Generalized Artin–Hasse and Iwasawa Formulas for the Hilbert Symbol in a High-Dimensional Local Field

The generalized Hilbert symbol in a cyclotomic extension of an absolutely unramified higher local field of characteristic 0 with a perfect last residue field of characteristic p > 2 is considered. Generalized Artin–Hasse and Iwasawa formulas are derived from an explicit Kummer Type Vostokov formula. Bibliography: 13 titles.     

On Lorentz invariant distributions

n-point Lorentz invariant tempered distributions with the supports for one-point only in are described.     

Flow fluctuation measurement in the flow-through path of continuous electric-discharge CO<Subscript>2</Subscript>-laser contour

We have experimentally proved the applicability of hot-wire anemometer method to measure fluctuations at low pressures and gas flow rates when medium is no longer considered as continuous. Use of individual calibration at working conditions serves to apply standard technique for determination of sensitivity coefficients of hot-wire probes. Additional investigations are necessary to take into account influence of Knudsen number and, possibly, other parameters in heat transfer law presented as criterial dependence of Nusselt number on Reynolds number. Hot-wire measurement of mean velocity distribution and their fluctuations in the flow-through path of electric-discharge CO₂ laser have been performed. The level of velocity fluctuations in the vicinity of anode is high that is caused by the zone of interacting flows from the ventilators rotating in opposite directions forming in the symmetry plane. Placing of the damping screen between the ventilator and air gap allows decreasing fluctuations intensity to the acceptable level. It has been ascertained that the vortex shedding with high intensity of fluctuations which influence is noticeable in significant part of air gap is formed in the wake behind the streamlined cathode. However, pulsation level in the proper gap does not exceed permissible values.     

Spontaneous Magnetization in the Two-Dimensional Ising Model

We suggest a new definition of the spontaneous magnetization. We calculate this spontaneous magnetization for the two-dimensional Ising model with free boundary conditions.     

Thermal explosion in multiple phase media

Thermal explosion (or self-ignition) is one of the mainly interesting and dangerous phenomena in the combustion. Homogeneous combustible gas mixtures with highly exothermic reaction reveal only two main types of the behaviour: explosive and slow. Transient regime between these main types of behaviour is traditionally called the thermal explosion limit. Addition of another phase (solid or liquid) into combustible gas mixture can lead to essential changes in the dynamical situation. This paper deals with an interesting phenomenon of thermal explosion in two-phase media, so-called thermal explosion with delay. Existence and type of temperature delay before the self-ignition depends on mass and thermal properties of the second phase. The analysis is based on qualitative theory of singular perturbed systems and duck techniques.     

Ising model on the generalized Bruhat-Tits tree

The partition function and the correlation functions of the Ising model on the generalized Bruhat-Tits tree are calculated. We computed also the averages of these correlation functions when the corresponding vertices are attached to the boundary of the generalized Bruhat-Tits tree.     

Relation of orbital integrals on SO(5) and PGL(2)

We relate the “Fourier” orbital integrals of corresponding spherical functions on thep-adic groups SO(5) and PGL(2). The correspondence is defined by a “lifting” of representations of these groups. This is a local “fundamental lemma” needed to compare the geometric sides of the global Fourier summation formulae (or relative trace formulae) on these two groups. This comparison leads to conclusions about a well known lifting of representations from PGL(2) to PGSp(4). This lifting produces counter examples to the Ramanujan conjecture.     

Single photon decay of double hole states in atoms

The probability of single-photon decay of highly excited double hole state of atoms is considered. The concrete calculations are performed in the lowest order of perturbation theory in interelectron interaction for states (2s)^-2 in Ne, (2s)^-2 and (2p)^-2 in double ionized Ar?Ar⁺⁺ and (4d)^-2 in Xe.     

Synthesis of unsymmetrical 2-(2-vinyl-4H-chromen-4-ylidene)malononitrile dyes via Knoevenagel reaction

A series of new chromophores based on 4-(dicyanomethylene)-2-vinyl 4H-chromenes was synthesized and their physical properties (UV–Vis, NMR, mass spectrometry) were studied. The chromophores show moderate or good solubility in organic solvents. Solvatochromism was investigated for compounds in solvents of low (1.4-dioxane, chlorobenzene, methylene chloride) and high polarity (acetone, acetonitrile). The calculations of the first hyperpolarizability (β) was performed using M052X/6-31+G(d) approximation. Chromophore’s molecules are characterized by high calculated values of the first hyperpolarizability (β) that is only 15% lower than that of FTC analogs.