Physical Sciences and Mathematics Commons^™

Local And Distributed Pib Accumulation Associated With Development Of Preclinical Alzheimer's Disease, Matthew R. Brier, John E. Mccarthy, Tammie L.S. Benzinger, Ari Stern, Yi Su, Karl A. Friedrichsen, John C. Morris, Beau M. Ances, Andrei G. Vlassenko

Mathematics Faculty Publications

Amyloid-beta plaques are a hallmark of Alzheimer's disease (AD) that can be assessed by amyloid imaging (e.g., Pittsburgh B compound [PiB]) and summarized as a scalar value. Summary values may have clinical utility but are an average over many regions of interest, potentially obscuring important topography. This study investigates the longitudinal evolution of amyloid topographies in cognitively normal older adults who had normal (N = 131) or abnormal (N = 26) PiB scans at baseline. At 3 years follow-up, 16 participants with a previously normal PiB scan had conversion to PiB scans consistent with preclinical AD. We investigated the multivariate …

Entropy Vs. Energy Waveform Processing: A Comparison Based On The Heat Equation, Michael S. Hughes, John E. Mccarthy, Paul J. Bruillard, Jon N. Marsh, Samuel A. Wickline

Mathematics Faculty Publications

Virtually all modern imaging devices collect electromagnetic or acoustic waves and use the energy carried by these waves to determine pixel values to create what is basically an “energy” picture. However, waves also carry “information,” as quantified by some form of entropy, and this may also be used to produce an “information” image. Numerous published studies have demonstrated the advantages of entropy, or “information imaging”, over conventional methods. The most sensitive information measure appears to be the joint entropy of the collected wave and a reference signal. The sensitivity of repeated experimental observations of a slowly-changing quantity may be defined …

Sobriety In Delta Not Sober, Joe Mashburn

Mathematics Faculty Publications

We will show that the space delta not sober defined by Coecke and Martin is sober in the Scott topology, but not in the weakly way below topology.

Qualitative Theory Of Functional Differential And Integral Equations, Muhammad Islam, Cemil Tunc, Mouffak Benchohra, Bingwen Lui, Samir H. Saker

Mathematics Faculty Publications

Functional differential equations arise in many areas of science and technology: whenever a deterministic relationship involving some varying quantities and their rates of change in space and/or time (expressed as derivatives or differences) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies. In some cases, this differential equation (called an equation of motion) may be solved explicitly. In fact, differential equations play an important role in modelling virtually every physical, technical, biological, ecological, and epidemiological process, from celestial motion, to bridge design, …

Bounded, Asymptotically Stable, And L^1 Solutions Of Caputo Fractional Differential Equations, Muhammad Islam

Mathematics Faculty Publications

The existence of bounded solutions, asymptotically stable solutions, and L¹ solutions of a Caputo fractional differential equation has been studied in this paper. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional differential equation. The kernel function of this integral equation is weakly singular and hence the standard techniques that are normally applied on Volterra integral equations do not apply here. This hurdle is overcomed using a resolvent equation and then applying some known properties of the resolvent. In the analysis Schauder's fixed point theorem and Liapunov's method have been employed. …

High-Order Short-Time Expansions For Atm Option Prices Of Exponential Lévy Models, José E. Figueroa-López, Ruoting Gong, Christian Houdré

Mathematics Faculty Publications

The short-time asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In this work, a novel second-order approximation for at-the-money (ATM) option prices is derived for a large class of exponential Lévy models with or without Brownian component. The results hereafter shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In the presence of a Brownian component, the second-order term, in time-t, is of the form , with d 2 only depending …

A Generalization Of Poincaré-Cartan Integral Invariants Of A Nonlinear Nonholonomic Dynamical System, Muhammad Usman, M. Imran

Mathematics Faculty Publications

Based on the d'Alembert-Lagrange-Poincar\'{e} variational principle, we formulate general equations of motion for mechanical systems subject to nonlinear nonholonomic constraints, that do not involve Lagrangian undetermined multipliers. We write these equations in a canonical form called the Poincar\'{e}-Hamilton equations, and study a version of corresponding Poincar\'{e}-Cartan integral invariant which are derived by means of a type of asynchronous variation of the Poincar\'{e} variables of the problem that involve the variation of the time. As a consequence, it is shown that the invariance of a certain line integral under the motion of a mechanical system of the type considered characterizes the …

Abstract Functional Stochastic Evolution Equations Driven By Fractional Brownian Motion, Mark A. Mckibben, Micah Webster

Mathematics Faculty Publications

We investigate a class of abstract functional stochastic evolution equations driven by a fractional Brownianmotion in a real separable Hilbert space.Global existence results concerningmild solutions are formulated under various growth and compactness conditions. Continuous dependence estimates and convergence results are also established. Analysis of three stochastic partial differential equations, including a second-order stochastic evolution equation arising in the modeling of wave phenomena and a nonlinear diffusion equation, is provided to illustrate the applicability of the general theory.

Lyapunov Functionals That Lead To Exponential Stability And Instability In Finite Delay Volterra Difference Equations, Catherine Kublik, Youssef Raffoul

Mathematics Faculty Publications

We use Lyapunov functionals to obtain sufficient conditions that guarantee exponential stability of the zero solution of the finite delay Volterra difference equation.

Also, by displaying a slightly different Lyapunov functional, we obtain conditions that guarantee the instability of the zero solution. The highlight of the paper is the relaxing of the condition |a(t)| < 1. Moreover, we provide examples in which we show that our theorems provide an improvement of some recent results.

Leslie Matrices For Logistic Population Modeling, Bruce Kessler

Mathematics Faculty Publications

Leslie matrices are taught as a method of modeling populations in a discrete-time fashion with more detail in the tracking of age groups within the population. Leslie matrices have limited use in the actual modeling of populations, since when the age groups are summed, it is basically equivalent to discrete-time modeling assuming exponential population growth. The logistic model of population growth is more realistic, since it takes into account a carrying capacity for the environment of the population. This talk will describe an adjustment to the Leslie matrix approach for population modeling that is both takes into account the carrying …

Peaklet Analysis: Software For Spectrum Analysis, Bruce Kessler

Mathematics Faculty Publications

This is the presentation I was invited to give at the Kentucky Innovation and Entrepreneurship Conference, regarding the software that I have developed and worked at commercializing with the help of Kentucky Science and Technology Corporation.

Long-Wave Model For Strongly Anisotropic Growth Of A Crystal Step, Mikhail Khenner

Mathematics Faculty Publications

A continuum model for the dynamics of a single step with the strongly anisotropic line energy is formulated and analyzed. The step grows by attachment of adatoms from the lower terrace, onto which atoms adsorb from a vapor phase or from a molecular beam, and the desorption is nonnegligible (the “one-sided” model). Via a multiscale expansion, we derived a long-wave, strongly nonlinear, and strongly anisotropic evolution PDE for the step profile. Written in terms of the step slope, the PDE can be represented in a form similar to a convective Cahn-Hilliard equation. We performed the linear stability analysis and computed …

Every Scattered Space Is Subcompact, William Fleissner, Vladimir Tkachuk, Lynne Yengulalp

Mathematics Faculty Publications

We prove that every scattered space is hereditarily subcompact and any finite union of subcompact spaces is subcompact. It is a long-standing open problem whether every Čech-complete space is subcompact. Moreover, it is not even known whether the complement of every countable subset of a compact space is subcompact. We prove that this is the case for linearly ordered compact spaces as well as for ω -monolithic compact spaces. We also establish a general result for Tychonoff products of discrete spaces which implies that dense G_δ-subsets of Cantor cubes are subcompact.

A Spectral Order For Infinite Dimensional Quantum Spaces, Joe Mashburn

Mathematics Faculty Publications

In this paper we extend the spectral order of Coecke and Martin to infinite-dimensional quantum states. Many properties present in the finite-dimensional case are preserved, but some of the most important are lost. The order is constructed and its properties analysed. Most of the useful measurements of information content are lost. Shannon entropy is defined on only a part of the model, and that part is not a closed subset of the model. The finite parts of the lattices used by Birkhoff and von Neumann as models for classical and quantum logic appear as subsets of the models for infinite …

Iterative Scheme For Solving Optimal Transportation Problems Arising In Reflector Design, Tilmann Glimm, Nick Henscheid

Mathematics Faculty Publications

We consider the geometric optics problem of finding a system of two reflectors that transform a spherical wavefront into a beam of parallel rays with prescribed intensity distribution. Using techniques from optimal transportation theory, it has been shown previously that this problem is equivalent to an infinite-dimensional linear programming (LP) problem. Here we investigate techniques for constructing the two reflectors numerically by considering the finite dimensional LP problems which arise as approximations to the infinite dimensional problem. A straightforward discretization has the disadvantage that the number of constraints increases rapidly with the mesh size, so only very coarse meshes are …

The Effects Of Variable Viscosity On The Peristaltic Flow Of Non-Newtonian Fluid Through A Porous Medium In An Inclined Channel With Slip Boundary Conditions, Ambreen Afsar Khan, R. Ellahi, Muhammad Usman

Mathematics Faculty Publications

The present paper investigates the peristaltic motion of an incompressible non-Newtonian fluid with variable viscosity through a porous medium in an inclined symmetric channel under the effect of the slip condition. A long wavelength approximation is used in mathematical modeling. The system of the governing nonlinear partial differential equation has been solved by using the regular perturbation method and the analytical solutions for velocity and pressure rise have been obtained in the form of stream function. In the obtained solution expressions, the long wavelength and low Reynolds number assumptions are utilized. The salient features of pumping and trapping phenomena are …

When Cp(X) Is Domain Representable, William Fleissner, Lynne Yengulalp

Mathematics Faculty Publications

Let M be a metrizable group. Let G be a dense subgroup of M^X . If G is domain representable, then G = MX . The following corollaries answer open questions. If X is completely regular and C_p(X) is domain representable, then X is discrete. If X is zero-dimensional, T₂ , and C_p(X;D) is subcompact, then X is discrete.

Mathematical Modelling Of Internal Heat Recovery In Flash Tank Heat Exchanger Cascades, Andrei Korobeinikov, John E. Mccarthy, Emma Mooney, Krum Semkov, James Varghese

Mathematics Faculty Publications

Flash tank evaporation combined with a condensing heat exchanger can be used when heat exchange is required between two streams and where at least one of these streams is difficult to handle (tends severely to scale, foul, causing blockages). To increase the efficiency of heat exchange, a cascade of these units in series can be used. Heat transfer relationships in such a cascade are very complex due to their interconnectivity, thus the impact of any changes proposed is difficult to predict. Moreover, the distribution of loads and driving forces in different stages and the number of designed stages faces tradeoffs …

Measure-Dependent Stochastic Nonlinear Beam Equations Driven By Fractional Brownian Motion, Mark A. Mckibben

Mathematics Faculty Publications

We study a class of nonlinear stochastic partial differential equations arising in themathematicalmodeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separableHilbert space and is studied using the tools …

An Implicit Interface Boundary Integral Method For Poisson’S Equation On Arbitrary Domains, Catherine Kublik, Nicolay M. Tanushev, Richard Tsai

Mathematics Faculty Publications

We propose a simple formulation for constructing boundary integral methods to solve Poisson’s equation on domains with smooth boundaries defined through their signed distance function. Our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain, where the integrations are carried out in the level set framework using an appropriate Jacobian. By the coarea formula, the algorithm operates in the Euclidean space and does not require any explicit parameterization of the boundaries. We present numerical results in two and three dimensions.

Analytic And Finite Element Solutions Of The Power-Law Euler-Bernoulli Beams, Dongming Wei, Yu Liu

Mathematics Faculty Publications

In this paper, we use Hermite cubic finite elements to approximate the solutions

of a nonlinear Euler-Bernoulli beam equation. The equation is derived

from Hollomon’s generalized Hooke’s law for work hardening materials with

the assumptions of the Euler-Bernoulli beam theory. The Ritz-Galerkin finite

element procedure is used to form a finite dimensional nonlinear program

problem, and a nonlinear conjugate gradient scheme is implemented to find

the minimizer of the Lagrangian. Convergence of the finite element approximations

is analyzed and some error estimates are presented. A Matlab finite

element code is developed to provide numerical solutions to the beam equation.

Some …

Controlling Nanoparticles Formation In Molten Metallic Bilayers By Pulsed-Laser Interference Heating, Mikhail Khenner, Sagar Yadavali, Ramki Kalyanaraman

Mathematics Faculty Publications

The impacts of the two-beam interference heating on the number of core-shell and embedded nanoparticles and on nanostructure coarsening are studied numerically based on the non-linear dynamical model for dewetting of the pulsed-laser irradiated, thin (< 20 nm) metallic bilayers. The model incorporates thermocapillary forces and disjoining pressures, and assumes dewetting from the optically transparent substrate atop of the reflective support layer, which results in the complicated dependence of light reflectivity and absorption on the thicknesses of the layers. Stabilizing thermocapillary effect is due to the local thickness-dependent, steady- state temperature profile in the liquid, which is derived based on the mean substrate temperature estimated from the elaborate thermal model of transient heating and melting/freezing. Linear stability analysis of the model equations set for Ag/Co bilayer predicts the dewetting length scales in the qualitative agreement with experiment.

Influence Of Damping On Hyperbolic Equations With Parabolic Degeneracy, Katarzyna Saxton, Ralph Saxton

Mathematics Faculty Publications

This paper examines the effect of damping on a nonstrictly hyperbolic 2 x 2 system. It is shown that the growth of singularities is not restricted as in the strictly hyperbolic case where dissipation can be strong enough to preserve the smoothness of solutions globally in time. Here, irrespective of the stabilizing properties of damping, solutions are found to break down in finite time on a line where two eigenvalues coincide in state space.

Critical Buckling Loads Of The Perfect Hollomon’S Power-Law Columns, Dongming Wei, Alejandro Sarria, Mohamed Elgindi

Mathematics Faculty Publications

In this work, we present analytic formulas for calculating the critical buckling states of some plastic axial columns of constant cross-sections. The associated critical buckling loads are calculated by Euler-type analytic formulas and the associated deformed shapes are presented in terms of generalized trigonometric functions. The plasticity of the material is defined by the Holloman’s power-law equation. This is an extension of the Euler critical buckling loads of perfect elastic columns to perfect plastic columns. In particular, critical loads for perfect straight plastic columns with circular and rectangular cross-sections are calculated for a list of commonly used metals. Connections and …

On The Global Solvability Of A Class Of Fourth-Order Nonlinear Boundary Value Problems, M.B.M. Elgindi, Dongming Wei

Mathematics Faculty Publications

In this paper we prove the global solvability of a class of fourth-order nonlinear boundary value problems that govern the deformation of a Hollomon’s power-law plastic beam subject to an axial compression and nonlinear lateral constrains. For certain ranges of the acting axial compression force, the solvability of the equations follows from the monotonicity of the fourth order nonlinear differential operator. Beyond these ranges the monotonicity of the operator is lost. It is shown that, in this case, the global solvability may be generated by the lower order nonlinear terms of the equations for a certain type of constrains.

Travelling Wave Solutions Of Burgers' Equation For Gee-Lyon Fluid Flows, Dongming Wei, Ken Holladay

Mathematics Faculty Publications

In this work we present some analytic and semi-analytic traveling wave solutions of generalized Burger' equation for isothermal unidirectional flow of viscous non-Newtonian fluids obeying Gee-Lyon nonlinear rheological equation. The solution of Burgers' equation for Newtonian flow as a special case. We also derive estimates of shock thickness for non-Newtonian flows.

An H1 Model For Inextensible Strings, Stephen C. Preston, Ralph Saxton

Mathematics Faculty Publications

We study geodesics of the H¹ Riemannian metric (see article for equation) on the space of inextensible curves (see article for equation). This metric is a regularization of the usual L² metric on curves, for which the submanifold geometry and geodesic equations have been analyzed already. The H¹ geodesic equation represents a limiting case of the Pochhammer-Chree equation from elasticity theory. We show the geodesic equation is C^∞ in the Banach topology C¹ ([0,1], R²), and thus there is a smooth Riemannian exponential map. Furthermore, if we hold one of the curves fixed, …

Blow-Up Of Solutions To The Generalized Inviscid Proudman-Johnson Equation, Alejandro Sarria, Ralph Saxton

Mathematics Faculty Publications

For arbitrary values of a parameter --- finite-time blowup of solutions to the generalized, inviscid Proudman Johnson equation is studied via a direct approach which involves the derivation of representation formulae for solutions to the problem.

Some Generalized Trigonometric Sine Functions And Their Applications, Dongming Wei, Yu Liu, Mohamed B. Elgindi

Mathematics Faculty Publications

In this paper, it is shown that D. Shelupsky's generalized sine function, and various general sine functions developed by P. Drabek, R. Manasevich and M. Otani, P. Lindqvist, including the generalized Jacobi elliptic sine function of S. Takeuchi can be defined by systems of first order nonlinear ordinary differential equations with initial conditions. The structure of the system of differential equations is shown to be related to the Hamilton System in Lagrangian Mechanics. Numerical solutions of the ODE systems are solved to demonstrate the sine functions graphically. It is also demonstrated that the some of the generalized sine functions can …

A Study Of The Gam Approach To Solve Laminar Boundary Layer Equations In The Presence Of A Wedge, Rahmat Ali Khan, Muhammad Usman

Mathematics Faculty Publications

We apply an easy and simple technique, the generalized ap- proximation method (GAM) to investigate the temperature field associated with the Falkner-Skan boundary-layer problem. The nonlinear partial differ- ential equations are transformed to nonlinear ordinary differential equations using the similarity transformations. An iterative scheme for the non-linear ordinary differential equations associated with the velocity and temperature profiles are developed via GAM. Numerical results for the dimensionless ve- locity and temperature profiles of the wedge flow are presented graphically for different values of the wedge angle and Prandtl number.