Other Mathematics Commons^™

Mathematical Models: The Lanchester Equations And The Zombie Apocalypse, Hailey Bauer

Undergraduate Theses and Capstone Projects

This research study used mathematical models to analyze and depicted specific battle situations and the outcomes of the zombie apocalypse. The original models that predicted warfare were the Lanchester models, while the zombie apocalypse models were fictional expansions upon mathematical models used to examine infectious diseases. In this paper, I analyzed and compared different mathematical models by examining each model’s set of assumptions and the impact of the change in variables on the population classes. The purpose of this study was to understand the basics of the discrete dynamical systems and to determine the similarities between imaginary and realistic models. …

Special Subset Vertex Multisubgraphs For Multi Networks, Florentin Smarandache, W.B. Vasantha Kandasamy, Ilanthenral K

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors study special type of subset vertex multi subgraphs; these multi subgraphs can be directed or otherwise. Another special feature of these subset vertex multigraphs is that we are aware of the elements in each vertex set and how it affects the structure of both subset vertex multisubgraphs and edge multisubgraphs. It is pertinent to record at this juncture that certain ego centric directed multistar graphs become empty on the removal of one edge, there by theorising the importance, and giving certain postulates how to safely form ego centric multi networks. Given any subset vertex multigraph we …

Conflict Free Connectivity And The Conflict-Free-Connection Number Of Graphs, Travis D. Wehmeier

Electronic Theses and Dissertations

We explore a relatively new concept in edge-colored graphs called conflict-free connectivity. A conflict-free path is a (edge-) colored path that has an edge with a color that appears only once. Conflict-free connectivity is the maximal number of internally disjoint conflict-free paths between all pairs of vertices in a graph. We also define the c-conflict-free-connection of a graph G. This is the maximum conflict-free connectivity of G over all c-colorings of the edges of G. In this paper we will briefly survey the works related to conflict-free connectivity. In addition, we will use the probabilistic method to achieve a bound …

On Coding For Partial Erasure Channels, Carolyn Mayer

Department of Mathematics: Dissertations, Theses, and Student Research

Error correcting codes have been essential to the technology we use in everyday life in digital storage, wireless communication, barcodes, and much more. Different channel models are used for different types of communication (for example, if information is sent to one person or to many people) and different types of errors. Partial erasure channels were recently introduced to model applications in which some information remains after an erasure event. These remnants of information may be used to increase the chances of successful decoding. We introduce a new partial erasure channel in which partial erasures correspond to individual bit erasures in …

Strong Degrees In Single Valued Neutrosophic Graphs, Florentin Smarandache, Said Broumi, Assia Bakali, Seema Mehra, Mohamed Talea, Manjeet Singh

Branch Mathematics and Statistics Faculty and Staff Publications

The concept of single valued neutrosophic graphs (SVNGs) generalizes the concept of fuzzy graphs and intuitionistic fuzzy graphs. The purpose of this research paper is to define different types of strong degrees in SVNGs and introduce novel concepts, such as the vertex truth-membership, vertex indeterminacy-membership and falsity-membership sequence in SVNG with proof and numerical illustrations.

Design And Analysis Of Graph-Based Codes Using Algebraic Lifts And Decoding Networks, Allison Beemer

Department of Mathematics: Dissertations, Theses, and Student Research

Error-correcting codes seek to address the problem of transmitting information efficiently and reliably across noisy channels. Among the most competitive codes developed in the last 70 years are low-density parity-check (LDPC) codes, a class of codes whose structure may be represented by sparse bipartite graphs. In addition to having the potential to be capacity-approaching, LDPC codes offer the significant practical advantage of low-complexity graph-based decoding algorithms. Graphical substructures called trapping sets, absorbing sets, and stopping sets characterize failure of these algorithms at high signal-to-noise ratios. This dissertation focuses on code design for and analysis of iterative graph-based message-passing decoders. The …

Summary Of The Special Issue “Neutrosophic Information Theory And Applications” At “Information” Journal, Florentin Smarandache, Jun Ye

Branch Mathematics and Statistics Faculty and Staff Publications

Over a period of seven months (August 2017–February 2018), the Special Issue dedicated to “Neutrosophic Information Theory and Applications” by the “Information” journal (ISSN 2078-2489), located in Basel, Switzerland, was a success. The Guest Editors, Prof. Dr. Florentin Smarandache from the University of New Mexico (USA) and Prof. Dr. Jun Ye from the Shaoxing University (China), were happy to select—helped by a team of neutrosophic reviewers from around the world, and by the “Information” journal editors themselves—and publish twelve important neutrosophic papers, authored by 27 authors and coauthors. There were a variety of neutrosophic topics studied and used by the …

Distributive Lattice Models Of The Type C One-Rowed Weyl Group Symmetric Functions, William Atkins

Murray State Theses and Dissertations

We present two families of diamond-colored distributive lattices – one known and one new – that we can show are models of the type C one-rowed Weyl symmetric functions. These lattices are constructed using certain sequences of positive integers that are visualized as filling the boxes of one-rowed partition diagrams. We show how natural orderings of these one-rowed tableaux produce our distributive lattices as sublattices of a more general object, and how a natural coloring of the edges of the associated order diagrams yields a certain diamond-coloring property. We show that each edge-colored lattice possesses a certain structure that is …

Self-Assembly Of Dna Graphs And Postman Tours, Katie Bakewell

UNF Graduate Theses and Dissertations

DNA graph structures can self-assemble from branched junction molecules to yield solutions to computational problems. Self-assembly of graphs have previously been shown to give polynomial time solutions to hard computational problems such as 3-SAT and k-colorability problems. Jonoska et al. have proposed studying self-assembly of graphs topologically, considering the boundary components of their thickened graphs, which allows for reading the solutions to computational problems through reporter strands. We discuss weighting algorithms and consider applications of self-assembly of graphs and the boundary components of their thickened graphs to problems involving minimal weight Eulerian walks such as the Chinese Postman Problem and …

Subset Vertex Graphs For Social Networks, 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 the notion of subset vertex graph using the vertex set as the subset of the power set P(S), S is assumed in this book to be finite; however it can be finite or infinite. We have defined two types of subset vertex graphs, one is directed and the other one is not directed. The most important fact which must be kept in record is that for a given set of vertices there exists one and only one subset vertex graph be it of type I or type II. Several important and …

Ns-K-Nn: Neutrosophic Set-Based K-Nearest Neighbors Classifier, Florentin Smarandache, Yaman Akbulut, Abdulkadir Sengur, Yanhui Guo

Branch Mathematics and Statistics Faculty and Staff Publications

k-nearest neighbors (k-NN), which is known to be a simple and efficient approach, is a non-parametric supervised classifier. It aims to determine the class label of an unknown sample by its k-nearest neighbors that are stored in a training set. The k-nearest neighbors are determined based on some distance functions. Although k-NN produces successful results, there have been some extensions for improving its precision. The neutrosophic set (NS) defines three memberships namely T, I and F. T, I, and F shows the truth membership degree, the false membership degree, and the indeterminacy membership degree, respectively. In this paper, the NS …

Efficiently Representing The Integer Factorization Problem Using Binary Decision Diagrams, David Skidmore

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Let p be a prime positive integer and let α be a positive integer greater than 1. A method is given to reduce the problem of finding a nontrivial factorization of α to the problem of finding a solution to a system of modulo p polynomial congruences where each variable in the system is constrained to the set {0,...,p − 1}. In the case that p = 2 it is shown that each polynomial in the system can be represented by an ordered binary decision diagram with size less than 20.25log₂(α)³ + 16.5log₂(α)² + …

Vertex Weighted Spectral Clustering, Mohammad Masum

Electronic Theses and Dissertations

Spectral clustering is often used to partition a data set into a specified number of clusters. Both the unweighted and the vertex-weighted approaches use eigenvectors of the Laplacian matrix of a graph. Our focus is on using vertex-weighted methods to refine clustering of observations. An eigenvector corresponding with the second smallest eigenvalue of the Laplacian matrix of a graph is called a Fiedler vector. Coefficients of a Fiedler vector are used to partition vertices of a given graph into two clusters. A vertex of a graph is classified as unassociated if the Fiedler coefficient of the vertex is close to …

Complex Neutrosophic Soft Set, Florentin Smarandache, Said Broumi, Assia Bakali, Mohamed Talea, Mumtaz Ali, Ganeshsree Selvachandran

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we propose the complex neutrosophic soft set model, which is a hybrid of complex fuzzy sets, neutrosophic sets and soft sets. The basic set theoretic operations and some concepts related to the structure of this model are introduced, and illustrated. An example related to a decision making problem involving uncertain and subjective information is presented, to demonstrate the utility of this model.

Complex Neutrosophic Graphs Of Type 1, Florentin Smarandache, Said Broumi, Assia Bakali, Mohamed Talea

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we introduced a new neutrosophic graphs called complex neutrosophic graphs of type1 (CNG1) and presented a matrix representation for it and studied some properties of this new concept. The concept of CNG1 is an extension of generalized fuzzy graphs of type 1 (GFG1) and generalized single valued neutrosophic graphs of type 1 (GSVNG1).

Six Septembers: Mathematics For The Humanist, Patrick Juola, Stephen Ramsay

Zea E-Books Collection

Scholars of all stripes are turning their attention to materials that represent enormous opportunities for the future of humanistic inquiry. The purpose of this book is to impart the concepts that underlie the mathematics they are likely to encounter and to unfold the notation in a way that removes that particular barrier completely. This book is a primer for developing the skills to enable humanist scholars to address complicated technical material with confidence. This book, to put it plainly, is concerned with the things that the author of a technical article knows, but isn’t saying. Like any field, mathematics operates …

Sudoku Variants On The Torus, Kira A. Wyld

HMC Senior Theses

This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played.

Combinatorial Polynomial Hirsch Conjecture, Sam Miller

HMC Senior Theses

The Hirsch Conjecture states that for a d-dimensional polytope with n facets, the diameter of the graph of the polytope is at most n-d. This conjecture was disproven in 2010 by Francisco Santos Leal. However, a polynomial bound in n and d on the diameter of a polytope may still exist. Finding a polynomial bound would provide a worst-case scenario runtime for the Simplex Method of Linear Programming. However working only with polytopes in higher dimensions can prove challenging, so other approaches are welcome. There are many equivalent formulations of the Hirsch Conjecture, one of which is the …

Generalized Interval Valued Neutrosophic Graphs Of First Type, Florentin Smarandache, Said Broumi, Mohamed Talea, Assia Bakali, Ali Hassan

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, motivated by the notion of generalized single valued neutrosophic graphs of first type, we defined a new neutrosophic graphs named generalized interval valued neutrosophic graphs of first type (GIVNG1) and presented a matrix representation for it and studied few properties of this new concept. The concept of GIVNG1 is an extension of generalized fuzzy graphs (GFG1) and generalized single valued neutrosophic of first type (GSVNG1).

Characterizations Of Families Of Rectangular, Finite Impulse Response, Para-Unitary Systems, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

We here study Finite Impulse Response (FIR) rectangular, not necessarily causal, systems which are (para)-unitary on the unit circle (=the class U). First, we offer three characterizations of these systems. Then, introduce a description of all FIRs in U, as copies of a real polytope, parametrized by the dimensions and the McMillan degree of the FIRs.

Finally, we present six simple ways (along with their combinations) to construct, from any FIR, a large family of FIRs, of various dimensions and McMillan degrees, so that whenever the original system is in U, so is the whole family.

A key role is …

Adaptive Orthonormal Systems For Matrix-Valued Functions, Daniel Alpay, Fabrizio Colombo, Tao Qian, Irene Sabadini, Tao Qian

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we consider functions in the Hardy space Hp×q2 defined in the unit disc of matrix-valued. We show that it is possible, as in the scalar case, to decompose those functions as linear combinations of suitably modified matrix-valued Blaschke product, in an adaptive way. The procedure is based on a generalization to the matrix-valued case of the maximum selection principle which involves not only selections of suitable points in the unit disc but also suitable orthogonal projections. We show that the maximum selection principle gives rise to a convergent algorithm. Finally, we discuss the case of real-valued signals.

Functions Of The Infinitesimal Generator Of A Strongly Continuous Quaternionic Group, Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, David P. Kimsey

Mathematics, Physics, and Computer Science Faculty Articles and Research

The analogue of the Riesz-Dunford functional calculus has been introduced and studied recently as well as the theory of semigroups and groups of linear quaternionic operators. In this paper we suppose that T is the infinitesimal generator of a strongly continuous group of operators (ZT (t))t2R and we show how we can define bounded operators f(T ), where f belongs to a class of functions which is larger than the class of slice regular functions, using the quaternionic Laplace-Stieltjes transform. This class will include functions that are slice regular on the S-spectrum of T but not necessarily at infinity. Moreover, …

On A Class Of Quaternionic Positive Definite Functions And Their Derivatives, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we start the study of stochastic processes over the skew field of quaternions. We discuss the relation between positive definite functions and the covariance of centered Gaussian processes and the construction of stochastic processes and their derivatives. The use of perfect spaces and strong algebras and the notion of Fock space are crucial in this framework.

Complex Valued Graphs For Soft Computing, Florentin Smarandache, W.B. Vasantha Kandasamy, Ilanthenral K

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce in a systematic way the notion of complex valued graphs, strong complex valued graphs and complex neutrosophic valued graphs. Several interesting properties are defined, described and developed. Most of the conjectures which are open in case of usual graphs continue to be open problems in case of both complex valued graphs and strong complex valued graphs. We also give some applications of them in soft computing and social networks. At this juncture it is pertinent to keep on record that Dr. Tohru Nitta was the pioneer to use complex valued graphs …

The Use Of The Pivot Pairwise Relative Criteria Importance Assessment Method For Determining The Weights Of Criteria, Florentin Smarandache, Dragisa Stanujkic, Edmundas Kazimieras Zavadskas, Darjan Karabasevic, Zenonas Turskis

Branch Mathematics and Statistics Faculty and Staff Publications

The weights of evaluation criteria could have a significant impact on the results obtained by applying multiple criteria decision-making methods. Therefore, the two extensions of the SWARA method that can be used in cases when it is not easy, or even is impossible to reach a consensus on the expected importance of the evaluation criteria are proposed in this paper. The primary objective of the proposed extensions is to provide an understandable and easy-to-use approach to the collecting of respondents’ real attitudes towards the significance of evaluation criteria and to also provide an approach to the checking of the reliability …

Math And Sudoku: Exploring Sudoku Boards Through Graph Theory, Group Theory, And Combinatorics, Kyle Oddson

Student Research Symposium

Encoding Sudoku puzzles as partially colored graphs, we state and prove Akman’s theorem [1] regarding the associated partial chromatic polynomial [5]; we count the 4x4 sudoku boards, in total and fundamentally distinct; we count the diagonally distinct 4x4 sudoku boards; and we classify and enumerate the different structure types of 4x4 boards.

Single Valued Neutrosophic Graphs, Florentin Smarandache, Said Broumi, Assia Bakali, Mohamed Talea

Branch Mathematics and Statistics Faculty and Staff Publications

The notion of single valued neutrosophic sets is a generalization of fuzzy sets, intuitionistic fuzzy sets. We apply the concept of single valued neutrosophic sets, an instance of neutrosophic sets, to graphs. We introduce certain types of single valued neutrosophic graphs (SVNG) and investigate some of their properties with proofs and examples.

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 …

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 …