Physical Sciences and Mathematics Commons^™

Mathematization And Modern Science, Calvin Jongsma

Faculty Work Comprehensive List

The discipline of mathematics has not been spared the sweeping critique of postmodernism. Is mathematical theory true for all time, or are mathematical constructs in fact fallible? This fascinating book examines the tensions that have arisen between modern and postmodern views of mathematics, explores alternative theories of mathematical truth, explains why the issues are important, and shows how a Christian perspective makes a difference.

This chapter continues the process of tracing Western mathematization.

Power Connector, Stephen L. Clark, Joseph B. Shuey, Jose L. Ortega, John B. Brown Iii

Mathematics and Statistics Faculty Research & Creative Works

A pair of mating connectors includes a receptacle having an insulative housing and at least one conductive receptacle contact with a pair of spaced walls forming a plug contact receiving space. The plug connector has an insulative housing and at least one conductive contact having a pair of spaced walls which converge to form a projection engageable in the plug receiving space of the receptacle contact. In each case, the spaced walls are joined by a bridging structure that unites the walls. The plug and receptacle contacts are retained in the respective housings by engagement of opposed lateral edge portions …

A Critical Look At Self-Dual Codes, Judy L. Walker

Department of Mathematics: Faculty Publications

We investigate self-dual codes from a structural point of view. In particular, we study properties of critical indecomposable codes which appear in the spectrum of a self-dual code. As an application of the results we obtain, we revisit the study of self-dual codes of dimension at most 10.

In the late 1950’s, Slepian [4] became the first to take an abstract approach to the study of error-correcting codes. He introduced a structure theory for binary linear codes, developing in particular the idea of an indecomposable code; that is, a code which is not isomorphic to a nontrivial direct sum of …

On The Evolution Of Probability-Weighting Function And Its Impact On Gambling, Steven Li, Yun Hsing Cheung

Research outputs pre 2011

It is well known that individuals treat losses and gains differently and there exists non-linearity in probability. The asymmetry between gains and losses is highlighted by the reflection effect. The non-linearity in probability is described by the curvature of the probability-weighting function. This paper studies the evolution of the probability-weighting function. It is assumed that the probability weighting for an individual follows a mean-reverting stochastic process. The Monte Carlo simulation technique is employed to study the evolution of the weighting function. The evolution of the probability- weighting function implies that an individual does not treat gains or losses consistently over …

A Class Of Monoids Embeddable In A Group, Ebru Keyman

Turkish Journal of Mathematics

In this paper, we develop a new method to show that a monoid, given by a certain kind of presentation, embeds in a group. A mathematical device called the diamond condition was used in [5] to prove that the singular braid monoid SB_n embeds. Motivated by this, we consider monoid presentations which have the basic properties of the presentation of the singular braid monoid. In the same way as in [5], we prove that the monoid embeds. The proof of the diamond condition is completely geometric in [5], but here we prove it by using elementary algebraic properties.

Optimal Token Allocations In Solitaire Knock 'M Down, Arthur Benjamin, Matthew T. Fluet, Mark L. Huber

All HMC Faculty Publications and Research

In the game Knock ’m Down, tokens are placed in N bins. At each step of the game, a bin is chosen at random according to a fixed probability distribution. If a token remains in that bin, it is removed. When all the tokens have been removed, the player is done. In the solitaire version of this game, the goal is to minimize the expected number of moves needed to remove all the tokens. Here we present necessary conditions on the number of tokens needed for each bin in an optimal solution, leading to an asymptotic solution. MR Subject Classifications: …

Transient Anomalous Diffusion In Poiseuille Flow, Marco Latini '01, Andrew J. Bernoff

All HMC Faculty Publications and Research

We revisit the classical problem of dispersion of a point discharge of tracer in laminar pipe Poiseuille flow. For a discharge at the centre of the pipe we show that in the limit of small non-dimensional diffusion, D, tracer dispersion can be divided into three regimes. For small times (t [double less-than sign] D^−1/3), diffusion dominates advection yielding a spherically symmetric Gaussian dispersion cloud. At large times (t [dbl greater-than sign] D⁻¹), the flow is in the classical Taylor regime, for which the tracer is homogenized transversely across the pipe and diffuses with …

Uber Potenzsummenpolynome (On Polynomials Of Sums Of Power), Jorg Feldvoss

University Faculty and Staff Publications

No abstract provided.

Proceedings Of The First International Conference On Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability And Statistics, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In 1960s Abraham Robinson has developed the non-standard analysis, a formalization of analysis and a branch of mathematical logic, that rigorously defines the infinitesimals. Informally, an infinitesimal is an infinitely small number. Formally, x is said to be infinitesimal if and only if for all positive integers n one has xxx < 1/n. Let &>0 be a such infinitesimal number. The hyper-real number set is an extension of the real number set, which includes classes of infinite numbers and classes of infinitesimal numbers. Let’s consider the non-standard finite numbers 1+ = 1+&, where “1” is its standard part and “&” its non-standard part, …

Enumeration Of Equicolourable Trees, Nicholas Pippenger

All HMC Faculty Publications and Research

A tree, being a connected acyclic graph, can be bicolored in two ways, which differ from each other by exchange of the colors. We shall say that a tree is equicolorable if these bicolorings assign the two colors to equal numbers of vertices. Labelled equicolored trees have been enumerated several times in the literature, and from this result it is easy to enumerate labelled equicolorable trees. The result is that the probability that a randomly chosen n-vertex labelled tree is equicolorable is asymptotically just twice the probability that its vertices would be equicolored if they were assigned colors by …

A Mathematical Tumor Model With Immune Resistance And Drug Therapy: An Optimal Control Approach, Lisette G. De Pillis, Ami E. Radunskaya

All HMC Faculty Publications and Research

We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four-population model that includes tumor cells, host cells, immune cells, and drug interaction. We analyze the stability of the drug-free equilibria with respect to the immune response in order to look for target basins of attraction. One of our goals was to simulate qualitatively the asynchronous tumor-drug interaction known as “Jeffs phenomenon.” The model we develop is successful in generating this asynchronous response behavior. Our other goal was to identify treatment protocols that could improve standard pulsed chemotherapy …

Unit Sum Numbers Of Abelian Groups And Modules, Christopher Meehan

Doctoral

We discuss some open questions regarding the unit sum numbers of free modules of arbitrary infinite rank over commutative rings and, in particular, over principal ideal domains. The unit sum numbers of rational groups are then investigated: the importance of the rational prime 2 being an automorphism of the rational group is discussed and other results are achieved considering the number and distribution of rational primes which are, or are not, automorphisms of the group. We next prove the existence of rational groups with unit sum numbers greater than 2 but of finite value and we estimate an upper bound …

Scintillation Behind The Collecting Lens Of A Receiver, Clarissa A. Fleming Russell

Retrospective Theses and Dissertations

One of the negative effects that a laser beam experiences as it propagates through the atmosphere is intensity fluctuations or scintillation. Because scintillation-- as it pertains to laser radar and laser satellite communication systems-- is the main subject of this research, the assumption of an optical element ( such as a Gaussian lens) along the propagation path in front of the detector is valid. The mathematical addition of optical elements to the propagation path is treated using the ABCD ray matrix method. The expression for scintillation is derived, analyzed, and numerically calculated for positions to the left and right of …

On Stochastic Comparisons Of Energy Functions With Applications, Broderick O. Oluyede

Department of Mathematical Sciences Faculty Publications

We develop simple methods for the stochastic comparisons of informational energy functions. We introduce modified informational energy functions and uncertainty of parameter functions are introduced for models with realistic parameter spaces. We present inequalities, comparisons, and applications including test procedures for testing the equality of informational energy functions. Some illustrative examples are also presented.

Heckman's Methodology For Correcting Selectivity Bias : An Application To Road Crash Costs, Margaret Giles

Research outputs pre 2011

Aggregate road crash costs are traditionally determined using average costs applied to incidence figures found in Police-notified crash data. Such data only comprise a non-random sample of the true population of road crashes, the bias being due to the existence of crashes that are not notified to the Police. The traditional approach is to label the Police-notified sample as 'non-random' thereby casting a cloud over data analyses using this sample. Heckman however viewed similar problems as 'omitted variables' problems in that the exclusion of some observations in a systematic manner (so-called selectivity bias) has inadvertently introduced the need for an …

Generalizing Bailey's Generalization Of The Catalan Numbers, Darrin D. Frey, James A. Sellers

Science and Mathematics Faculty Publications

No abstract provided.

Mathematics As Worship, David J. Stucki

Mathematics Faculty Scholarship

This paper treats worship in a broad, inclusive sense as the primary and necessary response of human beings to their creator. It provides a brief overview of the relationship between mathematics and theology from the Pythagoreans to the present. It argues that all knowledge is contingent upon faith and thus that contemplation of mathematical insights can lead to worship of God.

Jkarelrobot: A Case Study In Supporting Levels Of Cognitive Development In The Computer Science Curriculum, Duane Buck, David J. Stucki

Mathematics Faculty Scholarship

We introduce a new software tool, JKarelRobot, for supporting an Inside/Out pedagogy in introductory programming courses. Extending the original conception of "Karel the Robot", with Bloom's Taxonomy of Educational Objectives as a guiding principle, we have provided a mechanism for designing exercises that are cognitively appropriate to the developmental levels of our students. JKarelRobot is platform independent (written in Java) and language/paradigm independent, supporting Pascal, Java, and Lisp style environments.

[Introduction To] Basic Java Programming: A Laboratory Approach, Lewis Barnett

Bookshelf

For first- and second-year undergraduates, an introduction to programming with Java, an object-oriented programming language that is a popular choice for Web applications. Kent and Barnett (U. of Richmond) introduce algorithms and problem-solving approaches that are important to programming general.

Dialectics And The Dao: On Both, A And Non-A In Neutrosophy And Chinese Philosophy, Feng Liu, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper introduces readers to a new approach to dialectical logic: neutrosophy. Specifically it proposes a multi-valued logic in which the statement “both A and Non-A,” historically rejected as logically incoherent, is treated as meaningful. This unity of opposites constitutes both the objective world and the subjective world –a view with deep roots in Buddhism and Daoism, including the I-Ching. This leads in turn to the presentation of a framework for the development of a contradiction oriented learning philosophy inspired by the Later Trigrams of King Wen in the I-Ching. We show that although A and Non-A are logically inconsistent, …

Numerical Ranges Of Composition Operators, Valentin Matache

Mathematics Faculty Publications

Composition operators on the Hilbert Hardy space of the unit disk are considered. The shape of their numerical range is determined in the case when the symbol of the composition operator is a monomial or an inner function fixing 0. Several results on the numerical range of composition operators of arbitrary symbol are obtained. It is proved that 1 is an extreme boundary point if and only if 0 is a fixed point of the symbol. If 0 is not a fixed point of the symbol 1 is shown to be interior to the numerical range. Some composition operators whose …

Iteration Of Λ-Complete Forcing Notions Not Collapsing Λ+, Andrzej Roslanowski

Mathematics Faculty Publications

We look for a parallel to the notion of “proper forcing” among λ-complete forcing notions not collapsing λ+. We suggest such a definition and prove that it is preserved by suitable iterations.

Locally 1-To-1 Maps And 2-To-1 Retractions, Jo Heath, Van C. Nall

Department of Math & Statistics Faculty Publications

This paper considers the question of which continua are 2-to-1 retracts of continua.

Analysis Of A Non-Replicated Split-Split Plot Experiment, Emily Simmons Sim

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

A major obstacle in the analysis of experimental data, in many situations, is the lack of "true" or "complete" replication. In some disciplines, researchers are very aware of the importance of replication and the methods for correctly replicating an experiment. In other subject areas, however, researchers are less aware of what it means to properly replicate an experiment. Due to this lack of awareness, many non-replicated experiments are carried out every year. For many of these non-replicated experiments, there is no satisfactory statistical analysis.

The subject of this report is the analysis of two non-replicated experiments in environmental engineering. First, …

Ghost Forests, Global Warming And The Mountain Pine Beetle, J. A. Logan, James A. Powell

James A. Powell

No abstract provided.

Wholes And Parts In General Systems Methodology, Martin Zwick

Systems Science Faculty Publications and Presentations

Reconstructability analysis (RA) decomposes wholes, namely data in the form either of set-theoretic relations or multivariate probability distributions, into parts, namely relations or distributions involving subsets of variables. Data is modeled and compressed by variablebased decomposition, by more general state-based decomposition, or by the use of latent variables. Models, which specify the interdependencies among the variables, are selected to minimize error and complexity.

Nested Balanced Incomplete Block Designs, J. P. Morgan, D. A. Preece, D. H. Rees

Mathematics & Statistics Faculty Publications

If the blocks of a balanced incomplete block design (BIBD) with v treatments and with parameters (v; b₁;r;k₁) are each partitioned into sub-blocks of size k₂, and the b₂ =b₁k₁=k2 sub-blocks themselves constitute a BIBD with parameters (v; b₂;r;k₂), then the system of blocks, sub-blocks and treatments is, by de4nition, a nested BIBD (NBIBD). Whist tournaments are special types of NBIBD with k₁ =2k₂= 4. Although NBIBDs were introduced in the statistical literature in 1967 and have subsequently received occasional attention there, …

Modal Rules Are Co-Implications, Alexander Kurz

Engineering Faculty Articles and Research

In [13], it was shown that modal logic for coalgebras dualises—concerning definability— equational logic for algebras. This paper establishes that, similarly, modal rules dualise implications:It is shown that a class of coalgebras is definable by modal rules iff it is closed under H (images) and Σ (disjoint unions). As a corollary the expressive power of rules of infinitary modal logic on Kripke frames is characterised.

On Moment Inequalities And Stochastic Ordering For Weighted Reliability Measures, Broderick O. Oluyede, Mekki Terbeche

Department of Mathematical Sciences Faculty Publications

We obtain stochastic inequalities, error bounds, and classification probability for a general class of distributions. We introduce the notion of variability ordering via the probability functional and comparisons made for the weighted and the original distributions. We present moment inequalities, comparisons, and applications.

Eddies Induced In Cylindrical Containers By A Rotating End Wall, Christopher Hills

Articles

The flow generated in a viscous liquid contained in a cylindrical geometry by a rotating end wall is considered. Recent numerical and experimental work has established several distinct phases of the motion when fluid inertia plays a significant role. The current paper, however, establishes the nature of the flow in the thus far neglected low Reynolds number regime. Explicitly, by employing biorthogonality relations appropriate to the current geometry, it is shown that a sequence of exponentially decaying eddies extends outward from the rotating end wall. The cellular structure is a manifestation of the dominance of complex eigensolutions to the homogeneous …