Morphisms of Ball's Structures of Groups and Graphs

We introduce and study two kinds of morphisms between ball's structures related to groups and graphs.     

Detector of thermal radiation based on high-temperature superconductor matrix with magnetooptical image-visualization system

We propose a concept for constructing a thermal imaging system of a new type that is based on unique magnetic properties of high-temperature superconductors and on a noncontact magnetooptic information reading from a matrix sensing element.     

The GRANIT spectrometer

The existence of quantum states of matter in a gravitational field was demonstrated recently in the Institut Laue–Langevin (ILL), Grenoble, in a series of experiments with ultra cold neutrons (UCN). UCN in low quantum states is an excellent probe for fundamental physics, in particular for constraining extra short-range forces; as well as a tool in quantum optics and surface physics. The GRANIT is a follow-up project based on a second-generation spectrometer with ultra-high energy resolution, permanently installed in ILL. It has been constructed in framework of an ANR grant; and will become operational in 2011.     

Polytope Lyapunov Functions for Stable and for Stabilizable LSS

We present a new approach for constructing polytope Lyapunov functions for continuous-time linear switching systems (LSS). This allows us to decide the stability of LSS and to compute the Lyapunov exponent with a good precision in relatively high dimensions. The same technique is also extended for stabilizability of positive systems by evaluating a polytope concave Lyapunov function (“antinorm”) in the cone. The method is based on a suitable discretization of the underlying continuous system and provides both a lower and an upper bound for the Lyapunov exponent. The absolute error in the Lyapunov exponent computation is estimated from above and proved to be linear in the dwell time. The practical efficiency of the new method is demonstrated in several examples and in the list of numerical experiments with randomly generated matrices of dimensions up to 10 (for general linear systems) and up to 100 (for positive systems). The development of the method is based on several theoretical results proved in the paper: the existence of monotone invariant norms and antinorms for positively irreducible systems, the equivalence of all contractive norms for stable systems and the linear convergence theorem.     

Experimental study of optical gas-dynamic processes of polymeric material ablation by ultrashort laser pulses

Optical gas-dynamic processes occurring in polymeric targets ((CH₂O) _n , (C₂F₄) _n ) exposed to ultrashort laser pulses (τ _0.5 ∼ 45 − 70 fs; λ _I,II,III = 266, 400, 800 nm; and E/S ∼ 0.1 − 40 J/cm² at r ₀ ∼ 20 μm) were studied under normal conditions and in vacuum (p ∼ 10⁻² Pa). The dynamics of the mass flow from the target surface (m′ ∼ 10⁻⁵ − 10⁻⁴ g/J) was studied and the spectral-energy thresholds of laser ablation, the electron density distribution (n _e ∼ 10¹⁴ − 10¹⁸ cm⁻³), the mass-averaged velocity of the material flow from the target surface (∼ 10³ m/s), and the chemical composition and average temperature in the near-surface plasma formation (T ∼ 5000 K) were determined using interference microscopy, emission spectroscopy, and shadowgraphy.     

Bourbaki spaces of topological groups

A study is presented of the relationship between the topological and uniformity properties of a group G and the spaces (G), (G) of all nonempty closed subsets and closed subgroups of G. A base for the neighborhood system of a closed subset X of G is formed by the sets S(X, U)={Y Y XU, X YU}, where U ranges over all neighborhoods of the identity in G. Criteria are obtained for the space (G) and some of its subspaces to be totally bounded and locally totally bounded. Some classes of groups with compact spaces (G) are described. It is proved that the spaces (G), (G) are complete in the case of projective metrizable groups G.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 542–549, April, 1990.     

Asymptotics of products of nonnegative random matrices

Asymptotic properties of products of random matrices ξ _k = X _k …X ₁ as k → ∞ are analyzed. All product terms X _i are independent and identically distributed on a finite set of nonnegative matrices A = {A ₁, …, A _m }. We prove that if A is irreducible, then all nonzero entries of the matrix ξ _k almost surely have the same asymptotic growth exponent as k→∞, which is equal to the largest Lyapunov exponent λ(A). This generalizes previously known results on products of nonnegative random matrices. In particular, this removes all additional “nonsparsity” assumptions on matrices imposed in the literature.We also extend this result to reducible families. As a corollary, we prove that Cohen’s conjecture (on the asymptotics of the spectral radius of products of random matrices) is true in case of nonnegative matrices.