Mathematics Commons^™

On ΤΣ-Quasinormal Subgroups Of Finite Groups, James C. Beidleman, Alexander N. Skiba

Mathematics Faculty Publications

Let σ = {σ_i ∣ i ∈ I} be a partition of the set of all primes P and G a finite group. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σ_i-subgroup of G for some i ∈ I and H contains exactly one Hall σ_i-subgroup of G for every i such that σ_i ∩ π(G) ≠ ∅.

Let τ_H(A) = {σ_i ∈ σ …

Blow-Up Algebras, Determinantal Ideals, And Dedekind-Mertens-Like Formulas, Alberto Corso, Uwe Nagel, Sonja Petrović, Cornelia Yuen

Mathematics Faculty Publications

We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen–Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine …

Simulating Within-Vector Generation Of The Malaria Parasite Diversity, Lauren M. Childs, Olivia F. Prosper

Mathematics Faculty Publications

Plasmodium falciparum, the most virulent human malaria parasite, undergoes asexual reproduction within the human host, but reproduces sexually within its vector host, the Anopheles mosquito. Consequently, the mosquito stage of the parasite life cycle provides an opportunity to create genetically novel parasites in multiply-infected mosquitoes, potentially increasing parasite population diversity. Despite the important implications for disease transmission and malaria control, a quantitative mapping of how parasite diversity entering a mosquito relates to diversity of the parasite exiting, has not been undertaken. To examine the role that vector biology plays in modulating parasite diversity, we develop a two-part model framework …

Optical Tomography On Graphs, Francis J. Chung, Anna C. Gilbert, Jeremy G. Hoskins, John C. Schotland

Mathematics Faculty Publications

We present an algorithm for solving inverse problems on graphs analogous to those arising in diffuse optical tomography for continuous media. In particular, we formulate and analyze a discrete version of the inverse Born series, proving estimates characterizing the domain of convergence, approximation errors, and stability of our approach. We also present a modification which allows additional information on the structure of the potential to be incorporated, facilitating recovery for a broader class of problems.

Applications Of The Heine And Bauer-Muir Transformations To Rogers-Ramanujan Type Continued Fractions, Jongsil Lee, James Mclaughlin, Jaebum Sohn

Mathematics Faculty Publications

In this paper we show that various continued fractions for the quotient of general Ramanujan functions G(aq, b, λq)/G(a, b, λ) may be derived from each other via Bauer-Muir transformations. The separate convergence of numerators and denominators play a key part in showing that the continued fractions and their Bauer-Muir transformations converge to the same limit. We also show that these continued fractions may be derived from either Heine’s continued fraction for a ratio of 2φ1 functions, or other similar continued fraction expansions of ratios of 2φ1 functions. Further, by employing essentially the same methods, a new continued fraction for …

Classification Of Book Representations Of K6, Dana Rowland

Mathematics Faculty Publications

A book representation of a graph is a particular way of embedding a graph in three dimensional space so that the vertices lie on a circle and the edges are chords on disjoint topological disks. We describe a set of operations on book representations that preserves ambient isotopy, and apply these operations to K6, the complete graph with six vertices. We prove there are exactly 59 distinct book representations for K6, and we identify the number and type of knotted and linked cycles in each representation. We show that book representations of K6 contain between one and seven links, and …

Convergence Rates In Periodic Homogenization Of Systems Of Elasticity, Zhongwei Shen, Jinping Zhuge

Mathematics Faculty Publications

This paper is concerned with homogenization of systems of linear elasticity with rapidly oscillating periodic coefficients. We establish sharp convergence rates in L² for the mixed boundary value problems with bounded measurable coefficients.

Q-Stirling Identities Revisited, Yue Cai, Richard Ehrenborg, Margaret Readdy

Mathematics Faculty Publications

We give combinatorial proofs of q-Stirling identities using restricted growth words. This includes a poset theoretic proof of Carlitz's identity, a new proof of the q-Frobenius identity of Garsia and Remmel and of Ehrenborg's Hankel q-Stirling determinantal identity. We also develop a two parameter generalization to unify identities of Mercier and include a symmetric function version.

Signed Lozenge Tilings, D. Cook Ii, Uwe Nagel

Mathematics Faculty Publications

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a …

Removable Singularities In C*-Algebras Of Real Rank Zero, Lawrence A. Harris

Mathematics Faculty Publications

Let 𝕬 be a C*-algebra with identity and real rank zero. Suppose a complex- valued function is holomorphic and bounded on the intersection of the open unit ball of 𝕬 and the identity component of the set of invertible elements of 𝕬. We give a short transparent proof that the function has a holomorphic extension to the entire open unit ball of 𝕬. The author previously deduced this from a more general fact about Banach algebras.

Soil-Transmitted Helminthiasis And Schistosomiasis In Children Of Poor Families In Leyte, Philippines: Lessons For Disease Prevention And Control, Harvy Joy Liwanag, Jhanna Uy, Janis Ruth Gatchalian, Betty De La Calzada, Justine Alessandra Uy, Manuel Dayrit, Ramil T. Bataller

Mathematics Faculty Publications

Objective

Neglected tropical diseases (NTDs) continue to be a public health problem in the Philippines. We assessed the association of soil-transmitted helminthiasis (STH) and schistosomiasis with selected health-related and socioeconomic variables in four villages in Leyte, Philippines.

Methods

Stool specimens from 418 adults and 533 of their children from 209 families were examined through the Kato-Katz technique.

Results

STH and schistosomiasis were present in 64.6% and 12.5%, respectively, of study participants. Analysis through the generalized linear mixed model revealed a number of associations between infection in parents and their children. Findings indicate that years of disease prevention and control efforts …

Some Results On Almost Ricci Solitons And Geodesic Vector Fields., Ramesh Sharma

Mathematics Faculty Publications

We show that a compact almost Ricci soliton whose soliton vector field is divergence-free is Einstein and its soliton vector field is Killing. Next we show that an almost Ricci soliton reduces to Ricci soliton if and only if the associated vector field is geodesic. Finally, we prove that a contact metric manifold is K-contact if and only if its Reeb vector field is geodesic.

Mock Theta Function Identities Deriving From Bilateral Basic Hypergeometric Series, James Mclaughlin

Mathematics Faculty Publications

The bilateral series corresponding to many of the third-, fifth-, sixth- and eighth order mock theta functions may be derived as special cases of 2ψ2 series ∞ ∑n=−∞ (a, c;q)n (b,d;q)n z n . Three transformation formulae for this series due to Bailey are used to derive various transformation and summation formulae for both these mock theta functions and the corresponding bilateral series. New and existing summation formulae for these bilateral series are also used to make explicit in a number of cases the fact that for a mock theta function, say χ(q), and a root of unity in a …

An Optimal A Posteriori Error Estimates Of The Local Discontinuous Galerkin Method For The Second-Order Wave Equation In One Space Dimension, Mahboub Baccouch

Mathematics Faculty Publications

In this paper, we provide the optimal convergence rate of a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the second-order wave equation in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [W. Cao, D. Li and Z. Zhang, Commun. Comput. Phys. 21 (1) (2017) 211-236]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L 2 -norm to (p + 1)-degree right and left Radau interpolating polynomials under mesh refinement. The order of convergence is proved to be p + …

A Flexible Distribution Class For Count Data, Kimberly F. Sellers, Andrew W. Swift, Kimberly S. Weems

Mathematics Faculty Publications

The Poisson, geometric and Bernoulli distributions are special cases of a flexible count distribution, namely the Conway-Maxwell-Poisson (CMP) distribution – a two-parameter generalization of the Poisson distribution that can accommodate data over- or under-dispersion. This work further generalizes the ideas of the CMP distribution by considering sums of CMP random variables to establish a flexible class of distributions that encompasses the Poisson, negative binomial, and binomial distributions as special cases. This sum-of-Conway-Maxwell-Poissons (sCMP) class captures the CMP and its special cases, as well as the classical negative binomial and binomial distributions. Through simulated and real data examples, we demonstrate this …

Nonminimal Cyclic Invariant Subspaces Of Hyperbolic Composition Operators, Valentin Matache

Mathematics Faculty Publications

Operators on function spaces acting by composition to the right with a fixed self-map ϕ are called composition operators. We denote them Cϕ. Given ϕ, a hyperbolic disc automorphism, the composition operator Cϕ on the Hilbert Hardy space H2 is considered. The bilateral cyclic invariant subspaces Kf, f ∈ H2, of Cϕ are studied, given their connection with the invariant subspace problem, which is still open for Hilbert space operators. We prove that nonconstant inner functions u induce non–minimal cyclic subspaces Ku if they have unimodular, orbital, cluster points. Other results about Ku when u is inner are obtained. If …

Positive Solutions Of Boundary Value Problems For Ordinary Differential Equations With Dependence On Higher Order Derivatives, Ahlam Abid, Paul Eloe

Mathematics Faculty Publications

In this paper, the compression contraction fixed point theorem is applied to a right focal boundary value problem for a second order ordinary differential equation to provide sufficient conditions for existence of solutions in a cone. The nonlinear term is a function of three variables and singularities are allowed. A cone is defined in the Banach space C1[0, 1] and concavity of derivatives of solutions plays a key role. A family of examples is provided in which explicit sufficient conditions are exhibited.

Construction Of Weavings In The Plane, Eden Delight Miro, Aliw-Iw Zambrano, Agnes Garciano

Mathematics Faculty Publications

This work develops, in graph-theoretic terms, a methodology for systematically constructing weavings of overlapping nets derived from 2-colorings of the plane. From a 2-coloring, two disjoint simple, connected graphs called nets are constructed. The union of these nets forms an overlapping net, and a weaving map is defined on the intersection points of the overlapping net to form a weaving. Furthermore, a procedure is given for the construction of mixed overlapping nets and for deriving weavings from them.

Discovering New Tessellations Using Dynamic Geometry Software, Ma. Louise Antonette N. De Las Peñas, Eduard C. Taganap

Mathematics Faculty Publications

In this paper we use dynamic geometry software to investigate a class of tilings called k-uniform tilings or tessellations. A tiling consisting of regular polygons whose vertices belong to k-transitivity classes under the action of its symmetry group (vertex-k-transitive) is said to be k-uniform. We also present constructions of tilings consisting of irregular polygons that are vertex-k-transitive.

Going For Gold In The International Math Olympiad, Queena N. Lee-Chua

Mathematics Faculty Publications

In July 2016, two high school students in the Philippine team garnered gold at the 57th International Mathematics Olympiad (IMO) in Hong Kong, capping a three-decade long quest for the top prize in the most prestigious high school math competition in the world. The four other team members also brought home honors, boosting our country to its highest rank ever, 17th out of 109 countries. This article discusses the history of the Philippine participation in the IMO, and examines the critical factors that have led to the victory. For Philippine team participants in general, these include: institutionalized and refined search …

Density-Dependent Leslie Matrix Modeling For Logistic Populations With Steady-State Distribution Control, Bruce Kessler, Andrew Davis

Mathematics Faculty Publications

The Leslie matrix model allows for the discrete modeling of population age-groups whose total population grows exponentially. Many attempts have been made to adapt this model to a logistic model with a carrying capacity (see [1], [2], [4], [5], and [6]), with mixed results. In this paper we provide a new model for logistic populations that tracks age-group populations with repeated multiplication of a density-dependent matrix constructed from an original Leslie matrix, the chosen carrying capacity of the model, and the desired steady-state age-group distribution. The total populations from the model converge to a discrete logistic model with the same …

Integration Over Curves And Surfaces Defined By The Closest Point Mapping, Catherine Kublik, Richard Tsai

Mathematics Faculty Publications

We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wishes to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of …

Effects Of Cell Cycle Noise On Excitable Gene Circuits, Alan Veliz-Cuba, Chinmaya Gupta, Matthew R. Bennett, Krešimir Josić, William Ott

Mathematics Faculty Publications

We assess the impact of cell cycle noise on gene circuit dynamics. For bistable genetic switches and excitable circuits, we find that transitions between metastable states most likely occur just after cell division and that this concentration effect intensifies in the presence of transcriptional delay. We explain this concentration effect with a three-states stochastic model. For genetic oscillators, we quantify the temporal correlations between daughter cells induced by cell division. Temporal correlations must be captured properly in order to accurately quantify noise sources within gene networks.

Convolutions And Green’S Functions For Two Families Of Boundary Value Problems For Fractional Differential Equations, Paul W. Eloe, Jeffrey T. Neugebauer

Mathematics Faculty Publications

We consider families of two-point boundary value problems for fractional differential equations where the fractional derivative is assumed to be the Riemann-Liouville fractional derivative. The problems considered are such that appropriate differential operators commute and the problems can be constructed as nested boundary value problems for lower order fractional differential equations. Green's functions are then constructed as convolutions of lower order Green's functions. Comparison theorems are known for the Green's functions for the lower order problems and so, we obtain analogous comparison theorems for the two families of higher order equations considered here. We also pose a related open question …

Estimating Propensity Parameters Using Google Pagerank And Genetic Algorithms, David Murrugarra, Jacob Miller, Alex N. Mueller

Mathematics Faculty Publications

Stochastic Boolean networks, or more generally, stochastic discrete networks, are an important class of computational models for molecular interaction networks. The stochasticity stems from the updating schedule. Standard updating schedules include the synchronous update, where all the nodes are updated at the same time, and the asynchronous update where a random node is updated at each time step. The former produces a deterministic dynamics while the latter a stochastic dynamics. A more general stochastic setting considers propensity parameters for updating each node. Stochastic Discrete Dynamical Systems (SDDS) are a modeling framework that considers two propensity parameters for updating each node …

The Classification Of Zp -Modules With Partial Decomposition Bases In L∞Ω, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

Ulm’s Theorem presents invariants that classify countable abelian torsion groups up to isomorphism. Barwise and Eklof extended this result to the classification of arbitrary abelian torsion groups up to L∞ω-equivalence. In this paper, we extend this classification to a class of mixed Zp-modules which includes all Warfield modules and is closed under L∞ω-equivalence. The defining property of these modules is the existence of what we call a partial decomposition basis, a generalization of the concept of decomposition basis. We prove a complete classification theorem in L∞ω using invariants deduced from the classical Ulm and Warfield invariants.

Identification Of Control Targets In Boolean Molecular Network Models Via Computational Algebra, David Murrugarra, Alan Veliz-Cuba, Boris Aguilar, Reinhard Laubenbacher

Mathematics Faculty Publications

Background: Many problems in biomedicine and other areas of the life sciences can be characterized as control problems, with the goal of finding strategies to change a disease or otherwise undesirable state of a biological system into another, more desirable, state through an intervention, such as a drug or other therapeutic treatment. The identification of such strategies is typically based on a mathematical model of the process to be altered through targeted control inputs. This paper focuses on processes at the molecular level that determine the state of an individual cell, involving signaling or gene regulation. The mathematical model type …

Identification Of Control Targets In Boolean Molecular Network Models Via Computational Algebra, David Murrugarra, Alan Veliz-Cuba, Boris Aguilar, Reinhard Laubenbacher

Mathematics Faculty Publications

Many problems in biomedicine and other areas of the life sciences can be characterized as control problems, with the goal of finding strategies to change a disease or otherwise undesirable state of a biological system into another, more desirable, state through an intervention, such as a drug or other therapeutic treatment. The identification of such strategies is typically based on a mathematical model of the process to be altered through targeted control inputs. This paper focuses on processes at the molecular level that determine the state of an individual cell, involving signaling or gene regulation. The mathematical model type considered …

Theorems On Boundedness Of Solutions To Stochastic Delay Differential Equations, Youssef Raffoul, Dan Ren

Mathematics Faculty Publications

In this report, we provide general theorems about boundedness or bounded in probability of solutions to nonlinear delay stochastic differential systems. Our analysis is based on the successful construction of suitable Lyapunov functionals. We offer several examples as application of our theorems.

Logical Reduction Of Biological Networks To Their Most Determinative Components, Mihaela Teodora Matache, Valentin Matache

Mathematics Faculty Publications

Boolean networks have been widely used as models for gene regulatory networks, signal transduction networks, or neural networks, among many others. One of the main difficulties in analyzing the dynamics of a Boolean network and its sensitivity to perturbations or mutations is the fact that it grows exponentially with the number of nodes. Therefore, various approaches for simplifying the computations and reducing the network to a subset of relevant nodes have been proposed in the past few years. We consider a recently introduced method for reducing a Boolean network to its most determinative nodes that yield the highest information gain. …