Applied Mathematics Commons^™

On An Extension Of Abel-Gontscharoff's Expansion Formula, Tian-Xiao He, Leetsch Hsu, Peter Shiue

Scholarship

We present a constructive generalization of Abel-Gontscharoff’s series expansion to higher dimensions. A constructive application to a problem of multivariate interpolation is also investigated. In addition, two algorithms for the constructing the basis functions of the interpolants are given.

Modeling Basketball Free Throws, Andrew Lang, Joerg M. Gablonsky

College of Science and Engineering Faculty Research and Scholarship

This paper presents a mathematical model for basketball free throws. It is intended to be a supplement to an existing calculus course and could easily be used as a basis for a calculus project. Students will learn how to apply calculus to model an interesting real-world problem, from problem identification all the way through to interpretation and verification. Along the way we will introduce topics such as optimization (univariate and multiobjective), numerical methods, and differential equations.

An Explicit Mapping Between The Frequency Domain And The Time Domain Representations Of Nonlinear Systems, Marissa Condon, Rossen Ivanov

Articles

Explicit expressions are presented that describe the input-output behaviour of a nonlinear system in both the frequency and the time domain. The expressions are based on a set of coefficients that do not depend on the input to the system and are universal for a given system. The anharmonic oscillator is chosen as an example and is discussed for different choices of its physical parameters. It is shown that the typical approach for the determination of the Volterra Series representation is not valid for the important case when the nonlinear system exhibits oscillatory behaviour and the input has a pole …

Incompressible Finite Elements Via Hybridization. Part Ii: The Stokes System In Three Space Dimensions, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

We introduce a method that gives exactly incompressible velocity approximations to Stokes ow in three space dimensions. The method is designed by extending the ideas in Part I (http://archives.pdx.edu/ds/psu/10914) of this series, where the Stokes system in two space dimensions was considered. Thus we hybridize a vorticity-velocity formulation to obtain a new mixed method coupling approximations of tangential velocity and pressure on mesh faces. Once this relatively small tangential velocity-pressure system is solved, it is possible to recover a globally divergence-free numerical approximation of the fluid velocity, an approximation of the vorticity whose tangential component is continuous across …

Flocks And Formations, J. J. P. Veerman, Gerardo Lafferriere, John S. Caughman Iv, A. Williams

Mathematics and Statistics Faculty Publications and Presentations

Given a large number (the “flock”) of moving physical objects, we investigate physically reasonable mechanisms of influencing their orbits in such a way that they move along a prescribed course and in a prescribed and fixed configuration (or “in formation”). Each agent is programmed to see the position and velocity of a certain number of others. This flow of information from one agent to another defines a fixed directed (loopless) graph in which the agents are represented by the vertices. This graph is called the communication graph. To be able to fly in formation, an agent tries to match the …

Incompressible Finite Elements Via Hybridization. Part I: The Stokes System In Two Space Dimensions, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we introduce a new and efficient way to compute exactly divergence-free velocity approximations for the Stokes equations in two space dimensions. We begin by considering a mixed method that provides an exactly divergence-free approximation of the velocity and a continuous approximation of the vorticity. We then rewrite this method solely in terms of the tangential fluid velocity and the pressure on mesh edges by means of a new hybridization technique. This novel formulation bypasses the difficult task of constructing an exactly divergence-free basis for velocity approximations. Moreover, the discrete system resulting from our method has fewer degrees …

Nédélec Spaces In Affine Coordinates, Jay Gopalakrishnan, Luis E. García-Castillo, Leszek Demkowicz

Mathematics and Statistics Faculty Publications and Presentations

In this note, we provide a conveniently implementable basis for simplicial Nédélec spaces of any order in any space dimension. The main feature of the basis is that it is expressed solely in terms of the barycentric coordinates of the simplex.

A Note On Lattice Chains And Delannoy Numbers, John S. Caughman Iv, Clifford R. Haithcock, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

Fix nonnegative integers n₁,…,n_d and let L denote the lattice of integer points (a₁,…,a_d)∈Z^d satisfying 0⩽a_i⩽n_i for 1⩽i⩽d. Let L be partially ordered by the usual dominance ordering. In this paper we offer combinatorial derivations of a number of results concerning chains in L. In particular, the results obtained are established without recourse to generating functions or recurrence relations. We begin with an elementary derivation of the number of chains in L of a given size, from which one can deduce the classical expression for the total number …

Stable Motions Of Vehicle Formations, Anca Williams, Gerardo Lafferriere, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

We investigate stable maneuvers for a group of autonomous vehicles while moving in formation. The allowed decentralized feeback laws are factored through the Laplacian matrix of the communication graph. We show that such laws allow for stable circular or elliptical motions for certain vehicle dynamics. We find necessary and sufficient conditions on the feedback gains and the dynamic parameters for convergence to formation. In particular, we prove that for undirected graphs there exist feedback gains that stabilize rotational (or elliptical) motions of arbitrary radius (or eceentricity). In the directed graph case we provide necessary and sufficient conditions on the curvature …

Some Comments On: Existence Of Solutions Of Abstract Nonlinear Second-Order Neutral Functional Integrodifferential Equations, Eduardo Hernandez, Mark A. Mckibben

Mathematics Faculty Publications

We establish the existence of mild solutions for a class of abstract second-order partial neutral functional integro-differential equations with infinite delay in a Banach space using the theory of cosine families of bounded linear operators and Schaefer's fixed-point theorem.

Stability Properties Of Linear Volterra Integrodifferential Equations With Nonlinear Perturbation, Muhammad Islam, Youssef Raffoul

Mathematics Faculty Publications

A Lyapunov functional is employed to obtain conditions that guarantee stability, uniform stability and uniform asymptotic stability of the zero solution of a scalar linear Volterra integrodifferential equation with nonlinear perturbation.

Boundedness And Stability In Nonlinear Delay Difference Equations Employing Fixed Point Theory, Muhammad Islam, Ernest Yankson

Mathematics Faculty Publications

In this paper we study stability and boundedness of the nonlinear difference equation

x(t+1)=a(t)x(t)+c(t)Δx(t−g(t))+q(x(t),x(t−g(t))).

In particular we study equi-boundedness of solutions and the stability of the zero solution of this equation. Fixed point theorems are used in the analysis.

A Solvable Model For Gravity Driven Granular Dynamics, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

We discuss a toy model to study the dynamics of individual particles in avalanches. The model describes a particle launched from an inclined infinite staircase. The particle is not allowed to bounce when it collides with the staircase. During the collision, the particle loses some energy, and after that slides on to the end of the step it landed on. The process then repeats itself. The dynamics of this no-bounce model can essentially be completely understood. Partial versions of some results were stated and argued in previous work. Here we give a full description together with all the proofs. We …

The Simultaneous Onset And Interaction Of Taylor And Dean Instabilities In A Couette Geometry, C. P. Hills, A. P. Bassom

Articles

The fluid flow between a pair of coaxial circular cylinders generated by the uniform rotation of the inner cylinder and an azimuthal pressure gradient is susceptible to both Taylor and Dean type instabilities. The flow can be characterised by two parameters: a measure of the relative magnitude of the rotation and pressure effects and a non-dimensional Taylor number. Neutral curves associated with each instability can be constructed but it has been suggested that these curves do not cross but rather posses `kinks'. Our work is based in the small gap, large wavenumber limit and considers the simultaneous onset of Taylor …

On The Integrability Of A Class Of Nonlinear Dispersive Wave Equations, Rossen Ivanov

Articles

We investigate the integrability of a class of 1+1 dimensional models describing nonlinear dispersive waves in continuous media, e.g. cylindrical compressible hyperelastic rods, shallow water waves, etc. The only completely integrable cases coincide with the Camassa-Holm and Degasperis-Procesi equations.

The Asymptotic Z-Transform, Scott Jude Champagne

LSU Master's Theses

Sequences of numbers and transformations from sequences to functions have been studied extensively, including the multiplication of two sequences through convolution and the equivalent multiplication of functions. The focal points of this thesis are the convolution field of causal sequences and their Z-transforms. Classically, the treatment of the Z-transform has been limited to those causal sequences for which the power series has a nontrivial radius of convergence. In this thesis it is shown that the Z-transform can be extended to all causal sequences without compromising any of the operational properties of the classical Z-transform.

Stability Of Stochastic Pricing Models Under Volatility Fluctuations, Krassimir Zhivkov Nikolov

LSU Master's Theses

The standard theory of the stochastic models used to value financial derivatives contracts involves models whose input parameters are deterministic functions and often constants. Because of the random nature of the changes in the market prices of the financial instruments, the coefficients of these models are inevitably susceptible to random perturbation from their initial estimates. In this paper we will investigate the behavior of some of the most widely used models when small changes are applied to their volatility component. Starting with the Black-Scholes model for the price of a European call option, we will continue our analysis of the …

Modern Interpretation Of Euclid's Theory Of Ratio And Proportion, Mark Robert Stecher

LSU Master's Theses

Euclid’s Elements is the foundation for geometry. Book V of Euclid’s Elements, which is independent from the earlier books, focuses on multiples, ratios, and proportions. This paper presents a model of the conceptual content of Book V, but using carefully selected modern notation to represent Euclid’s ideas without changing them drastically. All of the propositions and proofs from Euclid have been restated using just enough modern language to make clear for a modern reader. We also present a modern theory that bears analogy, proposition by proposition, to Euclid’s theory, but uses rigorous modern methods of proof.

Quasicontinuous Derivatives And Viscosity Functions, Rodica Cazacu

LSU Doctoral Dissertations

In this work we demonstrate how the continuous domain theory can be applied to the theory of nonlinear optimization, particularly to the theory of viscosity solutions. We consider finding the viscosity solution for the Hamilton-Jacobi equation H(x, y) = g(x), with continuous hamiltonian, but with possibly discontinuous right-hand side. We begin by finding a new function space Q(X,L), the space of equivalence classes of quasicontinuous functions from a locally compact set X to a bicontinuous lattice L and we will define on Q(X,L) the qo-topology, which is a variant of classical order topology defined on complete lattices. On this …

On Cographic Matroids And Signed-Graphic Matroids, Dan Slilaty

Mathematics and Statistics Faculty Publications

We prove that a connected cographic matroid of a graph G is the bias matroid of a signed graph Σ iff G imbeds in the projective plane. In the case that G is nonplanar, we also show that Σ must be the projective-planar dual signed graph of an actual imbedding of G in the projective plane. As a corollary we get that, if G1, . . . , G29 denote the 29 nonseparable forbidden minors for projective-planar graphs, then the cographic matroids of G1, . . . , G29 are among the forbidden minors for the class of bias matroids …

Cyclic Difference Covers, K. T. Arasu, Surinder Sehgal

Mathematics and Statistics Faculty Publications

his paper studies cyclic difference covers in finite cyclic groups and develops several constructions and existence results for these combinatorial structures. The authors investigate conditions under which cyclic difference covers exist, derive new infinite families, and analyze connections with related objects such as difference sets and covering designs. Applications to combinatorial design theory and coding-related constructions are also discussed.

A Geometric Characterization: Complex Ellipsoids And The Bochner-Martinelli Kernel, Michael Bolt

University Faculty Publications and Creative Works

Boas' characterization of bounded domains for which the Bochner-Martinelli kernel is self-adjoint is extended to the case of a weighted measure. For strictly convex domains, this equivalently characterizes the ones whose Leray-Aǐzenberg kernel is self-adjoint with respect to weighted measure. In each case, the domains are complex linear images of a ball, and the measure is the Fefferman measure. The Leray-Aǐzenberg kernel for a strictly convex hypersurface in ℂn is shown to be Möbius invariant when defined with respect to Fefferman measure. © 2005 University of Illinois.

Fuzzy And Neutrosophic Analysis Of Women With Hiv/Aids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Fuzzy theory is one of the best tools to analyze data, when the data under study is an unsupervised one, involving uncertainty coupled with imprecision. However, fuzzy theory cannot cater to analyzing the data involved with indeterminacy. The only tool that can involve itself with indeterminacy is the neutrosophic model. Neutrosophic models are used in the analysis of the socio-economic problems of HIV/AIDS infected women patients living in rural Tamil Nadu. Most of these women are uneducated and live in utter poverty. Till they became seriously ill they worked as daily wagers. When these women got admitted in the hospital …

Fuzzy And Neutrosophic Analysis Of Periyar’S Views On Untouchability, Florentin Smarandache, Vasantha Kandasamy, K. Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

“Day in and day out we take pride in claiming that India has a 5000 year old civilization. But the way Dalits and those suppressed are being treated by the people who wield power and authority speaks volumes for the degradations of our moral structure and civilized standards.” Ex-President of India, the late K. R. Narayanan The New Indian Express, Saturday, 12 Nov. 2005 K.R.Narayanan was a lauded hero and a distinguished victim of his Dalit background. Even in an international platform when he was on an official visit to Paris, the media headlines blazed, ‘An Untouchable at Elysee’. He …

The Harmony Of The World, Chris Arthur

Faculty publications

Experimental music with mathematics and astronomy is discussed. Chord-like pitch arrangements are determined with geometric proportions arising in planetary movements. Rudimentary digital audio with Fourier series and the JPL on-line ephemeris is developed as a software solution. Computer musicians may listen to and select harmonies by specifying a date in time. A study and application of the ideas in The Harmony of the World by Johannes Kepler is presented with a software demonstration.

Positive Solutions For A Fourth Order Boundary Value Problem, Bo Yang

Faculty Articles

We consider a boundary value problem for the beam equation, in which the boundary conditions mean that the beam is embedded at one end and free at the other end. Some new estimates to the positive solutions to the boundary value problem are obtained. Some sufficient conditions for the existence of at least one positive solution for the boundary value problem are established. An example is given at the end of the paper to illustrate the main results.

Weak Solutions To The Cauchy Problem Of A Semilinear Wave Equation With Damping And Source Terms, Petronela Radu

Department of Mathematics: Faculty Publications

In this paper we prove local existence of weak solutions for a semilinear wave equation with power-like source and dissipative terms on the entire space ℝⁿ. The main theorem gives an alternative proof of the local in time existence result due to J. Serrin, G. Todorova and E. Vitillaro, and also some extension to their work. In particular, our method shows that sources that are not locally Lipschitz in L² can be controlled without any damping at all. If the semilinearity involving the displacement has a “good” sign, we obtain global existence of solutions.

Reversals And Transpositions Over Finite Alphabets, A. J. Radcliffe, A. D. Scott, E. L. Wilmer

Department of Mathematics: Faculty Publications

Extending results of Christie and Irving, we examine the action of reversals and transpositions on finite strings over an alphabet of size k. We show that determining reversal, transposition, or signed reversal distance between two strings over a finite alphabet is NP-hard, while for “dense” instances we give a polynomial-time approximation scheme. We also give a number of extremal results, as well as investigating the distance between random strings and the problem of sorting a string over a finite alphabet.

Bounds On Element Order In Rings Z(M) With Divisors Of Zero, C. H. Cooke

Mathematics & Statistics Faculty Publications

If p is a prime, integer ring Zp has exactly ¢¢(p) generating elements ω, each of which has maximal index Ip(ω) = (p) = p − 1. But, if m = ΠRJ = ₁p^αJ_J is composite, it is possible that Zm does not possess a generating element, and the maximal index of an element is not easily discernible. Here, it is determined when, in the absence of a generating element, one can still with confidence place bounds on the maximal index. Such a bound is usually less than ¢(m …

Application Of Meshless Methods For Thermal Analysis, Darrell Pepper, Bozidar Sarler

Mechanical Engineering Faculty Research

Many numerical and analytical schemes exist for solving heat transfer problems. The meshless method is a particularly attractive method that is receiving attention in the engineering and scientific modeling communities. The meshless method is simple, accurate, and requires no polygonalisation. In this study, we focus on the application of meshless methods using radial basis functions (RBFs) — which are simple to implement — for thermal problems. Radial basis functions are the natural generalization of univariate polynomial splines to a multivariate setting that work for arbitrary geometry with high dimensions. RBF functions depend only on the distance from some center point. …