Physical Sciences and Mathematics Commons^™

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 …

The Corona Problem For Kernel Multiplier Algebras, Eric T. Sawyer, Brett D. Wick

Mathematics Faculty Publications

We prove an alternate Toeplitz corona theorem for the algebras of pointwise kernel multipliers of Besov-Sobolev spaces on the unit ball in C, and for the algebra of bounded analytic functions on certain strictly pseudoconvex domains and polydiscs in higher dimensions as well. This alternate Toeplitz corona theorem extends to more general Hilbert function spaces where it does not require the complete Pick property. Instead, the kernel functions kx (y) of certain Hilbert function spaces H are assumed to be invertible multipliers on H and then we continue a research thread begun by Agler and McCarthy in 1999, and continued …

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 …

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 …

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.

Non-Commutative Automorphisms Of Bounded Non-Commutative Domains, John E. Mccarthy, Richard M. Timoney

Mathematics Faculty Publications

We establish rigidity (or uniqueness) theorems for non-commutative (NC) automorphisms that are natural extensions of classical results of H. Cartan and are improvements of recent results. We apply our results to NC domains consisting of unit balls of rectangular matrices.

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 …

Determinantal Representations Of Semihyperbolic Polynomials, Greg Knese

Mathematics Faculty Publications

We prove a generalization of the Hermitian version of the Helton–Vinnikov determinantal representation for hyperbolic polynomials to the class of semihyperbolic polynomials, a strictly larger class, as shown by an example. We also prove that certain hyperbolic polynomials affine in two out of four variables divide a determinantal polynomial. The proofs are based on work related to polynomials with no zeros on the bidisk and tridisk.

On Approximately Controlled Systems, Nazim I. Mahmudov, Mark A. Mckibben

Mathematics Faculty Publications

No abstract provided.

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. …

Optimal Control Analysis Of Ebola Disease With Control Strategies Of Quarantine And Vaccination, Muhammad Dure Ahmad, Muhammad Usman, Adnan Khan, Mudassar Imran

Mathematics Faculty Publications

The 2014 Ebola epidemic is the largest in history, affecting multiple countries in West Africa. Some isolated cases were also observed in other regions of the world.

Impartial Avoidance And Achievement Games For Generating Symmetric And Alternating Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben

Mathematics Faculty Publications

Anderson and Harary introduced two impartial games on finite groups. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. We determine the nim-numbers, and therefore the outcomes, of these games for symmetric and alternating groups.

Model For Computing Kinetics Of The Graphene Edge Epitaxial Growth On Copper, Mikhail Khenner

Mathematics Faculty Publications

A basic kinetic model that incorporates a coupled dynamics of the carbon atoms and dimers ona copper surface is used to compute growth of a single-layer graphene island. The speed of theisland's edge advancement on Cu[111] and Cu[100] surfaces is computed as a function of the growthtemperature and pressure. Spatially resolved concentration pro les of the atoms and dimers aredetermined, and the contributions provided by these species to the growth speed are discussed.Island growth in the conditions of a thermal cycling is studied.

Upper And Lower Solution Method For Boundary Value Problems At Resonance, Samerah Al Mosa, Paul W. Eloe

Mathematics Faculty Publications

We consider two simple boundary value problems at resonance for an ordinary differential equation. Employing a shift argument, a regular fixed point operator is constructed. We employ the monotone method coupled with a method of upper and lower solutions and obtain sufficient conditions for the existence of solutions of boundary value problems at resonance for nonlinear boundary value problems. Three applications are presented in which explicit upper solutions and lower solutions are exhibited for the first boundary value problem. Two applications are presented for the second boundary value problem. Of interest, the upper and lower solutions are easily and explicitly …

Comparing Local Constants Of Ordinary Elliptic Curves In Dihedral Extensions, Sunil Chetty

Mathematics Faculty Publications

We establish, for a substantial class of elliptic curves, that the arithmetic local constants introduced by Mazur and Rubin agree with quotients of analytic root numbers.

A Remark On The Multipliers On Spaces Of Weak Products Of Functions, Stefan Richter, Brett D. Wick

Mathematics Faculty Publications

Operators on function spaces of form Cɸf = f ∘ ɸ, where ɸ is a fixed map are called composition operators with symbol ɸ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.

Tau And Aβ Imaging, Csf Measures, And Cognition In Alzheimer's Disease, Matthew R. Brier, Brian Gordon, Karl Friedrichsen, John E. Mccarthy, Ari Stern, Jon Christensen, Christopher Owen, Patricia Aldea, Yi Su, Jason Hassenstab, Nigel J. Cairns, David M. Holtzman, Anne M. Fagan, John C. Morris, Tammie L.S. Benzinger, Beau M. Ances

Mathematics Faculty Publications

Alzheimer’s disease (AD) is characterized by two molecular pathologies: cerebral β-amyloidosis in the form of β-amyloid (Aβ) plaques and tauopathy in the form of neurofibrillary tangles, neuritic plaques, and neuropil threads. Until recently, only Aβ could be studied in humans using positron emission tomography (PET) imaging owing to a lack of tau PET imaging agents. Clinical pathological studies have linked tau pathology closely to the onset and progression of cognitive symptoms in patients with AD. We report PET imaging of tau and Aβ in a cohort of cognitively normal older adults and those with mild AD. Multivariate analyses identified unique …

On The Dimension Of Algebraic-Geometric Trace Codes, Phong Le, Sunil Chetty

Mathematics Faculty Publications

We study trace codes induced from codes defined by an algebraic curve X. We determine conditions on X which admit a formula for the dimension of such a trace code. Central to our work are several dimension reducing methods for the underlying functions spaces associated to X.

Uniform Stability In Nonlinear Infinite Delay Volterra Integro-Differential Equations Using Lyapunov Functionals, Youssef Raffoul, Habib Rai

Mathematics Faculty Publications

In [10] the first author used Lyapunov functionals and studied the exponential stability of the zero solution of nite delay Volterra Integro-dierential equation. In this paper, we use modified version of the Lyapunov functional that were used in [10] to obtain criterion for the stability of the zero solution of the infinite delay nonlinear Volterra integro-dierential equation

x′(t) = Px(t) + ^t∫ _−∞ C(t, s)g(x(s))ds.

General Existence Results For Abstract Mckean-Vlasov Stochastic Equations With Variable Delay, Mark A. Mckibben

Mathematics Faculty Publications

Results concerning the global existence and uniqueness of mild solutions for a class of first-order abstract stochastic integro-differential equations with variable delay in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time t, but also on the corresponding probability distribution at time t are established. The classical Lipschitz is replaced by a weaker so-called Caratheodory condition under which we still maintain uniqueness. The time-dependent case is discussed, as well as an extension of the theory to the case of a nonlocal …

Polynomials With No Zeros On A Face Of The Bidisk, Jeffrey S. Geronimo, Plamen Iliev, Greg Knese

Mathematics Faculty Publications

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement, namely no zeros on a face of the bidisk. Two different characterizations are given using a Hilbert space structure naturally associated to the trigonometric polynomial; one is in terms of a certain orthogonal decomposition the Hilbert space must possess called the “split-shift orthogonality condition” and another is an operator theoretic or matrix condition closely related to an earlier characterization due to the first two authors. This …

Cosmological Models Through Spatial Ricci Flow, Ramesh Sharma

Mathematics Faculty Publications

We consider the synchronization of the Einstein’s flow with the Ricci-flow of the standard spatial slices of the Robertson–Walker space–time and show that associated perfect fluid solution has a quadratic equation of state and is either spherical and collapsing, or hyperbolic and expanding.
Read More: http://www.worldscientific.com/doi/abs/10.1142/S0219887816500699

The Implicit Function Theorem And Free Algebraic Sets, Jim Agler, John E. Mccarthy

Mathematics Faculty Publications

We prove an implicit function theorem for non-commutative functions. We use this to show that if p ( X;Y ) is a generic non-commuting polynomial in two variables, and X is a generic matrix, then all solutions Y of p ( X;Y ) = 0 will commute with X

Acoustic Firearm Discharge Detection And Classification In An Enclosed Environment, Lorenzo Luzi, Eric Gonzalez, Paul Bruillard, Matthew Prowant, James Skorpik, Michael Hughes, Scott Child, Duane Kist, John E. Mccarthy

Mathematics Faculty Publications

Two different signal processing algorithms are described for detection and classification of acoustic signals generated by firearm discharges in small enclosed spaces. The first is based on the logarithm of the signal energy. The second is a joint entropy. The current study indicates that a system using both signal energy and joint entropy would be able to both detect weapon discharges and classify weapon type, in small spaces, with high statistical certainty.

Aspects Of Non-Commutative Function Theory, Jim Agler, John E. Mccarthy

Mathematics Faculty Publications

We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.