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.

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 …

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.

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.

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.

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.

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 …

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.

An Investigation Of Secondary Teachers’ Knowledge Of Rate Of Change In The Context Of Teaching A Standards-Based Curriculum, Jihwa Noh

Dissertations

This study investigated teachers' mathematical content knowledge and pedagogical content knowledge with respect to rate of change in the context of teaching a Standards-based high school mathematics curriculum that emphasizes rate of change as a central theme, the Core-Plus Mathematics Project (CPMP) materials. A framework was designed to provide a comprehensive guide for analyzing different aspects of rate of change knowledge incorporating existing frameworks relative to rate of change, NCTM recommendations described in Curriculum and Evaluation Standards for School Mathematics andPrinciples and Standards for School Mathematics (NCTM, 2000), and research related to pedagogical understanding of rate of change.

Data …

Strict Feedforward Form And Symmetries Of Nonlinear Control Systems, Witold Respondek, Issa Amadou Tall

Miscellaneous (presentations, translations, interviews, etc)

We establish a relation between strict feedforward form and symmetries of nonlinear control systems. We prove that a system is feedback equivalent to the strict feedforward form if and only if it gives rise to a sequence of systems, such that each element of the sequence, firstly, possesses an infinitesimal symmetry and, secondly, it is the factor system of the preceding one, i.e., is reduced from the preceding one by its symmetry. We also propose a strict feedforward normal form and prove that a smooth strict feedforward system can be smoothly brought to that form.

Weighted Canonical Forms Of Nonlinear Single-Input Control Systems With Noncontrollable Linearization, Issa Amadou Tall, Witold Respondek

Miscellaneous (presentations, translations, interviews, etc)

We propose a weighted canonical form for single-input systems with noncontrollable first order approximation under the action of formal feedback transformations. This weighted canonical form is based on associating different weights to the linearly controllable and linearly noncontrollable parts of the system. We prove that two systems are formally feedback equivalent if and only if their weighted canonical forms coincide up to a diffeomorphism whose restriction to the linearly controllable part is identity.

Optimal Control Of Semilinear Evolution Inclusions Via Discrete Approximations, Boris S. Mordukhovich, Dong Wang

Mathematics Research Reports

This paper studies a Mayer type optimal control problem with general endpoint constraints for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces. First, we construct a sequence of discrete approximations to the original optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions …

Variational Stability And Marginal Functions Via Generalized Differentiation, Boris S. Mordukhovich, Nguyen Mau Nam

Mathematics Research Reports

Robust Lipschitzian properties of set-valued mappings and marginal functions play a crucial role in many aspects of variational analysis and its applications, especially for issues related to variational stability and optimizatiou. We develop an approach to variational stability based on generalized differentiation. The principal achievements of this paper include new results on coderivative calculus for set-valued mappings and singular subdifferentials of marginal functions in infinite dimensions with their extended applications to Lipschitzian stability. In this way we derive efficient conditions ensuring the preservation of Lipschitzian and related properties for set-valued mappings under various operations, with the exact bound/modulus estimates, as …

Of Parts And Wholes: Self-Similarity And Synecdoche In Science, Culture And Literature, Dirk Vanderbeke

Reconstruction: Studies in Contemporary Culture

[Editors' Introduction]

The discussion here centers on how self-similarity, or parts resembling a whole, has been a recurring aspect in the most diverse fields of human thinking from antiquity to the present. Primary examples are drawn from physics, biology, cybernetics, alchemy, philosophy, myth and, of course, language and literature. The author argues that self-similar patterns are one of the persistent ways in which the human mind structures its experience and knowledge of the world, and then examines some of the questions that arise when an almost ubiquitous concept, structure or linguistic and literary phenomenon resurfaces in a new guise in …

Reconstruction Of Partially Conductive Cracks Using Boundary Data, David Mccune, Janine Haugh

Mathematical Sciences Technical Reports (MSTR)

This paper develops an algorithm for finding one or more non-insulated, pair-wise disjoint, linear cracks in a two dimensional region using boundary measurements.

Subgradients Of Distance Functions At Out-Of-Set Points, Boris S. Mordukhovich, Nguyen Mau Nam

Mathematics Research Reports

This paper deals with the classical distance function to closed sets and its extension to the case of set-valued mappings. It has been well recognized that the distance functions play a crucial role in many aspects of variational analysis, optimization, and their applications. One of the most remarkable properties of even the classical distance function is its intrinsic nonsmoothness, which requires the usage of generalized differential constructions for its study and applications. In this paper we present new results in theser directions using mostly the generalized differential constructions introduced earlier by the first author, as well as their recent modifications. …

On The Dressing Method For The Generalised Zakharov-Shabat System, Rossen Ivanov

Articles

The dressing procedure for the Generalised Zakharov-Shabat system is well known for systems, related to sl(N) algebras. We extend the method, constructing explicitly the dressing factors for some systems, related to orthogonal and symplectic Lie algebras. We consider ’dressed’ fundamental analytical solutions with simple poles at the prescribed eigenvalue points and obtain the corresponding Lax potentials, representing the soliton solutions for some important nonlinear evolution equations.

Reconstruction Of An Unknown Boundary Portion From Cauchy Data In N-Dimensions, Kurt M. Bryan, Lester Caudill

Mathematical Sciences Technical Reports (MSTR)

We consider the inverse problem of determining the shape of some inacces­ sible portion of the boundary of a region in n dimensions from Cauchy data for the heat equation on an accessible portion of the boundary. The inverse problem is quite ill-posed, and nonlinear. We develop a Newton-like algorithm for solving the problem, with a simple and efficient means for computing the required derivatives, develop methods for regularizing the process, and provide computational examples

Determining The Length Of A One-Dimensional Bar, Natalya Yarlikina, Holly Walrath

Mathematical Sciences Technical Reports (MSTR)

In this paper we examine the inverse problem of determining the length of a one-dimensional bar from thermal measurements (temperature and heat flux) at one end of the bar (the "accessible" end); the other inaccessible end of the bar is assumed to be moving. We develop two different approaches to estimating the length of the bar, and show how one approach can also be adapted to find unknown boundary conditions at the inaccessible end of the bar.

Asymptotic Solutions Of Semilinear Stochastic Wave Equations, Pao-Liu Chow

Mathematics Research Reports

Large-time asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are established. Under appropriate conditions, the existence theorem for a unique global solution is given. Next the questions of bounded solutions and the exponential stability of an equilibrium solution, in mean-square and the almost sure sense, are studied. Then, under some sufficient conditions, the existence of a unique invariant measure is proved. Two examples are presented to illustrate some applications of the theorems.

A Liouville-Gelfand Equation For K-Hessian Operators, Jon T. Jacobsen

All HMC Faculty Publications and Research

In this paper we establish existence and multiplicity results for a class of fully nonlinear elliptic equations of k-Hessian type with exponential nonlinearity. In particular, we characterize the precise dependence of the multiplicity of solutions with respect to both the space dimension and the value of k. The choice of exponential nonlinearity is motivated by the classical Liouville-Gelfand problem from combustible gas dynamics and prescribed curvature problems.

Absolutely Continuous Representations And A Kaplansky Density Theorem For Free Semigroup Algebras, Kenneth R. Davidson, Jiankui Li, David R. Pitts

Department of Mathematics: Faculty Publications

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra An. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a ∗-extendible represen- tation . A ∗-extendible representation of An is regular if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous func- tionals are intimately related to maps which intertwine a given ∗-extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is …