Mathematics Commons^™

A Mathematical Journey Through The Land Of The Maya, Philip Scalisi, Paul Fairbanks

Bridgewater Review

No abstract provided.

Dependence, Dispersiveness, And Multivariate Hazard Rate Ordering, Baha-Eldin Khaledi, Subhash C. Kochar

Mathematics and Statistics Faculty Publications and Presentations

To compare two multivariate random vectors of the same dimension, we define a new stochastic order called upper orthant dispersive ordering and study its properties. We study its relationship with positive dependence and multivariate hazard rate ordering as defined by Hu, Khaledi, and Shaked (Journal of Multivariate Analysis, 2002). It is shown that if two random vectors have a common copula and if their marginal distributions are ordered according to dispersive ordering in the same direction, then the two random vectors are ordered according to this new upper orthant dispersive ordering. Also, it is shown that the marginal distributions of …

Book Review: Across The Board: The Mathematics Of Chessboard Problems By John J. Watkins, Arthur T. Benjamin

All HMC Faculty Publications and Research

I think I became a mathematician because I loved to play games as a child. I learned about probability and expectation by playing games like backgammon, bridge, and Risk. But I experienced the greater thrill of careful deductive reasoning through games like Mastermind and chess. In fact, for many years I took the game of chess quite seriously and played in many tournaments. But I gave up the game when I started college and turned my attention to more serious pursuits, like learning real mathematics.

A Look At The Astrolabe, Emily Storrar

Honors Capstones

Capstone submitted as a graduation requirement for the BSU Honors program.

Some Sober Conceptions Of Mathematical Truth, Marco Panza

MPP Published Research

It is not sufficient to supply an instance of Tarski’s schema, ⌈“p” is true if and only if p⌉ for a certain statement in order to get a definition of truth for this statement and thus fix a truth-condition for it. A definition of the truth of a statement x of a language L is a bi-conditional whose two members are two statements of a meta-language L’. Tarski’s schema simply suggests that a definition of truth for a certain segment x of a language L consists in a statement of the form: ⌈v(x) is true if and only if τ(x)⌉, …

Convergence Analysis Of Mcmc Method In The Study Of Genetic Linkage With Missing Data, Diana Fisher

Theses, Dissertations and Capstones

Computational infeasibility of exact methods for solving genetic linkage analysis problems has led to the development of a new collection of stochastic methods, all of which require the use of Markov chains. The purpose of this work is to investigate the complexities of missing data in pedigree analysis using the Monte Carlo Markov Chain (MCMC) method as compared to the exact results. Also, we attempt to determine an association between missing data in a familial pedigree and the convergence to stationarity of a descent graph Markov chain implemented in the stochastic method for parametric linkage analysis.

In particular, we will …

Disko Solution In Braille, Jeremiah Farrell

Scholarship and Professional Work - LAS

A copy of a 4x4 DISKO solution in Braille. Constructed by students at the Indiana School for the Blind on magnetized squares on a "toasted", i.e. raised grid.

Farrell's Spider, Jeremiah Farrell, Ivan Moscovich

Scholarship and Professional Work - LAS

Puzzle game featured in Ivan Moscovich's magnetic puzzle pack:

Place the 18 discs on the web so that the sum of the numbers on each of the three hexagons and on each of the three ribs equals 57.

The Backward Shift On HP, William T. Ross

Department of Math & Statistics Faculty Publications

In this semi-expository paper, we examine the backward shift operator

Bf := (f-f(0)/z

on the classical Hardy space H^p. Through there are many aspects of this operator worthy of study [20], we will focus on the description of its invariant subspaces by which we mean the closed linear manifolds Ɛ ⊂ H^p for which BƐ ⊂ Ɛ. When 1 < p < ∞, a seminal paper of Douglas, Shapiro, and Shields [8] describes these invariant subspaces by using the important concept of a pseudocontinuation developed earlier by Shapiro [26]. When p = 1, the description is the same [1] except that in the proof, one must be mindful of some technical considerations involving the functions of bounded mean oscillation.

Flowers Of Ice- Beauty, Symmetry, And Complexity: A Review Of The Snowflake: Winter's Secret Beauty, John A. Adam

Mathematics & Statistics Faculty Publications

(First paragraph) Growing up as a child in southern England, my early memories of snow include trudging home from school with my father, gazing at the seemingly enormous snowdrifts that smoothed the hedgerows, fields and bushes, while listening to the soft “scrunch” of the snow under my Wellington boots. In the country, snow stretching as far as I could see was not a particularly uncommon sight. The quietness of the land under a foot of snow seemed eerie. I cannot remember the first time I looked at snowflakes per se; my interests as a small child were primarily in their …

Introduction To Bimatrices, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices find their applications in several real models. In fact it is not an exaggeration if one says that matrix theory and linear algebra (i.e. vector spaces) form an inseparable component of each other. The study of bialgebraic structures led to the invention of new notions like birings, Smarandache birings, bivector spaces, linear bialgebra, bigroupoids, bisemigroups, etc. But most of these are abstract algebraic concepts except, the bisemigroup being used in …

Weak Factorizations, Fractions And Homotopies, Alexander Kurz, Jiří Rosický

Engineering Faculty Articles and Research

We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel.

Fuzzy And Neutrosophic Analysis Of Women With Hiv/Aids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Fuzzy theory is one of the best tools to analyze data, when the data under study is an unsupervised one, involving uncertainty coupled with imprecision. However, fuzzy theory cannot cater to analyzing the data involved with indeterminacy. The only tool that can involve itself with indeterminacy is the neutrosophic model. Neutrosophic models are used in the analysis of the socio-economic problems of HIV/AIDS infected women patients living in rural Tamil Nadu. Most of these women are uneducated and live in utter poverty. Till they became seriously ill they worked as daily wagers. When these women got admitted in the hospital …

Fuzzy And Neutrosophic Analysis Of Periyar’S Views On Untouchability, Florentin Smarandache, Vasantha Kandasamy, K. Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

“Day in and day out we take pride in claiming that India has a 5000 year old civilization. But the way Dalits and those suppressed are being treated by the people who wield power and authority speaks volumes for the degradations of our moral structure and civilized standards.” Ex-President of India, the late K. R. Narayanan The New Indian Express, Saturday, 12 Nov. 2005 K.R.Narayanan was a lauded hero and a distinguished victim of his Dalit background. Even in an international platform when he was on an official visit to Paris, the media headlines blazed, ‘An Untouchable at Elysee’. He …

Introduction To Linear Bialgebra, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief economic models and so on. This book has for the first time, introduced a new algebraic structure called linear bialgebra, which is also a very powerful algebraic tool that can yield itself to applications. With the recent introduction of bimatrices (2005) we have ventured in this book to introduce new concepts like linear bialgebra and Smarandache neutrosophic linear bialgebra and also give the applications of these algebraic …

Applications Of Bimatrices To Some Fuzzy And Neutrosophic Models, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

Graphs and matrices play a vital role in the analysis and study of several of the real world problems which are based only on unsupervised data. The fuzzy and neutrosophic tools like fuzzy cognitive maps invented by Kosko and neutrosophic cognitive maps introduced by us help in the analysis of such real world problems and they happen to be mathematical tools which can give the hidden pattern of the problem under investigation. This book, in order to generalize the two models, has systematically invented mathematical tools like bimatrices, trimatrices, n-matrices, bigraphs, trigraphs and n-graphs and describe some of its properties. …

Rational Hyperholomorphic Functions In R4, Daniel Alpay, Michael Shapiro, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce the notion of rationality for hyperholomorphic functions (functions in the kernel of the Cauchy-Fueter operator). Following the case of one complex variable, we give three equivalent definitions: the first in terms of Cauchy-Kovalevskaya quotients of polynomials, the second in terms of realizations and the third in terms of backward-shift invariance. Also introduced and studied are the counterparts of the Arveson space and Blaschke factors.

Point Evaluation And Hardy Space On A Homogeneous Tree, Daniel Alpay, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We consider stationary multiscale systems as defined by Basseville, Benveniste, Nikoukhah and Willsky. We show that there are deep analogies with the discrete time non stationary setting as developed by the first author, Dewilde and Dym. Following these analogies we define a point evaluation with values in a C*–algebra and the corresponding “Hardy space” in which Cauchy’s formula holds. This point evaluation is used to define in this context the counterpart of classical notions such as Blaschke factors.

Teaching Fractions, Natalie Miles

Mahurin Honors College Capstone Experience/Thesis Projects

Because fractions are a vital part of mathematics instruction in schools and can be found in real life, it is important that all students understand and be able to utilize them. Teachers should choose strategies that will work best for their individual students, taking into account the various ways their students learn and the fraction knowledge students already possess. To effectively teach fractions, teachers must find ways to prompt student interest. Teachers can accomplish this through the use of technology. games. and manipulatives. Effective teachers must take into account the cultural diversity of their students. Teachers must be able to …

Pokémon® Cards And The Shortest Common Superstring, Mark Stamp, Austin Stamp

Faculty Publications, Computer Science

Evidence is presented that certain sequences of Pokémon cards are determined by selecting consecutive elements from a longer sequence. We then consider the problem of recovering the shortest common superstring (SCS), i.e., the shortest string that contains each of the Pokémon card sequences as a consecutive substring. The SCS problem arises in many applications, most notably in DNA sequencing.

Mathematical Magic, Arthur T. Benjamin

All HMC Faculty Publications and Research

In this paper, we present simple strategies for performing mathematical calculations that appear magical to most audiences. Specifically, we explain how to square large numbers, memorize pi to 100 places and determine the day of the week of any given date.

Preface, Thomas Hildebrandt, Alexander Kurz

Engineering Faculty Articles and Research

No abstract provided.

Tree-Like Continua And 2-To-1 Maps, Jo Heath, Van C. Nall

Department of Math & Statistics Faculty Publications

It is not known if there is a 2-to-1 map from a continuum onto a tree-like continuum. In fact, it is not known if there is a 2-to-1 map onto a hereditarily decomposable tree-like continuum. We show that the domain of such a map would have to contain an indecomposable continuum.

Prolongations And Cyclic Vectors, William T. Ross, Harold S. Shapiro

Department of Math & Statistics Faculty Publications

For functions belonging to invariant subspaces of the backward shift operator Bf = (f − f(0))/z on spaces of analytic functions on the unit disk D, we explore, in a systematic way, the continuation properties of these functions.

Algebraic Semantics For Coalgebraic Logics, Clemens Kupke, Alexander Kurz, Dirk Pattinson

Engineering Faculty Articles and Research

With coalgebras usually being defined in terms of an endofunctor T on sets, this paper shows that modal logics for T-coalgebras can be naturally described as functors L on boolean algebras. Building on this idea, we study soundness, completeness and expressiveness of coalgebraic logics from the perspective of duality theory. That is, given a logic L for coalgebras of an endofunctor T, we construct an endofunctor L such that L-algebras provide a sound and complete (algebraic) semantics of the logic. We show that if L is dual to T, then soundness and completeness of the algebraic semantics immediately yield the …

Coalgebras And Modal Expansions Of Logics, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

In this paper we construct a setting in which the question of when a logic supports a classical modal expansion can be made precise. Given a fully selfextensional logic S, we find sufficient conditions under which the Vietoris endofunctor V on S-referential algebras can be defined and we propose to define the modal expansions of S as the logic that arises from the V-coalgebras. As an example, we also show how the Vietoris endofunctor on referential algebras extends the Vietoris endofunctor on Stone spaces. From another point of view, we examine when a category of ‘spaces’ (X,A), ie sets X …

Basic Neutrosophic Algebraic Structures And Their Application To Fuzzy And Neutrosophic Models, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Study of neutrosophic algebraic structures is very recent. The introduction of neutrosophic theory has put forth a significant concept by giving representation to indeterminates. Uncertainty or indeterminacy happen to be one of the major factors in almost all real-world problems. When uncertainty is modeled we use fuzzy theory and when indeterminacy is involved we use neutrosophic theory. Most of the fuzzy models which deal with the analysis and study of unsupervised data make use of the directed graphs or bipartite graphs. Thus the use of graphs has become inevitable in fuzzy models. The neutrosophic models are fuzzy models that permit …

Sheep Updates 2003 - Wool, Richard Coole, Stephen Gherardi, Chris Oldham, K. Curtis, J. Stanton, Johan Greeff, Chris Oldham, Mike Hyder, Beth Pagamoni, Andrew Thompson, Tom Plaisted, Kazue Tanaka, Mike Ferguson, Darren Gordon, Andrew Peterson, Peter Sommerville

Sheep Updates

This session covers six papers from different authors:1.‘Pastures from space’ - an opportunity to increase the profitability of sheep production Richard Coole Farmer Kojonup W.A. Stephen Gherardi Chris Oldham Department of Agriculture Western Australia 2. K. Curtis Department of Agriculture WA J. Stanton Department of Agriculture WA and Curtin University 3. Is selection of ewe hogget replacement on measurement profitable? Johan Greeff Department of Agriculture of Western Australia 4. Optimising the nutrition/grazing management of ewe flocks Chris Oldham Mike Hyder Beth Paganoni Department of Agriculture of Western Australia Andrew Thompson Department of Primary Industries, Hamilton, Victoria Tom Plaisted Kazue Tanaka …

Developing Into Series And Returning From Series: A Note On The Foundations Of Eighteenth-Century Analysis, Giovanni Ferraro, Marco Panza

MPP Published Research

In this paper we investigate two problems concerning the theory of power series in 18th-century mathematics: the development of a given function into a power series and the inverse problem, the return from a given power series to the function of which this power series is the development. The way of conceiving and solving these problems closely depended on the notion of function and in particular on the conception of a series as the result of a formal transformation of a function. After describing the procedures considered acceptable by 18th-century mathematicians, we examine in detail the different strategies—both direct and …

Stone Coalgebras, Clemens Kupke, Alexander Kurz, Yde Venema

Engineering Faculty Articles and Research

In this paper we argue that the category of Stone spaces forms an interesting base category for coalgebras, in particular, if one considers the Vietoris functor as an analogue to the power set functor. We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen as Stone coalgebras in a natural way. This yields a duality between the category of modal algebras and that of coalgebras over the Vietoris functor. Building on this idea, we introduce the notion of a Vietoris polynomial functor over the category of Stone spaces. …