On The (Non)-Integrability Of The Perturbed Kdv Hierarchy With Generic Self-Consistent Sources,
2011
Bulgarian Academy of Sciences
On The (Non)-Integrability Of The Perturbed Kdv Hierarchy With Generic Self-Consistent Sources, Vladimir Gerdjikov, Georgi Grahovski, Rossen Ivanov
Conference papers
Non-holonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called "squared solutions" (squared eigenfunctions). Such deformations are equivalent to perturbed models with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. This allows one to formulate a set of conditions on the perturbation terms that preserve the integrability. The perturbation corrections to the scattering data and to the corresponding action-angle variables …
Modeling Diverse Physics Of Nanoparticle Self-Assembly In Pulsed Laser-Irradiated Metallic Films,
2011
Western Kentucky University
Modeling Diverse Physics Of Nanoparticle Self-Assembly In Pulsed Laser-Irradiated Metallic Films, Mikhail Khenner
Mathematics Faculty Publications
Presents physics behind dewetting of thin liquid films and mathematical/computational modeling tools (Educational/Research presentation for senior physics majors).
Stability Of A Strongly Anisotropic Thin Epitaxial Film In A Wetting Interaction With Elastic Substrate,
2011
Western Kentucky University
Stability Of A Strongly Anisotropic Thin Epitaxial Film In A Wetting Interaction With Elastic Substrate, Mikhail Khenner, Wondimu T. Tekalign, Margo S. Levine
Mathematics Faculty Publications
The linear dispersion relation for longwave surface perturbations, as derived by Levine et al. Phys. Rev. B 75, 205312 (2007) is extended to include a smooth surface energy anisotropy function with a variable anisotropy strength (from weak to strong, such that sharp corners and slightly curved facets occur on the corresponding Wulff shape). Through detailed parametric studies it is shown that a combination of a wetting interaction and strong anisotropy, and even a wetting interaction alone results in complicated linear stability characteristics of strained and unstrained films.
Stability Of A Strongly Anisotropic Thin Epitaxial Film In A Wetting Interaction With Elastic Substrate,
2011
Western Kentucky University
Stability Of A Strongly Anisotropic Thin Epitaxial Film In A Wetting Interaction With Elastic Substrate, Mikhail Khenner, Wondimu Tekalign, Margo Levine
Mathematics Faculty Publications
The linear dispersion relation for longwave surface perturbations, as derived by Levine et al. Phys. Rev. B 75, 205312 (2007) is extended to include a smooth surface energy anisotropy function with a variable anisotropy strength (from weak to strong, such that sharp corners and slightly curved facets occur on the corresponding Wulff shape). Through detailed parametric studies it is shown that a combination of a wetting interaction and strong anisotropy, and even a wetting interaction alone results in complicated linear stability characteristics of strained and unstrained films.
Neural Spike Renormalization. Part Ii — Multiversal Chaos,
2011
University of Nebraska - Lincoln
Neural Spike Renormalization. Part Ii — Multiversal Chaos, Bo Deng
Department of Mathematics: Faculty Publications
Reported here for the first time is a chaotic infinite-dimensional system which contains infinitely many copies of every deterministic and stochastic dynamical system of all finite dimensions. The system is the renormalizing operator of spike maps that was used in a previous paper to show that the first natural number 1 is a universal constant in the generation of metastable and plastic spike-bursts of a class of circuit models of neurons.
Neural Spike Renormalization. Part I — Universal Number 1,
2011
University of Nebraska-Lincoln
Neural Spike Renormalization. Part I — Universal Number 1, Bo Deng
Department of Mathematics: Faculty Publications
For a class of circuit models for neurons, it has been shown that the transmembrane electrical potentials in spike bursts have an inverse correlation with the intra-cellular energy conversion: the fewer spikes per burst the more energetic each spike is. Here we demonstrate that as the per-spike energy goes down to zero, a universal constant to the bifurcation of spike-bursts emerges in a similar way as Feigenbaum’s constant does to the period-doubling bifurcation to chaos generation, and the new universal constant is the first natural number 1.
On The (Non)-Integrability Of Kdv Hierarchy With Self-Consistent Sources,
2011
Bulgarian Academy of Sciences
On The (Non)-Integrability Of Kdv Hierarchy With Self-Consistent Sources, Vladimir Gerdjikov, Georgi Grahovski, Rossen Ivanov
Articles
Nonholonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called “squared solutions” (squared eigenfunctions). Such deformations are equivalent to a perturbed model with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV6 equation are analysed. This allows for a formulation of conditions on the perturbation terms that preserve its integrability. The perturbation corrections to the scattering data and to the corresponding action-angle (canonical) variables are studied. The analysis shows …
Rational Bundles And Recursion Operators For Integrable Equations On A.Iii-Type Symmetric Spaces,
2011
Bulgarian Academy of Sciences
Rational Bundles And Recursion Operators For Integrable Equations On A.Iii-Type Symmetric Spaces, Vladimir Gerdjikov, Georgi Grahovski, Alexander Mikhailov, Tihomir Valtchev
Articles
We analyze and compare the methods of construction of the recursion operators for a special class of integrable nonlinear differential equations related to A.III-type symmetric spaces in Cartan’s classification and having additional reductions.
Currency Trading Using The Fractal Market Hypothesis,
2011
Technological University Dublin
Currency Trading Using The Fractal Market Hypothesis, Jonathan Blackledge, Kieran Murphy
Articles
We report on a research and development programme in financial modelling and economic security undertaken in the Information and Communications Security Research Group (ICSRG, 2011) which has led to the launch of a new company - Currency Traders Ireland Limited - funded by Enterprise Ireland. Currency Traders Ireland Limited (CTI, 2011) has a fifty year exclusive license to develop a new set of indicators for analysing currency exchange rates (Forex trading). We consider the background to the approach taken and present examples of the results obtained to date.
A Generalized Nonlinear Model For The Evolution Of Low Frequency Freak Waves,
2011
Technological University Dublin
A Generalized Nonlinear Model For The Evolution Of Low Frequency Freak Waves, Jonathan Blackledge
Articles
This paper presents a generalized model for simulating wave fields 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 …
Nonlinear Behaviour Of Sea Surface Waves Based On Low-Gradient Phase-Only Scattering Effects,
2011
Technological University Dublin
Nonlinear Behaviour Of Sea Surface Waves Based On Low-Gradient Phase-Only Scattering Effects, Jonathan Blackledge, Eugene Coyle, Derek Kearney
Conference papers
Nonlinear sea waves generated by the wind, including freak waves, are considered to be phenomena that can be modelled using the nonlinear (cubic) Schrodinger equation, for example. However, there is a problem with this approach which is that sea surface waves, driven by wind speeds of varying strength, must be considered to be composed of two distinct types, namely, linear waves and nonlinear waves. In this paper, we consider a different approach to modelling ‘nonlinear’ waves that is based on a solution to the linear wave equation under a low-gradient, phase-only condition. This approach is entirely compatible with the fluid …
The Sir Model When S(T) Is A Multi-Exponential Function.,
2010
East Tennessee State University
The Sir Model When S(T) Is A Multi-Exponential Function., Teshome Mogessie Balkew
Electronic Theses and Dissertations
The SIR can be expressed either as a system of nonlinear ordinary differential equations or as a nonlinear Volterra integral equation. In general, neither of these can be solved in closed form. In this thesis, it is shown that if we assume S(t) is a finite multi-exponential, i.e. function of the form S(t) = a+ ∑nk=1 rke-σkt or a logistic function which is an infinite-multi-exponential, i.e. function of the form S(t) = c + a/b+ewt, then …
Neural Extensions To Robust Parameter Design,
2010
Air Force Institute of Technology
Neural Extensions To Robust Parameter Design, Bernard Jacob Loeffelholz
Theses and Dissertations
Robust parameter design (RPD) is implemented in systems in which a user wants to minimize the variance of a system response caused by uncontrollable factors while obtaining a consistent and reliable system response over time. We propose the use of artificial neural networks to compensate for highly non-linear problems that quadratic regression fails to accurately model. RPD is conducted under the assumption that the relationship between system response and controllable and uncontrollable variables does not change over time. We propose a methodology to find a new set of settings that will be robust to moderate system degradation while remaining robust …
Non-Classical Symmetry Solutions To The Fitzhugh Nagumo Equation.,
2010
East Tennessee State University
Non-Classical Symmetry Solutions To The Fitzhugh Nagumo Equation., Arash Mehraban
Electronic Theses and Dissertations
In Reaction-Diffusion systems, some parameters can influence the behavior of other parameters in that system. Thus reaction diffusion equations are often used to model the behavior of biological phenomena. The Fitzhugh Nagumo partial differential equation is a reaction diffusion equation that arises both in population genetics and in modeling the transmission of action potentials in the nervous system. In this paper we are interested in finding solutions to this equation. Using Lie groups in particular, we would like to find symmetries of the Fitzhugh Nagumo equation that reduce this non-linear PDE to an Ordinary Differential Equation. In order to accomplish …
Codes From Riemann-Roch Spaces For Y2 = Xp - X Over Gf(P),
2010
Gettysburg College
Codes From Riemann-Roch Spaces For Y2 = Xp - X Over Gf(P), Darren B. Glass, David Joyner, Amy Ksir
Math Faculty Publications
Let Χ denote the hyperelliptic curve y2 = xp - x over a field F of characteristic p. The automorphism group of Χ is G = PSL(2, p). Let D be a G-invariant divisor on Χ(F). We compute explicit F-bases for the Riemann-Roch space of D in many cases as well as G-module decompositions. AG codes with good parameters and large automorphism group are constructed as a result. Numerical examples using GAP and SAGE are also given.
Morphological Evolution Of Single-Crystal Ultrathin Solid Films,
2010
Western Kentucky University
Morphological Evolution Of Single-Crystal Ultrathin Solid Films, Mikhail Khenner
Mathematics Faculty Publications
An introduction to mathematical modeling of ultrathin solid films and the role of such modeling in nanotechnologies: Educational presentation for senior physics majors
Morphological Evolution Of Single-Crystal Ultrathin Solid Films,
2010
Western Kentucky University
Morphological Evolution Of Single-Crystal Ultrathin Solid Films, Mikhail Khenner
Mathematics Faculty Publications
An introduction to mathematical modeling of ultrathin solid films and the role of such modeling in nanotechnologies: Educational/Research presentation for senior physics majors
The Camassa-Holm Hierarchy And Soliton Perturbations,
2010
Technological University Dublin
The Camassa-Holm Hierarchy And Soliton Perturbations, Georgi Grahovski, Rossen Ivanov
Conference papers
The theory of soliton perturbations is considered. The focus is on the usefulness of the so-called Generalised Fourier Transform (GFT). This is a method that involves expansions over the complete basis of “squared solutions” of the spectral problem, associated to the soliton equation. The Inverse Scattering Transform for the corresponding hierarchy of soliton equations can be viewed as a GFT where the expansions of the solutions have generalised Fourier coefficients given by the scattering data. The GFT provides a natural setting for the analysis of small perturbations to an integrable equation: starting from a purely soliton solution one can ’modify’ …
Two Soliton Interactions Of Bd.I Multicomponent Nls Equations And Their Gauge Equivalent,
2010
Bulgarian Academy of Sciences
Two Soliton Interactions Of Bd.I Multicomponent Nls Equations And Their Gauge Equivalent, Vladimir Gerdjikov, Georgi Grahovski
Conference papers
Using the dressing Zakharov-Shabat method we re-derive the effects of the two-soliton interactions for the MNLS equations related to the BD.I-type symmetric spaces. Next we generalize this analysis for the Heisenberg ferromagnet type equations, gauge equivalent to MNLS.
Basic Social Math: A Linguistic Upgrade For Decision Analysis And Social Dynamics Research.,
2010
SIT Graduate Institute
Basic Social Math: A Linguistic Upgrade For Decision Analysis And Social Dynamics Research., Jared Hanson
MA TESOL Collection
There are foundational errors in the mathematical frameworks currently used in Economic and Decision Theories. Recent systemic failures in the interdependent business and educational sectors also show that many practices based on these theories are unsustainable in the changing dynamics of the global economy. A new approach is needed in social science research and systems engineering. This paper examines how the new understandings of complex systems, the role of emotion in cognition, and the core dynamics of decision making can help us correct these errors and to create a general framework for systemic innovation. It argues for the development of …
