Physical Sciences and Mathematics Commons^™

A Bijective Proof Of A Factorization Formula For Specialized Macdonald Polynomials, Nicholas A. Loehr, Elizabeth Niese

Mathematics Faculty Research

Let μ and ν = (ν 1, . . . , ν k ) be partitions such that μ is obtained from ν by adding m parts of sizer. Descouens and Morita proved algebraically that the modified Macdonald polynomials

H~μ(X;q,t)

satisfy the identity

H~μ=H~νH~(r^m)

when the parameter t is specialize to an mth root of unity. Descouens, Morita, and Numata proved this formula bijectively when r ≤ ν k and

r∈{1,2}.

This note gives a bijective proof of the formula for all r ≤ ν k .

Hamilton Decompositions Of Certain 6-Regular Cayley Graphs On Abelian Groups With A Cyclic Subgroup Of Index Two, Erik E. Westlund

Faculty Articles

Alspach conjectured that every connected Cayley graph of even valency on a finite Abelian group is Hamilton-decomposable. Using some techniques of Liu, this article shows that if A is an Abelian group of even order with a generating set {a,b}, and A contains a subgroup of index two, generated by c, then the 6-regular Cayley graph is Hamilton-decomposable.

The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, Katherine Victoria Johnson

Department of Mathematics: Dissertations, Theses, and Student Research

A linear extension of a partially ordered set is simply a total ordering of the poset that is consistent with the original ordering. The linear extension diameter is a measure of how different two linear extensions could be, that is, the number of pairs of elements that are ordered differently by the two extensions. In this dissertation, we calculate the linear extension diameter of grids. This also gives us a nice characterization of the linear extensions that are the farthest from each other, and allows us to conclude that grids are diametrally reversing.

A linear extension of a poset might …

Cyclic Matching Sequencibility Of Graphs, Richard A. Brualdi, Kathleen P. Kiernan, Seth A. Meyer, Michael W. Schroeder

Mathematics Faculty Research

We define the cyclic matching sequencibility of a graph to be the largest integer d such that there exists a cyclic ordering of its edges so that every d consecutive edges in the cyclic ordering form a matching. We show that the cyclic matching sequencibility of K_2m and K_2m+1 equals m − 1.

A Simple Bijection Between Standard 3×N Tableaux And Irreducible Webs For 𝔰𝔩3, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and 𝔰𝔩₂. Building on a result of Kuperberg’s, Khovanov-Kuperberg found a recursive algorithm giving a bijection between standard Young tableaux of shape 3 × n and irreducible webs for 𝔰𝔩₃whose boundary vertices are all sources. In this paper, we give a simple and explicit map from standard Young tableaux of shape 3 …

The Rook-Brauer Algebra, Elise G. Delmas

Mathematics, Statistics, and Computer Science Honors Projects

We introduce an associative algebra RB_k(x) that has a basis of rook-Brauer diagrams. These diagrams correspond to partial matchings on 2k vertices. The rook-Brauer algebra contains the group algebra of the symmetric group, the Brauer algebra, and the rook monoid algebra as subalgebras. We show that the basis of RB_k(x) is generated by special diagrams s_i, t_i (1 <= i < k) and p_j (1 <= j <= k), where the s_i are the simple transpositions that generated the symmetric group S_k, the t_i are the "contraction maps" which generate the …

Combinatorics Using Computational Methods, Derrick Stolee

Department of Mathematics: Dissertations, Theses, and Student Research

Computational combinatorics involves combining pure mathematics, algorithms, and computational resources to solve problems in pure combinatorics. This thesis provides a theoretical framework for combinatorial search, which is then applied to several problems in combinatorics. Some results in space-bounded computational complexity are also presented.

The 1, 2-Conjecture For Graphs With Relatively Small Chromatic Number, Sogol Jahanbekam, Douglas West

Faculty Publications

No abstract provided.

Fixed Points And Excedances In Restricted Permutations, Sergi Elizalde

Dartmouth Scholarship

Using an unprecedented technique involving diagonals of non-rational generating functions, we prove that among the permutations of length $n$ with $i$ fixed points and $j$ excedances, the number of 321-avoiding ones equals the number of 132-avoiding ones, for any given $i,j$. Our theorem generalizes a result of Robertson, Saracino and Zeilberger. Even though bijective proofs have later been found by the author jointly with Pak and with Deutsch, this paper contains the original analytic proof that was presented at FPSAC 2003.

Finite Factors Of Bernoulli Schemes And Distinguishing Labelings Of Directed Graphs, Andrew Lazowski, Stephen M. Shea

Mathematics Faculty Publications

A labeling of a graph is a function from the vertices of the graph to some finite set. In 1996, Albertson and Collins defined distinguishing labelings of undirected graphs. Their definition easily extends to directed graphs. Let G be a directed graph associated to the k -block presentation of a Bernoulli scheme X . We determine the automorphism group of G , and thus the distinguishing labelings of G . A labeling of G defines a finite factor of X . We define demarcating labelings and prove that demarcating labelings define finitarily Markovian finite factors of X . We use …

Complete Multipartite Graphs And The Relaxed Coloring Game, Charles Dunn

Faculty Publications

Let k be a positive integer, d be a nonnegative integer, and G be a finite graph. Two players, Alice and Bob, play a game on G by coloring the uncolored vertices with colors from a set X of k colors. At all times, the subgraph induced by a color class must have maximum degree at most d. Alice wins the game if all vertices are eventually colored; otherwise, Bob wins. The least k such that Alice has a winning strategy is called the d-relaxed game chromatic number of G, denoted χ _g^d (G). …

On A Pair Of Identities From Ramanujan's Lost Notebook, James Mclaughlin, Andrew Sills

Mathematics Faculty Publications

Using a pair of two variable series-product identities recorded by Ramanujan in the lost notebook as inspiration, we find some new identities of similar type. Each identity immediately implies an infinite family of Rogers-Ramanujan type identities, some of which are well-known identities from the literature. We also use these identities to derive some general identities for integer partitions.

Stochastic Processes Induced By Singular Operators, Daniel Alpay, Palle Jorgensen

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we study a general family of multivariable Gaussian stochastic processes. Each process is prescribed by a fixed Borel measure σ on Rn. The case when σ is assumed absolutely continuous with respect to Lebesgue measure was stud- ied earlier in the literature, when n = 1. Our focus here is on showing how different equivalence classes (defined from relative absolute continuity for pairs of measures) translate into concrete spectral decompositions of the corresponding stochastic processes under study. The measures σ we consider are typically purely singular. Our proofs rely on the theory of (singular) unbounded operators in …

Special Dual Like Numbers And Lattices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce a new type of dual numbers called special dual like numbers. These numbers are constructed using idempotents in the place of nilpotents of order two as new element. That is x = a + bg is a special dual like number where a and b are reals and g is a new element such that g2 =g. The collection of special dual like numbers forms a ring. Further lattices are the rich structures which contributes to special dual like numbers. These special dual like numbers x = a + bg; when a and b …

Schur Functions And Their Realizations In The Slice Hyperholomorphic Setting, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allows to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels, positive definite functions in this setting and we show how they can be obtained in our setting using the extension operator and the slice regular product. We define Schur multipliers, and find their co-isometric realization …

New Topological C-Algebras With Applications In Linear Systems Theory, Daniel Alpay, Guy Salomon

Mathematics, Physics, and Computer Science Faculty Articles and Research

Motivated by the Schwartz space of tempered distributions S′ and the Kondratiev space of stochastic distributions S−1 we define a wide family of nuclear spaces which are increasing unions of (duals of) Hilbert spaces H′p,p∈N, with decreasing norms |⋅|p. The elements of these spaces are functions on a free commutative monoid. We characterize those rings in this family which satisfy an inequality of the form |f∗g|p≤A(p−q)|f|q|g|p for all p≥q+d, where * denotes the convolution in the monoid, A(p−q) is a strictly positive number and d is a fixed natural number (in this case we obtain commutative topological C-algebras). Such an …

The Minimum Of The Maximum Rectilinear Crossing Numbers Of Small Cubic Graphs, Matthew Alpert, Jens-P. Bode, Elie Feder, Heiko Harborth

Publications and Research

Here we consider the minimum of the maximum rectilin­ear crossing numbers for all d-regular graphs of order n. The case of connected graphs only is investigated also. For d = 3 exact values are determined for n are less than or equal to 12 and some estimations are given in general.

White Noise Based Stochastic Calculus Associated With A Class Of Gaussian Processes, Daniel Alpay, Haim Attia, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using the white noise space setting, we define and study stochastic integrals with respect to a class of stationary increment Gaussian processes. We focus mainly on continuous functions with values in the Kondratiev space of stochastic distributions, where use is made of the topology of nuclear spaces. We also prove an associated Ito formula.

An Interpolation Problem For Functions With Values In A Commutative Ring, Daniel Alpay, Haim Attia

Mathematics, Physics, and Computer Science Faculty Articles and Research

It was recently shown that the theory of linear stochastic systems can be viewed as a particular case of the theory of linear systems on a certain commutative ring of power series in a countable number of variables. In the present work we study an interpolation problem in this setting. A key tool is the principle of permanence of algebraic identities.

On The Class Rsi Of J-Contractive Functions Intertwining Solutions Of Linear Differential Equations, Daniel Alpay, Andrey Melnikov, Victor Vinnikov

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we extend and solve in the class of functions RSI mentioned in the title, a number of problems originally set for the class RS of rational functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned self-adjoint matrix. The problems we consider include the Schur algorithm, the partial realization problem and the Nevanlinna-Pick interpolation problem. The arguments rely on the one-to-one correspondence between elements in a given subclass of RSI and elements in RS. Another important tool in the arguments is a new result pertaining to the classical tangential …

Semigroup As Graphs, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors study the zero divisor graph and unit graph of a semigroup. The zero divisor graphs of semigroups Zn under multiplication is studied and characterized.

Non Associative Linear Algebras, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce the notion of non associative vector spaces and non associative linear algebras over a field. We construct non associative space using loops and groupoids over fields. In general in all situations, which we come across to find solutions may not be associative; in such cases we can without any difficulty adopt these non associative vector spaces/linear algebras. Thus this research is a significant one.

This book has six chapters. First chapter is introductory in nature. The new concept of non associative semilinear algebras is introduced in chapter two. This structure is …

Innovative Uses Of Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy, Indra Venkatbabu

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors bring out the innovative applications of matrices defined, described and developed by them. Here they do not include the natural product on matrices newly described and defined by them in the book on ‘natural product ×n on matrices’.

This book is organized into seven chapters. The first one is introductory in nature. In the second chapter authors give the unique and new way of analyzing the data which is time dependent. We construct three types of matrices called Average Time Dependent data matrix (ATD matrix), Refined Time Dependent Data matrix (RTD matrix) and Combined Effective Time …