Discrete Mathematics and Combinatorics Commons^™

Special Type Of Fixed Point Pairs Using Mod Rectangular Matrix Operators, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time define a special type of fixed points using MOD rectangular matrices as operators. In this case the special fixed points or limit cycles are pairs which is arrived after a finite number of iterations. Such study is both new and innovative for it can find lots of applications in mathematical modeling. Since all these Zn or I nZ or 〈Zn ∪ g〉 or 〈Zn ∪ g〉I or C(Zn) or CI(Zn) are all of finite order we are sure to arrive at a MOD fixed point pair or a MOD limit cycle pair …

Neutrosophic Set Approach To Algebraic Structures, Florentin Smarandache, Madad Khan, Fazal Tahir

Branch Mathematics and Statistics Faculty and Staff Publications

Real world is featured with complex phenomenons. As uncertainty is inevitably involved in problems arise in various elds of life and classical methods failed to handle these type of problems. Dealing with imprecise, uncertain or imperfect information was a big task for many years. Many modelswerepresentedinordertoproperlyincorporateuncertaintyintosystem description, LotA.Zadeh in 1965 introduced the idea of a fuzzy set. Zadeh replaced conventional characteristic function of classical crisp sets which takes on its values in f0;1g by membership function which takes on its values in closed interval [0;1]. Fuzzy set theory is conceptually a very powerful technique to deal with another aspect or …

Single Valued Neutrosophic Graphs: Degree, Order And Size, Florentin Smarandache, Said Broumi, Mohamed Talea, Assia Bakali

Branch Mathematics and Statistics Faculty and Staff Publications

The single valued neutrosophic graph is a new version of graph theory presented recently as a generalization of fuzzy graph and intuitionistic fuzzy graph. The single valued neutrosophic graph (SVN-graph) is used when the relation between nodes (or vertices) in problems are indeterminate. In this paper, we examine the properties of various types of degrees, order and size of single valued neutrosophic graphs and a new definition for regular single valued neutrosophic graph is given.

The Spectral Theorem For Quaternionic Unbounded Normal Operators Based On The S-Spectrum, Daniel Alpay, Fabrizio Colombo, David P. Kimsey

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of spectral analysis of quaternionic operators. The S-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators. The fact that the correct notion of …

A New Realization Of Rational Functions, With Applications To Linear Combination Interpolation, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce the following linear combination interpolation problem (LCI): Given N distinct numbers w1,…wN and N+1 complex numbers a1,…,aN and c, find all functions f(z) analytic in a simply connected set (depending on f) containing the points w1,…,wN such that ∑u=1Nauf(wu)=c. To this end we prove a representation theorem for such functions f in terms of an associated polynomial p(z). We first introduce the following two operations, (i) substitution of p, and (ii) multiplication by monomials zj,0≤j

New Perspectives Of Quantum Analogues, Yue Cai

Theses and Dissertations--Mathematics

In this dissertation we discuss three problems. We first show the classical q-Stirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. We extend this enumerative result via a decomposition of a new poset which we call the Stirling poset of the second kind. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done. We also give a bijective …

Flag F-Vectors Of Polytopes With Few Vertices, Sarah A. Nelson

Theses and Dissertations--Mathematics

We may describe a polytope P as the convex hull of n points in space. Here we consider the numbers of chains of faces of P. The toric g-vector and CD-index of P are useful invariants for encoding this information. For a simplicial polytope P, Lee defined the winding number w_k in a Gale diagram corresponding to P. He showed that w_k in the Gale diagram equals g_k of the corresponding polytope. In this dissertation, we fully establish how to compute the g-vector for any polytope with few vertices from its …

Path-Factors Involving Paths Of Order Seven And Nine, Yoshimi Egawa, Michitaka Furuya

Theory & Applications of Graphs

In this paper, we show the following two theorems (here c_i(G-Χ) is the number of components C of G-Χ with |V(C)|=i): (i) If a graph G satisfies c₁(G-Χ)+⅓c₃(G-Χ)+⅓c₅(G-Χ) ≤ ⅔ |Χ| for all Χ ≤ V(G), then G has a {Ρ₂},Ρ₇}-factor. (ii)If a graph G satisfies c₁}(G-Χ)+c₃}(G-Χ)+⅔ c₅}(G-Χ)+⅓c₇}(G-Χ) ≤ ⅔|Χ| for all Χ⊆ V(G), then G has a {P₂,P₉}-factor.

Classifying Resolving Lists By Distances Between Members, Paul Feit

Theory & Applications of Graphs

Let G be a connected graph and let w₁,…w_r be a list of vertices. We refer the choice of a triple (r;G;w₁,…w_r), as a metric selection. Let ρ be the shortest path metric of G. We say that w₁,…w_r resolves G (metricly) if the function c:V(G) → ℤ^r given by

x → (ρ (w1,x),…,ρ(wr,x))

is injective. We refer to c as the code map and to its image as the codes of the triple (r;G;w₁,…,w_r).

This paper proves basic results …

Characterizations Of Rectangular (Para)-Unitary Rational Functions, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle:
(i) through the realization matrix of Schur stable systems,
(ii) the Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters,
(iii) through the (not necessarily reducible) Matrix Fraction Description (MFD).
In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the …

The H∞ Functional Calculus Based On The S-Spectrum For Quaternionic Operators And For N-Tuples Of Noncommuting Operators, Daniel Alpay, Fabrizio Colombo, Tao Qian, Irene Sabadini, Tao Qian

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we extend the H∞ functional calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called S-functional calculus. The S-functional calculus has two versions one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz-Dunford functional calculus based on slice hyperholomorphicity because it shares with it the most important properties.

The S-functional calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears …

Nidus Idearum. Scilogs, Ii: De Rerum Consectatione, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Welcome into my scientific lab! My lab[oratory] is a virtual facility with noncontrolled conditions in which I mostly perform scientific meditation and chats: a nest of ideas (nidus idearum, in Latin). I called the jottings herein scilogs (truncations of the words scientific, and gr. Λόγος – appealing rather to its original meanings "ground", "opinion", "expectation"), combining the welly of both science and informal (via internet) talks (in English, French, and Romanian). In this second book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, some of them already put at work, others …

Tridiagonal Matrices And Boundary Conditions, J. J. P. Veerman, David K. Hammond

Mathematics and Statistics Faculty Publications and Presentations

We describe the spectra of certain tridiagonal matrices arising from differential equations commonly used for modeling flocking behavior. In particular we consider systems resulting from allowing an arbitrary boundary condition for the end of a one-dimensional flock. We apply our results to demonstrate how asymptotic stability for consensus and flocking systems depends on the imposed boundary condition.

Two Problems Of Gerhard Ringel, Adrian Pastine

Dissertations, Master's Theses and Master's Reports

Gerhard Ringel was an Austrian Mathematician, and is regarded as one of the most influential graph theorists of the twentieth century. This work deals with two problems that arose from Ringel's research: the Hamilton-Waterloo Problem, and the problem of R-Sequences.

The Hamilton-Waterloo Problem (HWP) in the case of C_m-factors and C_n-factors asks whether K_v, where v is odd (or K_v-F, where F is a 1-factor and v is even), can be decomposed into r copies of a 2-factor made entirely of m-cycles and s copies of a 2-factor made entirely of …

The Diameter Of A Rouquier Block, Andrew Mayer

Williams Honors College, Honors Research Projects

For my Honors Research Project, I will be researching special properties of Rouquier blocks that represent the partitions of integers. This problem is motivated by ongoing work in representation theory of the symmetric group. For each integer n and each prime p, there is an object called a Rouquier block; this block can be visualized as a collection of points in a plane, each corresponding to a partition. In this group of points, we say a pair of points is “connected” if certain conditions on the partitions are met. We compare each partition with each other partition, add edges when …

Realizing The 2-Associahedron, Patrick N. Tierney

HMC Senior Theses

The associahedron has appeared in numerous contexts throughout the field of mathematics. By representing the associahedron as a poset of tubings, Michael Carr and Satyan L. Devadoss were able to create a gener- alized version of the associahedron in the graph-associahedron. We seek to create an alternative generalization of the associahedron by considering a particle-collision model. By extending this model to what we dub the 2- associahedron, we seek to further understand the space of generalizations of the associahedron.

Interval Graphs, Joyce C. Yang

HMC Senior Theses

We examine the problem of counting interval graphs. We answer the question posed by Hanlon, of whether the formal power series generating function of the number of interval graphs on n vertices has a positive radius of convergence. We have found that it is zero. We have obtained a lower bound and an upper bound on the number of interval graphs on n vertices. We also study the application of interval graphs to the dynamic storage allocation problem. Dynamic storage allocation has been shown to be NP-complete by Stockmeyer. Coloring interval graphs on-line has applications to dynamic storage allocation. The …

Classifying Coloring Graphs, Julie Beier, Janet Fierson, Ruth Haas, Heather M. Russell, Kara Shavo

Department of Math & Statistics Faculty Publications

Given a graph G, its k-coloring graph is the graph whose vertex set is the proper k-colorings of the vertices of G with two k-colorings adjacent if they differ at exactly one vertex. In this paper, we consider the question: Which graphs can be coloring graphs? In other words, given a graph H, do there exist G and k such that H is the k-coloring graph of G? We will answer this question for several classes of graphs and discuss important obstructions to being a coloring graph involving order, girth, and induced subgraphs.

Induced Subgraph Saturated Graphs, Craig M. Tennenhouse

Theory & Applications of Graphs

A graph G is said to be H-saturated if G contains no subgraph isomorphic to H but the addition of any edge between non-adjacent vertices in G creates one. While induced subgraphs are often studied in the extremal case with regard to the removal of edges, we extend saturation to induced subgraphs. We say that G is induced H-saturated if G contains no induced subgraph isomorphic to H and the addition of any edge to G results in an induced copy of H. We demonstrate constructively that there are non-trivial examples of saturated graphs for …

On Minimum Spanning Subgraphs Of Graphs With Proper Connection Number 2, Zhenming Bi, Gary Chartrand, Garry L. Johns, Ping Zhang

Theory & Applications of Graphs

An edge coloring of a connected graph G is a proper-path coloring if every two vertices of G are connected by a properly colored path. The minimum number of colors required of a proper-path coloring of G is called the proper connection number pc(G) of G. For a connected graph G with proper connection number 2, the minimum size of a connected spanning subgraph H of G with pc(H) = 2 is denoted by μ(G). It is shown that if s and t are integers such that t ≥ s + 2 ≥ 5 …

Special Type Of Fixed Points Of Mod Matrix Operators, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce a special type of fixed points using MOD square matrix operators. These special type of fixed points are different from the usual classical fixed points. A study of this is carried out in this book. Several interesting properties are developed in this regard. The notion of these fixed points find many applications in the mathematical models which are dealt systematically by the authors in the forth coming books. These special type of fixed points or special realized limit cycles are always guaranteed as we use only MOD matrices as operators with …

Mod Graphs, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

Zadeh introduced the degree of membership/truth (t) in 1965 and defined the fuzzy set. Atanassov introduced the degree of nonmembership/ falsehood (f) in 1986 and defined the intuitionistic fuzzy set. Smarandache introduced the degree of indeterminacy/ neutrality (i) as independent component in 1995 (published in 1998) and defined the neutrosophic set on three components (t, i, f) = (truth, indeterminacy, falsehood): http://fs.gallup.unm.edu/FlorentinSmarandache.htm Etymology. The words “neutrosophy” and “neutrosophic” were coined/ invented by F. Smarandache in his 1998 book. Neutrosophy: A branch of philosophy, introduced by F. Smarandache in 1980, which studies the origin, nature, and scope of neutralities, as well …

Nidus Idearum. Scilogs, I: De Neutrosophia, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Welcome into my scientific lab! My lab[oratory] is a virtual facility with noncontrolled conditions in which I mostly perform scientific meditation and chats: a nest of ideas (nidus idearum, in Latin). I called the jottings herein scilogs (truncations of the words scientific, and gr. Λόγος – appealing rather to its original meanings "ground", "opinion", "expectation"), combining the welly of both science and informal (via internet) talks (in English, French, and Romanian). In this first books of scilogs collected from my nest of ideas, one may find new and old questions and solutions, some of them already put at work, others …

Mathematics For Everything With Combinatorics On Nature – A Report On The Promoter Dr. Linfan Mao Of Mathematical Combinatorics, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

The science’s function is realizing the natural world, developing our society in coordination with natural laws and the mathematics provides the quantitative tool and method for solving problems helping with that understanding. Generally, understanding a natural thing by mathematical ways or means to other sciences are respectively establishing mathematical model on typical characters of it with analysis first, and then forecasting its behaviors, and finally, directing human beings for hold on its essence by that model.

New Facets Of The Balanced Minimal Evolution Polytope, Logan Keefe

Williams Honors College, Honors Research Projects

The balanced minimal evolution (BME) polytope arises from the study of phylogenetic trees in biology. It is a geometric structure which has a variant for each natural number n. The main application of this polytope is that we can use linear programming with it in order to determine the most likely phylogenetic tree for a given genetic data set. In this paper, we explore the geometric and combinatorial structure of the BME polytope. Background information will be covered, highlighting some points from previous research, and a new result on the structure of the BME polytope will be given.

Eternal Independent Sets In Graphs, Yair Caro, William F. Klostermeyer

Theory & Applications of Graphs

The use of mobile guards to protect a graph has received much attention in the literature of late in the form of eternal dominating sets, eternal vertex covers and other models of graph protection. In this paper, eternal independent sets are introduced. These are independent sets such that the following can be iterated forever: a vertex in the independent set can be replaced with a neighboring vertex and the resulting set is independent.

Group-Antimagic Labelings Of Multi-Cyclic Graphs, Dan Roberts, Richard M. Low

Theory & Applications of Graphs

Let Α be a non-trivial abelian group. A connected simple graph G = (V, E) is Α-antimagic if there exists an edge labeling f: Ε(G) → A \{0} such that the induced vertex labeling f+: V(G) → Α, defined by f+(v) = Σ{f(u,v): (u, v) ∈ E(G)}, is a one-to-one map. The integer-antimagic spectrum of a graph G is the set IAM(G) = {κ: G is ℤ_k - antimagic and κ ≥ 2}. In this paper, we analyze the integer-antimagic spectra for various classes of multi-cyclic graphs.

Computing Closed Forms For The Convergent Series $\Displaystyle\Sum_{N \In \Mathbb{Z}}\Frac{1}{(N^3+Bn^2+Cn+D)^K}$, Gagandeep K. Virk

Theses and Dissertations (Comprehensive)

In this thesis we discuss the various approaches that will be taken to evaluate and find a finite closed form for the sum $$\sum_{n \in \mathbb{Z}} \frac{1}{(n^3+Bn^2+Cn+D)^k}$$ where $B, C, D \in \mathbb{C}$ and $k$ is a positive integer. We begin this thesis by studying the cubic equations and discussing briefly various methods of finding their roots. Cardano's method (1545) for finding the roots of cubic polynomials is explored in detail as this method is used in later parts of the thesis to make calculations while evaluating the sums. Various tools and techniques from Fourier analysis are reviewed for these …

Wiener Algebra For The Quaternions, Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-Lévy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener-Hopf operators.

The Spectral Theorem For Unitary Operators Based On The S-Spectrum, Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem.

In this paper we prove the quaternionic spectral theorem for unitary operators using the S-spectrum. In the case of quaternionic matrices, the S-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of S-spectrum is relatively new, see [17], and has been used for …