# Vol 25, No 2 (2019)

**Year:**2019**Articles:**7**URL:**https://journals.ssau.ru/est/issue/view/404

## Full Issue

## Статьи

### ON SOME CLASS OF INTERPOLATION FUNCTORS

#### Abstract

As it is well known, the Gustavsson — Peetre construction, using the concept of unconditional convergence in Banach spaces, provides an important class of interpolation functors. In this paper, we deﬁne a new close construction, based on the use of the so-called random unconditional convergence. We ﬁnd necessary and suﬃcient conditions, which being imposed on a generating function give us an interpolation functor deﬁned on the category of Banach couples. It is shown that calculating the latter functor for a couple of Orlicz spaces results in the ”natural” interpolation theorem. Moreover, we obtain conditions that guarantee the coincidence of this functor with the corresponding Gustavsson — Peetre functor, as well as with the Calder´on — Lozanovskii method.

**Vestnik of Samara University. Natural Science Series**. 2019;25(2):7-20

### ON A CONSTRUCTION OF A FRAME IN THE HARDY SPACE DEFINED ON THE TWO-DIMENSIONAL POLYDISC

#### Abstract

In the article we present a construction of a representing system based on the discretized Szego kernel in the Hardy space deﬁned on the two-dimensional polydisc. An answer to the question on the existence of representing systems based on reproducing kernels depends signiﬁcantly on the space under consideration. It is well known that in the Hardy space there are no both bases and Duﬃn — Shaeﬀer frames, based on the discretized Szego kernel. We use a notion of a Banach frame which generalizes the concept of the Duﬃn — Shaeﬀer frame. Having constructed a Banach frame, we can say that any function from the Hardy space can be represented as a series of discretized kernels.

**Vestnik of Samara University. Natural Science Series**. 2019;25(2):21-29

### THEORETICAL AND EXPERIMENTAL INVESTIGATION OF CRACK PROPAGATION DIRECTION. PART I

#### Abstract

In the present paper the crack propagation direction angles on the basis of three diﬀerent fracture criteria are found. The maximum tangential stress criterion, the minimum strain energy density criterion and the deformation criterion are used and analysed. The generalized forms of these criteria have been used. It implies that the crack propagation direction angles are obtained with the Williams series expansion in which the higher order terms are kept. The calculations are performed in Waterloo Maple computer algebra software. The analysis of the crack propagation direction angles show that the inﬂuence of the higher order terms can’t be ignored. The angles diﬀer considerably when the higher order terms are taken into account.

**Vestnik of Samara University. Natural Science Series**. 2019;25(2):30-54

### THEORETICAL AND EXPERIMENTAL INVESTIGATION OF CRACK PROPAGATION DIRECTION. PART II

#### Abstract

The paper is devoted to experimental study of the crack propagation direction angles under mixed mode loading in the plate with the central crack inclined at diﬀerent angles. Fracture mechanics criteria are discussed and compared. In the present paper the crack propagation direction angles on the basis of three diﬀerent fracture criteria are found. The maximum tangential stress criterion, the minimum strain energy density criterion and the deformation criterion are used and analysed. The generalized forms of these criteria have been used. It implies that the crack propagation direction angles are obtained with the Williams series expansion in which the higher order terms are kept. The calculations are performed in Waterloo Maple computer algebra software. The analysis of the crack propagation direction angles show that the inﬂuence of the higher order terms can’t be ignored. The angles diﬀer considerably when the higher order terms are taken into account.

**Vestnik of Samara University. Natural Science Series**. 2019;25(2):55-74

### DIGITAL PROCESSING OF INTERFEROGRAMS OBTAINED BY THE PHOTOELASTICITY METHOD

#### Abstract

The article is devoted to the digital processing interferograms (isochromatic fringe patterns) obtained by the photoelasticity method. An application to interpretate the isochromatic fringe patterns is developed. It allows us to automate this procedure, gets to avoid the routine and time-consuming work. The features of the developed software package are described in detail by means of the classic problem of a diametrically compressed disk. The application’s algorithm includes following main steps: image pre-processing, localization interference fringe, fringe tracing. The application creates a text ﬁle containing all the necessary data to further determine the stress-strain state (isochromatic fringe’s number and locus of fringes).

**Vestnik of Samara University. Natural Science Series**. 2019;25(2):75-91

### FLOW CURVATURE APPLIED TO MODELLING OF CRITICAL PHENOMENA

#### Abstract

Modeling of critical phenomena is a very important problem, which has direct applied application in many branches of science and technology. In this paper we regard a modiﬁcation of the low curvature method applied to construction of invariant manifolds of autonomous fast-slow dynamic systems. We compared a new method with original ones via ﬁnding duck-trajectories and their multidimensional analogues surfaces with variable stability. Comparison was used a three-dimensional autocatalytic reaction model and a model of the burning problem.

**Vestnik of Samara University. Natural Science Series**. 2019;25(2):92-99

### MODELING THE GROWTH RATES OF ALIEN INSECTS SPECIFIED DIFFERENTIATED BY STAGES OF ONTOGENESIS

#### Abstract

Our studies are devoted to various aspects of invasive processes in biosystems. When invading aggressive insects, the resistance of the biotic environment is signiﬁcant, but the ﬁnal time may be completely absent. Under conditions of high speciﬁc fecundity, there will be non-stationary regimes of changes in the abundance of individuals in the population. An outbreak with a phase of explosive growth is realized, so we modeled one particular variant of an outbreak of populations earlier. Outbreaks are associated with a number of changes in physiological regulation that are observed in ecodynamics under extreme conditions of the species and environment. A classic example of changes is the appearance of migratory winged forms in the locust and the usually ﬂightless leaf beetle introduced in the Stavropol Territory *Zygogramma suturalis *during the formation of a population wave front with a huge density of individuals. Traditional mathematical models for describing the rates of average weight gain for individuals of a generation cannot consider situations of rapid invasions, where competition and survival factors at diﬀerent stages of insect development are very diﬀerent, since the density of generations varies by orders of magnitude. Eﬀect on growth of local congestion value may not be constant in ontogenesis. The factor of the total number is reﬂected diﬀerently at diﬀerent stages of life. The purpose of the simulation is to obtain a bistable and ﬂexible dynamic system. The article proposes a model for a diﬀerentiated description of weight gain at three stages of development of insects with an incomplete cycle of transformations by three diﬀerential equations conjugated by initial conditions. As a result, a continuously discrete dynamic system with a hybrid representation of time on the life cycle interval is implemented. We will use the model when supplementing the hybrid computing structure for calculating the rate of decline of individuals of generations. Mortality changes dramatically with the launch and attenuation of an invasive outbreak. Here, the deviation of growth rates from optimal values plays a role. The idea of the work is that all processes: changes in numbers or growth for insects should be simulated by the stages of development of their ontogenesis.

**Vestnik of Samara University. Natural Science Series**. 2019;25(2):100-109