Applied Mathematics Commons^™

Approximate Analytical Solutions For Fractional Space- And Time- Partial Differential Equations Using Homotopy Analysis Method, Subir, Das, R. Kumar, P. K. Gupta, Hossein Jafari

Applications and Applied Mathematics: An International Journal (AAM)

This article presents the approximate analytical solutions of first order linear partial differential equations (PDEs) with fractional time- and space- derivatives. With the aid of initial values, the explicit solutions of the equations are solved making use of reliable algorithm like homotopy analysis method (HAM). The speed of convergence of the method is based on a rapidly convergent series with easily computable components. The fractional derivatives are described in Caputo sense. Numerical results show that the HAM is easy to implement and accurate when applied to space- time- fractional PDEs.

On The Eigenvalue And Inertia Problems For Descriptor Systems, Asadollah Aasaraai, Kameleh N. Pirbazari

Applications and Applied Mathematics: An International Journal (AAM)

The present study is intended to demonstrate that for a descriptor system with matrix pencil there exists a matrix such that matrix and matrix pencil have the same positive and negative eigenvalues. It is also shown that matrix can be calculated as a contour integral. On the other hand, different representations for matrix are introduced.

A Mathematical Approach For Optimizing The Casino Slot Floor: A Linear Programming Application, Kasra Christopher Ghaharian

UNLV Theses, Dissertations, Professional Papers, and Capstones

Linear programming is a tool that has been successfully applied to various problems across many different industries and businesses. However, it appears that casino operators may have overlooked this useful and proven method. At most casino properties the bulk of gaming revenues are derived from slot machines. It is therefore imperative for casino operators to effectively manage and cultivate the performance of this department. A primary task for the casino operator is planning and deciding the mix of slot machines in order to maximize performance.

This paper presents the task of optimizing the casino slot floor as a linear programming …

The Carter Constant For Inclined Orbits About A Massive Kerr Black Hole: I. Circular Orbits, Peter G. Komorowski, Sree Ram Valluri, Martin Houde

Physics and Astronomy Publications

In an extreme binary black hole system, an orbit will increase its angle of inclination (ι) as it evolves in Kerr spacetime. We focus our attention on the behaviour of the Carter constant (Q) for near-polar orbits, and develop an analysis that is independent of and complements radiation-reactionmodels. For a Schwarzschild black hole, the polar orbits represent the abutment between the prograde and retrograde orbits at whichQis at its maximum value for given values of the latus rectum (˜l ) and the eccentricity (e). The introduction of spin (S˜ = |J|/M2) to themassive black hole causes this boundary, or abutment, …

Computational Biology, Harvey Greenberg, Allen Holder

Mathematical Sciences Technical Reports (MSTR)

Computational biology is an interdisciplinary field that applies the techniques of computer science, applied mathematics, and statistics to address biological questions. OR is also interdisciplinary and applies the same mathematical and computational sciences, but to decision-making problems. Both focus on developing mathematical models and designing algorithms to solve them. Models in computational biology vary in their biological domain and can range from the interactions of genes and proteins to the relationships among organisms and species.

Lipchitzian Stability Of Parametric Variational Inequalities Over Generalized Polyhedra In Banach Spaces, Liqun Ban, Boris S. Mordukhovich, Wen Song

Mathematics Research Reports

This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solutions maps entirely via their initial data. This is done on the base of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. …

Engineering Flow States With Localized Forcing In A Thin, Marangoni-Driven Inclined Film, Rachel Levy, Stephen Rosenthal '09, Jeffrey Wong '11

All HMC Faculty Publications and Research

Numerical simulations of lubrication models provide clues for experimentalists about the development of wave structures in thin liquid films. We analyze numerical simulations of a lubrication model for an inclined thin liquid film modified by Marangoni forces due to a thermal gradient and additional localized forcing heating the substrate. Numerical results can be explained through connections to theory for hyperbolic conservation laws predicting wave fronts from Marangoni-driven thin films without forcing. We demonstrate how a variety of forcing profiles, such as Gaussian, rectangular, and triangular, affect the formation of downstream transient structures, including an N wave not commonly discussed in …

The Positive Solutions Of The Matukuma Equation And The Problem Of Finite Radius And Finite Mass, Jurgen Batt, Yi Li

Mathematics and Statistics Faculty Publications

This work is an extensive study of the 3 different types of positive solutions of the Matukuma equation 1r2(r2ϕ′)′=−rλ−2(1+r2)λ/2ϕp,p>1,λ>0 : the E-solutions (regular at r = 0), the M-solutions (singular at r = 0) and the F-solutions (whose existence begins away from r = 0). An essential tool is a transformation of the equation into a 2-dimensional asymptotically autonomous system, whose limit sets (by a theorem of H. R. Thieme) are the limit sets of Emden–Fowler systems, and serve as a characterization of the different solutions. The emphasis lies on the study of the M …

Curvedland: An Applet For Illustrating Curved Geometry Without Embedding, Gary Felder, Stephanie Erickson

Physics: Faculty Publications

We have written a Java applet to illustrate the meaning of curved geometry. The applet provides a mapping interface similar to MapQuest or Google Maps; features include the ability to navigate through a space and place permanent point objects and/or shapes at arbitrary positions. The underlying two-dimensional space has a constant, positive curvature, which causes the apparent paths and shapes of the objects in the map to appear distorted in ways that change as you view them from different relative angles and distances.

On The Lqg Theory With Bounded Control, D. V. Iourtchenko, J. L. Menaldi, A. S. Bratus

Mathematics Faculty Research Publications

We consider a stochastic optimal control problem in the whole space, where the corresponding HJB equation is degenerate, with a quadratic running cost and coeffcients with linear growth. In this paper we provide a full mathematical details on the key estimate relating the asymptotic behavior of the solution as the space variable goes to infinite.

Existence Of Solutions For A Semilinear Wave Equation With Non-Monotone Nonlinearity, Alfonso Castro, Benjamin Preskill '09

All HMC Faculty Publications and Research

For double-periodic and Dirichlet-periodic boundary conditions, we prove the existence of solutions to a forced semilinear wave equation with asymptotically linear nonlinearity, no resonance, and non-monotone nonlinearity when the forcing term is not flat on characteristics. The solutions are in L^∞ when the forcing term is in L^∞ and continous when the forcing term is continuous. This is in contrast with the results in [4], where the non-enxistence of continuous solutions is established even when forcing term is of class C^∞ but is flat on a characteristic.

Generalized Newton's Method Based On Graphical Derivatives, T Hoheisel, C Kanzow, Boris S. Mordukhovich, Hung M. Phan

Mathematics Research Reports

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness assumption, which is illustrated by examples. The algorithm and main results obtained in the paper …

A Class Of Discontinuous Petrov–Galerkin Methods. Part Iv: The Optimal Test Norm And Time-Harmonic Wave Propagation In 1d., Jeffrey Zitelli, Leszek Demkowicz, Jay Gopalakrishnan, D. Pardo, V. M. Calo

Mathematics and Statistics Faculty Publications and Presentations

The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test …

On Circulant And Two-Circulant Weighing Matrices, K. T. Arasu, I. S. Kotsireast, C. Koukouvinos, Jennifer Seberry

Mathematics and Statistics Faculty Publications

We employ theoretical and computational techniques to construct new weighing matrices constructed from two circulants. In particular, we con- struct W (148, 144), W (152, 144), W (156, 144) which are listed as open in the second edition of the Handbook of Combinatorial Designs. We also fill a missing entry in Strassler’s table with answer “YES”, by constructing a circulant weighing matrix of order 142 with weight 100.

Deformation Waves In Microstructured Materials: Theory And Numerics, Juri Engelbrecht, Arkadi Berezovski, Mihhail Berezovski

Publications

A linear model of the microstructured continuum based on Mindlin theory is adopted which can be represented in the framework of the internal variable theory. Fully coupled systems of equations for macro-motion and microstructure evolution are represented in the form of conservation laws. A modification of wave propagation algorithm is used for numerical calculations. Results of direct numerical simulations of wave propagation in periodic medium are compared with similar results for the continuous media with the modelled microstructure. It is shown that the proper choice of material constants should be made to match the results obtained by both approaches

Neural Extensions To Robust Parameter Design, Bernard Jacob Loeffelholz

Theses and Dissertations

Robust parameter design (RPD) is implemented in systems in which a user wants to minimize the variance of a system response caused by uncontrollable factors while obtaining a consistent and reliable system response over time. We propose the use of artificial neural networks to compensate for highly non-linear problems that quadratic regression fails to accurately model. RPD is conducted under the assumption that the relationship between system response and controllable and uncontrollable variables does not change over time. We propose a methodology to find a new set of settings that will be robust to moderate system degradation while remaining robust …

A Generalised Kummer's Conjecture, M. J.R. Myers

University Faculty Publications and Creative Works

Kummer's conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott-Halberstam conjecture implies that this generalised Kummer's conjecture is true for almost all n but is false for infinitely many n. Copyright © 2010 Glasgow Mathematical Journal Trust.

Coarser Connected Metrizable Topologies, Lynne Yengulalp

Mathematics Faculty Publications

We show that every metric space, X, with w(⩾) c has a coarser connected metrizable topology.

A Generalised Kummer's Conjecture, M. J.R. Myers

University Faculty Publications and Creative Works

Kummer's conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott-Halberstam conjecture implies that this generalised Kummer's conjecture is true for almost all n but is false for infinitely many n.

Analytic Treatment Of Vortex States In Cylindrical Superconductors In Applied Axial Magnetic Field, Andrei Ludu, J. Van Deun, M. V, Milosevic, A. Cuyt, F. M. Peeters

Publications

We solve the linear Ginzburg–Landau GL equation in the presence of a uniform magnetic field with cylindrical symmetry and we find analytic expressions for the eigenfunctions in terms of the confluent hypergeometric functions. The discrete spectrum results from an implicit equation associated to the boundary conditions and it is resolved in analytic form using the continued fractions formalism. We study the dependence of the spectrum and the eigenfunctions on the sample size and the surface conditions for solid and hollow cylindrical superconductors. Finally, the solutions of the nonlinear GL formalism are constructed as expansions in the linear GL eigenfunction basis …

Non-Classical Symmetry Solutions To The Fitzhugh Nagumo Equation., Arash Mehraban

Electronic Theses and Dissertations

In Reaction-Diffusion systems, some parameters can influence the behavior of other parameters in that system. Thus reaction diffusion equations are often used to model the behavior of biological phenomena. The Fitzhugh Nagumo partial differential equation is a reaction diffusion equation that arises both in population genetics and in modeling the transmission of action potentials in the nervous system. In this paper we are interested in finding solutions to this equation. Using Lie groups in particular, we would like to find symmetries of the Fitzhugh Nagumo equation that reduce this non-linear PDE to an Ordinary Differential Equation. In order to accomplish …

G-Lattices For An Unrooted Perfect Phylogeny, Monica Grigg

Mathematical Sciences Technical Reports (MSTR)

We look at the Pure Parsimony problem and the Perfect Phylogeny Haplotyping problem. From the Pure Parsimony problem we consider structures of genotypes called g-lattices. These structures either provide solutions or give bounds to the pure parsimony problem. In particular, we investigate which of these structures supports an unrooted perfect phylogeny, a condition that adds biological interpretation. By understanding which g-lattices support an unrooted perfect phylogeny, we connect two of the standard biological inference rules used to recreate how genetic diversity propagates across generations.

A Spectral Approach To Protein Structure Alignment, Yosi Shibberu, Allen Holder

Mathematical Sciences Technical Reports (MSTR)

We present two algorithms that use spectral methods to align protein folds. One of the algorithms is suitable for database searches, the other for difficult alignments. We present computational results for 780 pairwise alignments used to classify 40 proteins as well as results for a separate set of 36 protein alignments used for comparison to four other alignment algorithms. We also provide a mathematically rigorous development of the intrinsic geometry underlying our spectral approach.

Mutation Size Optimizes Speciation In An Evolutionary Model, Nathan Dees, Sonya Bahar

Physics Faculty Works

The role of mutation rate in optimizing key features of evolutionary dynamics has recently been investigated in various computational models. Here, we address the related question of how maximum mutation size affects the formation of species in a simple computational evolutionary model. We find that the number of species is maximized for intermediate values of a mutation size parameter μ; the result is observed for evolving organisms on a randomly changing landscape as well as in a version of the model where negative feedback exists between the local population size and the fitness provided by the landscape. The same result …

Bilinear Programming And Protein Structure Alignment, J. Cain, D. Kamenetsky, N. Lavine

Mathematical Sciences Technical Reports (MSTR)

Proteins are a primary functional component of organic life, and understanding their function is integral to many areas of research in biochemistry. The three-dimensional structure of a protein largely determines this function. Protein structure alignment compares the structure of a protein with known function to that of a protein with unknown function. A protein’s three-dimensional structure can be transformed through a smooth piecewise-linear sigmoid function to a real symmetric contact matrix that represents the functional significance of certain parts of the protein. We address the protein alignment problem as a minimization of the 2-norm difference of two proteins’ contact matrices. …

Multistage Homotopy Analysis Method For Solving Nonlinear Integral Equations, H. Jafari, M. A. Firoozjaee

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we present an efficient modification of the homotopy analysis method (HAM) that will facilitate the calculations. We then conduct a comparative study between the new modification and the homotopy analysis method. This modification of the homotopy analysis method is applied to nonlinear integral equations and mixed Volterra-Fredholm integral equations, which yields a series solution with accelerated convergence. Numerical illustrations are investigated to show the features of the technique. The modified method accelerates the rapid convergence of the series solution and reduces the size of work.

A Global Characterization Of Tubed Surfaces In ℂ2, Michael Bolt

University Faculty Publications and Creative Works

Let M3 S C2 be a three times differentiable real hypersurface. The Levi form of M transforms under biholomorphism, and when restricted to the complex tangent space, the skew-Hermitian part of the second fundamental form transforms under Möbius transformations. The surfaces for which these forms are constant multiples of each other were identified in previous work, provided the constant is not unimodular. Here it is proved that if the surface is assumed to be complete and if the constant is unimodular, then the surface is tubed over a strongly convex curve. The converse statement is true, too, and is easily …

On The Solution Of The Vibration Equation By Means Of The Homotopy Perturbation Method, Ahmet Yıldırım, Canan Ünlü, Syed T. Mohyud-Din

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we present a reliable algorithm, the homotopy perturbation method, to solve the well-known vibration equation for very large membrane which is given initial conditions. By using initial value, the explicit solutions of the equation for different cases have been derived, which accelerate the rapid convergence of the series solution. Numerical results show that the homotopy perturbation method is easy to implement and accurate when applied to differential equations. Numerical results for different particular cases of the problem are presented graphically.

Latest Developments In Nonlinear Sciences, Syed T. Mohyud-Din, Ahmet Yildirim

Applications and Applied Mathematics: An International Journal (AAM)

This paper outlines a detailed study of some latest trends and developments in nonlinear sciences. The major focus of our study will be variational iteration (VIM) and its modifications, homotopy perturbation (HPM), parameter expansion and exp-function methods. The above mentioned schemes are highly accurate, extraordinary efficient, capable to cope with the versatility of the physical problems and are being used to solve a wide class of nonlinear problems. Several examples are given which reveal the justification of our claim.

On Numerical Solutions Of Two-Dimensional Boussinesq Equations By Using Adomian Decomposition And He's Homotopy Perturbation Method, Syed T. Mohyud-Din, Mustafa Inc, Ebru Cavlak

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we obtain the approximate solution for 2-dimensional Boussinesq equation with initial condition by Adomian's decomposition and homotopy perturbation methods and numerical results are compared with exact solutions.