Thurston conjectured that a closed triangulated 3-manifold in which every edge has degree 5 or 6, and no two edges of degree 5 lie in a common 2-cell, has word-hyperbolic fundamental group. We establish Thurston’s conjecture by proving that such a manifold admits a piecewise Euclidean metric of non-positive curvature and the universal cover contains no isometrically embedded flat planes. The pr...

We consider the problems of completing a low-rank positive semidefinite square matrix M or a low-rank rectangular matrix N from a given subset of their entries. Following the approach initiated by Singer and Cucuringu [20] we study the local and global uniqueness of such completions by analysing the structure of the graphs determined by the positions of the known entries of M or N . We present ...

1 Flajolet, P. and M. Soria, General combinatorial schemas: Gaussian limit distributions and exponential tails, Discrete Mathematics 114 (1993) 159-180. Under general conditions, the number of components in combinatorial structures defined as sequences, cycles or sets of components admits a Gaussian limit distribution together with an exponential tail. The results are valid, assuming simple ana...

This article substantiates the importance of introducing topic “Raman and giant Raman scattering light” into course “Nanophysics”, taught to students Faculty Physics National University Uzbekistan named after Mirzo Ulugbek. The essence light is explained from point view quantum mechanics. Detailed information given on polarization, dipole moment molecules, stokes anti-stokes components spectrum...

Here, issues connected with characteristic stochastic practices are considered. In the first part, the plausibility of covering the arrangements of an improvement issue on subjective subgraphs is studied. The impulse for this strategy is a state where an advancement issue must be settled as often as possible for discretionary illustrations. Then, a preprocessing stage is considered that would q...

We consider the homogeneous and the non-homogeneous convex relaxations for combinatorial penalty functions defined on support sets. Our study identifies key differences in the tightness of the resulting relaxations through the notion of the lower combinatorial envelope of a set-function along with new necessary conditions for support identification. We then propose a general adaptive estimator ...

In the design of branch and bound methods for NP-hard combinatorial optimization problems, dominance conditions have always been applied. In this work we show how the use of dominance conditions within search tree algorithms can lead to non trivial worst-case upper time bounds for the considered algorithms on bounded combinatorial optimization problems. We consider here the MIN 3-SET COVERING p...

In the design of branch and bound methods for NP-hard combinatorial optimization problems, dominance conditions have always been applied. In this work we show how the use of dominance conditions within search tree algorithms can lead to non trivial worst-case upper time bounds for the considered algorithms on bounded combinatorial optimization problems. We consider here the MIN 3-SET COVERING p...