Bounding Stochastic Dependence, Complete Mixability of Matrices, and Multidimensional Bottleneck Assignment Problems

We call a matrix completely mixable if the entries in its columns can be permuted so that all row sums are equal. If it is not completely mixable, we want to determine the smallest maximal and largest minimal row sum attainable. These values provide a discrete approximation of of minimum variance problems for discrete distributions, a problem motivated by the question how to estimate the $\alpha$-quantile of an aggregate random variable with unknown dependence structure given the marginals of the constituent random variables. We relate this problem to the multidimensional bottleneck assignment problem and show that there exists a polynomial $2$-approximation algorithm if the matrix has only $3$ columns. In general, deciding complete mixability is $\mathcal{NP}$-complete. In particular the swapping algorithm of Puccetti et al. is not an exact method unless $\mathcal{NP}\subseteq\mathcal{ZPP}$. For a fixed number of columns it remains $\mathcal{NP}$-complete, but there exists a PTAS. The problem can be solved in pseudopolynomial time for a fixed number of rows, and even in polynomial time if all columns furthermore contain entries from the same multiset

Dynamics on resonant clusters for the quintic non linear Schr\"odinger equation

We construct solutions to the quintic nonlinear Schr\"odinger equation on the circle with initial conditions supported on arbitrarily many different resonant clusters. This is a sequel of a work of Beno\^it Gr\'ebert and the second author.Comment: 11 page

Growth of Sobolev norms for the quintic NLS on T2\mathbb T^2

We study the quintic Non Linear Schr\"odinger equation on a two dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one used in the case of the cubic NLS. This requires an accurate combinatorial analysis.Comment: 41 pages, 5 figures. arXiv admin note: text overlap with arXiv:0808.1742 by other author

Logic Integer Programming Models for Signaling Networks

We propose a static and a dynamic approach to model biological signaling networks, and show how each can be used to answer relevant biological questions. For this we use the two different mathematical tools of Propositional Logic and Integer Programming. The power of discrete mathematics for handling qualitative as well as quantitative data has so far not been exploited in Molecular Biology, which is mostly driven by experimental research, relying on first-order or statistical models. The arising logic statements and integer programs are analyzed and can be solved with standard software. For a restricted class of problems the logic models reduce to a polynomial-time solvable satisfiability algorithm. Additionally, a more dynamic model enables enumeration of possible time resolutions in poly-logarithmic time. Computational experiments are included

Theory of sub-10 fs Generation in Kerr-lens Mode-locked Solid-State Lasers with a Coherent Semiconductor Absorber

The results of the study of ultra-short pulse generation in continuous-wave Kerr-lens mode-locked (KLM) solid-state lasers with semiconductor saturable absorbers are presented. The issues of extremely short pulse generation are addressed in the frames of the theory that accounts for the coherent nature of the absorber-pulse interaction. We developed an analytical model that bases on the coupled generalized Landau-Ginzburg laser equation and Bloch equations for a coherent absorber. We showed, that in the absence of KLM semiconductor absorber produces 2pi - non-sech-pulses of self-induced transparency, while the KLM provides an extremely short sech-shaped pulse generation. 2pi- and pi-sech-shaped solutions and variable-area chirped pulses have been found. It was shown, that the presence of KLM removes the limitation on the minimal modulation depth in absorber. An automudulational stability and self-starting ability were analyzed, too.Comment: revised version, 18 pages, 6 figures, LaTeX, Maple program is available on http://www.geocities.com/optomaple

Controllability of quasi-linear Hamiltonian NLS equations

We prove internal controllability in arbitrary time, for small data, for quasi-linear Hamiltonian NLS equations on the circle. We use a procedure of reduction to constant coefficients up to order zero and HUM method to prove the controllability of the linearized problem. Then we apply a Nash-Moser-H\"ormander implicit function theorem as a black box