3,315 research outputs found
Control And Inverse Problems For One Dimensional Systems
Thesis (Ph.D.) University of Alaska Fairbanks, 2009The thesis is devoted to control and inverse problems (dynamical and spectral) for systems on graphs and on the half line. In the first part we study the boundary control problems for the wave, heat, and Schrodinger equations on a finite graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge an equation is defined. The control is acting through the Dirichlet condition applied to all or all but one boundary vertices. The exact controllability in L2-classes of controls is proved and sharp estimates of the time of controllability are obtained for the wave equation. The null controllability for the heat equation and exact controllability for the Schrodinger equation in arbitrary time interval are obtained. In the second part we consider the in-plane motion of elastic strings on a tree-like network, observed from the 'leaves.' We investigate the inverse problem of recovering not only the physical properties, i.e. the 'optical lengths' of each string, but also the topology of the tree which is represented by the edge degrees and the angles between branching edges. It is shown that under generic assumptions the inverse problem can be solved by applying measurements at all leaves, the root of the tree being fixed. In the third part of the thesis we consider Inverse dynamical and spectral problems for the Schrodinger operator on the half line. Using the connection between dynamical (Boundary Control method) and spectral approaches (due to Krein, Gelfand-Levitan, Simon and Remling), we improved the result on the representation of so-called A---amplitude and derive the "local" version of the classical Gelfand-Levitan equations
Soliton self-modulation of the turbulence amplitude and plasma rotation
The space-uniform amplitude envelope of the Ion Temperature Gradient driven
turbulence is unstable to small perturbations and evolves to nonuniform,
soliton-like modulated profiles. The induced poloidal asymmetry of the
transport fluxes can generate spontaneous poloidal spin-up of the tokamak
plasma.Comment: Latex file, 66 pages, 24 postscript figures included. New section on
rotation five new figures, comparison with magnetic pumping dampin
Probabilistic methods for discrete nonlinear Schr\"odinger equations
We show that the thermodynamics of the focusing cubic discrete nonlinear
Schrodinger equation are exactly solvable in dimensions three and higher. A
number of explicit formulas are derived.Comment: 30 pages, 2 figures. To appear in Comm. Pure Appl. Mat
Unitarity, Crossing Symmetry and Duality of the S-matrix in large N Chern-Simons theories with fundamental matter
We present explicit computations and conjectures for scattering
matrices in large {\it } Chern-Simons theories coupled to fundamental
bosonic or fermionic matter to all orders in the 't Hooft coupling expansion.
The bosonic and fermionic S-matrices map to each other under the recently
conjectured Bose-Fermi duality after a level-rank transposition. The S-matrices
presented in this paper may be regarded as relativistic generalization of
Aharonov-Bohm scattering. They have unusual structural features: they include a
non analytic piece localized on forward scattering, and obey modified crossing
symmetry rules. We conjecture that these unusual features are properties of
S-matrices in all Chern-Simons matter theories. The S-matrix in one of the
exchange channels in our paper has an anyonic character; the parameter map of
the conjectured Bose-Fermi duality may be derived by equating the anyonic phase
in the bosonic and fermionic theories.Comment: 66 pages+ 45 pages appendices, 20 figures, Few typos corrected and
few references adde
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