Discrete Mathematics and Combinatorics Commons^™

On Representations Of Semigroups Having Hypercube-Like Cayley Graphs, Cody Cassiday, G. Stacey Staples

SIUE Faculty Research, Scholarship, and Creative Activity

The $n-dimensional hypercube, or n-cube, is the Cayley graph of the Abelian group Z₂ⁿ. A number of combinatorially-interesting groups and semigroups arise from modified hypercubes. The inherent combinatorial properties of these groups and semigroups make them useful in a number of contexts, including coding theory, graph theory, stochastic processes, and even quantum mechanics. In this paper, particular groups and semigroups whose Cayley graphs are generalizations of hypercubes are described, and their irreducible representations are characterized. Constructions of faithful representations are also presented for each semigroup. The associated semigroup algebras are realized within the context …

Special Type Of Topological Spaces Using [0, N), 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 special type of topological spaces using the interval [0, n). They are very different from the usual topological spaces. Algebraic structure using the interval [0, n) have been systemically dealt by the authors. Now using those algebraic structures in this book authors introduce the notion of special type of topological spaces. Using the super subset interval semigroup special type of super interval topological spaces are built. Several interesting results in this direction are obtained. Next six types of topological spaces using subset interval pseudo ring semiring of type …

Mod Functions: A New Approach To Function Theory, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the notion of MOD functions are defined on MOD planes. This new concept of MOD functions behaves in a very different way. Even very simple functions like y = nx has several zeros in MOD planes where as they are nice single line graphs with only (0, 0) as the only zero. Further polynomials in MOD planes do not in general follows the usual or classical laws of differentiation or integration. Even finding roots of MOD polynomials happens to be very difficult as they do not follow the fundamental theorem of algebra, viz a nth degree polynomial …

Mod Planes: A New Dimension To Modulo Theory, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book for the first time authors study mod planes using modulo intervals [0, m); 2 ≤ m ≤ ∞. These planes unlike the real plane have only one quadrant so the study is carried out in a compact space but infinite in dimension. We have given seven mod planes viz real mod planes (mod real plane) finite complex mod plane, neutrosophic mod plane, fuzzy mod plane, (or mod fuzzy plane), mod dual number plane, mod special dual like number plane and mod special quasi dual number plane. These mod planes unlike real plane or complex plane or neutrosophic …

Mod Pseudo Linear Algebras, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time elaborately study the notion of MOD vector spaces and MOD pseudo linear algebras. This study is new, innovative and leaves several open conjectures. In the first place as distributive law is not true we can define only MOD pseudo linear algebras. Secondly most of the classical theorems true in case of linear algebras are not true in case of MOD pseudo linear algebras. Finding even eigen values and eigen vectors happens to be a challenging problem. Further the notion of multidimensional MOD pseudo linear algebras are defined using the notion of MOD …

Fuzzy Abel Grassmann Groupoids, Florentin Smarandache, Madad Khan, Tariq Aziz

Branch Mathematics and Statistics Faculty and Staff Publications

Usually the models of real world problems in almost all disciplines like engineering, medical sciences, mathematics, physics, computer science, management sciences, operations research and articial intelligence are mostly full of complexities and consist of several types of uncertainties while dealing them in several occasion. To overcome these di¢ culties of uncertainties, many theories have been developed such as rough sets theory, probability theory, fuzzy sets theory, theory of vague sets, theory of soft ideals and the theory of intuitionistic fuzzy sets, theory of neutrosophic sets, Dezert-Smarandache Theory (DSmT), etc. Zadeh introduced the degree of membership/truth (t) in 1965 and dened …

Theory Of Abel Grassmann's Groupoids, Florentin Smarandache, Madad Khan

Branch Mathematics and Statistics Faculty and Staff Publications

It is common knowledge that common models with their limited boundaries of truth and falsehood are not su¢ cient to detect the reality so there is a need to discover other systems which are able to address the daily life problems. In every branch of science problems arise which abound with uncertainties and impaction. Some of these problems are related to human life, some others are subjective while others are objective and classical methods are not su¢ cient to solve such problems because they can not handle various ambiguities involved. To overcome this problem, Zadeh [67] introduced the concept of …

Boundary Interpolation For Slice Hyperholomorphic Schur Functions, Khaled Abu-Ghanem, Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers κ1,…,κN, quaternions p1,…,pN all of modulus 1, so that the 2-spheres determined by each point do not intersect and pu≠1 for u=1,…,N, and quaternions s1,…,sN, we wish to find a slice hyperholomorphic Schur function s so that
limr→1r∈(0,1)s(rpu)=suforu=1,…,N,
and
limr→1r∈(0,1)1−s(rpu)su¯¯¯¯¯1−r≤κu,foru=1,…,N.
Our arguments relies on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.

Unimodality Questions In Ehrhart Theory, Robert Davis

Theses and Dissertations--Mathematics

An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal h^*-vector.Although various sufficient conditions have been found, necessary conditions remain a challenge. Highly-structured polytopes, such as the polytope of real doubly-stochastic matrices, have been proven to possess unimodal h^*-vectors, but the same is unknown even for small variations of it.

In this dissertation, we mainly consider two particular classes of polytopes: reflexive simplices and the polytope of symmetric real doubly-stochastic matrices. For the first class, we discuss an operation that preserves reflexivity, integral closure, and unimodality of the h^{* …}

Properly Colored Notions Of Connectivity - A Dynamic Survey, Xueliang Li, Colton Magnant

Theory & Applications of Graphs

A path in an edge-colored graph is properly colored if no two consecutive edges receive the same color. In this survey, we gather results concerning notions of graph connectivity involving properly colored paths.

Dynamic Approach To K-Forcing, Yair Caro, Ryan Pepper

Theory & Applications of Graphs

The k-forcing number of a graph is a generalization of the zero forcing number. In this note, we give a greedy algorithm to approximate the k-forcing number of a graph. Using this dynamic approach, we give corollaries which improve upon two theorems from a recent paper of Amos, Caro, Davila and Pepper [2], while also answering an open problem posed by Meyer [9].

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 …

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: …

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.

Modeling The Progress And Retention Of International Students Using Markov Chains, Lucas Gagne

Williams Honors College, Honors Research Projects

International students are a small and diverse student population present in any sizable American university. One of the greatest obstacles in their path is the acquisition of the English language. English for Academic Purposes (EAP) programs, such as the English Language Institute (ELI) at the University of Akron, attempt to address this problem. By studying how this student population progresses in their academic studies, EAP programs and their associated universities can make well-informed decisions on how best to serve their English Language Learners. One way to study International students is through the use of a Markov model based on university …

Multidimensional Mod Planes, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors name the interval [0, m); 2 ≤ m ≤ ∞ as mod interval. We have studied several properties about them but only here on wards in this book and forthcoming books the interval [0, m) will be termed as the mod real interval, [0, m)I as mod neutrosophic interval, [0,m)g; g2 = 0 as mod dual number interval, [0, m)h; h2 = h as mod special dual like number interval and [0, m)k, k2 = (m − 1) k as mod special quasi dual number interval. However there is only one real interval (∞, ∞) but …

Neutrosophic Graphs: A New Dimension To Graph Theory, Florentin Smarandache, Wb. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time have made a through study of neutrosophic graphs. This study reveals that these neutrosophic graphs give a new dimension to graph theory. The important feature of this book is it contains over 200 neutrosophic graphs to provide better understanding of this concepts. Further these graphs happen to behave in a unique way inmost cases, for even the edge colouring problem is different from the classical one. Several directions and dimensions in graph theory are obtained from this study. Finally certainly these new notions of neutrosophic graphs in general and in particular the …

Some Properties Of The Exchange Operator With Respect To Structured Matrices Defined By Indefinite Scalar Product Spaces, Hanz Martin C. Cheng, Roden Jason David

Mathematics Faculty Publications

The properties of the exchange operator on some types of matrices are explored in this paper. In particular, the properties of exc(A,p,q), where A is a given structured matrix of size (p+q)Ã(p+q) and exc : M ÃNÃN â M is the exchange operator are studied. This paper is a generalization of one of the results in [N.J. Higham. J-orthogonal matrices: Properties and generation. SIAM Review, 45:504â519, 2003.].

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.

Second Hamiltonian Cycles In Claw-Free Graphs, Hossein Esfandiari, Colton Magnant, Pouria Salehi Nowbandegani, Shirdareh Haghighi

Theory & Applications of Graphs

Sheehan conjectured in 1975 that every Hamiltonian regular simple graph of even degree at least four contains a second Hamiltonian cycle. We prove that most claw-free Hamiltonian graphs with minimum degree at least 3 have a second Hamiltonian cycle and describe the structure of those graphs not covered by our result. By this result, we show that Sheehan’s conjecture holds for claw-free graphs whose order is not divisible by 6. In addition, we believe that the structure that we introduce can be useful for further studies on claw-free graphs.