Oscillatory And Monotonic Modes Of Long-Wave Marangoni Convection In A Thin Film, 2010 Western Kentucky University

#### Oscillatory And Monotonic Modes Of Long-Wave Marangoni Convection In A Thin Film, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

*Mathematics Faculty Publications*

We study long-wave Marangoni convection in a layer heated from below. Using the scaling k=OBi, where k is the wave number and Bi is the Biot number, we derive a set of amplitude equations. Analysis of this set shows presence of monotonic and oscillatory modes of instability. Oscillatory mode has not been previously found for such direction of heating. Studies of weakly nonlinear dynamics demonstrate that stable steady and oscillatory patterns can be found near the stability threshold.

Oscillatory And Monotonic Modes Of Long-Wave Marangoni Convection In A Thin Film, 2010 Institute of Continuous Media Mechanics

#### Oscillatory And Monotonic Modes Of Long-Wave Marangoni Convection In A Thin Film, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

*Mathematics Faculty Publications*

We study long-wave Marangoni convection in a layer heated from below. Using the scaling k=OBi, where k is the wave number and Bi is the Biot number, we derive a set of amplitude equations. Analysis of this set shows presence of monotonic and oscillatory modes of instability. Oscillatory mode has not been previously found for such direction of heating. Studies of weakly nonlinear dynamics demonstrate that stable steady and oscillatory patterns can be found near the stability threshold.

A Generalized Nonlinear Model For The Evolution Of Low Frequency Freak Waves, 2010 Dublin Institute of Technology

#### A Generalized Nonlinear Model For The Evolution Of Low Frequency Freak Waves, Jonathan Blackledge

*Articles*

This paper presents a generalized model for simulating wavefields associated with the sea surface. This includes the case when `freak waves' may occur through an effect compounded in the nonlinear (cubic) Schrodinger equation. After providing brief introductions to linear sea wave models, `freak waves' and the linear and nonlinear Schrodinger equations, we present a unified model that provides for a piecewise continuous transition from a linear to a nonlinear state. This is based on introducing a fractional time derivative to develop a fractional nonlinear partial differential equation with a stochastic source function. In order to explore the characteristics of this ...

Positive Solutions Of The (N-1, 1) Conjugate Boundary Value Problem, 2010 Kennesaw State University

#### Positive Solutions Of The (N-1, 1) Conjugate Boundary Value Problem, Bo Yang

*Faculty Publications*

We consider the (n - 1, 1) conjugate boundary value problem. Some upper estimates to positive solutions for the problem are obtained. We also establish some explicit sufficient conditions for the existence and nonexistence of positive solutions of the problem.

On The Accuracy Of Explicit Finite-Volume Schemes For Fluctuating Hydrodynamics, 2010 Lawrence Livermore National Laboratory

#### On The Accuracy Of Explicit Finite-Volume Schemes For Fluctuating Hydrodynamics, Aleksandar Donev, Eric Vanden-Eijnden, Alejandro Garcia, John B. Bell

*Faculty Publications*

This paper describes the development and analysis of finite-volume methods for the Landau–Lifshitz Navier–Stokes (LLNS) equations and related stochastic partial differential equations in fluid dynamics. The LLNS equations incorporate thermal fluctuations into macroscopic hydrodynamics by the addition of white noise fluxes whose magnitudes are set by a fluctuation-dissipation relation. Originally derived for equilibrium fluctuations, the LLNS equations have also been shown to be accurate for nonequilibrium systems. Previous studies of numerical methods for the LLNS equations focused primarily on measuring variances and correlations computed at equilibrium and for selected nonequilibrium flows. In this paper, we introduce a more ...

A Class Of Discontinuous Petrov–Galerkin Methods. Ii. Optimal Test Functions, 2010 University of Texas at Austin

#### A Class Of Discontinuous Petrov–Galerkin Methods. Ii. Optimal Test Functions, Leszek Demkowicz, Jay Gopalakrishnan

*Mathematics and Statistics Faculty Publications and Presentations*

We lay out a program for constructing discontinuous Petrov–Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well posedness of the undiscretized problem. Although the question of stable test space choice had attracted the attention of many previous authors, the novelty in our approach lies in the fact we identify a discontinuous Galerkin (DG) framework wherein test functions, arbitrarily close to the optimal ones, can be locally computed. The idea is presented abstractly and its feasibility illustrated ...

Hybridization And Postprocessing Techniques For Mixed Eigenfunctions, 2010 University of Minnesota - Twin Cities

#### Hybridization And Postprocessing Techniques For Mixed Eigenfunctions, Bernardo Cockburn, Jay Gopalakrishnan, F. Li, Ngoc Cuong Nguyen, Jaume Peraire

*Mathematics and Statistics Faculty Publications and Presentations*

We introduce hybridization and postprocessing techniques for the Raviart–Thomas approximation of second-order elliptic eigenvalue problems. Hybridization reduces the Raviart–Thomas approximation to a condensed eigenproblem. The condensed eigenproblem is nonlinear, but smaller than the original mixed approximation. We derive multiple iterative algorithms for solving the condensed eigenproblem and examine their interrelationships and convergence rates. An element-by-element postprocessing technique to improve accuracy of computed eigenfunctions is also presented. We prove that a projection of the error in the eigenspace approximation by the mixed method (of any order) superconverges and that the postprocessed eigenfunction approximations converge faster for smooth eigenfunctions. Numerical ...

A Projection-Based Error Analysis Of Hdg Methods, 2010 Portland State University

#### A Projection-Based Error Analysis Of Hdg Methods, Jay Gopalakrishnan, Bernardo Cockburn, Francisco-Javier Sayas

*Mathematics and Statistics Faculty Publications and Presentations*

We introduce a new technique for the error analysis of hybridizable discontinuous Galerkin (HDG) methods. The technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods. This renders the analysis of the projections of the discretization errors simple and concise. By showing that these projections of the errors are bounded in terms of the distance between the solution and its projection, our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection. We illustrate the technique on a specific HDG ...

A Class Of Discontinuous Petrov–Galerkin Methods. Part I: The Transport Equation, 2010 University of Texas at Austin

#### A Class Of Discontinuous Petrov–Galerkin Methods. Part I: The Transport Equation, Leszek Demkowicz, Jay Gopalakrishnan

*Mathematics and Statistics Faculty Publications and Presentations*

Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG and streamline diffusion methods, in that it uses a space of discontinuous trial functions tailored for stability. The new method, unlike the older approaches, yields optimal estimates for the primal variable in both the element size *h* and polynomial degree *p*, and outperforms the standard upwind DG method.

Parity Periodicity: An Eliminative Approach To The Collatz Conjecture, 2010 Ouachita Baptist University

#### Parity Periodicity: An Eliminative Approach To The Collatz Conjecture, Austin J. Phillips

*Honors Theses*

The 3n + l Conjecture states that when the Collatz function is applied repeatedly to an initial value, the sequence of values generated always converges to 1, regardless of the starting value. This paper strengthens the claim that all such sequences are convergent by showing that certain types of nonconvergent sequences cannot exist. Specifically, no sequence with parity-periodic values can exist This eliminates all possible nontrivially periodic sequences and all divergent sequences with periodic parity. Therefore, if a counterexample to the conjecture exists, It must be a divergent sequence whose values display no parity periodicity.

Temporal Scales For Transport Patterns In The Gulf Of Finland, 2010 Tallinn University of Technology

#### Temporal Scales For Transport Patterns In The Gulf Of Finland, Bert Viikmae, Tarmo Soomere, Mikk Viidebaum, Mihhail Berezovski

*Publications*

The basic time scales for current-induced net transport of surface water and associated time scales of reaching the nearshore in the Gulf of Finland, the Baltic Sea, are analysed based on Lagrangian trajectories of water particles reconstructed from three-dimensional velocity fields by the Rossby Centre circulation model for 1987–1991. The number of particles reaching the nearshore exhibits substantial temporal variability whereas the rate of leaving the gulf is almost steady. It is recommended to use an about 3 grid cells wide nearshore area as a substitute to the coastal zone and about 10–15 day long trajectories for calculations ...

Waves In Materials With Microstructure: Numerical Simulation, 2010 Tallinn University of Technology

#### Waves In Materials With Microstructure: Numerical Simulation, Mihhail Berezovski, Arkadi Berezovski, Juri Engelbrecht

*Publications*

Results of numerical experiments are presented in order to compare direct numerical calculations of wave propagation in a laminate with prescribed properties and corresponding results obtained for an effective medium with the microstructure modelling. These numerical experiments allowed us to analyse the advantages and weaknesses of the microstructure model.

Periodic Solutions Of Neutral Delay Integral Equations Of Advanced Type, 2010 University of Dayton

#### Periodic Solutions Of Neutral Delay Integral Equations Of Advanced Type, Muhammad Islam, Nasrin Sultana, James Booth

*Mathematics Faculty Publications*

We study the existence of continuous periodic solutions of a neutral delay integral equation of advanced type. In the analysis we employ three fixed point theorems: Banach, Krasnosel'skii, and Krasnosel'skii-Schaefer. Krasnosel'skii-Schaefer fixed point theorem requires an a priori bound on all solutions. We employ a Liapunov type method to obtain such bound.

Coarsening In High Order, Discrete, Ill-Posed Diffusion Equations, 2010 University of Dayton

#### Coarsening In High Order, Discrete, Ill-Posed Diffusion Equations, Catherine Kublik

*Mathematics Faculty Publications*

We study the discrete version of a family of ill-posed, nonlinear diffusion equations of order 2n. The fourth order (n=2) version of these equations constitutes our main motivation, as it appears prominently in image processing and computer vision literature. It was proposed by You and Kaveh as a model for denoising images while maintaining sharp object boundaries (edges). The second order equation (n=1) corresponds to another famous model from image processing, namely Perona and Malik's anisotropic diffusion, and was studied in earlier papers. The equations studied in this paper are high order analogues of the Perona-Malik equation ...

Linearly Ordered Topological Spaces And Weak Domain Representability, 2010 University of Dayton

#### Linearly Ordered Topological Spaces And Weak Domain Representability, Joe Mashburn

*Mathematics Faculty Publications*

It is well known that domain representable spaces, that is topological spaces that are homeomorphic to the space of maximal elements of some domain, must be Baire. In this paper it is shown that every linearly ordered topological space (LOTS) is homeomorphic to an open dense subset of a weak domain representable space. This means that weak domain representable spaces need not be Baire.

A Characterization Of Near Outer-Planar Graphs, 2010 Louisiana State University and Agricultural and Mechanical College

#### A Characterization Of Near Outer-Planar Graphs, Tanya Allen Lueder

*LSU Master's Theses*

This thesis focuses on graphs containing an edge whose removal results in an outer-planar graph. We present partial results towards the larger goal of describing the class of all such graphs in terms of a finite list of excluded graphs. Specifically, we give a complete description of those members of this list that are not 2-connected or do not contain a subdivision of a three-spoke wheel. We also show that no members of the list contain a five-spoke wheel.

Closed-Form Solutions To Discrete-Time Portfolio Optimization Problems, 2010 Missouri University of Science and Technology

#### Closed-Form Solutions To Discrete-Time Portfolio Optimization Problems, Mathias Christian Goeggel

*Masters Theses*

"In this work, we study some discrete time portfolio optimization problems. After a brief introduction of the corresponding continuous time models, we introduce the discrete time financial market model. The change in asset prices is modeled in contrast to the continuous time market by stochastic difference equations. We provide solutions for these stochastic difference equations. Then we introduce the discrete time risk-measure and the portfolio optimization problems. We provide closed form solutions to the discrete time problems. The main contribution of this thesis are the closed form solutions to the discrete time portfolio models. For simulation purposes the discrete time ...

A Cross Section Of Oscillator Dynamics, 2010 University of Colorado, Boulder

#### A Cross Section Of Oscillator Dynamics, Jason A. Desalvo

*Applied Mathematics Graduate Theses & Dissertations*

The goal of this research is to explore criteria sufficient to produce oscillations, sample some dynamical systems that oscillate, and investigate synchronization. A discussion on linear oscillators attempts to demonstrate why autonomous oscillators are inherently nonlinear in nature. After describing some criteria on second-order dynamics that ensure periodic orbits, we explore the dynamics of two second-order oscillators in both autonomous and periodically driven fashion. Finally, we investigate the phenomena of synchronization with the nonlinear phase-locked loop. Methods of analysis are exemplified as they become relevant including Poincaré; maps and the Zero-One test for chaos.

The Poincaré-Bendixson theorem is used to ...

High Accuracy Multiscale Multigrid Computation For Partial Differential Equations, 2010 University of Kentucky

#### High Accuracy Multiscale Multigrid Computation For Partial Differential Equations, Yin Wang

*University of Kentucky Doctoral Dissertations*

Scientific computing and computer simulation play an increasingly important role in scientific investigation and engineering designs, supplementing traditional experiments, such as in automotive crash studies, global climate change, ocean modeling, medical imaging, and nuclear weapons. The numerical simulation is much cheaper than experimentation for these application areas and it can be used as the third way of science discovery beyond the experimental and theoretical analysis. However, the increasing demand of high resolution solutions of the Partial Differential Equations (PDEs) with less computational time has increased the importance for researchers and engineers to come up with efficient and scalable computational techniques ...

Skyrmions, Rational Maps & Scaling Identities, 2010 Aristotle University of Thessaloniki

#### Skyrmions, Rational Maps & Scaling Identities, E. G. Charalampidis, T. A. Ioannidou, N. S. Manton

*Mathematics and Statistics Department Faculty Publication Series*

Starting from approximate Skyrmion solutions obtained using the rational map ansatz, improved approximate Skyrmions are constructed using scaling arguments. Although the energy improvement is small, the change of shape clarifies whether the true Skyrmions are more oblate or prolate.