Blow-up of solutions of nonlinear Schr\"odinger equations with oscillating nonlinearities

The finite time blow-up of solutions for 1-D NLS with oscillating nonlinearities is shown in two domains: (1) the whole real line where the nonlinear source is acting in the interior of the domain and (2) the right half-line where the nonlinear source is placed at the boundary point. The distinctive feature of this work is that the the initial energy is allowed to be non-negative and the momentum is allowed to be infinite in contrast to the previous literature on the blow-up of solutions with time dependent nonlinearities. The common finite momentum assumption is removed by using a compactly supported or rapidly decaying weight function in virial identities - an idea borrowed from Ogawa-Tsutsumi (1991). At the end of the paper, a numerical example satisfying the theory is provided.Comment: 19 page

Finite-parameter feedback control for stabilizing the complex Ginzburg-Landau equation

In this paper, we prove the exponential stabilization of solutions for complex Ginzburg-Landau equations using finite-parameter feedback control algorithms, which employ finitely many volume elements, Fourier modes or nodal observables (controllers). We also propose a feedback control for steering solutions of the Ginzburg-Landau equation to a desired solution of the non-controlled system. In this latter problem, the feedback controller also involves the measurement of the solution to the non-controlled system.Comment: 20 page

Structure of conflict graphs in constraint alignment problems and algorithms

We consider the constrained graph alignment problem which has applications in biological network analysis. Given two input graphs $G_1=(V_1,E_1), G_2=(V_2,E_2)$, a pair of vertex mappings induces an {\it edge conservation} if the vertex pairs are adjacent in their respective graphs. %In general terms The goal is to provide a one-to-one mapping between the vertices of the input graphs in order to maximize edge conservation. However the allowed mappings are restricted since each vertex from $V_1$ (resp. $V_2$) is allowed to be mapped to at most $m_1$ (resp. $m_2$) specified vertices in $V_2$ (resp. $V_1$). Most of results in this paper deal with the case $m_2=1$ which attracted most attention in the related literature. We formulate the problem as a maximum independent set problem in a related {\em conflict graph} and investigate structural properties of this graph in terms of forbidden subgraphs. We are interested, in particular, in excluding certain wheals, fans, cliques or claws (all terms are defined in the paper), which corresponds in excluding certain cycles, paths, cliques or independent sets in the neighborhood of each vertex. Then, we investigate algorithmic consequences of some of these properties, which illustrates the potential of this approach and raises new horizons for further works. In particular this approach allows us to reinterpret a known polynomial case in terms of conflict graph and to improve known approximation and fixed-parameter tractability results through efficiently solving the maximum independent set problem in conflict graphs. Some of our new approximation results involve approximation ratios that are function of the optimal value, in particular its square root; this kind of results cannot be achieved for maximum independent set in general graphs.Comment: 22 pages, 6 figure

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Taha Toros Arşivi, Dosya No: 41-Halide Edip-A. Adnan Adıva

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Taha Toros Arşivi, Dosya No: 476-Şevket Süreyya Aydemir. Not: Gazetenin "Yeni Yayınlar" köşesinde yayımlanmıştır