Discrete Mathematics and Combinatorics Commons^™

Inner Product Spaces And Krein Spaces In The Quaternionic Setting, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we provide a study of quaternionic inner product spaces. This includes ortho-complemented subspaces, fundamental decompositions as well as a number of results of topological nature. Our main purpose is to show that a closed uniformly positive subspace in a quaternionic Krein space is ortho-complemented, and this leads to our choice of the results presented in the paper.

Quaternionic Hardy Spaces In The Open Unit Ball And Half Space And Blaschke Products, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

The Hardy spaces H2(B) and H2(H+), where B and H+ denote, respectively, the open unit ball of the quaternions and the half space of quaternions with positive real part, as well as Blaschke products, have been intensively studied in a series of papers where they are used as a tool to prove other results in Schur analysis. This paper gives an overview on the topic, collecting the various results available.

Self-Mappings Of The Quaternionic Unit Ball: Multiplier Properties, Schwarz-Pick Inequality, And Nevanlinna-Pick Interpolation Problem, Daniel Alpay, Vladimir Bolotnikov, Fabrizio Colombo, Irene Sabadini, Fabrizio Colombo

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study several aspects concerning slice regular functions mapping the quaternionic open unit ball B into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space H2(B). In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.

Infinite Product Representations For Kernels And Iteration Of Functions, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz, Itzik Marziano

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study infinite products of reproducing kernels with view to their use in dynamics (of iterated function systems), in harmonic analysis, and in stochastic processes. On the way, we construct a new family of representations of the Cuntz relations. Then, using these representations we associate a fixed filled Julia set with a Hilbert space. This is based on analysis and conformal geometry of a fixed rational mapping R in one complex variable, and its iterations.

Realizations Of Infinite Products, Ruelle Operators And Wavelet Filters, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using the system theory notion of state-space realization of matrix-valued rational functions, we describe the Ruelle operator associated with wavelet filters. The resulting realization of infinite products of rational functions have the following four features: 1) It is defined in an infinite-dimensional complex domain. 2) Starting with a realization of a single rational matrix-function M, we show that a resulting infinite product realization obtained from M takes the form of an (infinitedimensional) Toeplitz operator with the symbol that is a reflection of the initial realization for M. 3) Starting with a subclass of rational matrix functions, including scalar-valued ones corresponding …

Wiener-Chaos Approach To Optimal Prediction, Daniel Alpay, Alon Kipnis

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this work we combine Wiener chaos expansion approach to study the dynamics of a stochastic system with the classical problem of the prediction of a Gaussian process based on part of its sample path. This is done by considering special bases for the Gaussian space G generated by the process, which allows us to obtain an orthogonal basis for the Fock space of G such that each basis element is either measurable or independent with respect to the given samples. This allows us to easily derive the chaos expansion of a random variable conditioned on part of the sample …

Spectral Theory For Gaussian Processes: Reproducing Kernels, Random Functions, Boundaries, And L2-Wavelet Generators With Fractional Scales, Daniel Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

A recurrent theme in functional analysis is the interplay between the theory of positive definite functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. This central theme is motivated by a host of applications, e.g., in mathematical physics, and in stochastic differential equations, and their use in financial models. In this paper, we show that, for three classes of cases in the correspondence, it is possible to obtain explicit formulas which are amenable to computations of the respective Gaussian stochastic processes. For achieving this, we first develop two functional analytic tools. They are: …

Community Detection Detailed For Online Social Networks, Christopher J. Hogan

Theses and Dissertations (Comprehensive)

Ever since the internet became publicly available it has allowed users to interact with each other across virtual networks. With this large amounts of data being collected the clustering of this information has become an even more powerful tool for recognize patterns and trends in a network. In this research we look build a model for Community Detection in these online social networks. We combine the ideas from both discrete mathematics and sociology, to build an algorithm with the specific intent on discovering communities that exist in an online social network. We present many of the sociology theories behind the …

Combinatorial Game Theory: An Introduction To Tree Topplers, John S. Ryals Jr.

College of Graduate Studies: Theses & Dissertations

The purpose of this thesis is to introduce a new game, Tree Topplers, into the field of Combinatorial Game Theory. Before covering the actual material, a brief background of Combinatorial Game Theory is presented, including how to assign advantage values to combinatorial games, as well as information on another, related game known as Domineering. Please note that this document contains color images so please keep that in mind when printing.

Extremal Theorems For Degree Sequence Packing And The Two-Color Discrete Tomography Problem, Jennifer Diemunsch, Michael Ferrara, Sogol Jahanbekam, James Shook

Faculty Publications

No abstract provided.

Lucky Choice Number Of Planar Graphs With Given Girth, Axel Brandt, Jennifer Diemunsch, Sogol Jahanbekam

Faculty Publications

No abstract provided.

I Don't Play Chess: A Study Of Chess Piece Generating Polynomials, Stephen R. Skoch

Senior Independent Study Theses

This independent study examines counting problems of non-attacking rook, and non-attacking bishop placements. We examine boards for rook and bishop placement with restricted positions and varied dimensions. In this investigation, we discuss the general formula of a generating function for unrestricted, square bishop boards that relies on the Stirling numbers of the second kind. We discuss the maximum number of bishops we can place on a rectangular board, as well as a brief investigation of non-attacking rook placements on three-dimensional boards, drawing a connection to latin squares.

Domination Numbers Of Semi-Strong Products Of Graphs, Stephen R. Cheney

Theses and Dissertations

This thesis examines the domination number of the semi-strong product of two graphs G and H where both G and H are simple and connected graphs. The product has an edge set that is the union of the edge set of the direct product of G and H together with the cardinality of V(H), copies of G. Unlike the other more common products (Cartesian, direct and strong), the semi-strong product is neither commutative nor associative.

The semi-strong product is not supermultiplicative, so it does not satisfy a Vizing like conjecture. It is also not submultiplicative so it shares these two …

Coloring The Square Of Planar Graphs Without 4-Cycles Or 5-Cycles, Robert Jaeger

Theses and Dissertations

The famous Four Color Theorem states that any planar graph can be properly colored using at most four colors. However, if we want to properly color the square of a planar graph (or alternatively, color the graph using distinct colors on vertices at distance up to two from each other), we will always require at least \Delta + 1 colors, where \Delta is the maximum degree in the graph. For all \Delta, Wegner constructed planar graphs (even without 3-cycles) that require about \frac{3}{2} \Delta colors for such a coloring.

To prove a stronger upper bound, we consider only planar graphs …

Modeling Human Gaming Playing Behavior And Reward/Penalty Mechanism Using Discrete Event Simulation (Des), Christina M. Frederick, Michael Fitzgerald, Dahai Liu, Yolanda Ortiz, Christopher Via, Shawn Doherty, Jason P. Kring

Publications

Humans are remarkably complex and unpredictable; however, while predicting human behavior can be problematic, there are methods such as modeling and simulation that can be used to predict probable futures of human decisions. The present study analyzes the possibility of replacing human subjects with data resulting from pure models. Decisions made by college students in a multi-level mystery-solving game under 3 different gaming conditions are compared with the data collected from a predictive sequential Markov-Decision Process model. In addition, differences in participants’ data influenced by the three different conditions (additive, subtractive, control) were analyzed. The test results strongly suggest that …

Decomposing The Blocks Of A Steiner Triple System Of Order 4v-3 Into Partial Parallel Classes Of Size V-1, Leah C. Tollefson

Dissertations, Master's Theses and Master's Reports

In this report we present a summary and our new results on finding partial parallel classes of uniform size of Steiner triple systems, STS(v). We show several results for STS(4v - 3), where v = 3 mod 12 and v = 9 mod 12. In Chapter 1 we provide background knowledge and introduce the problem. In Chapter 2 we discuss some important known results to the problem, introduce the needed ingredients, and explain the methodology of the construction. Finally, in Chapter 3, we conclude with a summary and discuss possibilities for future work.

Labeled Trees And Spanning Trees: Computational Discrete Mathematics And Applications, Demet Yalman

College of Graduate Studies: Theses & Dissertations

In this thesis, we examine two topics. In the first part, we consider Leech tree which is a tree of order n with positive integer edge weights such that the weighted distances between pairs of vertices are exactly from 1 to n choose 2. Only five Leech trees are known and some non-existence results have been presented through the years. Variations of Leech trees such as the minimal distinct distance trees and modular Leech trees have been considered in recent years. In this thesis, such Leech-type questions on distances between leaves are studied as well as some other labeling questions …

Domination Integrity Of Some Path Related Graphs, S. K. Vaidya, N. H. Shah

Applications and Applied Mathematics: An International Journal (AAM)

The stability of a communication network is one of the important parameters for network designers and users. A communication network can be considered to be highly vulnerable if the destruction of a few elements cause large damage and only few members are able to communicate. In a communication network several vulnerability measures like binding number, toughness, scattering number, integrity, tenacity, edge tenacity and rupture degree are used to determine the resistance of network to the disruption after the failure of certain nodes (vertices) or communication links (edges). Domination theory also provides a model to measure the vulnerability of a graph …

Private Out-Domination Number Of Generalized De Bruijn Digraphs, G. Marimuthu, B. Johnson

Applications and Applied Mathematics: An International Journal (AAM)

Dominating sets are widely applied in the design and efficient use of computer networks. They can be used to decide the placement of limited resources, so that every node has access to the resource through neighbouring node. The most efficient solution is one that avoids duplication of access to the resources. This more restricted version of minimum dominating set is called an private dominating set. A vertex v in a digraph D is called a private out-neighbor of the vertex u in S (subset of V(D)) if u is the only element in the intersection of in-neighborhood set of v …

Difference Cordial Labeling Of Graphs Obtained From Triangular Snakes, R. Ponraj, S. S. Narayanan

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we investigate the difference cordial labeling behavior of corona of triangular snake with the graphs of order one and order two and also corona of alternative triangular snake with the graphs of order one and order two.