Applied Mathematics Commons^™

Mathematical Methods In Composing Melodies, Thomas Brown

Undergraduate Theses and Capstone Projects

This thesis, “Mathematical Methods in Composing Melodies,” explores the different ways in which mathematics can be used to create music. Some research has been done in this field already. Richard F. Voss and John Clarke used fractals and different frequencies of noise to create music. The Greek composer Iannis Xenakis used Markovian Stochastic trees to create some of his compositions. Explored in this thesis are seven different methods to compose melodies. After compiling the different melodies, they were categorized by different musical periods based on the musical characteristics found in the melody. This thesis differs from other research that deals …

Lanchester's Equations In Three Dimensions, Christina Spradlin, Greg Spradlin

Publications

This paper generalizes Lanchester’s equations of warfare to partial differential equations involving time and two spatial variables. Unlike in Lanchester’s original ordinary differential equations, the distribution of armies over the battlefield must be considered. Four different modes of attack are introduced, generalizing Lanchester’s equations for area fire and for direct fire. The effect of the distribution of forces and their movement on the outcome is considered, and numerical simulations given.

Turing Patterns On Growing Spheres: The Exponential Case, Julijana Gjorgjieva, Jon T. Jacobsen

All HMC Faculty Publications and Research

We consider Turing patterns for reaction-diffusion systems on the surface of a growing sphere. In particular, we are interested in the effect of dynamic growth on the pattern formation. We consider exponential isotropic growth of the sphere and perform a linear stability analysis and compare the results with numerical simulations.

Hamiltonian Formulation, Nonintegrability And Local Bifurcations For The Ostrovsky Equation, S. Roy Choudhury, Rossen Ivanov, Yue Liu

Articles

The Ostrovsky equation is a model for gravity waves propagating down a channel under the influence of Coriolis force. This equation is a modification of the famous Korteweg-de Vries equation and is also Hamiltonian. However the Ostrovsky equation is not integrable and in this contribution we prove its nonintegrability. We also study local bifurcations of its solitary waves.

Water Waves And Integrability, Rossen Ivanov

Articles

The Euler’s equations describe the motion of inviscid fluid. In the case of shallow water, when a perturbative asymtotic expansion of the Euler’s equations is taken (to a certain order of smallness of the scale parameters), relations to certain integrable equations emerge. Some recent results concerning the use of integrable equation in modeling the motion of shallow water waves are reviewed in this contribution.

Generalised Fourier Transform For The Camassa-Holm Hierarchy, Adrian Constantin, Vladimir Gerdjikov, Rossen Ivanov

Articles

The squared eigenfunctions of the spectral problem associated to the Camassa-Holm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform for the Camassa-Holm hierarchy as a Generalised Fourier transform. The main result of this work is the derivation of the completeness relation for the squared solutions of the Camassa-Holm spectral problem. We show that all the fundamental properties of the Camassa-Holm equation such as the integrals of motion, the description of the equations of the whole hierarchy and their Hamiltonian structures can be naturally expressed making use of the completeness relation and the recursion …

Conformal And Geometric Properties Of The Camassa-Holm Hierarchy, Rossen Ivanov

Articles

Integrable equations with second order Lax pair like KdV and Camassa-Holm (CH) exhibit interesting conformal properties and can be written in terms of the so-called conformal invariants (Schwarz form). These properties for the CH hierarchy are discussed in this ontribution. The squared eigenfunctions of the spectral problem, associated to the Camassa-Holm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform (IST) for the Camassa-Holm hierarchy as a Generalised Fourier Transform (GFT). Using GFT we describe explicitly some members of the CH hierarchy, including integrable deformations for the CH equation. Also we show that solutions …

Multiphoton Response Of Retinal Rod Photoreceptors, Vasilios Alexiades, Harihar Khanal

Publications

Phototransduction is the process by which light is converted into an electrical response in retinal photoreceptors. Rod photoreceptors contain a stack of (about 1000) disc membranes packed with photopigment rhodopsin molecules, which absorb the photons. We present computational experiments which show the profound effect on the response of the distances (how many discs apart) photons happen to be absorbed at. This photon-distribution effect alone can account for much of the observed variability in response.

Oscillations Of Hyperbolic Systems With Functional Arguments, Yutaka Shoukaku, Norio Yoshida

Applications and Applied Mathematics: An International Journal (AAM)

Hyperbolic systems with functional arguments are studied, and sufficient conditions are obtained for every solution of boundary value problems to be weakly oscillatory (that is, at least one of its components is oscillatory) in a cylindrical domain. Robin-type boundary condition is considered. The approach used is to reduce the multi-dimensional oscillation problems to one-dimensional oscillation problems by using some integral means of solutions.

A Partially Discretized Age-Dependent Population Model With An Additional Stucture, Jean Tchuenche

Applications and Applied Mathematics: An International Journal (AAM)

A semi-discretization method for solving an age-dependent population dynamics model with an additional structure is proposed. This method, unlike previous ones, considers the partial discretization which reduces the model equation into a first order ordinary differential equation. The latter is then solved explicitly and conditions under which second order accuracy arises are given. While the approach adopted is basically analytical, the main result shows that the sum of errors is bounded. An extension to the non-trivial case where growth depends on the additional parameter leads to a Riccati equation, and the existence and
convergence of solutions are proved.

Global Stability Results And Well Posedness Of An Si Age-Structured Epidemic Model With Vertical Transmission, M. El-Doma

Applications and Applied Mathematics: An International Journal (AAM)

An SI age-structured epidemic model for a vertically as well as horizontally transmitted disease is investigated when the fertility and mortality rates depend on age and the force of infection of proportionate mixing assumption type. We prove the well posedness of the model as well as the global stability for endemic equilibriums.

Strings, Chains, And Ropes, Darryl H. Yong

All HMC Faculty Publications and Research

Following Antman [Amer. Math. Mon., 87 (1980), pp. 359–370], we advocate a more physically realistic and systematic derivation of the wave equation suitable for a typical undergraduate course in partial differential equations. To demonstrate the utility of this derivation, three applications that follow naturally are described: strings, hanging chains, and jump ropes.

Abstract Semilinear Itó-Volterra Integro-Differential Stochastic Evolution Equations, David N. Keck, Mark A. Mckibben

Mathematics Faculty Publications

We consider a class of abstract semilinear stochastic Volterra integrodifferential equations in a real separable Hilbert space. The global existence and uniqueness of a mild solution, as well as a perturbation result, are established under the so-called Caratheodory growth conditions on the nonlinearities. An approximation result is then established, followed by an analogous result concerning a so-called McKean-Vlasov integrodi fferential equation, and then a brief commentary on the extension of the main results to the time-dependent case. The paper ends with a discussion of some concrete examples to illustrate the abstract theory.

Interval - Mtype Oscillation Criteria For Half - Linear Pde With Damping, Robert Marik

Applications and Applied Mathematics: An International Journal (AAM)

Using the Riccati substitution we derive new sufficient conditions which ensure that the half-linear partial differential equation with p-Laplacian and damping in the form of Equation (E) in the paper is oscillatory. These criteria, called interval criteria in theory of ODE's, allow to eliminate “bad parts” of the potential function c(x) from our considerations. Some of the results are new even in the case when (E) becomes linear ordinary differential equation.

A Discrete Time Counterpart Of The Black-Scholes Bond Replication Portfolio, Andrzej Korzeniowski

Applications and Applied Mathematics: An International Journal (AAM)

We construct a discrete time self-financing portfolio comprised of call options short and stock shares long which is riskless and grows at a fixed rate of return. It is also shown that when shorting periods tend to zero then so devised portfolio turns into the Black-Scholes bond replication. Unlike in standard approach the analysis presented here requires neither Ito Calculus nor solving the Heat Equation for option pricing.

Analysis Of An Sirs Age-Structured Epidemic Model With Vaccination And Vertical Transmission Of Disease, Mohammed El-Doma

Applications and Applied Mathematics: An International Journal (AAM)

An SIRS age-structured epidemic model for a vertically as well as horizontally transmitted disease under vaccination is investigated when the fertility, mortality and removal rates depend on age and the force of infection of proportionate mixing assumption type, and vaccination wanes over time. We prove the existence and uniqueness of solution to the model equations, and show that solutions of the model equations depend continuously on the initial age-distributions. Furthermore, we determine the steady states and obtain an explicitly computable threshold condition, in terms of the demographic and epidemiological parameters of the model; we then study the stability of the …

Remarks On The Stability Of Some Size-Structured Population Models I: Changes In Vital Rates Due To Population Only, Mohammed El-Doma

Applications and Applied Mathematics: An International Journal (AAM)

We consider a size-structured population model that has been studied in Calsina et al. (2003). We propose a different approach that provides direct stability results, and we correct a stability result given therein. In addition, we obtain global stability results that have not been given in Calsina et al. (2003).

Abstract Second-Order Damped Mckean-Vlasov Stochastic Evolution Equations, N. I. Mahmudov, Mark A. Mckibben

Mathematics Faculty Publications

We establish results concerning the global existence, uniqueness, approximate and exact controllability of mild solutions for a class of abstract second-order stochastic evolution equations 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. First-order equations of McKean-Vlasov type were first analyzed in the finite dimensional setting when studying diffusion processes, and then subsequently extended to the Hilbert space setting. The current manuscript provides a formulation of such …

Numerical Simulation Of Nonlinear Elastic Wave Propagation In Piecewise Homogeneous Media, Arkadi Berezovski, Mihhail Berezovski, Juri Engelbrecht

Publications

Systematic experimental work [S. Zhuang, G. Ravichandran, D. Grady, J. Mech. Phys. Solids 51 (2003) 245–265] on laminated composites subjected to high velocity impact loading exhibits the dispersed wave field and the oscillatory behavior of waves with respect to a mean value. Such a behavior is absent in homogeneous solids. An approximate solution to the plate impact in layered heterogeneous solids has been developed in [X. Chen, N. Chandra, A.M. Rajendran, Int. J. Solids Struct. 41 (2004) 4635–4659]. The influence of the particle velocity on many process characteristics was demonstrated. Based on earlier results [A. Berezovski, J. Engelbrecht, G.A. Maugin, …

The Motion Of A Thin Liquid Film Driven By Surfactant And Gravity, Michael Shearer, Rachel Levy

All HMC Faculty Publications and Research

We investigate wave solutions of a lubrication model for surfactant-driven flow of a thin liquid film down an inclined plane. We model the flow in one space dimension with a system of nonlinear PDEs of mixed hyperbolic-parabolic type in which the effects of capillarity and surface diffusion are neglected. Numerical solutions reveal distinct patterns of waves that are described analytically by combinations of traveling waves, some with jumps in height and surfactant concentration gradient. The various waves and combinations are strikingly different from what is observed in the case of flow on a horizontal plane. Jump conditions admit new shock …

Poisson Structure And Action-Angle Variables For The Camassa-Holm Equation, Adrian Constantin, Rossen Ivanov

Articles

The Poisson brackets for the scattering data of the Camassa-Holm equation are computed. Consequently, the action-angle variables are expressed in terms of the scattering data.

Inverse Scattering Transform For The Camassa-Holm Equation, Adrian Constantin, Vladimir Gerdjikov, Rossen Ivanov

Articles

An Inverse Scattering Method is developed for the Camassa-Holm equation. As an illustration of our approach the solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data. The main difference with respect to the standard Inverse Scattering Transform lies in the fact that we have a weighted spectral problem. We therefore have to develop different asymptotic expansions.

The Stable Manifold Theorem For Semilinear Stochastic Evolution Equations And Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang, Huaizhong Zhao

Articles and Preprints

The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts 1, 2.

In Part 1, we prove general existence and compactness theorems for C^k-cocycles of semilinear see’s and spde’s. Our results cover a large class of semilinear see’s as well as certain semilinear spde’s with Lipschitz and non-Lipschitz terms such as stochastic reaction diffusion equations and the …

On State-Dependent Delay Partial Neutral Functional–Differential Equations, Eduardo Hernandez M., Mark A. Mckibben

Mathematics Faculty Publications

No abstract provided.

Extended Camassa-Holm Hierarchy And Conserved Quantities, Rossen Ivanov

Articles

An extension of the Camassa-Holm hierarchy is constructed in this letter. The conserved quantities of the hierarchy are studied and a recurrent formula for the integrals of motion is derived.

On Doubly Periodic Solutions Of Quasilinear Hyperbolic Equations Of The Fourth Order, T. Kiguradze, T. Smith

Publications

The problem on doubly periodic solutions is considered for a class of quasilinear hyperbolic equations. Effective sufficient conditions of solvability and unique solvability of this problem are established.

Lattice Quantum Algorithm For The Schrodinger Wave Equation In 2+1 Dimensions With A Demonstration By Modeling Soliton Instabilities, Jeffrey Yepez, George Vahala, Linda L. Vahala

Electrical & Computer Engineering Faculty Publications

A lattice-based quantum algorithm is presented to model the non-linear Schrödinger-like equations in 2 + 1 dimensions. In this lattice-based model, using only 2 qubits per node, a sequence of unitary collide (qubit-qubit interaction) and stream (qubit translation) operators locally evolve a discrete field of probability amplitudes that in the long-wavelength limit accurately approximates a non-relativistic scalar wave function. The collision operator locally entangles pairs of qubits followed by a streaming operator that spreads the entanglement throughout the two dimensional lattice. The quantum algorithmic scheme employs a non-linear potential that is proportional to the moduli square of the wave function. …

Multivalued Logic, Neutrosophy And Schrodinger Equation, Florentin Smarandache, Victor Christianto

Branch Mathematics and Statistics Faculty and Staff Publications

This book was intended to discuss some paradoxes in Quantum Mechanics from the viewpoint of Multi-Valued-logic pioneered by Lukasiewicz, and a recent concept Neutrosophic Logic. Essentially, this new concept offers new insights on the idea of ‘identity’, which too often it has been accepted as given. Neutrosophy itself was developed in attempt to generalize Fuzzy-Logic introduced by L. Zadeh. While some aspects of theoretical foundations of logic are discussed, this book is not intended solely for pure mathematicians, but instead for physicists in the hope that some of ideas presented herein will be found useful. The book is motivated by …

Thermal Imaging Of Circular Inclusions Within A Two-Dimensional Region, Shannon Talbott, Hilary Spring

Mathematical Sciences Technical Reports (MSTR)

The ability to study the interior of an object without destroying it is an important industrial tool. One method of recent interest is steady-state thermal or impedance imaging. In this paper we will use the steady-state heat equation to locate one or more circular inclusions within a two-dimensional region, where the boundaries of the inclusions have partially disbonded from the surrounding material; this disbond is modelled as between the heat flux and jump discontinuity at the disbonded interface.

Time-Dependent Thermal Imaging Of Circular Inclusions, Donald L. Brouwn, Mark Hubenthal

Mathematical Sciences Technical Reports (MSTR)

This paper considers the inverse problem of locating one or more circular inclusions in a two-dimensional domain using thermal boundary data, specifically, the input heat flux and measured boundary temperature. The forward problem is governed by the heat equation. We show how the position and size of such defects can be recovered using the boundary data and various approximations of the solution to the forward problem. We also consider the stability of the algorithm involved to recover the defects.