Mathematics Commons^™

Interval Linear Algebra, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

This Interval arithmetic or interval mathematics developed in 1950’s and 1960’s by mathematicians as an approach to putting bounds on rounding errors and measurement error in mathematical computations. However no proper interval algebraic structures have been defined or studies. In this book we for the first time introduce several types of interval linear algebras and study them. This structure has become indispensable for these concepts will find applications in numerical optimization and validation of structural designs. In this book we use only special types of intervals and introduce the notion of different types of interval linear algebras and interval vector …

Rank Distance Bicodes And Their Generalization, Florentin Smarandache, W.B. Vasantha Kandasamy, N. Suresh Babu, R.S. Selvaraj

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce the new notion of rank distance bicodes and generalize this concept to Rank Distance n-codes (RD n-codes), n, greater than or equal to three. This definition leads to several classes of new RD bicodes like semi circulant rank bicodes of type I and II, semicyclic circulant rank bicode, circulant rank bicodes, bidivisible bicode and so on. It is important to mention that these new classes of codes will not only multitask simultaneously but also they will be best suited to the present computerised era. Apart from this, these codes are best suited in cryptography. …

Interval Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy, Moon Kumar Chetry

Branch Mathematics and Statistics Faculty and Staff Publications

This book introduces several new classes of groupoid, like polynomial groupoids, matrix groupoids, interval groupoids, polynomial interval groupoids, matrix interval groupoids and their neutrosophic analogues.

Methods In Environmental Biotechnology For Environmentalists, Florentin Smarandache, W.B. Vasantha Kandasamy, S.R. Kannan, S. Ramathilagam

Branch Mathematics and Statistics Faculty and Staff Publications

The study of environmental pollution by chemicals used in agriculture as pesticide or as fertilizers or pollution caused by industries and chemical plants which use chemicals have not been analysed systematically. This book has five chapters. First chapter is introductory in nature. Here we just study chemical pollution caused by garment industries in chapter two of this book using fuzzy associative memories. In chapter three we give ways to control pollution by improving the system performance using hierarchical genetic fuzzy control algorithm. This study is carried out using the past data reported by Shimada et al (1995). Health hazards suffered …

Proposed Problems Of Mathematics (Vol. Ii), Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

The first book of “Problèmes avec et sans … problèmes!” was published in Morocco in 1983. I collected these problems that I published in various Romanian or foreign magazines (amongst which: “Gazeta Matematică”, magazine which formed me as problem solver, “American Mathematical Monthly”, “Crux Mathematicorum” (Canada), “Elemente der Mathematik” (Switzerland), “Gaceta Matematica” (Spain), “Nieuw voor Archief” (Holland), etc. while others are new proposed problems in this second volume.

These have been created in various periods: when I was working as mathematics professor in Romania (1984-1988), or co-operant professor in Morocco (1982-1984), or emigrant in the USA (1990-1997). I thank to …

Fusion Of Sources Of Evidence With Different Importances And Reliabilities, Florentin Smarandache, Jean Dezert, J.M. Tacnet

Branch Mathematics and Statistics Faculty and Staff Publications

This paper presents a new approach for combining sources of evidences with different importances and reliabilities. Usually, the combination of sources of evidences with different reliabilities is done by the classical Shafer’s discounting approach. Therefore, to consider unequal importances of sources, if any, a similar reliability discounting process is generally used, making no difference between the notion of importance and reliability. In fact, in multicriteria decision context, these notions should be clearly distinguished. This paper shows how this can be done and we provide simple examples to show the differences between both solutions for managing importances and reliabilities of sources. …

Single Valued Neutrosophic Sets, Haibin Wang, Florentin Smarandache, Yanqing Zhang, Rajshekhar Sunderraman

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic set is a part of neutrosophy which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Neutrosophic set is a powerful general formal framework that has been recently proposed. However, neutrosophic set needs to be specified from a technical point of view. To this effect, we define the settheoretic operators on an instance of neutrosophic set, we call it single valued neutrosophic set (SVNS). We provide various properties of SVNS, which are connected to the operations and relations over SVNS.

Properties Of The Generalized Laplace Transform And Transport Partial Dynamic Equation On Time Scales, Chris R. Ahrendt

Department of Mathematics: Dissertations, Theses, and Student Research

In this dissertation, we first focus on the generalized Laplace transform on time scales. We prove several properties of the generalized exponential function which will allow us to explore some of the fundamental properties of the Laplace transform. We then give a description of the region in the complex plane for which the improper integral in the definition of the Laplace transform converges, and how this region is affected by the time scale in question. Conditions under which the Laplace transform of a power series can be computed term-by-term are given. We develop a formula for the Laplace transform for …

On Hyperplanes And Semispaces In Max-Min Convex Geometry, Viorel Nitica, Sergeĭ Sergeev

Mathematics Faculty Publications

No abstract provided.

Some Implications Of Chu's 10Ψ10 Generalization Of Bailey's 6Ψ6 Summation Formula, James Mclaughlin, Andrew Sills, Peter Zimmer

Mathematics Faculty Publications

Lucy Slater used Bailey's 6Ã6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type.

In the present paper we apply the same techniques to Chu's 10Ã10 generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs.

In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new …

General Wp-Bailey Chains, James Mclaughlin, Peter Zimmer

Mathematics Faculty Publications

Motivated by a recent paper of Liu and Ma, we describe a number of general WP-Bailey chains. We show that many of the existing WP-Bailey chains (or branches of the WP-Bailey tree), including chains found by Andrews, Warnaar and Liu and Ma, arise as special cases of these general WP-Bailey chains. We exhibit three new branches of the WP-Bailey tree, branches which also follow as special cases of these general WP-Bailey chains. Finally, we describe a number of new transformation formulae for basic hypergeometric series which arise as consequences of these new WP-Bailey chains.

Continued Fraction Proofs Of M-Versions Of Some Identities Of Rogers-Ramanujan-Slater Type, Douglas Bowman, James Mclaughlin, Nancy Wyshinksi

Mathematics Faculty Publications

We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two q-continued fractions previously investigated by the authors. By then specializing certain free parameters in these transformations, and employing various identities of Rogers-Ramanujan type, we derive m-versions of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods. By applying certain transformations due to Watson, Heine and Ramanujan, we derive still more examples of such m-versions of Rogers Ramanujan-type identities.

An Identity Motivated By An Amazing Identity Of Ramanujan, James Mclaughlin

Mathematics Faculty Publications

Ramanujan stated an identity to the effect that if three sequences {an}, {bn} and {cn} are defined by r1(x) =: ∑∞ n=0 anx n , r2(x) =: ∑∞ n=0 bnx n and r3(x) =: ∑∞ n=0 cnx n (here each ri(x) is a certain rational function in x), then a 3 n + b 3 n − c 3 n = (−1)n , ∀ n ≥ 0. Motivated by this amazing identity, we state and prove a more general identity involving eleven sequences, the new identity being ”more general” in the sense that equality holds not just for the power …

Some New Transformations For Bailey Pairs And Wp-Bailey Pairs, James Mclaughlin

Mathematics Faculty Publications

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding relations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.

Some Applications Of A Bailey-Type Transformation, James Mclaughlin, Peter Zimmer

Mathematics Faculty Publications

If k is set equal to aq in the definition of a WP Bailey pair, βn(a, k) = Xn j=0 (k/a)n−j (k)n+j (q)n−j (aq)n+j αj (a, k), this equation reduces to βn = Pn j=0 αj . This seemingly trivial relation connecting the αn’s with the βn’s has some interesting consequences, including several basic hypergeometric summation formulae, a connection to the Prouhet-Tarry-Escott problem, some new identities of the Rogers-Ramanujan-Slater type, some new expressions for false theta series as basic hypergeometric series, and new transformation formulae for poly-basic hypergeometric series.

Homomorphic Images Of Progenitors Of Order Three, Mark Gutierrez

Theses Digitization Project

The main purpose of this thesis is to construct finite groups as homomorphic images of infinite semi-direct products, 2*n : N, 3*n : N, and 3*n :m N, where 2*n and 3*n are free products of n copies of the cyclic group C₂ extended by N, a group of permutations on n letters.

If Mowat And Davis Are Correct, Then Teaching Is Hard: A Response To Elizabeth Mowat & Brent Davis, Kris H. Green, Bernard P. Ricca

Mathematical and Computing Sciences Faculty/Staff Publications

In lieu of an abstract, below is the article's first paragraph.

Mowat & Davis (this issue) present a model of learning mathematics that relies heavily on ideas from network (or graph) theory. The important questions (to us, at least) concern the dynamics of the nodes and links. Answers – even tentative ones such as we present here – to these questions lead to a second set of questions concerning the implications of these answers to teachers and researchers.

2010 Vol. 4 Table Of Contents, University Of Dayton. Department Of Mathematics

Undergraduate Mathematics Day: Past Content

No abstract provided.

On The Construction Of Order Six Multilevel Hadamard Matrices, Keli Parker

Undergraduate Mathematics Day: Past Content

The existence of multilevel Hadamard matrices (MHMs) of all orders as well as a construction for full-rate circulant MHMs of all orders n 6= 4 is known. We use computer search methods to look for previously unknown full-rate circulant MHMs of orders 5, 6, and 7 and find solutions that potentially do not follow from the known construction. We then give an alternate construction to explain some order six MHMs.

Uniqueness Of Solutions Implies Existence And Uniqueness Of Solutions Of Boundary Value Problems For Third Order Differential Equations, Veronica Respress

Undergraduate Mathematics Day: Past Content

In this paper we are concerned with uniqueness implies uniqueness and uniqueness implies existence questions for solutions of a class of boundary value problems for the third order ordinary differential equation (ODE). First we show uniqueness of solutions of a class of two-point problems implies the uniqueness of solutions of an associated class of three-point problems. Then we establish uniqueness of solutions of the class of two-point problems implies the existence of solutions of the class of two point problems and the associated class of three-point problems.