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Control and inverse problems for partial differential equations on graphs
by
Sergei Avdonin
University of Alaska Fairbanks
Differential equations on graphs are used to describe many physical processes such as mechanical vibrations of multi-linked flexible structures usually composed of flexible beams or strings, propagation of electro-magnetic waves in networks of optical fibers, heat flow in multi-link networks, and also electron flow in quantum mechanical circuits. In this talk we discuss some known and new controllability results for partial differential equations on graphs and their relations to boundary inverse problems.
Date received: October 29, 2007
Stochastic wave equation driven by a fractional Brownian motion
by
Boris P Belinskiy
University of Tennessee at Chattanooga
Coauthors: Peter Caithamer, Indiana University Northwest, USA
We consider a linear stochastic wave equation driven by fractional-in-time noise. We prove the existence and uniqueness of the weak solution. We also study the expected energy associated with wave equation and improve our previous results on that matter. Specifically, we find the iff condition of the convergence of the series representing the expected energy. We discuss the smoothness of the solution. We consider both cases H > [1/2] and H < [1/2] for the Hurst parameter.
Date received: September 21, 2007
Quasi-stationary solitons for Langmuir waves in plasmas
by
Anjan Biswas
Delaware State University
The multiple-scale perturbation analysis is used to study the perturbed nonlinear Schrödinger’s equation, that describes the Langmuir waves in plasmas. The perturbation terms include the non-local term due to nonlinear Landau damping. The WKB type ansatz is used to define the phase of the soliton that captures the corrections to the pulse where the standard soliton perturbation theory fails.
Date received: October 17, 2007
The equivariant index theorem for Dirac operators
by
Jochen Brüning
Department of Mathematics, Humboldt-Universität Berlin
Coauthors: Franz Kamber, Ken Richardson
We consider a Dirac operator, D, on the sections of a hermitian vector bundle over a compact manifold M which is odd with respect to a given supersymmetry and equivariant with respect to the action of a compact Lie group G. Fixing a unitary representation, r, of G we derive an index formula for the restriction of D to the sections transforming like r. This formula is local on the strata of M naturally defined by the G-action and also involves h-invariants of the links. It generalizes previous work of Atiyah, for tori, and of Kawasaki, for orbifolds; it extends to transversally elliptic first order differential operators with only small changes.
Date received: October 31, 2007
Cable Formation for Finite-Gap Solutions of the Vortex Filament Flow
by
Annalisa Calini
Department of Mathematics, College of Charleston, Charleston SC 29424, USA
Coauthors: Thomas Ivey, Department of Mathematics, College of Charleston
The simplest model of self-induced dynamics of a vortex filament in an ideal fluid leads to an integrable nonlinear evolution equation closely related to the cubic focussing nonlinear Schroedinger equation. Closed finite-gap solutions of the vortex filament flow provide examples of evolving curves whose topological features can be related to their algebro-geometric description. We describe how the theory of isoperiodic deformations (developed by Grinevich and Schmidt, after Krichever) can be used to generate a family of closed finite-gap solutions of increasingly higher genus close to a multiply covered circle. Each step of the deformation process corresponds to constructing a cable on the previous filament, whose knot type is determined from the deformation scheme, and is invariant under the time evolution.
Date received: September 25, 2007
Inequalities of Hardy-Sobolev and Hardy-Gagliardo-Nirenberg type
by
William Desmond Evans
Cardiff University, Wales, UK.
Coauthors: A. Balinsky, D. Hundertmark, R. T. Lewis
The lecture will report on recent joint work with A. Balinsky, D. Hundertmark and R. T. Lewis on Sobolev and Gagliardo-Nirenberg inequalities in Rn which also relate to a modified Hardy-type inequality involving the operator L: = x·∇. A pseudo-Poincaré inequality with respect to the operator semigroup {e-tL*L}t > 0 has a central role in the proof, the approach being reminiscent of that of M. Ledoux in establishing an improved Sobolev inequality which highlights a connection between Sobolev embeddings and heat kernel bounds.
Date received: August 31, 2007
Semi-analytic spectral methods
by
Colin Fox
Physics, University of Otago
Coauthors: Hyuck Chung (Acoustics Research Centre, Auckland University)
Analytic techniques allow explicit solution of wave propagation and scattering in simple geometries, or for composite geometries typically limited to asymptotic regimes of 'large' or 'small' lengths. These solutions provide scaling laws that aid engineering and design, and explicit formulas for the inverse problem. Semi-analytic methods provide these tools in complex composite geometries by augmenting analytic spectral methods with numerical calculations that a computer can perform essentially exactly. We examine these methods in the setting of ocean wave scattering. There, removal of exponentials allows exact evaluation of solutions, while application of Liouville's theorem reduces the Dirichlet-to-Neumann map to an operator between low-dimensional spaces. Scattering in composite structures is then easily characterized. These methods are applied to determining low-frequency sound transmission through lightweight timber-framed construction, which is typical in New Zealand. Those solutions agree closely with measurements, and were recently used in the design of a timber floor with excellent sound insulation properties.
Date received: October 26, 2007
Absolutely Continuous Spectrum for the Anderson Model on More General Trees
by
Florina Halasan
University of British Columbia
We study the Anderson Model on trees that have a variation in their coordination number. Using geometric tools, we prove that the Anderson Hamiltonian has absolutely continuous spectrum for small disorder.
Date received: October 29, 2007
Quasi-intersections of Isoenergetic Surfaces: Description in Terms of Determinants.
by
Yulia Karpeshina
UAB
When the Laplacian is perturbed by a periodic potential, self-intersections of isoenergetic surfaces get transformed into quasi-intersections. We define quasi-intersections in terms of determinants and use Rouche's Theorem to establish results on stability of quasi-intersections.
Date received: October 9, 2007
On unitary conformal holonomy
by
Felipe Leitner
University of Stuttgart/Uni Auckland
To any space with conformal structure there is an invariant notion of holonomy which is defined via the canonical Cartan connection. I will discuss in my talk CR-spaces and their Fefferman construction from the view point of conformal holonomy. This is the case of unitary conformal holonomy. The conformal Einstein condition can also be characterised in terms of holonomy. This will allow me to present a construction and characterisation result about transversally symmetric pseudo-Einstein and Fefferman Einstein spaces.
Date received: October 22, 2007
Hardy and Rellich Inequalities with Remainders
by
Roger T Lewis
University of Alabama at Birmingham
Coauthors: W Desmond Evans
In this talk our primary concern is with the establishment of weighted Hardy inequalities in Lp(W) and Rellich inequalities in L2(W) depending upon the distance to the boundary of domains W ⊂ Rn with a finite diameter D(W). Improved constants are presented in most cases.
Date received: October 1, 2007
Optical tomography in media with varying index of refraction.
by
Stephen McDowall
Western Washington University, USA
Optical tomography refers to the use of near-infrared light to determine the optical absorption and scattering properties of a medium. In the stationary Euclidean setting the dynamics are modeled by the radiative transport equation, which assumes that in the absence of interaction particles follow straight lines. Here we shall study the problem in the presence of a (simple) Riemannian metric where particles follow the geodesic flow of the metric. This non-Euclidean geometry models a medium which has a continuously varying refractive index. We will present results for all dimensions, in the case of full angular-dependent measurements and in the case where the information available at the boundary is averaged over angle. We show that knowledge of the albedo operator, that which maps incoming flux to outgoing flux at the boundary, uniquely determines the absorption and scattering properties of the medium. In dimensions three and higher we assume prior knowledge of the metric while in dimension two it can be shown that the albedo operator also determines the metric. When the measurements are averaged over angle, we are able to determine the absorption, and spatial dependence of the scattering assuming a-priori knowledge of its angular dependence.
Date received: September 27, 2007
An ill-posed problem in scattering theory
by
Boris Pavlov
The University of Auckland
Coauthors: J. Bruning
Scattered waves in the scattering problem for Helmholtz resonator are obtained via breeding of the standing waves in the inner domain and plain running waves in the outer domain. Breeding them with a help of an appropriate Dirichlet-to-Neumann map requires solution of an ill-posed problem with compact integral operators on the common boundary of the inner and outer domain. We suggest a regularization method for this ill-posed problem.
Date received: October 7, 2007
Spectral properties of a magnetic quantum Hamiltonian on a strip
by
Georgi Raikov
Pontificia Universdidad Catolica de Chile
Coauthors: Philippe Briet (CPT, Marseilles, France),
Eric Soccorsi (CPT, Marseilles, France)
Date received: October 7, 2007
Bubbles tend to the boundary
by
Gunter Stolz
University of Alabama at Birmingham
Coauthors: Jeff Baker, Michael Loss
How should a given compactly supported potential be placed into a bounded domain so as to minimize or maximize the first Neumann eigenvalue of the Schrodinger operator on this domain? We answer this question for rectangular domains and reflection symmetric potentials. As an application we determine the spectral minimum of the so-called random displacement model of an electron in a deformed lattice.
Date received: October 2, 2007
Spectral properties of non-local eigenvalue problems
by
Graeme Wake
Centre for Mathematics in Industry, Massey University Auckland, New Zealand
Coauthors: Ronald Begg, Massey and Canterbury Universities
In the course of developing generic models of the evolution of cell cohorts, simultaneously undergoing growth and fission, which are structured by size (usually taken as DNA content), we have encountered an unusual class of functional differential equations. The solution of these functional partial differential equations possess the behaviour described as a "steady-size distribution (SSD)", where the size distribution is constant in shape but not magnitude, as time evolves. The solutions are candidates for probability distributions, scaled by a time-factor, whose Lyapunov exponent satifies a non-local singular Sturm-Liouville eigenvalue problem (NLSSLEVP). The SSD-like behaviour is usually globally-attracting, but this is established only for some special cases. This paper will outline some interesting properties of these and some similar problems. The support of the NZ Institute of Mathematics and its Applications by the award of a Maclaurin Fellowship for the current part of this work is gratefully acknowledged.
Date received: September 20, 2007
The mathematics of Imaging in Magnetic Resonance Elastography
by
David Wall
University of Canterbury
Coauthors: Peter Olsson, School of Engineering, Jönköping University
Early and accurate detection of tumours in the female breast is of clinical importance. Use of a combination of low frequency acoustic waves excitation and magnetic resonance imaging (MRI) of the displacement field within the tissue is thought to result in better image reconstruction of the tumour than from standard scattering measurements. Breast cancer tumours are about ten times stiffer than normal tissue.
This elastography imaging problem is an inverse problem. The nature of an inverse problem is that it is ill-conditioned. We consider properties of the mathematical map which describes how the elastic properties of the tissue being reconstructed vary with the field measured by MRI. This map is a nonlinear mapping and our interest is in proving certain conditioning and regularity results for this operator which occurs naturally in this problem of imaging in elastography. In this treatment we consider the tissue to be linearly elastic, anisotropic and spatially heterogeneous. The emphasize on anisotropy in this problem should provide better contrast of the tumour to the background tissue.
Date received: October 28, 2007