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Comment: Part of my Laurea thesis. REVTeX4. Minor changes from previous version
Comment: 30 pages, 7 figures, work has been presented at COST workshop ``Networks, Topology, dynamics and Risk'' and submitted to Physica A. Text has been edited, figures have been cleaned up and a new section, a new Appendix, new references and one additional figure have been added
Comment: 18 pages, simplified proof
Karabulut and Sibert (\textit{J. Math. Phys}. \textbf{38} (9), 4815 (1997)) have constructed an orthogonal set of functions from linear combinations of equally spaced Gaussians. In this paper we show that they are actually eigenfunctions of a q-oscillator in coordinate representation. We also reinterpret the coordinate representation example of q-oscillator given by Macfarlane as the functions orthogonal with respect to an unusual inner product definition. It is shown that the eigenfunctions in both q-oscillator examples are infinitely degenerate.
Adapting a method developed for the study of quantum chaos on {\it quantum (metric)} graphs \cite {KS}, spectral $\zeta$ functions and trace formulae for {\it discrete} Laplacians on graphs are derived. This is achieved by expressing the spectral secular equation in terms of the periodic orbits of the graph, and obtaining functions which belongs to the class of $\zeta$ functions proposed originally by Ihara \cite {Ihara}, and expanded by subsequent authors \cite {Stark,Sunada}. Finally, a model of "classical dynamics" on the discrete graph is proposed. It is analogous to the corresponding classical dynamics derived for quantum graphs \cite {KS}.
Skew critical problems occur in continuous and discrete nonholonomic Lagrangian systems. They are analogues of constrained optimization problems, where the objective is differentiated in directions given by an apriori distribution, instead of tangent directions to the constraint. We show semiglobal existence and uniqueness for nondegenerate skew critical problems, and show that the solutions of two skew critical problems have the same contact as the problems themselves. Also, we develop some infrastructure that is necessary to compute with contact order geometrically, directly on manifolds.
Comment: Rewritten version of math-ph/0601040
Comment: 38 pages; main results have been reported earlier on international conferences
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