ArticlesCopyright (c) 2015 Dublin Institute of Technology All rights reserved.
http://arrow.dit.ie/scschmatart
Recent documents in Articlesen-usSat, 16 May 2015 01:46:04 PDT3600Eigenfunction Expansions Associated with the One-Dimensional Schrödinger Operator
http://arrow.dit.ie/scschmatart/185
http://arrow.dit.ie/scschmatart/185Thu, 14 May 2015 08:44:39 PDT
We consider the form of eigenfunction expansions associated with the time-independent Schrödinger operator on the line, under the assumption that the limit point case holds at both of the infinite endpoints. It is well known that in this situation the multiplicity of the operator may be one or two, depending on properties of the potential function. Moreover, for values of the spectral parameter in the upper half complex plane, there exist Weyl solutions associated with the restrictions of the operator to the negative and positive half-lines respectively, together with corresponding Titchmarsh-Weyl functions.
]]>
Daphne GilbertSYMMETRY AND REDUCTIONS OF INTEGRABLE DYNAMICAL SYSTEMS: PEAKON AND THE TODA CHAIN SYSTEMS
http://arrow.dit.ie/scschmatart/184
http://arrow.dit.ie/scschmatart/184Thu, 14 May 2015 04:22:26 PDT
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.
]]>
Vladimir Gerdjikov et al.Higher Derivatives of Spectral Functions Associated with One-Dimensional Schrödinger Operators II
http://arrow.dit.ie/scschmatart/183
http://arrow.dit.ie/scschmatart/183Thu, 14 May 2015 00:44:00 PDTDaphne Gilbert et al.ALGEBRAIC AND NUMERICAL EXPLORATION OF FREE ENERGIES FOR MATERIALS WITH MEMORY
http://arrow.dit.ie/scschmatart/182
http://arrow.dit.ie/scschmatart/182Thu, 09 Apr 2015 06:37:23 PDT
Abstract. We study the forms of a range of free energy functionals for materials with memory for two types of strain history, namely sinusoidal and ex- ponential behaviours. The work deals with discrete spectrum materials, which are those with relaxation functions given by sums of decaying exponentials.
]]>
Murrough Golden et al.Derivatives Pricing with Accelerated Trinomial Trees
http://arrow.dit.ie/scschmatart/181
http://arrow.dit.ie/scschmatart/181Mon, 26 Jan 2015 09:08:35 PST
Accelerated Trinomial Trees (ATTs) are a derivatives pricing lattice method that circumvent the restrictive time step condition inherent in standard trinomial trees and explicit finite difference methods (FDMs) in which the time step must scale with the square of the spatial step. ATTs consist of L uniform supersteps each of which contains an inner lattice/trinomial tree with N non-uniform subtime steps. Similarly to implicit FDMs, the size of the superstep in ATTs, a function of N, are constrained primarily by accuracy demands. ATTs can price options up to N times faster than standard trinomial trees (explicit FDMs). ATTs can be interpreted as using risk neutral extended probabilities; extended in the sense that values can lie outside the range [0; 1] on the substep scale but aggregate to probabilities within the range [0; 1] on the superstep scale. Hence it is only strictly at the end of each superstep that a practically meaningful solution may be extracted from the tree. We demonstrate that ATTs with L supersteps are more efficient than competing implicit methods which use L time steps in pricing Black-Scholes American put options and 2-dimensional American basket options. Crucially this performance is achieved using an algorithm that requires only a modest modification of a standard trinomial tree. This is in contrast to implicit FDMs which may be relatively complex in their implementation.
]]>
Conall O'Sullivan et al.A Numerical Method for a Nonlinear Singularly Perturbed Interior Layer Problem Using an Approximate Layer Location
http://arrow.dit.ie/scschmatart/180
http://arrow.dit.ie/scschmatart/180Tue, 16 Dec 2014 02:32:22 PST
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 values of the perturbation parameter. A numerical experiment is presented to demonstrate the convergence rate established.
]]>
Jason QuinnA Linearised Singularly Perturbed Convection-Diffusion Problem with an Interior Layer
http://arrow.dit.ie/scschmatart/179
http://arrow.dit.ie/scschmatart/179Tue, 16 Dec 2014 02:12:45 PST
A linear time dependent singularly perturbed convection-diffusion problem is examined. The convective coefficient contains an interior layer (with a hyperbolic tangent profile), which in turn induces an interior layer in the solution. A numerical method consisting of a monotone finite difference operator and a piecewise-uniform Shishkin mesh is constructed and analysed. Neglecting logarithmic factors, first order parameter uniform convergence is established.
]]>
Eugene O'Riordan et al.Hamiltonian Approach to the Modeling of Internal Geophysical Waves with Vorticity
http://arrow.dit.ie/scschmatart/178
http://arrow.dit.ie/scschmatart/178Thu, 11 Dec 2014 00:37:48 PST
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, determine the equations of motion of the linearized system and calculate the dispersion relation. Finally, limiting cases are examined to recover irrotational and single medium systems as well as an infinite 2 media system.
]]>
Alan CompelliDressing Method and Quadratic Bundles Related to Symmetric spaces: Vanishing Boundary Conditions
http://arrow.dit.ie/scschmatart/177
http://arrow.dit.ie/scschmatart/177Wed, 03 Dec 2014 01:38:32 PST
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.
]]>
Tihomir I. ValchevHamiltonian Formulation of 2 Bounded Immiscible Media with Constant Non-Zero Vorticities and a Common Interface
http://arrow.dit.ie/scschmatart/176
http://arrow.dit.ie/scschmatart/176Tue, 02 Dec 2014 06:07:46 PST
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.
]]>
Alan CompelliHadamard Renormalization of the Stress Energy Tensor in a Spherically Symmetric Black Hole Space-Time with an Application to Lukewarm Black Holes
http://arrow.dit.ie/scschmatart/175
http://arrow.dit.ie/scschmatart/175Thu, 18 Sep 2014 01:52:40 PDT
We consider a quantum field which is in a Hartle-Hawking state propagating in a spherically symmetric black hole space-time. We calculate the components of the stress tensor, renormalized using the Hadamard form of the Green's function, in the exterior region of this space-time. We then specialize these results to the case of the `lukewarm' Riessner-Nordstrom-de Sitter black hole.
]]>
Cormac Breen et al.Hadamard Renormalisation of the Stress Energy Tensor on the Horizons of a Spherically Symmetric Black Hole Space-Time
http://arrow.dit.ie/scschmatart/174
http://arrow.dit.ie/scschmatart/174Thu, 18 Sep 2014 01:45:42 PDT
We consider a quantum field which is in a Hartle-Hawking state propagating in a general spherically symmetric black hole space-time. We make use of uniform approximations to the radial equation to calculate the components of the stress tensor, renormalized using the Hadamard form of the Green's function, on the horizons of this space-time. We then specialize these results to the case of the `lukewarm' Reissner-Nordstrom-de Sitter black hole and derive some conditions on the stress tensor for the regularity of the Hartle-Hawking state
]]>
Cormac Breen et al.Extended Green-Liouville asymptotics and vacuum polarization for lukewarm black holes
http://arrow.dit.ie/scschmatart/173
http://arrow.dit.ie/scschmatart/173Thu, 18 Sep 2014 01:22:56 PDT
We consider a quantum field on a lukewarm black hole spacetime. We introduce a new uniform approximation to the radial equation, constructed using an extension of Green-Liouville asymptotics. We then use this new approximation to construct the renormalized vacuum polarization in the Hartle-Hawking vacuum. Previous calculations of the vacuum polarization rely on the WKB approximation to the solutions of the radial equation, however the nonuniformity of the WKB approximations obscures the results of these calculations near both horizons. The use of our new approximation eliminates these obscurities, enabling us to obtain explicitly finite and easily calculable values of the vacuum polarization on the two horizon
]]>
Cormac Breen et al.Free Energies for Materials with Memory in Terms of State Functionals
http://arrow.dit.ie/scschmatart/172
http://arrow.dit.ie/scschmatart/172Tue, 24 Jun 2014 06:45:35 PDT
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 uncover new free energy functionals. However, while, in principle, the same method should apply to free energieswhich can be given by quadratic forms in terms of It , it emerges that this requirement is very restrictive; indeed, only the minimum free energy can be expressed in such a manner
]]>
Murrough GoldenMatrix G-strands
http://arrow.dit.ie/scschmatart/171
http://arrow.dit.ie/scschmatart/171Fri, 13 Jun 2014 02:03:05 PDT
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 in examples in the language of vectors in R^{3}.
]]>
Darryl Holm et al.On Mikhailov's Reduction Group
http://arrow.dit.ie/scschmatart/170
http://arrow.dit.ie/scschmatart/170Tue, 03 Jun 2014 07:08:17 PDT
We present a generalization of the notion of reduction group which allows one to study in a uniform way certain classes of nonlocal $S$-integrable equations like Ablowitz-Musslimani's nonlocal Schr\"odinger equation. Another aspect of the proposed generalization is the possibility to derive in a systematic way solutions to S-integrable equations with prescribed symmetries.
]]>
Tihomir I. ValchevNumerical Simulations of Steady and Pulsed Non-Adiabatic Magnetised Jets From Young Stars
http://arrow.dit.ie/scschmatart/169
http://arrow.dit.ie/scschmatart/169Thu, 22 May 2014 05:02:15 PDTStephen O'Sullivan et al.An Explicit Scheme for Multifluid Magnetohydrodynamics
http://arrow.dit.ie/scschmatart/168
http://arrow.dit.ie/scschmatart/168Thu, 22 May 2014 04:52:25 PDTStephen O'Sullivan et al.A Three‐Dimensional Numerical Method for Modelling Weakly Ionized Plasmas
http://arrow.dit.ie/scschmatart/167
http://arrow.dit.ie/scschmatart/167Thu, 22 May 2014 04:47:11 PDTStephen O'Sullivan et al.Interacting Jets From Binary Protostars
http://arrow.dit.ie/scschmatart/166
http://arrow.dit.ie/scschmatart/166Thu, 22 May 2014 04:42:53 PDT
Aims. We investigate potential models that could explain why multiple proto-stellar systems predominantly show single jets. During their formation, stars most frequently produce energetic outﬂows and jets. However, binary jets have only been observed in a very small number of systems. Methods. We model numerically 3D binary jets for various outﬂow parameters. We also model the propagation of jets from a speciﬁc source, namely L1551 IRS 5, known to have two jets, using recent observations as constraints for simulations with a new MHD code. We examine their morphology and dynamics, and produce synthetic emission maps. Results. We ﬁnd that the two jets interfere up to the stage where one of them is almost destroyed or engulfed into the second one. We are able to reproduce some of the observational features of L1551 such as the bending of the secondary jet. Conclusions. While the eﬀects of orbital motion are negligible over the jets dynamical timeline, their interaction has signiﬁcant impact on their morphology. If the jets are not strictly parallel, as in most observed cases, we show that the magnetic ﬁeld can help the collimation and refocusing of both of the two jets.
]]>
Gareth Murphy et al.