On convexity of a planar domain with a pair of concave tomography projections

We describe simple sufficient conditions on tomography-type measurements of a planar set which imply convexity of this set. The cases of partial convexity and higher-dimensional sets are considered as well.     

Estimates of the motion of detonation waves in a gravitating gas

According to present ideas [1, 2], the light elements contained in the outer layers of stars may be detonated when there is gravitational collapse of the core of the star accompanied by neutrino emission that initiates the detonation. The main attention in investigations has been devoted to the physical processes associated with the thermonuclear reactions and the propagation of the radiation, less attention being paid to the gas dynamics as a whole. The present paper considers the spherically symmetric problem of the adiabatic motion of a gravitating perfect gas in the presence of a detonation wave resulting from an inhomogeneous gravitational collapse of gas at zero pressure or a loss of equilibrium. Following a method developed by Golubyatnikov [3, 4], the authors construct a system of integrodifferential inequalities that, in particular, determine the law of motion of a detonation wave through a known initial state of the gas. The corresponding self-similar problems are investigated as examples.Translated from Izvestiya Akademii nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 140–145, March–April, 1984.     

Stability problems in certain inverse problems of reconstruction of convex compacta from their projections

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 3, pp. 50–57, May–June, 1992.     

A three-cell model of the initial stage of development of a proneural cluster

We construct a 9-dimensional nonlinear dynamical system that simulates the initial stage of interaction of three adjacent cells in the proneural cluster of Drosophila melanogaster. We describe the conditions of existence of three stable equilibrium points in the phase space of this system, list its other equilibrium points, and provide a biological interpretation.     

Integral inequalities in the dynamics of a gravitating gas

In the framework of Newtonian mechanics, a study is made of the spherically symmetric problem of the adiabatic motion of a gravitating perfect gas in the presence of a shock wave produced by inhomogeneous gravitational collapse or a point explosion. The method of Golubyatnikov [1, 2] is used to construct a system of integro—differential inequalities that determine, in particular, the law of motion of the shock wave if the initial state of the gas is known. The investigated examples include some self-similar and nearly self-similar solutions to the problem of the gravitational contraction of dust with the formation of a strong shock wave, possibly with the release of energy; the self-similar problem of a point explosion in a gas at rest; and also the problem of the equilibrium of a gas sphere for =4/3 and arbitrary distribution of the entropy. In these cases, the inequalities reduce to algebraic relations and can be solved numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 169–173, September–October, 1980.     

On the acceleration of shock waves and concentration of energy

By a series of simple examples related to exact solutions of problems in gas dynamics and magnetohydrodynamics, possible mechanisms of acceleration of shock waves and concentration of energy are elucidated. The acceleration of a shock wave is investigated in the problem of motion of a plane piston at a constant velocity in the case when the initial density of the medium drops in the presence of constant counterpressure. It is shown that in this situation a “blow-up” regime is induced by a shock wave going to infinity in finite time even for limited work of the piston. A simple spherically symmetric solution with a converging shock wave is constructed and shown to lead to the concentration of energy. A general method for solving one-dimensional non-self-similar problems related to matching the equilibrium state to a motion with homogeneous deformation on a shock wave is discussed; this method leads to a solution in quadratures.     

On the Nonexistence of Distance-Regular Graphs with Intersection Array $$ \{ 53,40,28,16; 1,4,10,28\}$$

Journal of Applied and Industrial Mathematics - We consider $$Q$$ -polynomial graphs of diameter $$4 $$ . Alongside the infinite series of intersection arrays $$ \{m(2m+1),(m-1)(2m+1),m^2, m;...     

Self-similar gas motions in a gravity field

The decelerating effect of a homogeneous gravity field on the plane shock wave acceleration near the outside surface of a gas layer, initially in equilibrium, is analyzed within the framework of the self-similar formulation. A qualitative investigation is performed, the cases in which the first integrals exist are noted, and certain exact solutions of the problem are obtained at different power laws of the initial density variation.