Physical Sciences and Mathematics Commons^™

Online Resource Platform For Mathematics Education, Marisa Llorens, Edmund Nevin, Eileen Mageean

Conference papers

Engineering education is facing many challenges: a decline in core mathematical skills; lowering entry requirements; and the diversity of the student cohort. One approach to confronting these challenges is to make subject content appropriate to the communication styles of today’s student. To achieve this, a pedagogical shift from the traditional hierarchical approach to learning to one that embraces the use of technology as a tool to enhance the student learning experience is required. By including the student as co-creator of course content, a greater sense of engagement is achieved and a change to one where students become agents of their …

Dressing Method And Quadratic Bundles Related To Symmetric Spaces: Vanishing Boundary Conditions, Tihomir Valchev

Articles

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m) x U(n)). The simplest representative of the corresponding integrable hierarchy is given by a multi-component Kaup-Newell derivative nonlinear Schroedinger equation which serves as a motivational example for our general considerations. We extensively discuss how one can apply Zakharov-Shabat's dressing procedure to derive reflectionless potentials obeying zero boundary conditions. Those could be used for one to construct fast decaying solutions to any nonlinear equation belonging to the same hierarchy. One can distinguish between generic soliton type solutions and rational solutions.

One-Dimensional Weakly Nonlinear Model Equations For Rossby Waves, David Henry, Rossen Ivanov

Articles

In this study we explore several possibilities for modelling weakly nonlinear Rossby waves in fluid of constant depth, which propagate predominantly in one direction. The model equations obtained include the BBM equation, as well as the integrable KdV and Degasperis-Procesi equations.

Work In Progress: Online Resource Platform For Mathematics Education, Marisa Llorens, Edmund Nevin, Eileen Mageean

Conference papers

Mathematics is intrinsic to engineering and as such plays an integral role in the education of engineers. New challenges are being faced in higher education particularly in the areas of student motivation, engagement and attainment. As a result mathematics is often the focus of engineering education research. Traditional methods of delivery such as lectures and tutorials need to evolve to counter these challenges with new pedagogical approaches explored including the use of new technologies. Today’s students are immersed in an increasingly technological world and are willing to adapt to new technological advances. This paper describes a study being undertaken in …

Free Energies In A General Non-Local Theory Of A Material With Memory, Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Articles

A general theory of non-local materials, with linear constitutive equations and memory effects, is developed within a thermodynamic framework. Several free energy and dissipation functionals are constructed and explored. These include an expression for the minimum free energy and a functional that is a free energy for important categories of memory kernels and is explicitly a functional of the minimal state. The functionals discussed have a similar general form to the corresponding expressions for simple materials. A number of new results are derived for them, most of which apply equally to both types of material. In particular, detailed formulae are …

Free Energies For Materials With Memory In Terms Of State Functionals, John Murrough Golden

Articles

Abstract The aim of thiswork is to determinewhat free energy functionals are expressible as quadratic forms of the state functional It which is discussed in earlier papers. The single integral form is shown to include the functional wF proposed a few years ago, and also a further category of functionals which are easily described but more complicated to construct. These latter examples exist only for certain types of materials. The double integral case is examined in detail, against the background of a newsystematic approach developed recently for double integral quadratic forms in terms of strain history, which was used to …

Euler-Poincar´E Equations For G-Strands, Darryl Holm, Rossen Ivanov

Conference papers

The G-strand equations for a map R×R into a Lie group G are associated to a G-invariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The G-strand itself is the map g(t,s):R×R→G, where t and s are the independent variables of the G-strand equations. The Euler-Poincar'e reduction of the variational principle leads to a formulation where the dependent variables of the G-strand equations take values in the corresponding Lie algebra g and its co-algebra, g^∗ with respect to the pairing provided by the variational derivatives of the Lagrangian. We review examples of different G-strand …

Fully Inert Subgroups Of Abelian P-Groups, Brendan Goldsmith, Luigi Salce, Paolo Zanardo

Articles

A subgroup H of an Abelian group G is said to be fully inert in G, if for every endomorphism ϕ of G, the factor group (H+ϕ(H))/H" role="presentation"> is finite. This notion arises in the study of the dynamical properties of endomorphisms (entropy). The principal result of this work is that fully inert subgroups of direct sums of cyclic p-groups are commensurable with fully invariant subgroups of the direct sum.

On The Modelling Of Tsunami Generation And Tsunami Inundation, Frédéric Dias, Denys Dutykh, Laura Cooke, Emiliano Renzi, Themistoklis Stefanakis

Conference papers

While the propagation of tsunamis is well understood and well simulated by numerical models, there are still a number of unanswered questions related to the generation of tsunamis or the subsequent inundation. We review some of the basic generation mechanisms as well as their simulation. In particular, we present a simple and computationally inexpensive model that describes the seabed displacement during an underwater earthquake. This model is based on the finite fault solution for the slip distribution under some assumptions on the kinematics of the rupturing process. We also consider an unusual source for tsunami generation: the sinking of a …

Integrability, Recursion Operators And Soliton Interactions, Boyka Aneva, Georgi Grahovski, Rossen Ivanov, Dimitar Mladenov

Book chapter/book

This volume contains selected papers based on the talks,presentedat the Conference Integrability, Recursion Operators and Soliton Interactions, held in Sofia, Bulgaria (29-31 August 2012) at the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences. Included are also invited papers presenting new research developments in the thematic area. The Conference was dedicated to the 65-th birthday of our esteemed colleague and friend Vladimir Gerdjikov. The event brought together more than 30 scientists, from 6 European countries to celebrate Vladimir's scientific achievements. All participants enjoyed a variety of excellent talks in a friendly and stimulating atmosphere. …

Hamiltonian Approach To The Modeling Of Internal Geophysical Waves With Vorticity, Alan Compelli

Articles

We examine a simplified model of internal geophysical waves in a rotational 2-dimensional water-wave system, under the influence of Coriolis forces and with gravitationally induced waves. The system consists of a lower medium, bound underneath by an impermeable flat bed, and an upper lid. The 2 media have a free common interface. Both media have constant density and constant (non-zero) vorticity. By examining the governing equations of the system we calculate the Hamiltonian of the system in terms of its conjugate variables and perform a variable transformation to show that it has canonical Hamiltonian structure. We then linearize the system, …

Symmetry And Reductions Of Integrable Dynamical Systems: Peakon And The Toda Chain Systems, Vladimir Gerdjikov, Rossen Ivanov, Gaetano Vilasi

Articles

We are analyzing several types of dynamical systems which are both integrable and important for physical applications. The first type are the so-called peakon systems that appear in the singular solutions of the Camassa-Holm equation describing special types of water waves. The second type are Toda chain systems, that describe molecule interactions. Their complexifications model soliton interactions in the adiabatic approximation. We analyze the algebraic aspects of the Toda chains and describe their real Hamiltonian forms.

Hamiltonian Formulation Of 2 Bounded Immiscible Media With Constant Non-Zero Vorticities And A Common Interface, Alan Compelli

Articles

We examine a 2-dimensional water-wave system, with gravitationally induced waves, consisting of a lower medium bound underneath by an impermeable flat bed and an upper medium bound above by an impermeable lid such that the 2 media have a free common interface. Both media have constant density and constant (non-zero) vorticity. By examining the governing equations of the system we calculate the Hamiltonian of the system in terms of it's conjugate variables and per- form a variable transformation to show that it has canonical Hamiltonian structure.

A Numerical Method For A Nonlinear Singularly Perturbed Interior Layer Problem Using An Approximate Layer Location, Jason Quinn

Articles

A class of nonlinear singularly perturbed interior layer problems is examined in this paper. Solutions exhibit an interior layer at an a priori unknown location. A numerical method is presented that uses a piecewise uniform mesh refined around approximations to the first two terms of the asymptotic expansion of the interior layer location. The first term in the expansion is used exactly in the construction of the approximation which restricts the range of problem data considered. The method is shown to converge point-wise to the true solution with a first order convergence rate (overlooking a logarithmic factor) for sufficiently small …

Complex Absorbing Potential Method For Dirac Operators. Clusters Of Resonances, Jimmy Kungsman, Michael Melgaard

Articles

For both nonrelativistic and relativistic Hamiltonians, the Complex Absorbing Potential (CAP) method has been applied extensively to calculate resonances in Physics and Chemistry. We study clusters of resonances for the perturbed Dirac operator near the real axis and, in the semiclassical limit, we establish the CAP method rigorously by showing that resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa.

Evaporation Screening, Scattering, And Quantum Tunneling Near The Horizons Of Schwarzschild And Reissner-Nordstroem Black Holes, Emil Prodanov

Articles

For the radial motion of massive particles with large angular momenta in Schwarzschild geometry and that of massive charged particles with large
angular momenta or energy in a particular range in Reissner-Nordstroem geometry, there exist classically forbidden regions on the outside of
the respective event horizons which scatter certain in-falling geodesics or screen some of the black holes' evaporation by reflecting the emitted

particles back into the black holes. Quantum tunneling across this forbidden regions is studied.

Delayed Rejection Algorithm To Estimate Bayesian Social Networks, Alberto Caimo, Antonietta Mira

Articles

Statistical social network analysis has become a very active and fertile area of research in the recent past. Recent developments in Bayesian computational methods have been successfully applied to estimate social network models. The Delayed rejection (DR) strategy is a modification of the Metropolis-Hastings (MH) algorithms that reduces the variance of the resulting Markov chain Monte Carlo estimators and allows partial adaptation of the proposal distribution. In this paper we show how the DR strategy can be exploited to estimate dyadic independence social network models leading to an average 40% variance reduction relative to the competing MH algorithm, confirming that …

Bergm: Bayesian Exponential Random Graphs In R, Alberto Caimo, Nial Friel

Articles

In this paper we describe the main featuress of the Bergm package for the open-source R software which provides a comprehensive framework for Bayesian analysis for exponential random graph models: tools for parameter estimation, model selection and goodness-of-fit diagnostics. We illustrate the capabilities of this package describing the algorithms through a tutorial analysis of three network datasets.

New Insights On Free Energies And Saint-Venant’S Principle In Viscoelasticity, L. Diseri, G. Gentili, John Murrough Golden

Articles

This work was conceived in 1999 and brought near completion by 2003. Giorgio Gentili was deeply involved in this research until his untimely death. He is greatly missed. Work pressures on the other authors forced a postponement of research on this topic, originally envisaged as lasting a few months but in the event it turned out to be nearly ten years. We now dedicate this work to the memory of Giorgio and to his Family.

On Commutator Socle-Regular Abelian P-Groups, Brendan Goldsmith, Peter Danchev

Articles

We define the notion of a commutator socle-regular Abelian p-group. After establishing some crucial properties of commutator socle-regularity, we investigate its relationship with socle-regularity, strong socle-regularity and projection socle-regularity.

Matrix G-Strands, Darryl Holm, Rossen Ivanov

Articles

We discuss three examples in which one may extend integrable Euler–Poincare ordinary differential equations to integrable Euler–Poincare partial differential
equations in the matrix G-Strand context. After describing matrix G-Strand examples for SO(3) and SO(4) we turn our attention to SE(3) where the matrix G-Strand equations recover the exact rod theory in the convective representation. We then find a zero curvature representation of these equations and establish the conditions under which they are completely integrable. Thus, the G-Strand equations turn out to be a rich source of integrable systems. The treatment is meant to be expository and most concepts are explained …