Capstones

In this manuscript, we will examine several methods of interpolation, with an emphasis on Chebyshev polynomials and the removal of the Gibbs Phenomenon. Included as an appendix are the author’s Mat- Lab implementations of Lagrange, Chebyshev, and rational interpolation methods.

Introduction To Linear Bialgebra, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief economic models and so on. This book has for the first time, introduced a new algebraic structure called linear bialgebra, which is also a very powerful algebraic tool that can yield itself to applications. With the recent introduction of bimatrices (2005) we have ventured in this book to introduce new concepts like linear bialgebra and Smarandache neutrosophic linear bialgebra and also give the applications of these algebraic …

Applications Of Bimatrices To Some Fuzzy And Neutrosophic Models, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

Graphs and matrices play a vital role in the analysis and study of several of the real world problems which are based only on unsupervised data. The fuzzy and neutrosophic tools like fuzzy cognitive maps invented by Kosko and neutrosophic cognitive maps introduced by us help in the analysis of such real world problems and they happen to be mathematical tools which can give the hidden pattern of the problem under investigation. This book, in order to generalize the two models, has systematically invented mathematical tools like bimatrices, trimatrices, n-matrices, bigraphs, trigraphs and n-graphs and describe some of its properties. …

Point Evaluation And Hardy Space On A Homogeneous Tree, Daniel Alpay, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We consider stationary multiscale systems as defined by Basseville, Benveniste, Nikoukhah and Willsky. We show that there are deep analogies with the discrete time non stationary setting as developed by the first author, Dewilde and Dym. Following these analogies we define a point evaluation with values in a C*–algebra and the corresponding “Hardy space” in which Cauchy’s formula holds. This point evaluation is used to define in this context the counterpart of classical notions such as Blaschke factors.

Introduction To Bimatrices, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices find their applications in several real models. In fact it is not an exaggeration if one says that matrix theory and linear algebra (i.e. vector spaces) form an inseparable component of each other. The study of bialgebraic structures led to the invention of new notions like birings, Smarandache birings, bivector spaces, linear bialgebra, bigroupoids, bisemigroups, etc. But most of these are abstract algebraic concepts except, the bisemigroup being used in …

Rational Hyperholomorphic Functions In R4, Daniel Alpay, Michael Shapiro, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce the notion of rationality for hyperholomorphic functions (functions in the kernel of the Cauchy-Fueter operator). Following the case of one complex variable, we give three equivalent definitions: the first in terms of Cauchy-Kovalevskaya quotients of polynomials, the second in terms of realizations and the third in terms of backward-shift invariance. Also introduced and studied are the counterparts of the Arveson space and Blaschke factors.

Generating Sequences Of Clique-Symmetric Graphs Via Eulerian Digraphs, John P. Mcsorley, Thomas D. Porter

Articles and Preprints

Let {G_p1,G_p2, . . .} be an infinite sequence of graphs with G_pn having pn vertices. This sequence is called K_p-removable if G_p1 ≅ K_p, and G_pn − S ≅ G_p(n−1) for every n ≥ 2 and every vertex subset S of G_pn that induces a K_p. Each graph in such a sequence has a high degree of symmetry: every way of removing the vertices of any fixed number of disjoint K_p’s yields the same …

Two New Criteria For Comparison In The Bruhat Order, Brian Drake, Sean Gerrish, Mark Skandera

Dartmouth Scholarship

We give two new criteria by which pairs of permutations may be compared in defining the Bruhat order (of type $A$). One criterion utilizes totally nonnegative polynomials and the other utilizes Schur functions.

On Classifying Finite Edge Colored Graphs With Two Transitive Automorphism Groups, Thomas Q. Sibley

Mathematics Faculty Publications

This paper classifies all finite edge colored graphs with doubly transitive automorphism groups. This result generalizes the classification of doubly transitive balanced incomplete block designs with λ=1 and doubly transitive one-factorizations of complete graphs. It also provides a classification of all doubly transitive symmetric association schemes.

A Theorem On Divergence In The General Sense For Continued Fractions, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of such continued fractions. We apply this theorem to two general classes of q continued fraction to show, that if G(q) is one of these continued fractions and |q| > 1, then either G(q) converges or does not converge in the general sense. We also show that if the odd and even parts of the continued fraction K∞n=1an/1 converge to …

On The Divergence Of The Rogers-Ramanujan Continued Fraction On The Unit Circle, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

This paper is an intensive study of the convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number t ∈ (0, 1) be denoted by [0, a1(t), a2(t), · · · ] and let the i-th convergent of this continued fraction expansion be denoted by ci(t)/di(t). Let S = {t ∈ (0, 1) : ai+1(t) ≥ φ di(t) infinitely often}, where φ = (√ 5 + 1)/2. Let YS = {exp(2πit) : t ∈ S}. It is shown that if y ∈ YS then the Rogers-Ramanujan continued fraction, R(y), diverges at y. S is an …

Tree Diagrams For String Links Ii: Determining Chord Diagrams, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations).

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.

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 …

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 …

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.

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.

Two Quick Combinatorial Proofs, Arthur T. Benjamin, Michael E. Orrison

All HMC Faculty Publications and Research

Presentation of two simple combinatorial proofs.

Fixed Point And Two-Cycles Of The Discrete Logarithm, Joshua Holden

Mathematical Sciences Technical Reports (MSTR)

We explore some questions related to one of Brizolis: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? We extend this question to ask about not only fixed points but also two-cycles. Campbell and Pomerance have not only answered the fixed point question for sufficiently large p but have also rigorously estimated the number of such pairs given certain conditions on g and h. We attempt to give heuristics for similar estimates given other conditions on g and h and also in the case …

Totally Magic Graphs, Geoffrey Exoo, Alan C. H. Ling, John P. Mcsorley, Nicholas C. Phillips, Walter D. Wallis

Articles and Preprints

A total labeling of a graph with v vertices and e edges is defined as a one-to-one map taking the vertices and edges onto the integers 1, 2, · · · , v+e. Such a labeling is vertex magic if the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex, and edge magic if the sum of an edge label and the labels of the endpoints of the edge is constant. In this paper we examine graphs possessing a labeling that is simultaneously …