Dirac and Plateau billiards in domains with corners

Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C ²-smooth Riemannian metrics g on a smooth manifold X, such that scal _g (x) ≥ κ(x), is closed under C ⁰-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below.     

Decay of electronic excitations in CdS and CdS/ZnS colloidal quantum dots: spectral and kinetic investigations

Using the spectral methods of induced absorption, luminescence, and photostimulated luminescence flash, we have experimentally investigated processes of decay of electronic excitations in CdS colloidal quantum dots and in CdS/ZnS “core-shell” systems synthesized in gelatin by the sol-gel method. It has been shown that the decay of electronic excitations in colloidal quantum dots of this type is predominantly related to a fast localization of nonequilibrium charge carriers on surface defects and their subsequent recombination during times on the order of units and tens of picoseconds. The passage to core-shell systems eliminates, to a large extent, surface defects of the core, some of which are luminescence centers. However, upon using the sol-gel synthesis, a noticeable fraction of luminescence centers are formed in the interior of the CdS quantum dot, which, as well as in the case of CdS/ZnS systems, ensures localization of exciton, blocks its direct annihilation, and maintains recombination radiation.     

Beale-Kato-Majda type condition for Burgers equation

We consider a multidimensional Burgers equation on the torus Td and the whole space Rd. We show that, in case of the torus, there exists a unique global solution in Lebesgue spaces. For a torus we also provide estimates on the large time behaviour of solutions. In the case of Rd we establish the existence of a unique global solution if a Beale-Kato-Majda type condition is satisfied. To prove these results we use the probabilistic arguments which seem to be new.     

Nonlinear Instability in Two Dimensional Ideal Fluids: The Case of a Dominant Eigenvalue

It is proved that any steady 2 dimensional ideal fluid flow is nonlinearly unstable with respect to L ² growth in the velocity, provided there exists an eigenvalue for the linearised Euler equation with Re>. Here is the maximal Lyapunov exponent of the steady flow.     

The functions operating on multiplier algebras

Let 1 < p < ∞ with p ≠ 2. Let G denote one of the groups $\text{T}$ⁿ, $\text{R}$ⁿ, or $\text{Z}$ⁿ. We show that only entire functions operate in certain algebras of multipliers on L_p(G).     

Investigation of the temperature field in a turbulent air flow in the channels with structured packing

Temperature distribution and intensity of temperature pulsations in the airflow in a complex heat exchanger of “Frenkel packing” type were studied experimentally. Measurements were carried out at the airflow between two corrugated plates with triangular embossing, directed at 90° relative to each other. The temperature in the flow was measured by a special thermocouple probe. The hot junction of the thermocouple did not exceed 10 μm. The effect of contact points and Reynolds number on static characteristics of temperature in a turbulent airflow is analysed. The main attention is paid to temperature distribution in an elementary cell. According to the studies, there is a considerable difference between temperature distributions in the flow at the back and front sides of the channel.