Mathematics Commons^™

Mass Transfer From A Vertical Flat Plate Due To A Constant Upward Flow, David Mcdonnell, Brendan Redmond, Lawrence J. Crane

Articles

This paper examines the mass transfer from a vertical flat surface of a soluble material due to a constant upward flow. The mass transfer rate due to this upward flow is calculated and used to obtain the distance along the surface at which the boundary layer separates. For relatively large velocities no separation will occur and the solution approaches that of forced convection on a horizontal surface.

Dark Solitons Of The Qiao's Hierarchy, Rossen Ivanov, Tony Lyons

Articles

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'.

G-Strands, Darryl Holm, Rossen Ivanov, James Percival

Articles

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) …

Cyclic Universe With An Inflationary Phase From A Cosmological Model With Real Gas Quintessence, Rossen Ivanov, Emil Prodanov

Articles

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.

The 2-Component Dispersionless Burger's Equation Arising In The Modelling Of Bloodflow, Tony Lyons

Articles

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.

Hadamard Renormalization Of The Stress Energy Tensor In A Spherically Symmetric Black Hole Space-Time With An Application To Lukewarm Black Holes, Cormac Breen, Adrian Ottewill

Articles

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.

Stiefel And Grassmann Manifolds In Quantum Chemistry, Eduardo Chiumiento, Michael Melgaard

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We establish geometric properties of Stiefel and Grassmann manifolds which arise in relation to Slatertype variational spaces in many-particle Hartree-Fock theory and beyond. In particular, we prove thatthey are analytic homogeneous spaces and submanifolds of the space of bounded operators on the single-particle Hilbert space. As a by-product we obtain that they are complete Finsler manifolds. These geometric properties underpin state-of-the-art results on existence of solutions to Hartree-Fock type equations.

Multiple Solutions Of The Quasi Relativistic Choquard Equation, Michael Melgaard, Frederic D. Y. Zongo

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We prove existence of multiple solutions to the quasi relativistic Choquard equations with a scalar potential.

Hadamard Renormalisation Of The Stress Energy Tensor On The Horizons Of A Spherically Symmetric Black Hole Space-Time, Cormac Breen, Adrian Ottewill

Articles

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

On Soliton Interactions For A Hierarchy Of Generalized Heisenberg Ferromagnetic Models On Su(3)/S(U(1) $\Times$ U(2)) Symmetric Space, Vladimir Gerdjikov, Georgi Grahovski, Alexander Mikhailov, Tihomir Valchev

Articles

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 …

Existence Of A Minimizer For The Quasi-Relativistic Kohn-Sham Model, Carlos Argáez García, Michael Melgaard

Articles

We study the standard and extended Kohn-Sham models for quasi-relativistic N-electron Coulomb systems; that is, systems where the kinetic energy of the electrons is given by the quasi-relativistic operator (see article) . For spin-unpolarized systems in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge Z of K nuclei is greater than N-1 and that Z is smaller than a critical charge (see article).

Second Gradient Viscoelastic Fluids: Dissipation Principle And Free Energies, G. Amendola, M. Fabrizio, John Murrough Golden

Articles

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 …

On Super And Hereditarily Hopfian And Co-Hopfian Abelian Groups, Brendan Goldsmith, Katao Kong

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The notions of Hopfian and co-Hopfian groups have been of interest for some time. In this present work we characterize the more restricted classes of hereditarily Hopfian (co-Hopfian) and super Hopfian (co-Hopfian) groups in the case where the groups are Abelian.

On Maximal Relatively Divisible Submodules, Brendan Goldsmith, Paolo Zanardo

Articles

In recent work Goebel and Goldsmith investigated the spectrum of maximal pure subgroups of certain Abelian groups. Here the situation relating to maximal pure submodules of a torsion-free module over an integral domain R is investigated. Connections to the level of coherency are established along with a detailed investigation of the situation where all maximal pure submodules are isomorphic to a product of copies of the ring R.

The Generalised Zakharov-Shabat System And The Gauge Group Action, Georgi Grahovski

Articles

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 Schrodinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type …

Integrable Models For Shallow Water With Energy Dependent Spectral Problems, Rossen Ivanov, Tony Lyons

Articles

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 Z₂ reduction group. The soliton solutions are explicitly obtained.

On Projection-Invariant Subgroups Of Abelian P-Groups, Brendan Goldsmith

Articles

A subgroup P of an Abelian p-group G is said to be projection-invariant in G if Pf is contained in P for all idempotent endomorphisms f. Clearly fully invariant subgroups are projection invariant, but the converse is not true in general. Hausen and Megibben have shown that in many familiar situations these two concepts coincide. In a different direction, the authors have previously introduced the notions of socle-regular and strongly socle-regular groups by focussing on the socles of fully invariant and characteristic subgroups of p-groups. In the present work the authors examine the socles of projection-invariant subgroups of Abelian p-groups.

The Generalised Zakharov-Shabat System And The Gauge Group Action, Georgi Grahovski

Articles

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 …

Abstract Criteria For Multiple Solutions To Nonlinear Coupled Equations Involving Magnetic Schrodinger Operators, Mattias Enstedt, Michael Melgaard

Articles

We consider a system of nonlinear coupled equations involving magnetic Schrodinger

operators and general potentials. We provide a criteria for the existence of multiple

solutions to these equations. As special cases we get the classical results on

existence of innitely many distinct solutions within Hartree and Hartree-Fock

theory of atoms and molecules subject to an external magnetic fields. We also

extend recent results within this theory, including Coulomb system with a constant

magnetic field, a decreasing magnetic field and a "physically measurable" magnetic field.

On The N-Wave Equations And Soliton Interactions In Two And Three Dimensions, Vladimir S. Gerdjikov, Rossen Ivanov, Assen V. Kyuldjiev

Articles

Several important examples of the N-wave equations are studied. These integrable equations can be linearized by formulation of the inverse scattering as a local Riemann–Hilbert problem (RHP). Several nontrivial reductions are presented. Such reductions can be applied to the generic N-wave equations but mainly the 3- and 4-wave interactions are presented as examples. Their one and two-soliton solutions are derived and their soliton interactions are analyzed. It is shown that additional reductions may lead to new types of soliton solutions. In particular the 4-wave _{equations with} Z₂xZ₂ reduction group allow breather-like solitons. Finally it is demonstrated that …

Euler Equations On A Semi-Direct Product Of The Diffeomorphisms Group By Itself, Joachim Escher, Rossen Ivanov, Boris Kolev

Articles

The geodesic equations of a class of right invariant metrics on the semi-direct product of two Diff(S) groups are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra are found.

Solutions To Quasi-Relativistic Multi-Configurative Hartree-Fock Equations In Quantum Chemistry, Carlos Argáez García, Michael Melgaard

Articles

We establish existence of infinitely many distinct solutions to the multi-configurative Hartree-Fock type equations for N-electron Coulomb systems with quasi-relativistic kinetic energy for the n th electron. Finitely many of the solutions are interpreted as excited states of the molecule. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge Z of K nuclei is greater than N-1 and that Z is smaller than a critical charge. The proofs are based on a new application of the Lions-Fang-Ghoussoub critical point approach to nonminimal solutions on a complete analytic Hilbert-Riemann manifold.

On The Acceleration Of Explicit Finite Difference Methods For Option Pricing, Stephen O'Sullivan, Conall O'Sullivan

Articles

Implicit finite difference methods are conventionally preferred over their explicit counterparts for the numerical valuation of options. In large part the reason for this is a severe stability constraint known as the Courant–Friedrichs–Lewy (CFL) condition which limits the latter class’s efficiency. Implicit methods, however, are difficult to implement for all but the most simple of pricing models, whereas explicit techniques are easily adapted to complex problems. For the first time in a financial context, we present an acceleration technique, applicable to explicit finite difference schemes describing diffusive processes with symmetric evolution operators, called Super-Time-Stepping. We show that this method can …

Smooth And Peaked Solitons Of The Camassa-Holm Equation And Applications, Darryl Holm, Rossen Ivanov

Articles

The relations between smooth and peaked soliton solutions are reviewed for the Camassa- Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is expressed for solitons by using the scattering data for its associated isospectral eigenvalue problem, rephrased as a Riemann- Hilbert problem. The momentum map from the action-angle scattering variables T^∗(T^N) to the flow momentum (X^∗) provides the Eulerian representation of the N-soliton solution of CH in terms of the scattering data and squared eigenfunctions of its isospectral eigenvalue problem. The …

A Stability Study Of A New Explicit Numerical Scheme For A System Of Differential Equations With A Large Skew-Symmetric Component, Katharine Gurski, Stephen O'Sullivan

Articles

Abstract. Explicit numerical methods for the solution of a system of stiff differential equations suffer from a time step size that approaches zero in order to satisfy stability conditions. Implicit schemes allow a larger time-step, but require more computations. When the differential equations are dominated by a skew-symmetric component, the problem is not stiffness in the sense that the size of the eigenvalues are unequal, rather the that the real eigenvalues are dominated by imaginary eigenvalues. We present and compare analytical results for stable time step limits for several explicit methods including the super-time-stepping method of Alexiades, Amiez, and Gremaud …

Daemon Decay And Cosmic Inflation, Emil Prodanov

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Quantum tunneling in Reissner--Nordstroem geometry is studied and the tunneling rate is determined. A possible scenario for cosmic inflation, followed by reheating phases and subsequent radiation-domination expansion, is proposed.

On The Singular Weyl-Titchmarsh Function Of Perturbed Spherical Schrödinger Operators, Aleksey Kostenko, Gerald Teschl

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We investigate the singular Weyl–Titchmarsh m-function of perturbed spherical Schrödinger operators (also known as Bessel operators) under the assumption that the perturbation q(x) satisfies xq(x) ∈ L1(0, 1). We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular mfunction belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.

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, 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.

Polynomial Bundles And Generalised Fourier Transforms For Integrable Equations On A.Iii-Type Symmetric Spaces, Vladimir Gerdjikov, Georgi Grahovski, Alexander V. Mikhailov, Tihomir Valchev

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A certain class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions is analysed via the Inverse Scattering Method (ISM). The class contains systems of nonlinear evolution equations (NLEEs) associated with a Lax operator (for the time-evolution) polynomial in the spectral parameter. Using the Wronskian relations, the mapping between the potential and the minimal sets of scattering data is constructed. Furthermore, completeness relations for the `squared solutions' (generalised exponentials) are derived. Next, expansions of Q and its variation are obtained. This demonstrates that the interpretation of the inverse scattering method as a generalized Fourier transform …