Artificial neural network for solving multi-parameter optimization problems

An artificial neural network (ANN) is a mathematical or computational model that simulates the structure and function of biological neural networks used to evaluate or approximate functions at given points. After developing the training algorithm, the resulting model will be used to solve image recognition problems, control problems, optimization, etc. In the process of ANN training, the algorithm of backpropagation is used in the case of convex optimization functions. The article is analyzed test functions for experiments and also study the effect of the number of ANN layers on the quality of approximation in cases one-, two- and three-dimensional. The backpropagation method is improved during the experiments with the help of adaptive gradient, as a result of which more accurate approximations of the functions are obtained. This article also presents the numerical results of test functions.

Estimation of the growth rate of wild ungulate populations in the territory of the Saratov region by the dynamic series of their numbers

The present paper is devoted to the study of the possibility of estimating the reproductive potentials of wild ungulate populations, and possibly other large mammals, by the time series of their numbers. We have found out that this is possible, which is confirmed by the high quality of approximation of the time series of abundance by logistic curves, and the corresponding coefficients of their determination for different species ranged from 75 to 96%. For such calculations, one circumstance is necessary, which is that the population of the studied species has been briefly exposed to some unfavorable factor causing a significant reduction in its numbers with subsequent restoration to the previous level, or the time series should contain a well-expressed and extended section of the transition of the population from some lower level to the upper level of the population, passing into a stationary state. The values of the maximum exponential growth rates of ungulate populations that we obtained do not fundamentally differ from the data available in other researchers’ works. In addition, it should be borne in mind that our method for assessing the reproductive potentials of ungulates is statistical, with features accompanying all such methods, for example, in the presence of statistical errors in all determined parameters. However, the evaluation of the magnitude of these errors is a topic for a separate study.

Signal processing of nonlinear dynamic systems

The paper considers Hermite polynomials that act as a self-similar basis for the decomposition of functions in phase space. It is shown that the equations of behavior of nonlinear dynamical systems are simplified. It is also noted that the wavelet decomposition over Hermite polynomials reduces the number of approximation coefficients and improves the quality of approximation.

Constructing the models of programmed flight for path calculation in designing tactical and anti-aircraft guided missiles

Several models of programmed flight have been constructed to perform calculations on flight path optimization in designing tactical and anti-aircraft-guided missiles. The developed models are based on the determination of interrelated programmed values of altitude and the flight path angle depending on the range which have a differential relationship. The combination of flight altitude and flight-path angle programs allows the users to simulate the steady flight of a guided missile to the calculated endpoint using the methods of proportional control. Good correspondence of the developed models to the physics of flight was shown by assessing the quality of approximation of the developed models of flight paths of anti-aircraft guided missiles obtained using other known models. The obtained approximation error was less than 5 % which indicates a good correspondence of the developed models to the physics of flight. Compliance of the developed models of programmed flight with the intended purpose and the advantage over the most common known models were proved by optimizing the flight paths of the anti-aircraft-guided missile. In most of the considered calculation cases, the value of the objective function was improved to 2.9 %. The flight path was optimized using a genetic algorithm. The developed models have a simple algebraic form and a small number of control parameters are presented in a ready-to-use form and do not require refinement for a concrete task. This allows them to be implemented in design practice without spending much time to speed up the calculation of optimal design variables and optimal flight paths of tactical and anti-aircraft-guided missiles

Dualism of Approximation Principle in Linguadidactics

Dualism is a specific quality of approximation principle. The present research featured the potential of approximation principle for linguadidactics, namely to what degree it can be used to teach Russian as a foreign language and perform correction work in class. The research objective was to assess the efficiency of this principle. The study was based on the method of observation. The article introduces analyses of scientific linguadidactic literature and some typical situations of educational process. The author separates correction work from speech activity, i.e. talking, writing, and reading. The author believes that speech mistakes must be corrected immediately, involving the student in the correction process. Graphic facilities should be used to illustrate the norm. Therefore, in practical linguadidactics, approximation principle should be minimal. Yet, approximation is impossible to avoid in the abovementioned types of speech activity, which hints at the dualism of this principle. Therefore, all errors must be corrected using graphic means, if possible, by both the teacher and the student. Students should be encouraged to participate in the correction process, while the teacher maintains supervisory control.

Walking on Kendall’s Shape Space: Understanding Shape Spaces and Their Coordinate Systems

More and more analyses of biological shapes are using the techniques of geometric morphometrics based on configurations of landmarks in two or three dimensions. A fundamental concept at the core of these analyses is Kendall’s shape space and local approximations to it by shape tangent spaces. Kendall’s shape space is complex because it is a curved surface and, for configurations with more than three landmarks, multidimensional. This paper uses the shape space for triangles, which is the surface of a sphere, to explore and visualize some properties of shape spaces and the respective tangent spaces. Considerations about the dimensionality of shape spaces are an important step in understanding them, and can offer a coordinate system that can translate between positions in the shape space and the corresponding landmark configurations and vice versa. By simulation studies “walking” along that are great circles around the shape space, each of them corresponding to the repeated application of a particular shape change, it is possible to grasp intuitively why shape spaces are curved and closed surfaces. From these considerations and the available information on shape spaces for configurations with more than three landmarks, the conclusion emerges that the approach using a tangent space approximation in general is valid for biological datasets. The quality of approximation depends on the scale of variation in the data, but existing analyses suggest this should be satisfactory to excellent in most empirical datasets.

“SPOCU”: scaled polynomial constant unit activation function

We address the following problem: given a set of complex images or a large database, the numerical and computational complexity and quality of approximation for neural network may drastically differ from one activation function to another. A general novel methodology, scaled polynomial constant unit activation function “SPOCU,” is introduced and shown to work satisfactorily on a variety of problems. Moreover, we show that SPOCU can overcome already introduced activation functions with good properties, e.g., SELU and ReLU, on generic problems. In order to explain the good properties of SPOCU, we provide several theoretical and practical motivations, including tissue growth model and memristive cellular nonlinear networks. We also provide estimation strategy for SPOCU parameters and its relation to generation of random type of Sierpinski carpet, related to the [pppq] model. One of the attractive properties of SPOCU is its genuine normalization of the output of layers. We illustrate SPOCU methodology on cancer discrimination, including mammary and prostate cancer and data from Wisconsin Diagnostic Breast Cancer dataset. Moreover, we compared SPOCU with SELU and ReLU on large dataset MNIST, which justifies usefulness of SPOCU by its very good performance.

Small and Medium-Sized Enterprises in the Russian Federation: Regularities of Spatial Distribution

Small and medium-sized business enterprises (SMEs) have been operating in the Russian Federation since 1991. The study is devoted to the development of methods and tools for assessing the current structure of production volumes, the number of employees and the number of small and medium enterprises, as well as individual entrepreneurs through: economic and mathematical modeling; analysis of statistics for all SMEs of each of the regions in Russia; modeling of the weights of small, medium-sized enterprises, individual entrepreneurs in the overall indicators of SMEs and their distribution by regions of Russia is based on the functions of the density of normal distribution. Association of regions of the country with similar indicators is based on cluster analysis using the k-means method. The nine functions of the normal distribution density obtained in the course of the computational experiment have a high quality of approximation of the empirical data, which was confirmed by the Kolmogorov-Smirnov, Pierson and Shapiro-Wilk tests. Clusters have been formed that unite the regions of the country with similar indicators, namely, the specific weights of production, the number of employees and the number of business entities. The results can be used to solve the problems of institutional, financial and infrastructural support for the development of entrepreneurship in the regions of Russia, and the proposed methodology is applicable for studying the activities of territorial aggregates of enterprises of any state.

Production Functions Small Enterprises and Microenterprises: Russian Regions Data for 2018

The entrepreneurship started in Russia during the process transformation of state economy into market economy, beginning from 1992. The aim of the study was to assess the production functions that describe the volume of production small enterprises in all Russian regions. This functions explain dependence of the amount of release of small enterprises and microenterprises on the salary of their workers and investments in fixed capital. During the research was made assess of two-factor degree production functions that describe the dependence of the volume of production small enterprises on the wages of their employees and investments in fixed assets. Evaluation of functions was made on methods of regression analysis and linearization. The research was based on official empirical territorial data describing the work of small enterprises and microenterprises. In the paper we used statistical information on 82 regions, territories, areas of Russian Federation for 2018. We proved the high quality of approximation of the initial data by the two-factor production functions. Evaluated production functions have increasing return to scale. Proved that economy of Russian regions has not reached saturation with products of small enterprises and microenterprises. They have significant reserves for further development. That is, in all regions there are opportunities to increase the number of enterprises and the number of employees in them. Results of the study and tools for evaluation production functions can be applied in the studying of Russian entrepreneurship, in explanation of the development plans for this economic sector. Such information can be used by governments, regional authorities and municipal management. The methodic and software that were applied in the study process can be used in making analogous studies in various countries with a considerable amount of territories.

Development of an Autoclave Thermal Processes Model for the Simulator of Canned Food Sterilization Process

The article describes an autoclave thermal processes model, which is used for the simulator of canned food sterilization process. The simulator is based on a simulation model that adequately describes the reaction of the autoclave to the actions of the control system and the operator of the sterilization unit. The model’s parameters were obtained by means of experimental data processing. The computer program ”autoclave Model” for simulating sterilization process in the steam and water environment is described. The examples of the canned food’s manual control sterilization modeling are shown. The results of numerical mathematical modeling of canned food sterilization processes in the autoclave showed a high degree of the implemented process models quality of approximation. The calculation schemes done as a result of the mathematical models creation were used to develop a hardwaresoftware complex of the sterilization process simulator. The increase of training level on carrying out process of canned goods sterilization will be provided as a result of designing the simulator of sterilization process in educational process. Consequently reducing defects in production and improving the quality of canned products are expected.