Dissertations / Theses on the topic 'Applied mathematics'

For matrix population models with nonnegative, irreducible and primitive inherent projection matrices, the stability of the branch of positive equilibria that bifurcates from the extinction equilibrium as the dominant eigenvalue of the inherent projection matrix increases through one is determined by the direction of bifurcation. However, if the inherent projection matrix is imprimitive this bifurcation becomes more complicated. This is the result of the simultaneous departure of multiple eigenvalues from the unit complex circle. Matrix models with imprimitive projection matrices commonly appear in models of semelparous species, which are characterized by one reproductive event that is often followed by death.

Due to the imprimitivity of the projection matrix, semelparous Leslie models exhibit two contrasting dynamics, either equilibria in which all age classes are present or synchronized cycles in which age classes are separated temporally. The two-stage semelparous Leslie model has index of imprimitivity two, meaning that two eigenvalues simultaneously leave the unit circle when the dominant eigenvalue increases past one. This model exhibits a dynamic dichotomy in which the two steady states have opposite stability properties.

We show that this dynamic dichotomy is a general feature of synchrony models which are characterized by the simultaneous creation of a branch of positive equilibria and a branch of synchronous 2-cycles when the extinction equilibrium destabilizes (Chapter 3). A synchrony model must, necessarily, have index of imprimitivity two but is not limited to models of semelparous species. We provide a specific example of a synchrony model for an iteroparous species which is motivated by observations of a cannibalistic gull population (Chapter 2). We also extend the study of the synchrony model to a Darwinian model which couples population dynamics with the dynamics of a suite of evolving phenotypic traits (Chapter 4). For the evolutionary synchrony model, we show that the dynamic dichotomy occurs provided that fitness, as measured by the spectral radius, is maximized. In addition, we examine the dynamic dichotomy for semelparous species in a continuous-time setting (Chapter 5).

The main subject of this dissertation is smooth incompressible fluids. The emphasis is on the incompressible Euler equations in all of R ² or R³, but many of the ideas and results can also be adapted to other hydrodynamic systems, such as the Navier-Stokes or surface quasi-geostrophic (SQG) equations. A second subject is the modeling of moving contact lines and dynamic contact angles in inviscid liquid-vapor-solid systems under surface tension.

The dissertation is divided into three independent parts: First, we introduce notation and prove useful identities for studying incompressible fluids in a pointwise Lagrangian sense. The main purpose is to provide a unified treatment of results scattered across the literature. Furthermore, we prove several analogs of Constantin’s local pressure formula for other nonlocal operators, such as the Biot-Savart law and Leray projection. Also, we define and study properties of a Lagrangian locally compact Abelian group in terms of which some nonlocal formulas encountered in fluid dynamics may be interpreted as convolutions.

Second, we apply the algebraic theory of scalar polynomial orthogonal invariants to the incompressible Euler equations in two and three dimensions. Using this framework, we give simplified proofs of results of Chae and Vieillefosse. We also investigate other uses of orthogonal transformations, such as diagonalizing the deformation tensor along a particle trajectory, and comment on relative advantages and disadvantages. These techniques are likely to be useful in other orthogonally invariant PDE systems as well.

Third, we propose an idealized inviscid liquid-vapor-solid model for the macroscopic study of moving contact lines and dynamic contact angles. Previous work mostly addresses viscous systems and frequently ignores a singular stress present when the contact angle is not at its equilibrium value. We also examine and clarify the role that disjoining pressure plays and outline a program for further research.

Historically, electrostatic forces and capillary surfaces have been a main focus of scientific inquiry. Recently, with the move towards miniaturization in technology, systems that include the interplay of these two phenomena have become more relevant than ever. This is because at small scales, capillary and electrostatic forces come to dominate familiar macro scale forces and consequently, govern the behavior of many components used in modern technology. In particular, these electro-capillary systems have been applied to areas such self-assembly, “lab-on-a-chip” devices, microelectromechanical systems and mass spectrometry.

In this dissertation, we study two such systems. The first system involves subjecting a planar soap film to a vertically directed electric field. The second is an extension of the first that includes the small effect of gravity (or, similarly, a constant external pressure). Mathematical models for these systems are developed via variational techniques to describe the equilibrium deflection of the soap-film. In contrast to the standard theory, these models include the full effect of capillarity, yielding two prescribed mean curvature problems. These problems are then studied for general and specific domains, using a combination of analytic, asymptotic and numerical techniques. A detailed analysis of the solution set reveals several interesting bifurcation structures. Highlighted areas include a blow-up in the gradient, which occurs at the onset of strictly parametric solutions, and a prediction of the so-called pull-in instability with respect to the aspect ratio of the system, which provides an update to the standard theory. The work here illustrates the effect of including the mean curvature operator in such models and starts to build a general theory of electro-capillary surfaces.

A new graph partitioning algorithm which makes use of a novel objective function and seeding strategy, Product Cut, frequently outperforms standard clustering methods. The solution strategy on solving this objective depends on developing a fast solution method for the systems of graph--based analogues of the screened Poisson equation, which is a well-studied problem in the special case of structured graphs arising from PDE discretization.

In this work, we attempt to improve the powerful Algebraic Multigrid (AMG) method and build upon the recently introduced Product Cut algorithm. Specifically, we study the consequences of incorporating a dynamic determination of the diffusion parameter by introducing a prior to the objective function. This culminates in an algorithm which seems to partially eliminate an advantage present in the original Product Cut algorithm's slower implementation.

In 2010 it was found that God’s number is 20 in the face turn metric. That is, if the Rubik’s cube hasn’t been disassembled, it can always be solved in 20 twists or fewer, but sometimes requires 20 twists. However, the face turn metric only allows one face to be turned at a time for a total of 18 generators, or 18 possible twists at any time. This dissertation allows opposing, parallel faces to be twisted independent amounts at the same time and still get counted as 1 twist for a total of 45 generators. A new optimal-solving program was constructed, and the results so far show that God’s number is at least 16 for the simultaneously-possible turn metric.

I note that in 3 dimensions the simultaneously-possible turn metric is the same as the axial turn metric (or robot turn metric), but not in 4 dimensions nor higher (e.g. 2×2×2×2, 3×3×3×3, 4×4×4×4, etc.—not to be confused with the 3-dimensional 4×4×4 cube). This difference is also described.

A full heap is a labeled infinite partially ordered set with labeling taken from the vertices of an underlying Dynkin diagram, satisfying certain conditions intended to capture the structure of that diagram. The notion of full heaps was introduced by R. Green as an affine extension of the minuscule heaps of J. Stembridge. Both authors applied these constructions to make observations of the Lie algebras associated to the underlying Dynkin diagrams. The main result of this thesis, Theorem 4.7.1, is a complete classification of all full heaps over Dynkin diagrams with a finite number of vertices, using only the general notion of Dynkin diagrams and entirely elementary methods that rely very little on the associated Lie theory. The second main result of the thesis, Theorem 5.1.7, is an extension of the Fundamental Theorem of Finite Distributive Lattices to locally finite posets, using a novel analogue of order ideal posets. We apply this construction in an analysis of full heaps to find our third main result, Theorem 5.5.1, an ADE classification of the full heaps over simply laced affine Dynkin diagrams.

In this dissertation we present various results pertaining to the Parabolic Anderson Model. First we show that the Lyapunov exponent, λ(κ), of the Parabolic Anderson Model in continuous space with Stratonovich differential is O(κ^1/3) near 0. We prove the required upper bound, the lower bound having been proven in (Cranston & Mountford 2006).

Second, we prove the existence of stationary measures for the Parabolic Anderson Model in continuous space with Ito differential. Furthermore, we prove that these measures are associated and determined by the average mass of the initial configuration.

Finally we present progress towards computing the Lyapunov exponent of the Quasi-Stationary Parabolic Anderson Model. We prove a smaller upper bound on λ(κ), improving on the work in (Boldrighini, Molchanov, & Pellegrinotti 2007), but our bound is not sharp. Computing λ(κ) in this model remains an open problem.

Edwards curves are a new normal form for elliptic curves that exhibit some cryptographically desirable properties and advantages over the typical Weierstrass form. Because the group law on an Edwards curve (normal, twisted, or binary) is complete and unified, implementations can be safer from side channel or exceptional procedure attacks. The different types of Edwards provide a better platform for cryptographic primitives, since they have more security built into them from the mathematic foundation up.

Of the three types of Edwards curves—original, twisted, and binary—there hasn't been as much work done on binary curves. We provide the necessary motivation and background, and then delve into the theory of binary Edwards curves. Next, we examine practical considerations that separate binary Edwards curves from other recently proposed normal forms. After that, we provide some of the theory for binary curves that has been worked on for other types already: pairing computations. We next explore some applications of elliptic curve and pairing-based cryptography wherein the added security of binary Edwards curves may come in handy. Finally, we finish with a discussion of e2c2, a modern C++11 library we've developed for Edwards Elliptic Curve Cryptography.

The Bloch representation of an n-qubit channel provides a way to represent quantum channels as certain affine transformations on [special characters omitted]. In higher dimensions (n > 1), the correspondence between quantum channels and their Bloch representations is not well-understood. Partly motivated by the ability to simplify the calculation of information theoretic quantities of a qubit channel using the Bloch representation, in this thesis we investigate the correspondence between a channel and its Bloch representation with an emphasis on unital n-qubit channels, in which case the Bloch representation is linear.

The thesis is divided into three main sections. First we focus our attention on qubit channels. For certain sets of quantum channels, we establish the surprising existence of a special isomorphism into the set of classical channels. We classify the sets of qubit channels with this property and show that information theoretic quantities are preserved by such classical representations. In a natural progression, we prove some well-known facts about SO(3), the proofs of which are either nonexistent or difficult to find in the literature. Some of this work is based on [12, 13].

In the next section, we consider the multi-qubit channels and show that every finite group can be realized as a subgroup of the quantum channels; this approach allows for the construction of a quantum representation for the free affine monoid over any finite group and gives a classical representation for it. We extend some fundamental results from [26, 28] to the multi-qubit case, including that the set of diagonal Bloch matrices is equal to the free affine monoid over the involution group [special characters omitted]. Some of this work appeared in [10].

Lastly, we study the extreme points for the set of n-qubit channels. There are two types of extreme points: invertible and non-invertible; invertible channels are non-singular maps for which the inverse is also a channel. We briefly study the non-invertible extreme points and then parameterize and analyze the invertiblen-qubit Bloch matrices, which form a compact connected Lie group. We calculate the Lie algebra and give an explicit generating set for the invertible Bloch matrices and a maximal torus.

For a field F and the polynomial ring F [x] in a single indeterminate, we define [special characters omitted] = {α ∈ End_F(F [x]) : α(f) ∈ f F [x ] for all f ∈ F [ x]}. Then [special characters omitted] is naturally isomorphic to F [x] if and only if F is infinite. If F is finite, then [special characters omitted] has cardinality continuum. We study the ring [special characters omitted] for finite fields F. For the case that F is finite, we discuss many properties and the structure of [special characters omitted].

The alcove model of Cristian Lenart and Alexander Postnikov describes highest weight crystals of semisimple Lie algebras in terms of so-called alcove walks. We present a generalization, called the quantum alcove model, which has been related to tensor products of column shape Kirillov-Reshetikhin crystals, for all untwisted affine types.

We also investigate Ram's version of Schwer's formula for Hall-Littlewood P-polynomials in type A, which is expressed in terms of the alcove model. We connect it to a formula similar in flavor to the Haglund-Haiman-Loehr formula, which is expressed in terms of fillings of Young diagrams.

A closed three manifold invariant Heegaard Floer homology was generalized to bordered Heegaard Floer homology, defined by Robert Lipshitz, Peter Ozsváth and Dylan Thurston. Bordered Heegaard Floer homology is an invariant of three manifold with connected boundary, and its variant doubly bordered Floer homology is a bimodule defined on three manifold with two disconnected boundary components. In this thesis, we compute bordered Floer homology of (2,2n)-torus link complement.

We show the existence and uniqueness of an invariant measure for the kick-forced Navier-Stokes system on the 2-dimensional sphere, first without deterministic force and then with a time-independent deterministic force. The existence and uniqueness of an invariant measure for the white noise forced Navier-Stokes system on the 2- dimensional sphere without a deterministic forcing is also shown.

We examine the support of the invariant measure and give a description of the support of the measure in general, and in several special cases, for the kick-forced flow. The support of the invariant measure for the white noise forced equations is shown to be the entire space of admissible vector fields of the sphere.

We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method successively extracts information from the previous hierarchical residual to the current hierarchical term at a finer hierarchical scale. As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of finitely many hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.

The solution for sequential Caputo linear fractional differential equations with variable coefficients of order q, 0 < q < 1 can be obtained from symbolic representation form. Since the iterative method developed in Chapter 2 is time-consuming even for the simple linear fractional differential equations with variable coefficients, the direct numerical approximation developed in Chapter 3 is very useful tool when computing the linear and non-linear fractional differential equations of a specific type. This direct numerical method is useful in developing the monotone method and the quasilinearization method for non-linear problems. As an application of this result, we have obtained the numerical solution for a special Ricatti, type of differential equation which blows up in finite time. The generalized monotone iterative method with coupled lower and upper solutions yields monotone natural sequence which converges uniformly and monotonically to coupled minimal and maximal solutions of Caputo fractional boundary value problem. We obtain the existence and uniqueness of sequential Caputo fractional boundary value problems with mixed boundary conditions with the Green's function representation.

The motion of a viscous fluid flow is described by the well-known Navier-Stokes equations. The Navier-Stokes equations contain the conservation laws of mass and momentum, and describe the spatial-temporal change of the fluid velocity field. This thesis aims to investigate numerical solvers for the incompressible Navier-Stokes equations in two and three space dimensions. In particular, we focus on the second-order projection method introduced by Kim and Moin, which was extended from Chorin’s first-order projection method. We apply Fourier-Spectral methods for the periodic boundary condition. Numerically, we discretize the system using central differences scheme on Marker and Cell (MAC) grid spatially and the Crank-Nicolson scheme temporally. We then apply the fast Fourier transform to solve the resulting Poisson equations as sub-steps in the projection method. We will verify numerical accuracy and provide the stability analysis using von Neumann. In addition, we will simulate the particles' motion in the 2D and 3D fluid flow.

Abstract

The aim of this paper is to implement and create a Java applet that performs the simulation of Fu and Hu model .The graphical result is presented on how investor can handle an American call option with discrete dividends paying stock. The technical of stochastic approximation algorithm is used to obtain the gradient, step size and observation length. The thesis is based on Fu and Hu model (2005).