Physical Sciences and Mathematics Commons^™

Expression Of Wnt-Signaling Pathway Genes And Their Associations With Mirnas In Colorectal Cancer, Martha L. Slattery, Lila E. Mullany, Lori C. Sakoda, Wade S. Samowitz, Roger K. Wolff, John R. Stevens, Jennifer S. Herrick

Mathematics and Statistics Faculty Publications

The Wnt-signaling pathway functions in regulating cell growth and thus is involved in the carcinogenic process of several cancers, including colorectal cancer. We tested the hypothesis that multiple genes in this signaling pathway are dysregulated and that miRNAs are associated with these dysregulated genes. We used data from 217 colorectal cancer (CRC) cases to evaluate differences in Wnt-signaling pathway gene expression between paired CRC and normal mucosa and identify miRNAs that are associated with these genes. Gene expression data from RNA-Seq and miRNA expression data from Agilent Human miRNA Microarray V19.0 were analyzed. We focused on genes most strongly associated …

Reconstructing Results From Voting Theory Using Linear Algebra, Brian Camara

Honors Program Theses and Projects

For many undergraduate students, achieving an understanding of upper-level mathematics can be extremely challenging. For us, it helps to connect these new concepts with material we are familiar with. This will be the central theme of this thesis. We will introduce some basic components of algebraic voting theory, along with briey discussing how (Daugherty, Eustis, Minton, & Orrison, 2009) used representation theory to achieve their results. We will then provide an alternative proof to the main result of the (Daugherty et al., 2009) article using linear algebra, which should be much more familiar to my peers. We will carry out …

New Implementations For Tabulating Pseudoprimes And Liars, Wuyang Liu

Honors Projects

Whether it is applied to primality test or cryptography, pseudoprimes are one of the most important topics in number theory. Regarding the study of strong pseudoprimes, there are two problems which mathematicians have been working on:
1. Given a, b, find all a-spsp up to b.
2. Given an odd composite n, find all a -n such that n is an a-spsp.
where n = a-spsp means n is a strong pseudoprime to base a, and a is a strong liar of n.

The two problems are respectively referred to as …

Hodge Theory On Transversely Symplectic Foliations, Yi Lin

Department of Mathematical Sciences Faculty Publications

In this paper, we develop symplectic Hodge theory on transversely symplectic foliations. In particular, we establish the symplectic dδ-lemma for any such foliations with the (transverse) s-Lefschetz property. As transversely symplectic foliations include many geometric structures, such as contact manifolds, co-symplectic manifolds, symplectic orbifolds, and symplectic quasi-folds as special examples, our work provides a unifying treatment of symplectic Hodge theory in these geometries.

As an application, we show that on compact K-contact manifolds, the s-Lefschetz property implies a general result on the vanishing of cup products, and that the cup length of a 2n+1 dimensional compact K-contact manifold with the …

Using Mixed Effects Modeling To Quantify Difference Between Patient Groups With Diabetic Foot Ulcers, Rachel French

Mahurin Honors College Capstone Experience/Thesis Projects

When diabetes progresses, many patients suffer from chronic foot ulcers. In a study described in Matrix Metalloproteinases and Diabetic Foot Ulcers (Muller et al., 2008), sixteen patients with diabetic foot ulcers were examined throughout a twelve week healing period. During this period, levels of matrix metalloproteinases (MMP-1), their inhibitors (TIMP-1), and the extracellular matrix in a wound area were measured at distinct time intervals for each patient. The ratios of these healing components are vital in determining whether a wound will heal or become chronic and never properly heal. Connecting Local and Global Sensitivities in a Mathematical Model for Wound …

Degree And Neighborhood Conditions For Hamiltonicity Of Claw-Free Graphs, Zhi-Hong Chen

Scholarship and Professional Work - LAS

For a graph H , let σ t ( H ) = min { Σ i = 1 t d H ( v i ) | { v 1 , v 2 , … , v t } is an independent set in H } and let U t ( H ) = min { | ⋃ i = 1 t N H ( v i ) | | { v 1 , v 2 , ⋯ , v t } is an independent set in H } . We show that for a given number ϵ and given integers …

Introduction To The Usu Library Of Solutions To The Einstein Field Equations, Ian M. Anderson, Charles G. Torre

Tutorials on... in 1 hour or less

This is a Maple worksheet providing an introduction to the USU Library of Solutions to the Einstein Field Equations. The library is part of the DifferentialGeometry software project and is a collection of symbolic data and metadata describing solutions to the Einstein equations.

Quantum Econometrics: How To Explain Its Quantitative Successes And How The Resulting Formulas Are Related To Scale Invariance, Entropy, Fuzzy, And Copulas, Hung T. Nguyen, Kittawit Autchariyapanitkul, Olga Kosheleva, Vladik Kreinovich, Songsak Sriboonchitta

Departmental Technical Reports (CS)

Many aspects of human behavior seem to be well-described by formulas of quantum physics. In this paper, we explain this phenomenon by showing that the corresponding quantum-looking formulas can be derived from the general ideas of scale invariance, fuzziness, and copulas. We also use these ideas to derive a general family of formulas that include non-quantum and quantum probabilities as particular cases -- formulas that may be more adequate for describing human behavior than purely non-quantum or purely quantum ones.

Why Rectified Linear Neurons Are Efficient: Symmetry-Based, Complexity-Based, And Fuzzy-Based Explanations, Olac Fuentes, Justin Parra, Elizabeth Y. Anthony, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditionally, neural networks used a sigmoid activation function. Recently, it turned out that piecewise linear activation functions are much more efficient -- especially in deep learning applications. However, so far, there have been no convincing theoretical explanation for this empirical efficiency. In this paper, we show that, by using different uncertainty techniques, we can come up with several explanations for the efficiency of piecewise linear neural networks. The existence of several different explanations makes us even more confident in our results -- and thus, in the efficiency of piecewise linear activation functions.

How To Make A Proof Of Halting Problem More Convincing: A Pedagogical Remark, Benjamin W. Robertson, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

As an example of an algorithmically undecidable problem, most textbooks list the impossibility to check whether a given program halts on given data. A usual proof of this result is based on the assumption that the hypothetical halt-checker works for all programs. To show that a halt-checker is impossible, we design an auxiliary program for which the existence of such a halt-checker leads to a contradiction. However, this auxiliary program is usually very artificial. So, a natural question arises: what if we only require that the halt-checker work for reasonable programs? In this paper, we show that even with such …

Statistics As Unbiased Estimators: Exploring The Teaching Of Standard Deviation, Nicholas H. Wasserman, Stephanie Casey, Joe Champion, Maryann Huey

Mathematics Faculty Publications and Presentations

This manuscript presents findings from a study about the knowledge for and planned teaching of standard deviation. We investigate how understanding variance as an unbiased (inferential) estimator – not just a descriptive statistic for the variation (spread) in data – is related to teachers’ instruction regarding standard deviation, particularly around the issue of division by n-1. In this regard, the study contributes to our understanding about how knowledge of mathematics beyond the current instructional level, what we refer to as nonlocal mathematics, becomes important for teaching. The findings indicate that acquired knowledge of nonlocal mathematics can play a role …

Facing The Sun, Frank Prendergast, Muiris O'Sullivan, Ken Williams, Gabriel Cooney

Articles

December 2017 marked 50 years since archaeologist Michael J. O’Kelly first observed the solar illumination of the burial chamber in the Neolithic passage tomb at Newgrange during the period of the winter solstice. O’Kelly subsequently recorded direct sunlight entering Newgrange through the ‘especially contrived slit which lies under the roof-box at the outer end of the passage roof’ on 21 December 1969. The discovery of this historic phenomenon, dating back over 5,000 years, captured the public interest and imagination at that time and ever since. In this major article published in the Winter 2017 edition of Archaeology Ireland (date of …

Some Remarks On Ricci Solitons, Ramesh Sharma, S Balasubramanian, N. Uday Kiran

Mathematics Faculty Publications

We obtain an intrinsic formula of a Ricci soliton vector field and a differential condition for the non-steady case to be gradient. Next we provide a condition for a Ricci soliton on a Kaehler manifold to be a Kaehler–Ricci soliton. Finally we give an example supporting the first result.

Do It Today Or Do It Tomorrow: Empirical Non-Exponential Discounting Explained By Symmetry And Fuzzy Ideas, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich, Thongchai Dumrongpokaphan

Departmental Technical Reports (CS)

At first glance, it seems to make sense to conclude that when a 1 dollar reward tomorrow is equivalent to a D < 1 dollar reward today, the day-after-tomorrow's 1 dollar reward would be equivalent to D * D = D² dollars today, and, in general, a reward after time t is equivalent to D(t) = D^t dollars today. This exponential discounting function D(t) was indeed proposed by the economists, but it does not reflect the actual human behavior. Indeed, according to this formula, the effect of distant future events is negligible, and thus, it would be reasonable for a person to take on huge loans or get engaged in unhealthy behavior even when the long-term consequences …

Neutrosophic Hough Transform, Florentin Smarandache, Umit Budak, Yanhui Guo, Abdulkadir Sengur

Branch Mathematics and Statistics Faculty and Staff Publications

Hough transform (HT) is a useful tool for both pattern recognition and image processing communities. In the view of pattern recognition, it can extract unique features for description of various shapes, such as lines, circles, ellipses, and etc. In the view of image processing, a dozen of applications can be handled with HT, such as lane detection for autonomous cars, blood cell detection in microscope images, and so on. As HT is a straight forward shape detector in a given image, its shape detection ability is low in noisy images. To alleviate its weakness on noisy images and improve its …

Variational Geometric Approach To Generalized Differential And Conjugate Calculi In Convex Analysis, Boris S. Mordukhovich, Nguyen Mau Nam, R. Blake Rector, T. Tran

Mathematics and Statistics Faculty Publications and Presentations

This paper develops a geometric approach of variational analysis for the case of convex objects considered in locally convex topological spaces and also in Banach space settings. Besides deriving in this way new results of convex calculus, we present an overview of some known achievements with their unified and simplified proofs based on the developed geometric variational schemes. Key words. Convex and variational analysis, Fenchel conjugates, normals and subgradients, coderivatives, convex calculus, optimal value functions.

Solving The Rubik's Cube Using Group Theory, Courtney Cooke

Honors Projects

While he was working in his mother's apartment in 1974, the professor of architecture from Budapest, Erno Rubik, had no idea he was inventing one of the most popular toys in history, the Rubik's Cube. As an estimated 350 million Rubik's cubes have been sold, and approximately one in every seven people have played with one (which is about 1 billion people) it is not surprising to see that the algorithm of solving the Rubik's cube has been applied to the eld of mathematics. By using abstract algebra and more specially, group theory, the Rubik's Cube, no matter what the …

Tangled Up: Women’S Experiences In Mathematics, Lori Loftin

Honors College

This thesis is a bridge between two disciplines: Women's, Gender, and Sexuality Studies, and Mathematics. The first portion of the work synthesizes both theory and previously done studies to describe the state of women in mathematics as a whole, as well as historicizing the role of women in mathematics. Obstacles to the full and equal participation of women in mathematics are examined through a feminist lens. The second part of the thesis is a feminist biography crafted from an interview with a professor of mathematics, Dr. Erica Flapan. This provides information about her personal experiences as a woman in mathematics …

Classification Of Minimal Separating Sets In Low Genus Surfaces, J. J. P. Veerman, William Maxwell, Victor Rielly, Austin K. Williams

Mathematics and Statistics Faculty Publications and Presentations

Consider a surface S and let M ⊂ S. If S \ M is not connected, then we say M separates S, and we refer to M as a separating set of S. If M separates S, and no proper subset of M separates S, then we say M is a minimal separating set of S. In this paper we use computational methods of combinatorial topology to classify the minimal separating sets of the orientable surfaces of genus g = 2 and g = 3. The classification for genus 0 and 1 was done …

A Universal Genus-Two Curve From Siegel Modular Forms, Andreas Malmendier, Tony Shaska

Mathematics and Statistics Faculty Publications

Let p be any point in the moduli space of genus-two curves M₂ and K its field of moduli. We provide a universal equation of a genus-two curve C_α,β defined over K(α, β), corresponding to p, where α and β satisfy a quadratic α² + bβ² = c such that b and c are given in terms of ratios of Siegel modular forms. The curve C_α,β is defined over the field of moduli K if and only if the quadratic has a K-rational point (α, β). We discover some interesting …

Mapping Class Group Orbits Of Curves With Self-Intersections, Patricia Cahn, Federica Fanoni, Bram Petri

Mathematics Sciences: Faculty Publications

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity.

We also consider the minimal genus of a subsurface that contains the curve. We determine the asymptotic number of orbits of curves with a fixed minimal genus and a bounded self-intersection number, as the complexity of the surface tends to infinity.

As a corollary of our methods, we obtain that most curves that are homotopic are also isotopic. …

Infinite-Dimensional Measure Spaces And Frame Analysis, Palle Jorgensen, Myung-Sin Song

SIUE Faculty Research, Scholarship, and Creative Activity

We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp distinction between stochastic analysis involving frames in finite vs. infinite dimensions. For the case of infinite-dimensional Hilbert space ℋ, we study three cases of measures. We first show that, for ℋ infinite dimensional, one must resort to infinite dimensional measure spaces which properly contain ℋ. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures.

Mode-Sum Prescription For Vacuum Polarization In Black Hole Spacetimes In Even Dimensions, Peter Taylor, Cormac Doran

Articles

We present a mode-sum regularization prescription for computing the vacuum polarization of a scalar field in static spherically symmetric black hole spacetimes in even dimensions. This is the first general and systematic approach to regularized vacuum polarization in higher even dimensions, building upon a previous scheme we developed for odd dimensions. Things are more complicated here since the even-dimensional propagator possesses logarithmic singularities which must be regularized. However, in spite of this complication, the regularization parameters can be computed in closed form in arbitrary even dimensions and for arbitrary metric function f(r). As an explicit example of our method, we …

Mode-Sum Prescription For Vacuum Polarization In Black Hole Spacetimes In Even Dimensions, Peter Taylor, Cormac Breen

Articles

We present a mode-sum regularization prescription for computing the vacuum polarization of a scalar field in static spherically symmetric black hole spacetimes in even dimensions. This is the first general and systematic approach to regularized vacuum polarization in higher even dimensions, building upon a previous scheme we developed for odd dimensions. Things are more complicated here since the even-dimensional propagator possesses logarithmic singularities which must be regularized. However, in spite of this complication, the regularization parameters can be computed in closed form in arbitrary even dimensions and for arbitrary metric function f(r). As an explicit example of our method, we …

Stability Of Equilibria In Quantitative Genetic Models Based On Modified-Gradient Systems, Benjamin J. Ridenhour, Jerry R. Ridenhour

Mathematics and Statistics Faculty Publications

Motivated by questions in biology, we investigate the stability of equilibria of the dynamical system x′ = P(t)∇f(x) which arise as critical points of f, under the assumption that P(t) is positive semi-definite. It is shown that the condition ∫^∞λ₁(P(t)) dt = ∞, where λ₁(P(t)) is the smallest eigenvalue of P(t), plays a key role in guaranteeing uniform asymptotic stability and in providing information on the basis of attraction of those equilibria.

A Class Of Transformations Of A Quadratic Integral Generating Dynamical Systems, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

A class of transformation is investigated which maps a quadratic integral back to its original form but under a redefinition of free parameters. When this process is iterated, a dynamical system is generated in the form of recursive sequences which involve the parameters of the integrand.

The creation of this dynamical system and some of its convergence properties are investigated.

MR3724632

A Geometric Formulation Of Lax Integrability For Nonlinear Equationsin Two Independent Variables, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

A geometric formulation of Lax integrability is introduced which makes use of a Pfaffan formulation of Lax integrability. The Frobenius theorem gives a necessary and suffcient condition for the complete integrability of a distribution, and provides a powerful way to study nonlinear evolution equations. This permits an examination of the relation between complete integrability and Lax integrability. The prolongation method is formulated in this context and gauge transformations can be examined in terms of differential forms as well as the Frobenius theorem.

Borges And The Subjective-Idealism In Relativity Theory And Quantum Mechanics, Victor Christianto, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper is intended to be a follow-up to our previous paper with title: "Reinterpreting Tlon, Uqbar, Orbis Tertius: On the antirealism tendency in modern physics." We will give more background for our propositions in the previous paper. Our message here is quite simple: allow us to remind fellow physicists and cosmologists to become more aware of Berkeley-idealism tendency, which can lead us to so many distractions instead of bringing us closer to the truth. We observe that much of the progress of modern physics in the last few decades only makes us as confused as before, but at a …

The Classification Of Infinite Abelian Groups With Partial Decomposition Bases In L∞Ω, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

We consider the class of abelian groups with partial decomposition bases, which includes groups classified by Ulm, Warfield, Stanton and others. We define an invariant and classify these groups in the language L∞ω, or equivalently, up to partial isomorphism. This generalizes a result of Barwise and Eklof and builds on Jacoby's classification of local groups with partial decomposition bases in L∞ω.

Combustion-Derived Nanoparticles, The Neuroenteric System, Cervical Vagus, Hyperphosphorylated Alpha Synuclein And Tau In Young Mexico City Residents, Lilian Calderón-Garcidueñas, Rafael Reynoso-Robles, Beatriz Pérez-Guillé, Partha S. Mukherjee, Angélica Gónzalez-Maciel

Mathematics Faculty Publications and Presentations

Mexico City (MC) young residents are exposed to high levels of fine particulate matter (PM_2.5), have high frontal concentrations of combustion-derived nanoparticles (CDNPs), accumulation of hyperphosphorylated aggregated α-synuclein (α-Syn) and early Parkinson's disease (PD). Swallowed CDNPs have easy access to epithelium and submucosa, damaging gastrointestinal (GI) barrier integrity and accessing the enteric nervous system (ENS). This study is focused on the ENS, vagus nerves and GI barrier in young MC v clean air controls. Electron microscopy of epithelial, endothelial and neural cells and immunoreactivity of stomach and vagus to phosphorylated ɑ-synuclein Ser129 and Hyperphosphorylated-Tau (Htau) …