Applied Mathematics Commons^™

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 The Generalized Möbius Inversion Formulas, Tian-Xiao He, Peter Shiue3, Leetsch Hsu

Scholarship

We provide a wide class of M¨obius inversion formulas in terms of the generalized M¨obius functions and its application to the setting of the Selberg multiplicative functions.

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

Mathematics Faculty Publications

No abstract provided.

Index Future Pricing Under Imperfect Market And Stochastic Volatility, Wei-Hsien Li

LSU Master's Theses

Financial markets in emerging countries are volatile and imperfect, so pricing model under traditional perfect-market frameset may not give reliable price of financial derivatives. The most famous pricing model for stock index future is the cost of carry model. The mis-pricing of cost of carry model inspires lots of following researches. Even transaction costs, dividends, stochastic interest rate, stochastic volatility, market imperfection, and other factors are considered, we still do not obtain a model price consistently better than cost of carry model. But these researches offer important insights, for example, the market needs time to mature and the more complex …

The Asymptotics Of Neutral Curve Crossing In Taylor–Dean Flow, 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. This work considers the small gap, large wavenumber limit for linear perturbations when the onset of the Taylor and Dean instabilities is concurrent. A consistent, matched asymptotic solution is found across the whole annular domain and identifies five regions of interest: two …

Hamiltonian Formulation And Integrability Of A Complex Symmetric Nonlinear System, Rossen Ivanov

Articles

The integrability of a complex generalisation of the ’elegant’ system, proposed by D. Fairlie and its relation to the Nahm equation and the Manakov top is discussed.

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.

Using Elimination To Describe Maxwell Curves, Lucas P. Beverlin

LSU Master's Theses

Cartesian ovals are curves in the plane that have been studied for hundreds of years. A Cartesian oval is the set of points whose distances from two fixed points called foci satisfy the property that a linear combination of these distances is a fixed constant. These ovals are a special case of what we call Maxwell curves. A Maxwell curve is the set of points with the property that a specific weighted sum of the distances to n foci is constant. We shall describe these curves geometrically. We will then examine Maxwell curves with two foci and a special case …

Characterization Of The Dependency Across Foreign Exchange Markets Using Copulas, Ryan Coelho

LSU Master's Theses

Though Pearson's correlation coefficient provides a convenient approach to measuring the dependency between two variables, in the last few years, there has been a significant amount of literature cautioning against the use of Pearson's correlation coefficient, as it does not remain invariant under monotone transformations of the underlying distribution functions. Since we are interested in examining the dependency pattern observed by the return on the Sterling Pound with that of the Japanese Yen, we will use the notion of a copula to approximate the joint density function between the daily returns on the Sterling Pound and the Japanese Yen. In …

Algebraic Characterizations Of Graph Imbeddability In Surfaces And Pseudosurfaces, Lowell Abrams, Dan Slilaty

Mathematics and Statistics Faculty Publications

Given a finite connected graph G and specifications for a closed, connected pseudosurface, we characterize when G can be imbedded in a closed, connected pseudosurface with the given specifications. The specifications for the pseudosurface are: the number of face-connected components, the number of pinches, the number of crosscaps and handles, and the dimension of the first Z2-homology group. The characterizations are formulated in terms of the existence of a dual graph G ∗ on the same set of edges as G which satisfies algebraic conditions inspired by homology groups and their intersection products.

On Adaptive Testing In Orthogonal Saturated Designs, Daniel T. Voss, Weizhen Wang

Mathematics and Statistics Faculty Publications

Adaptive, size-a step-down tests are provided for the analysis of orthogonal saturated designs. The tests work effectively under effect sparsity, and include as special cases the individual nonadaptive tests of Berk and Picard (1991) and the simultaneous nonadaptive tests of Voss (1988). The approach is similar to that used by Wang and Voss (2003) to construct adaptive confidence intervals, but testing is simpler because one can use the same denominator for all statistics. Step-down tests also have a clear power advantage over simultaneous confidence intervals and analogous single-step tests, as is demonstrated theoretically and assessed via simulation.

Optimal Binary Trees With Height Restrictions On Left And Right Branches, Song Ding

LSU Master's Theses

We begin with background definitions on binary trees. Then we review known algorithms for finding optimal binary search trees. Knuth's famous algorithm, presented in the second chapter, is the cornerstone for our work. It depends on two important results: the Quadrangle Lemma and the Monoticity Theorem. These enabled Knuth to achieve a time complexity of O(n²), while previous algorithms had been O(n³) (n = size of input). We present the known generalization of Knuth's algorithm to trees with a height restriction. Finally, we consider the previously unexamined case of trees with different restrictions on left and …

A System Equivalence Related To Dulac's Extension Of Bendixson's Negative Theorem For Planar Dynamical Systems, Charlie H. Cooke

Mathematics & Statistics Faculty Publications

Bendixson's Theorem [H. Ricardo, A Modem Introduction to Differential Equations, Houghton-Mifflin, New York, Boston, 2003] is useful in proving the non-existence of periodic orbits for planar systems

dx/dt = F(x, y), dy/dt = G (x, y)

in a simply connected domain D, where F, G are continuously differentiable. From the work of Dulac [M. Kot, Elements of Mathematical Ecology, 2nd printing, University Press, Cambridge, 2003] one suspects that system (1) has periodic solutions if and only if the more general system

dx/d tau = B(x, y)F(x, y), dy/d tau = B(x, y)G(x, y)

does, which makes the subcase (1) more …

On The Existence Of Strong Solutions To Autonomous Differential Equations With Minimal Regularity, Charlie H. Cooke

Mathematics & Statistics Faculty Publications

For proving the existence and uniqueness of strong solutions to

dY/dt = F(Y), Y(0) = C,

the most quoted condition seen in elementary differential equations texts is that F(Y) and its first derivative be continuous. One wonders about the existence of a minimal regularity condition which allows unique strong solutions. In this note, a bizarre example is seen where F(Y) is not differentiable at an equilibrium solution; yet unique non-global strong solutions exist at each point, whereas global non-unique weak solutions are allowed. A characterizing theorem is obtained.

Classifying Quadratic Number Fields Up To Arf Equivalence, Jeonghun Kim

LSU Doctoral Dissertations

Two number fields K and L are said to be Arf equivalent if there exists a bijection T : ­Ω_K → Ω_­L of places of K and of L such that K_P and L_TP are locally Arf equivalent for every place P ε Ω_K. That is, |K^*_p/K^*2_p| = |L^*_TP/L^*2_TP|, type[( , )_P] = type[( , )_TP], and Arf(r_P ) = Arf(r_TP ) for every place P ε Ω_K, where r_P is the local …

Limit Theorems For Weighted Stochastic Systems Of Interacting Particles, Jie Wu

LSU Doctoral Dissertations

The goal of this dissertation is to (a) establish the weak convergence of empirical measures formed by a system of stochastic differential equations, and (b) prove a comparison result and compactness of support property for the limit measure. The stochastic system of size n has coefficients that depend on the empirical measure determined by the system. The weights for the empirical measure are determined by a further n-system of stochastic equations. There is a random choice among N types of weights. The existence and uniqueness of solutions of the interacting system, weak convergence of the empirical measures, and the identification …

Representation Properties Of Definite Lattices In Function Fields, Jean Edouard Bureau

LSU Doctoral Dissertations

This work is made of two different parts. The first contains results concerning isospectral quadratic forms, and the second is about regular quadratic forms. Two quadratic forms are said to be isospectral if they have the same representation numbers. In this work, we consider binary and ternary definite integral quadratic form defined over the polynomial ring F[t], where F is a finite field of odd characteristic. We prove that the class of such a form is determined by its representation numbers. Equivalently, we prove that there is no nonequivalent definite F[t]-lattices of rank 2 or 3 having the same theta …

An Inverse Homogenization Design Method For Stress Control In Composites, Michael Stuebner

LSU Doctoral Dissertations

This thesis addresses the problem of optimal design of microstructure in composite materials. The work involves new developments in homogenization theory and numerical analysis. A computational design method for grading the microstructure in composite materials for the control of local stress in the vicinity of stress concentrations is developed. The method is based upon new rigorous multiscale stress criteria connecting the macroscopic or homogenized stress to local stress fluctuations at the scale of the microstructure. These methods are applied to three different types of design problems. The first treats the problem of optimal distribution of fibers with circular cross section …

Filippov's Operator And Discontinuous Differential Equations, Khalid Abdulaziz Alshammari

LSU Doctoral Dissertations

The thesis is mainly concerned about properties of the so-called Filippov operator that is associated with a differential inclusion x'(t) ε F(x(t)) a.e. t ε [0,T], where F : Rⁿ → Rⁿ is given set-valued map. The operator F produces a new set-valued map F[F], which in effect regularizes F so that F[F] has nicer properties. After presenting its definition, we show that F[F] is always upper-semicontinuous as a map from Rⁿ to the metric space of compact subsets of Rⁿ endowed with the Hausdorff metric. Our main approach is to study the …

Integral Cohomology Of The Siegel Modular Variety Of Degree Two And Level Three, Mustafa Arslan

LSU Doctoral Dissertations

In this thesis work Deligne's spectral sequence E^p,q_r with integer coefficients for the embedding of the Siegel modular variety of degree two and level three, A₂(3) into its Igusa compactification, A₂(3)*, is investigated. It is shown that E₃ = E_∞ and this information is applied to compute the cohomology groups of A₂(3) over the integers.

Topics In Quantum Topology, Khaled Moham Qazaqzeh

LSU Doctoral Dissertations

In chapter 1, which represents joint work with Gilmer, we define an index two subcategory of a 3-dimensional cobordism category. The objects of the category are surfaces equipped with Lagrangian subspaces of their real first homology. This generalizes the result of [9] where surfaces are equipped with Lagrangian subspaces of their rational first homology. To define such subcategory, we give a formula for the parity of the Maslov index of a triple of Lagrangian subspaces of a skew symmetric bilinear form over R. In chapter 2, we find two bases for the lattices of the SU(2)-TQFT-theory modules of the torus …

On Moment Conditions For The Girsanov Theorem, See Keong Lee

LSU Doctoral Dissertations

In this dissertation, the well-known Girsanov Theorem will be proved under a set of moment conditions on exponential processes. Our conditions are motivated by the desire to avoid using the local martingale theory in the proof of the Girsanov Theorem. Namely, we will only use the martingale theory to prove the Girsanov Theorem. Many sufficient conditions for the validity of the Girsanov Theorem have been found since the publication of the result by Girsanov in 1960. We will compare our conditions with some of these conditions. As an application of the Girsanov Theorem, we will show the nonexistence of an …

Markov Switching And Jump Diffusion Models With Applications In Mathematical Finance, Shengkun Xie

Theses and Dissertations (Comprehensive)

In this thesis, we study some jump diffusion models with Markov switching and transition densities for Markov switching diffusion processes with and without an absorbing barrier. We work out some analytical results, which have useful applications in mathematical finance and other related fields. The first-passage time problem for a Markov switching model is also studied and European type options and lookback options are computed in closed-form as examples to show that these models can be applied in practice. We apply optimization methods and kernel smoothing techniques to produce some important numerical results that show that jump diffusion with Markov switching …

Traveling Waves In A Suspension Bridge System, Robert A. Ain

UNLV Theses, Dissertations, Professional Papers, and Capstones

In this thesis, we study the traveling waves in a Lazer-McKenna suspension bridge system which is governed by two coupled nonlinear beam and wave equations. The Lazer-McKenna suspension bridge system describes the vertical deflections in the roadbed and supporting cable of a suspension bridge. Based on some basic analysis on the system, we are able to compute traveling waves numerically by using MAPLE. Multiple large-amplitude traveling waves are obtained. The graphs of various traveling waves are displayed and discussed.

Methods For The Estimation Of Missing Values In Time Series, David S. Fung

Theses: Doctorates and Masters

Time Series is a sequential set of data measured over time. Examples of time series arise in a variety of areas, ranging from engineering to economics. The analysis of time series data constitutes an important area of statistics. Since, the data are records taken through time, missing observations in time series data are very common. This occurs because an observation may not be made at a particular time owing to faulty equipment, lost records, or a mistake, which cannot be rectified until later. When one or more observations are missing it may be necessary to estimate the model and also …

An Euler-Type Formula For Ζ(2k +1), Tian-Xiao He, Michael Dancs

Scholarship

In this short paper, we give several new formulas for ζ(n) when n is an odd positive integer. The method is based on a recent proof, due to H. Tsumura, of Euler’s classical result for even n. Our results illuminate the similarities between the even and odd cases, and may give some insight into why the odd case is much more difficult.

Numerical Approximation To Ζ(2n+1), Tian-Xiao He, Michael Dancs

Scholarship

In this short paper, we establish a family of rapidly converging series expansions ζ(2n +1) by discretizing an integral representation given by Cvijovic and Klinowski [3] in Integral representations of the Riemann zeta function for odd-integer arguments, J. Comput. Appl. Math. 142 (2002) 435–439. The proofs are elementary, using basic properties of the Bernoulli polynomials.

Multivariate Expansion Associated With Sheffer-Type Polynomials And Operators, Tian-Xiao He, Leetsch Hsu, Peter Shiue

Scholarship

With the aid of multivariate Sheffer-type polynomials and differential operators, this paper provides two kinds of general expansion formulas, called respectively the first expansion formula and the second expansion formula, that yield a constructive solution to the problem of the expansion of A(ˆt)f([g(t)) (a composition of any given formal power series) and the expansion of the multivariate entire functions in terms of multivariate Sheffer-type polynomials, which may be considered an application of the first expansion formula and the Sheffer-type operators. The results are applicable to combinatorics and special function theory.

On The Convergence Of The Summation Formulas Constructed By Using A Symbolic Operator Approach, Tian-Xiao He, Leetsch Hsu, Peter Shiue

Scholarship

This paper deals with the convergence of the summation of power series of the form Σ_{a ≤ k ≤ b}f(k)x^k, where 0 ≤ a ≤ b < ∞, and {f(k)} is a given sequence of numbers with k ∈ [a, b) or f(t) a differentiable function defined on [a, b). Here, the summation is found by using the symbolic operator approach shown in [1]. We will give a different type of the remainder of the summation formulas. The convergence of the corresponding power series will be determined consequently. Several examples such as the generalized Euler's transformation series will also be given. In addition, we will compare the convergence of the given series transforms.