# Vol 22, No 3-4 (2016)

**Year:**2016**Articles:**7**URL:**https://journals.ssau.ru/est/issue/view/211

## Full Issue

## Статьи

### ON THE CLASSIFICATION OF FUNCTION GERMS OF TWO VARIABLES THAT ARE EQUIVARIANT SIMPLE WITH RESPECT TO AN ACTION OF THE CYCLIC GROUP OF ORDER THREE

#### Abstract

We consider the problem to classify function germs (C^{2} , 0) → (C, 0), that are equivariant simple with respect to nontrivial actions of the group Z_{3} on C^{2} and on C up to equivariant automorphism germs (C^{2} , 0) → (C^{2} , 0). The complete classification of such germs is obtained in the case of nonscalar action of Z_{3} on C^{2} that is nontrivial in both coordinates. Namely, a germ is equivariant simple with respect to such a pair of actions if and only if it is equivalent to ine of the following germs:

- (x, y) → x
^{3k+1}+ y^{2}, k ≥ 1; - (x, y) → x
^{2}y + y^{3k−1}, k ≥ 2; - (x, y) → x
^{4}+ xy^{3} - (x, y) → x
^{4}+ y^{5}.

**Vestnik of Samara University. Natural Science Series**. 2016;22(3-4):7-13

### ON A MODEL OF OPTIMAL TEMPERATURE CONTROL IN HOTHOUSES

#### Abstract

While growing plants in industrial hothouses it needs to keep the temperature according to round-the-clock graph at the point of growth of plant located at the fixed height. Only small deviations are admitted. To obtain this it is possible to increase the temperature by heating the floor and to decrease the temperature by opening the ventilator windows at the ceiling. We propose and analyse the model based on the heat equation. Physical sense of this problem is that at one end of the infinitely thin rod of length l (the height of the hothouse) we keep during the time T the temperature ϕ(t) (control function), while at the other end we have the given heat flow ψ(t). It requires to find the control function ϕ_{0}(t) such that the temperature at the fixed point c be maximally closed to the given temperature z(t). For the estimation of the control quality we use a quadratic integral functional.

**Vestnik of Samara University. Natural Science Series**. 2016;22(3-4):14-23

### DEZIN NONLOCAL PROBLEM FOR A MIXED-TYPE EQUATION WITH POWER DEGENERATION

#### Abstract

In this article, for the equation of mixed elliptic - hyperbolic type with a power degeneracy on the transition line in a rectangular area are studied the problem Dezin with periodicity conditions and non-local condition, binding values of the normal derivative on the lower base of the rectangle with the value of the target solution on the line type of study. Necessary and sufficient conditions for the uniqueness of the solution were settled, and the uniqueness of the solution was proved problem on the based on completeness of the system of peculiar functions of one-problem or the peculiar.

**Vestnik of Samara University. Natural Science Series**. 2016;22(3-4):24-31

### A NONLOCAL PROBLEM WITH INTEGRAL CONDITION FOR A FOURTH ORDER EQUATION

#### Abstract

In this paper we consider initial-boundary problems with integral conditions for certain fourth order equation. Unique solvability of posed problems is proved. The proof is based on apriori estimates, regularization method, auxiliary problems method, embedding theorems.

**Vestnik of Samara University. Natural Science Series**. 2016;22(3-4):32-50

### THE BOUNDARY VALUE PROBLEM FOR A HYPERBOLIC EQUATION WITH BESSEL OPERATOR IN A RECTANGULAR DOMAIN WITH INTEGRAL BOUNDARY VALUE CONDITION OF THE FIRST KIND

#### Abstract

We consider a boundary value problem with integral nonlocal boundary condition of the first kind for a hyperbolic equation with Bessel differential operator in a rectangular domain. The equivalence of this problem and a local problem with boundary conditions of the second kind is established. The existence and uniqueness of solution of the equivalent problem are proved by means of the spectral method. The solution of the problem is obtained in the form of the Fourier-Bessel series. Convergence is proved in the class of regular solutions.

**Vestnik of Samara University. Natural Science Series**. 2016;22(3-4):51-62

### RESTORING THE SIGNAL BY MODULES OF MEASUREMENT

#### Abstract

The questions of the selection of vectors in finite-dimensional real and complex spaces so that the modules of scalar products of these vectors with the unknown vector were possible to restore it, are observed. Also, the injectivity of nonlinear mappings is studied.

**Vestnik of Samara University. Natural Science Series**. 2016;22(3-4):63-74

### CASES OF INTEGRABILITY CORRESPONDING TO THE PENDULUM MOTION IN THREE-DIMENSIONAL SPACE

#### Abstract

In this article, we systemize some results on the study of the equations of spatial motion of dynamically symmetric fixed rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of a spatial motion of a free rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. The obtained results are systematized and served in the invariant form. We also show the nontrivial topological and mechanical analogies.

**Vestnik of Samara University. Natural Science Series**. 2016;22(3-4):75-97