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Recent documents in Articlesen-usThu, 17 Apr 2014 01:58:34 PDT3600$h$-Vectors of Generalized Associahedra and Noncrossing Partitions
http://arrow.dit.ie/scschmatart/147
http://arrow.dit.ie/scschmatart/147Tue, 15 Apr 2014 03:07:47 PDT
A uniform proof is given that the entries of the $h$-vector of the cluster complex $\Delta (\Phi)$, associated by S. Fomin and A. Zelevinsky to a finite root system $\Phi$, count elements of the lattice $\mathbf{L}$ of noncrossing partitions of corresponding type by rank. Similar interpretations for the $h$-vector of the positive part of $\Delta (\Phi)$ are provided. The proof utilizes the appearance of the complex $\Delta (\Phi)$ in the context of the lattice $\mathbf{L}$ in recent work of two of the authors, as well as an explicit shelling of $\Delta (\Phi)$.
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Colum Watt et al.Spectral multiplicity of selfadjoint Schrodinger operators on star-graphs with standard interface conditions
http://arrow.dit.ie/scschmatart/146
http://arrow.dit.ie/scschmatart/146Thu, 03 Apr 2014 05:17:08 PDT
We analyze the singular spectrum of selfadjoint operators which arise from pasting a finite number of boundary relations with a standard interface condition. A model example for this situation is a Schroedinger operator on a star-shaped graph with continuity and Kirchhoff conditions at the interior vertex. We compute the multiplicity of the singular spectrum in terms of the spectral measures of the Weyl functions associated with the single (independently considered) boundary relations. This result is a generalization and refinement of Theorem of I.S. Kac.
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Sergey Simonov et al.Free Energies in a General Non-Local Theory of a Material with Memory
http://arrow.dit.ie/scschmatart/145
http://arrow.dit.ie/scschmatart/145Thu, 03 Apr 2014 05:00:28 PDT
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 given for these quantities in the case of sinusoidal histories. Read More: http://www.worldscientific.com/doi/abs/10.1142/S0218202513500760
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Giovambattista Amendola et al.Weyl-Titchmarsh Type Formula for Periodic Schrodinger Operator with Wigner-von Neumann Potential
http://arrow.dit.ie/scschmatart/144
http://arrow.dit.ie/scschmatart/144Tue, 11 Mar 2014 05:47:44 PDT
Schroedinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner-von Neumann type is considered. The asymptotics of generalized eigenvectors for the values of the spectral parameter from the upper half-plane and on the absolutely continuous spectrum is established. Weyl-Titchmarsh type formula for this operator is proven.
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Sergey Simonov et al.One-Dimensional Weakly Nonlinear Model Equations for Rossby Waves
http://arrow.dit.ie/scschmatart/143
http://arrow.dit.ie/scschmatart/143Thu, 06 Feb 2014 00:37:26 PST
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.
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David Henry et al.Mathematical Modelling at Secondary School: the MACSI-Clongowes Wood College Experience
http://arrow.dit.ie/scschmatart/142
http://arrow.dit.ie/scschmatart/142Tue, 21 Jan 2014 04:57:10 PST
In Ireland, to encourage the study of STEM (science, technology, engineering and mathematics) subjects and particularly mathematics, the Mathematics Applications Consortium for Science and Industry (MACSI) and Clongowes Wood College (County Kildare, Ireland) organized a mathematical modelling workshop for senior cycle secondary school students. Participants developed simple mathematical models for everyday life problems with an open-ended answer. The format and content of the workshop are described and feedback from both students and participating teachers is provided. For nearly all participants, this workshop was an enjoyable experience which showed mathematics and other STEM components in a very positive way.
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J.P.F. Charpin et al.Particle Trajectories in Extreme Stokes Waves Over Inifinte Depth
http://arrow.dit.ie/scschmatart/141
http://arrow.dit.ie/scschmatart/141Fri, 20 Dec 2013 01:42:30 PST
We investigate the velocity field of fluid particles in an extreme water wave over infinite depth. It is shown that the trajectories of the particles within the fluid and along the free surface do not form closed paths over the course of one period, but rather undergo a positive drift in the direction of wave propagation. In addition it is shown that the wave crest cannot form a stagnation point despite the velocity of the fluid being zero there.
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Tony LyonsOn the Quadratic Bundles Related to Hermitian Symmetric Spaces
http://arrow.dit.ie/scschmatart/140
http://arrow.dit.ie/scschmatart/140Thu, 12 Dec 2013 01:32:49 PST
We develop the direct scattering problem for quadratic bundles associated to Hermitian symmetric spaces. We adapt the dressing method for quadratic bundles which allows us to find special solutions to multicomponent derivative Schrodinger equation for instance. The latter is an infinite dimensional Hamiltonian system possessing infinite number of integrals of motion. We demonstrate how one can derive them by block diagonalization of the corresponding Lax pair.
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Tihomir Ilchev ValchevHolomorphic Basis for Families of Subspaces of a Banach Space
http://arrow.dit.ie/scschmatart/139
http://arrow.dit.ie/scschmatart/139Tue, 26 Nov 2013 04:26:54 PST
In this article we investigate the connection between a family of com-plemented subspaces of a Banach space having a holomorphic basis, and being holomorphically complemented.
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Milena Venkova et al.On the Persistence Properties of the Cross-Coupled Camassa-Holm System
http://arrow.dit.ie/scschmatart/138
http://arrow.dit.ie/scschmatart/138Tue, 26 Nov 2013 00:52:07 PST
In this paper we examine the evolution of solutions, that initially have compact support, of a recently-derived system of cross-coupled Camassa-Holm equations. The analytical methods which we employ provide a full picture for the persistence of compact support for the momenta. For solutions of the system itself, the answer is more convoluted, and we determine when the compactness of the support is lost, replaced instead by an exponential decay rate.
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David Henry et al.Phase Transitions in Materials with Thermal Memory: the Case of Unequal Conductivities
http://arrow.dit.ie/scschmatart/137
http://arrow.dit.ie/scschmatart/137Fri, 12 Apr 2013 06:41:36 PDT
A model for thermally induced phase transitions in materials with thermal memory was recently proposed, where the equations determining heatflow were assumed to be the same in both phases. In this work, the model is generalized to the case of phase dependent heatflow relations. The temperature (or coldness) gradient is decomposed into two parts, each zero on one phase and equal to the temperature (or coldness) gradient on the other. However, they vary smoothly over the transition zone. These are treated as separate independent quantities in the derivation of field equations from thermodynamics. Heat flux is given by an integral over the history of the temperature gradient, with different kernels on each phase. Asymptotic analysis is carried out to obtain generalizations of previous results. These involve the jump in temperature across the transition zone and the normal derivatives of the temperature on each phase boundary, which are related to the velocity of the transition zone and a latent heat dependent on this velocity, as well as the speeds of thermal disturbances in the two phases.
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Murrough GoldenG-Strands and Peakon Collisions on Diff(R)
http://arrow.dit.ie/scschmatart/136
http://arrow.dit.ie/scschmatart/136Tue, 09 Apr 2013 07:17:12 PDT
A G-strand is a map g : R x R --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. Some G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that G-strands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of G-strands when G = Diff(R) is the group of diffeomorphisms of the real line R, for which the group product is composition of smooth invertible functions. In the case of peakon-antipeakon collisions, the solution reduces to solving either Laplace's equation or the wave equation (depending on a sign in the Lagrangian) and is written in terms of their solutions. We also consider the complexified systems of G-strand equations for G = Diff(R) corresponding to a harmonic map g : C --> Diff(R) and find explicit expressions for its peakon-antipeakon solutions, as well.
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Darryl Holm et al.Second Gradient Viscoelastic Fluids: Dissipation Principle and Free Energies
http://arrow.dit.ie/scschmatart/135
http://arrow.dit.ie/scschmatart/135Thu, 04 Apr 2013 08:31:51 PDT
We consider a generalization of the constitutive equation for an incompressible second order fluid, by including thermal and viscoelastic effects in the expression for the stress tensor. The presence of the histories of the strain rate tensor and its gradient yields a non-simple material, for which the laws of thermodynamics assume a modified form. These laws are expressed in terms of the internal mechanical power which is evaluated, using the dynamical equation for the fluid. Generalized thermodynamic constraints on the constitutive equation are presented. The required properties of free energy functionals are discussed. In particular, it is shown that they differ from the standard Graffi conditions. Various free energy functionals, which are well-known in relation to simple materials, are generalized so that they apply to this fluid. In particular, expressions for the minimum free energy and a more recently introduced explicit functional of the minimal state are proposed. Derivations of various formulae are abbreviated if closely analogous proofs already exist in the literature.
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G. Amendola et al.Integrable Models for Shallow Water With Energy Dependent Spectral Problems
http://arrow.dit.ie/scschmatart/134
http://arrow.dit.ie/scschmatart/134Thu, 07 Mar 2013 09:51:49 PST
We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependence on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup–Boussinesq equation. We formulate the inverse problem as a Riemann–Hilbert problem with a Z_{2} reduction group. The soliton solutions are explicitly obtained.
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Rossen Ivanov et al.G-Strands
http://arrow.dit.ie/scschmatart/133
http://arrow.dit.ie/scschmatart/133Wed, 12 Dec 2012 01:37:01 PST
A G-strand is a map g(t,s): RxR --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. The SO(3)-strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3)_{K}-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincar'e system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3)_{K}-strand is mapped into a completely integrable generalization of the classical chiral model for the SO(3)-strand. Analogous results are obtained for the Sp(2)-strand. The Sp(2)-strand is the G-strand version of the Sp(2) Bloch-Iserles ordinary differential equation, whose solutions exhibit dynamical sorting. Numerical solutions show nonlinear interactions of coherent wave-like solutions in both cases. Diff(R)-strand equations on the diffeomorphism group G=Diff(R) are also introduced and shown to admit solutions with singular support (e.g., peakons).
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Darryl Holm et al.The 2-Component Dispersionless Burger's Equation Arising In The Modelling Of Bloodflow
http://arrow.dit.ie/scschmatart/132
http://arrow.dit.ie/scschmatart/132Tue, 27 Nov 2012 01:06:59 PST
This article investigates the properties of the solutions of the dispersionless two-component Burgers (B2) equation, derived as a model for blood- ow in arteries with elastic walls. The phenomenon of wave breaking is investigated as well as applications of the model to clinical conditions.
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Tony LyonsDark Solitons of the Qiao's Hierarchy
http://arrow.dit.ie/scschmatart/131
http://arrow.dit.ie/scschmatart/131Mon, 26 Nov 2012 02:33:03 PST
We obtain a class of soliton solutions of the integrable hierarchy which has been put forward in a series of works by Z. Qiao. The soliton solutions are in the class of real functions approaching constant value fast enough at infinity, the so-called 'dark solitons'.
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Rossen Ivanov et al.The Generalised Zakharov-Shabat System and the Gauge Group Action
http://arrow.dit.ie/scschmatart/130
http://arrow.dit.ie/scschmatart/130Tue, 06 Nov 2012 01:41:44 PST
The generalized Zakharov–Shabat systems with complex-valued non-regular Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studied. This study includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations, solvable by the inverse scattering method, and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures. The results are illustrated on the example of the multi-component nonlinear Schr¨odinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type models, related to so(5,C) algebra.
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Georgi GrahovskiOn Soliton Interactions for a Hierarchy of Generalized Heisenberg Ferromagnetic Models on SU(3)/S(U(1) $\times$ U(2)) Symmetric Space
http://arrow.dit.ie/scschmatart/129
http://arrow.dit.ie/scschmatart/129Mon, 15 Oct 2012 03:06:35 PDT
We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator L. The Lax representation is Z2 \times Z2 reduced and is naturally associated with the symmetric space SU(3)/S(U(1) \times U(2)). The simplest nontrivial equation in the hierarchy is a generalization of Heisenberg ferromagnetic model. We construct the N-soliton solutions for an arbitrary member of the hierarchy by using the Zakharov-Shabat dressing method with an appropriately chosen dressing factor. Two types of soliton solutions: quadruplet and doublet solitons are found. The one-soliton solutions of NLEEs with even and odd dispersion laws have different properties. In particular, the one-soliton solutions for NLEEs with even dispersion laws are not traveling waves; their velocities and their amplitudes are time dependent. Calculating the asymptotics of the N-soliton solutions for t \rightarrow \pm \infty we analyze the interactions of quadruplet solitons.
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Vladimir Gerdjikov et al.Cyclic Universe with an Inflationary Phase from a Cosmological Model with Real Gas Quintessence
http://arrow.dit.ie/scschmatart/128
http://arrow.dit.ie/scschmatart/128Mon, 08 Oct 2012 02:16:15 PDT
Phase-plane stability analysis of a dynamical system describing the Universe as a two-fraction uid containing baryonic dust and real virial gas quintessence is presented. Existence of a stable periodic solution experiencing in ationary periods is shown. A van der Waals quintessence model is revisited and cyclic Universe solution again found.
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Rossen Ivanov et al.