Informatics, computer technology and management. Рубрика в журнале - Siberian Aerospace Journal
Статья научная
Failures of elements during the operation of technical and many other systems are, as a rule, random in nature. This leads to various models of the recovery process, studied in probability theory and mathematical reliability theory. During the restoration process, failed elements are restored or replaced with new ones, and there is often a change in the costs and quality of the restored elements (time-to-failure distribution functions). The work examines the cost function (average cost of restoration) in the process of restoration of order k1, k2 , in which, according to a certain rule, the costs of each restoration and the distribution functions of operating time change. Considering, that the recovery function (average number of failures) is well studied in reliability theory, a solution to the integral equation for the cost function is obtained through the recovery function of the model under consideration. For the order restoration process 1 2 k , k , a formula is obtained for calculating the cost function through the restoration function of a simple process formed by the convolution of all distribution functions of the periodic part. For practical application, explicit formulas are obtained for the cost function during the restoration process, in which the periodic part is distributed according to an exponential law or an Erlang law of order m with the same exponent α. The resulting formulas can be used to study the properties of the cost function and solve optimization problems in strategies for carrying out the restoration process in terms of price, quality, risk, if, for example, the average number of failures is taken as quality, the average cost of restorations as price, the dispersion of the number of failures as the risk, or cost of restoration.
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Статья научная
The plane problem for elasticity equations is well studied. It can be explained by its importance for applications and by the fact that the equations can be reduced to the Cauchy-Riemann system. In spite of this importance, exact solutions that would describe the stress-strain state of bodies of finite dimensions are not numerous. Conservation laws for differential equations have been appeared more than a hundred years ago, but, as a rule, they were not used to solve specific problems, but were of purely academic interest. The situation changed with the development of the technique of construction of conservation laws for arbitrary systems of differential equations, and then with the use of conservation laws to solve boundary value problems of the theory of plasticity and elastic-plasticity. In this article, new conservation laws are constructed for the equations of the plane theory of elasticity in the stationary case. These laws form an infinite series, which is closely related to the elasticity equations solving. This fact made possible to reduce solving of boundary value problems, in terms of displacements, to the calculation of contour integrals along the boundary of a domain bounded by the studying elastic body. As it follows from the proposed technique, the studied area can be multiply connected, and the considered boundary can be piecewise-smooth.
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The article presents the problem of optimizing the structure of information processing computer appli-ances for real-time control systems used, among other things, in the rocket and space industry. In addition, the features of this problem that affect the choice of optimization methods are studied. It’s concluded that this problem can be effectively solved using evolutionary optimization methods. Existing performance models allow you to determine the minimum hardware configuration of a multi-processor computing system. The approach proposed in this article allows us to find configurations that have hardware redundancy (compared to the minimum configuration), but, due to this, have a greater probability of being in states that provide performance sufficient to achieve the goals of functioning of the designed real-time control system. The described approach is more flexible than simply duplicating all hardware components of the minimum configuration, which can be used to reduce the cost of creating and operating the designed control system. The proposed model can be used to optimize the performance of multiprocessor hardware and software complexes of real-time control systems. At the same time, it should be taken into account that the resources allocated for the creation and operation of the hardware and software complex are always limited. There-fore, it is advisable to consider the problem of performance optimization as a multi-criterion: one criterion will be performance, and the other-the cost of creating a hardware and software complex.
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Статья научная
The S-box is the most important component of modern cryptographic algorithms which largely determines the quality of cryptographic transformation. The modern method of estimating the S-boxes quality employs their representation as component Boolean functions to which cryptographic quality criteria are applied. Such criteria include: nonlinearity, correlation immunity, an error propagation criterion, and a strict avalanche criterion. Nevertheless, it is obvious that a cryptanalyst is not constrained in the ways of representing the cipher components, in particular, using the functions of many-valued logic. The design features of modern cryptographic algorithms allow their representation in the form of 4-logic functions, which determines the need to research cryptographic properties of the S-boxes represented as component 4-functions. In the literature today there are methods for measuring the nonlinearity of 4-functions; nevertheless, there are no similar methods for researching the differential properties of 4-functions, in particular, involving their compliance with the strict avalanche criterion. In this paper the strict avalanche criterion is generalized to the case of 4-functions and the compliance of the S-boxes component 4-functions of the “Magma” cryptoalgorithm to the strict avalanche criterion has been researched. All balanced 4-functions of length N = 16 satisfying the strict avalanche criterion were synthesized using the restricted brute-force method. The basic properties of the constructed class of 4-functions are determined, and bijective S-boxes based on them are constructed. It has been established that S-boxes of length N = 16 satisfying the strict avalanche criterion, both in terms of component Boolean functions and in terms of 4-functions, also possess optimal nonlinear properties. This circumstance allows us to recommend S-boxes satisfying the strict avalanche criterion of component 4-functions for use in modern cryptographic algorithms.
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Structure, network protocols of the internet of things for quality production control
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The subject of research is the model and structure of the Internet of things (IoT) network for product quality control in industry and agriculture. The purpose of the article is to analyze communication protocols and structures of IoT networks. The method of analysis and structural design of IoT networks is applied. The field of application is automation of monitoring products of enterprises of the aerospace industry. The article provides an overview and analysis of existing IoT technology; it considers the protocols and composition of IoT networks, and provides variations in the structures of building such networks. 4 levels of IoT architecture are described, as well as the communication protocols are used. The directions of building the Internet of things network for product quality control are defined. A multi-agent model of such system is presented, for the implementation of which the structure of the IoT network is given. The structure of a multi-agent system (MAS) for monitoring product quality in industry and agriculture includes many agents, such as product quality agents, communication agents, database agents, agents for analyzing information received from sensor agents, and decision-making agents. This MAS implements functions to ensure the required class of product quality and it is based on building a local network of the Internet of things. The research proposes an algorithm for processing information in such an IoT network. Analyzers (sensors) product qualities will be periodically polled and their values will be recorded in the server database. The decision-making subsystem sends data on product quality compliance to the enterprise administrator on a mobile device. The server structure is implemented using cloud IoT platforms, for which a brief overview is provided. The one IoT network implementation is developed using LTE NBIoT technology. This approach can be used in the aerospace industry for product quality control within automation 4.0.
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Studying dynamics and classification of attacks to corporate network web services
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The article presents a study of the dynamics of attacks onto the web services using the classification of cyber threats by types due to the example of the corporate network of the Krasnoyarsk Scientific Center of the Siberian Branch of the Russian Academy of Sciences. The analysis is carried out on the basis of web services logs and it allows to solve urgent problems of ensuring the integrated security of web services, including identifying both existing and potential cybersecurity threats. The article demonstrates a review of the main approaches to the processing and analyzing logs. The authors describe the type and composition of data sources and provide a list of the software used. A feature of the study is a long observation period. The structure of the processing system is proposed and software tools for attack analysis and classification are implemented. The research shows that the use of classified samples allows to detect periodicity and reveal trends of certain types of attacks. Unclassified attacks have similar distribution parameters for different years, while in the case of classification, the distribution parameters have changed significantly, which makes it possible to track risks in automated intrusion prevention systems. A correlation matrix by type of attack is constructed. The analysis shows that most attack types have weak correlation, with the exception of the attacks “command injection”, “directory browsing”, “Java code injection”, which can be aggregated. The authors propose a heuristic method of risk comparison based on cyber threat classification. The method uses statistical parameters of sample distributions and permits to deal with different time intervals. The paper georeferences the IP addresses from which the attacks are carried out, builds attack profiles for different countries, and provides a list of countries with a stable attack profile. The conclusion indicates the features of the proposed method and outlines the prospects for its use in other areas.
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System analysis of dynamic problems of anisotropic plasticity theory
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Dynamic problems are the least studied area of plasticity theory. These problems arise in various fields of engineering and science, but the complexity of the original differential equations do not allow to develop accurate solutions and correctly solve numerical boundary value problems. This is even more typical of dynamic equations of anisotropic plasticity. Anisotropy reduces the group of symmetries allowed by the equations, and therefore narrows the number of invariant solutions. One-dimensional dynamic plasticity problems are well studied, but two-dimensional problems cause insurmountable mathematical difficulties due to the nonlinearity of the basic equations, even in the isotropic case. The study of the symmetries of the plasticity equations allowed us to find some exact solutions. The most known solution was found by B. D. Annin, who described the unsteady compression of a plastic layer made of isotropic material by rigid plates. Annin's solution is linear in two spatial variables, however, it includes arbitrary functions of time. Symmetries are also used in the proposed work. Point symmetries are first calculated for dynamic plasticity equations in the anisotropic case and are presented in the paper. The Lie algebra generated by the found symmetries appeared to be infinite-dimensional. This circumstance made it possible to apply the method of constructing new classes of nonstationary solutions. Symmetry can transform the exact solution of stationary dynamic equations in non-stationary solutions. The framed solutions include arbitrary functions and arbitrary constants. The outline of the article is as follows: according to the method of Lie group of point symmetries allowed by the equations of anisotropic plasticity is calculated. Two classes of new stationary invariant solutions are framed. These stationary solutions, by means of transformations generated by point symmetries, are transformed into new non-stationary solutions. In conclusion, a new selfsimilar solution of unsteady equations of anisotropic plasticity is framed; Annin's solution is generalized for the anisotropic case. The framed solutions can be used to describe the compression of plastic material between rigid plates, as well as to test programs, designed to solve anisotropic plastic problems.
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Статья научная
Dynamic multi-objective optimization problems are challenging and currently not-well understood class of optimization problems but this class is of great practical value. In such problems, the objective functions, their parameters and restrictions imposed on the search space can change over time. This fact means that solutions of the problem change too. When changes appear in the problem, an optimization algorithm needs to adapt to the changes in such a way that the convergence rate is sufficiently high. The work is devoted to the comparison of the different approaches to formation of a new population when changes in the dynamic multi-objective optimization problem appear: using solution, which obtained in the previous step; using a random generating of the population; partial using solutions which obtained in the previous step. In the first part of the article the classification of the changes in the problems is provided; the currently existing approaches to solving the problems based on evolutionary algorithms are considered. During the research NSGA-2 and SPEA2 algorithms are used to solving the dynamic optimization problems, the benchmark problems set is used to the comparison of the approaches. Obtained results being processed by Mann–Whitney U-test. It was obtained that changes rate in the problem affect the efficiency of the application of the solutions obtained in the previous step of new population the forming.
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Статья научная
In practice problems, which consist in the search of the best (optimal) solution according to the different irredundant and contradictory (conflicting) criteria, called multi-objective problems, are of frequent occurrence. One of the most commonly used methods for solving this kind of problems consists in combination of all criteria into the single one by using some linear relation. However, despite the simplicity of this method, solving problems with its help may cause other problems related to the determination of the mentioned linear combination, namely related to the determination of the weight coefficients for each criterion. The incorrect selection of these coefficients may lead to non-optimal solutions (according to the Pareto theory). In this regard, recently various population-based algorithms have been proposed for solving the described problems, which are the modifications of these population-based algorithms for solving singleobjective optimization problems. This article describes the developed modifications of the Differential Evolution algorithm (DE) for solving multi-objective unconstrained optimization problems based on the well-known NSGA (Nondominated Sorting Genetic Algorithm) and MOEA/D (Multiobjective Evolutionary Algorithm Based on Decomposition) schemes, which use the Pareto theory. The investigation into the efficiency of the Differential Evolution algorithm for solving multi-objective optimization problems in relation to the chosen mutation operator of the original DE algorithm and to the multi-objective scheme was conducted. The developed modifications were tested by using some well-known multi-objective real-valued optimization problems with 30 variables, such as ZDT1, ZDT2, ZDT3, etc. The practical problem of spacecraft control contour variant choice was solved as well. The experimental results show that better results were achieved by the Differential Evolution algorithm with the simplest mutation operators combined with the NSGA scheme. Thus, the applicability of the described modification for solving practical multi-objective optimization problems was demonstrated.
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The paper proposes the method for the numerical solution of Poisson kinematic equations system determining the evolution of the spacecraft position. The system of Poisson kinematic equations is used to designate the transition matrix from the coordinate system associated with the spacecraft at the selected time t1 to the coordinate system associated with the spacecraft at the current time t2. This matrix is used in the process of solving problems of determining a three-axis orientation of the spacecraft from the readings of the magnetometer using information about its angular velocities. The proposed method is based on replacing the derivatives of the desired functions in the Poisson kinematic equations by partial sums of series in the scaled Haar system. The partial sums of these series are generalized polynomials in the scaled Haar system. Hence, these sums are step (piecewise constant) functions. The estimates of the proposed method error are derived, which reveal that in the case of the coefficients of the equations which are functions matching the Lipschitz condition, the absolute error in calculating each of the elements of the transition matrix from one coordinate system to another is the value O(N–1) at N , where N is the number of partitions of the segment [t1, t2] when constructing a grid of nodes involved in this method. It is proved that the complexity of constructing an algorithm for approximating the system of Poisson kinematic properties insignificantly exceeds the complexity of solving this system by Euler method, which has the first order of accuracy. The research presents the results of numerical experiments, showing that in certain cases the Haar sums method gives an error that is much smaller than the Euler method, and is almost identical to the errors of the Euler – Cauchy and Runge – Kutta methods of the 2nd order, the complexity of which is approximately two times greater than the complexity of the Haar sums method.
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Статья научная
Plates, beams and shells with non-uniform and micro-inhomogeneities regular structure are widely used in aviation and rocket and space technology. At the preliminary design stage, it is initially important to know whether the design safety factor meets the specified strength conditions. To determine the margin factor, it is necessary to solve the elasticity problem for the designed structure by the finite element method (FEM), taking into account its inhomogeneous structure, which requires large computer resources. In this paper, we propose a method of equivalent strength conditions (MESC) for calculating the static strength of elastic structures with a inhomogeneous regular structure. The proposed method is reduced to the calculation of the strength of isotropic homogeneous bodies using equivalent strength conditions. The MESC is based on the following statement. For any composite body V0 , there exists such an isotropic homogeneous body Vb and such a number p (equivalence coefficient) that if the body Vb stock coefficient satisfies 0 nb the equivalent strength conditions 0 pn1 nb pn2 , then the body V0 stock coefficient satisfies n0 the given strength conditions n1 n0 n2 , and Vice versa, n1 , n2 – given, the coefficients 0 nb , n0 , meet the exact solutions of elasticity problems constructed for bodies V0 , Vb . The method under consideration is reduced to FEM strength calculation of isotropic homogeneous bodies, which is the easiest to implement and requires less computer memory than a similar calculation of composite bodies taking into account their inhomogeneous structure. The procedure for determining the equivalence coefficients for a number of composite plates, beams and shells of rotation is described. High-precision multigrid finite elements generating discrete models of small dimension and solutions with small error are used in the construction of elastic solutions according to FEM for isotropic homogeneous bodies. The adjusted equivalent strength conditions are of the form pn1(11) nb pn2 (12 ) , where nb is the body Vb reserve coefficient and the values 1 , 2 correspond to the approximate solution constructed for the body Vb . Implementation of FEM for multigrid discrete models requires several 103-106 times less computer memory than for basic models. The calculation of the strength of a beam with a micro-homogeneous regular structure with the help of MESC is given.
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Статья научная
When the strength of elastic composite structures (plates, beams, shells) widely used in aviation, rocket and space technology is calculated with the finite element method (FEM), it is important to know the solu-tion error. To analyze the solution error, it is necessary to use a sequence of approximate solutions con-structed according to the FEM using the grinding procedure for basic discrete models (BMs), which take into account an inhomogeneous microheterogeneous structure of bodies within the microapproach. Dis-crete models obtained by grinding BMs have a high dimension, which makes it difficult to use the FEM for them. In addition, there are BMs of composite solids (CSs), for example, BMs of bodies with a microhet-erogeneous structure, which have such a high dimension that the implementation of the FEM for such BMs is practically impossible due to limited computer resources. To solve these problems, it is proposed to use fictitious discrete models in the calculations of CSs according to the FEM. In this paper we propose a method of fictitious discrete models (MFDM) for calculating the strength of elastic bodies with an inhomogeneous microheterogeneous regular structure. The MFDM is implemented with the help of the FEM using corrected strength conditions, which take into account the error of ap-proximate solutions. The method is based on the following provision. We believe that BMs of CSs generate solutions that slightly differ from the exact ones. Such BMs always exist for CSs due to the convergence of the FEM. The calculation of CSs according to the MFDM is reduced to the construction and calculation of the strength of fictitious discrete models (FMs), the dimensions of which are smaller than the dimension of the BMs. FMs reflect: the shape, characteristic dimensions, fastening, loading and the type of the inhomogeneous structure of CSs and the distribution of the elastic moduli corresponding to the BM of the CS. The sequence consisting of the FM converges to the BM, i.e., the limiting FM coincides with the BM. The convergence of such a sequence ensures uniform convergence of the FM stresses to the corresponding BM stresses. The implementation of the FEM for FMs with the use of multigrid finite elements leads to a large saving of computer resources, which makes it possible to use the MFDM for strength calculations of bodies with a microheterogeneous regular structure. Calculation of the CS strength according to the MFDM requires times less computer memory volume than a similar calculation using the BM of the CS, and does not contain the procedure for grinding the BM. The given example of calculating the strength of a beam with an inhomogeneous regular fibrous structure according to the MFDM shows its high efficiency. Applying the adjusted strength conditions allows using approximate solutions with larger errors in CS strength calculations, which leads to improving the efficiency of the MFDM.
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The use of the inverse transformation method for time series analysis
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In modern conditions of technology development, signs of systemacity are manifested to one degree or another in all areas, so the use of system analysis is an urgent task. In this case, the main factors in this situation are data processing and prediction of the state of a system. Mathematical modeling is used as a prediction method for a given subject area. A mathematical model is a universal tool for describing complex systems representing the approximate description of the class of phenomena of the external world expressed by mathematical concepts and language. The mathematical model can be represented as a set of systematic components and a random component. In this article, the object of prediction is the irregular random component of a model, which reflects the impact of numerous random factors. The origin, nature and laws of variation of the random variable are known, therefore, to simulate its behavior or predict its future value, one needs high degree of certainty to establish the form of continuous distribution function of the random variable. The empirical distribution function is calculated using the sample of random variable values. This empirical function is close to the values of the desired unknown function of distribution. The resulting empirical function is discrete, therefore it is necessary to apply piecewise linear interpolation to obtain a continuous distribution function. The predicted random component of time series has been included in the initial regression model. In order to compare augmented and initial regression models, several values were excluded from the time series and new prediction was built. The value of the average approximation error for assessing the quality of the model is calculated. The augmented regression model proved to be more effective than the original one.
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To the task of controlling a group of objects on the basis of information technologies
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To participate the TPP with cross-section communications in the general primary frequency control, it is necessary to have a working main regulator. The main regulator is designed to maintain the steam pressure in the major steam line of the TPP at a given level, which is a difficult task. At the TPP with cross-connections, the steam produced by the boilers enters the major steam line. To maintain the pressure in the major steam line, it is necessary to control the heat load of the working boilers. Traditional solutions to construct the main regulator found no use, as have a number of disadvantages, not allowing exploiting a system of automatic control. Looking at the steam pressure control system in the major steam line from the bottom to up, it is possible to identify disadvantages that prevent the effective operation of the main regulator at each level. At the lower level of the main regulator, there are controllers of heat load of boilers, built according to the scheme task-heat. Heat load controllers are designed to maintain heat release in the boiler furnace at the required level. The heat signal is the sum of the signals for the steam flow of the boiler and the rate of change in the steam pressure in the boiler drum. Such a structure does not allow maintaining the invariance of the heat signal under external disturbances effectively, as sharp changes of the steam pressure in the major steam line lead to a "false" operation of the controllers. At the upper level there is the main regulator itself, which maintains the steam pressure in the major steam line at a given level and corrects the tasks to the controllers of the heat load of the boilers. The simultaneous identical effect on the heat load of the boilers cannot be optimal from the point of view of the criteria for assessing the quality of regulation, since the dynamic properties of the boilers, such as the gain, the transition time constant and the transport delay are individual for each boiler. However, in 2006–2008, the attempt to build an updated main regulator that takes into account the shortcomings of the traditional scheme was made. The basis of the structure of the main regulator is still parametric and, as a result of ten-years’ experience, shortcomings in the operation of the updated main regulator were identified. The shortcomings, in most cases, consist in need of frequent corrections of adjusting coefficients of system because of the change of dynamic properties of an object during the operation. In fact, the same problems related to the parametric structure of the regulator remain. Up-to-date information technologies made it possible to introduce adaptive process control systems that allow to count an extended number of signals entering the system and to form control actions, based on both current and historical data of the technological process. The use of the latest information technologies and modern hardware in the control of complex multi-connected units that solve not only the problems of process control, but also the problem of improving the economic and environmental performance of enterprises, should become a new step in the development of automatic control systems.
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Torsion of prismatic orthotropic elastoplastic rods
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Conservation laws were introduced into the theory of differential equations by E. Noether more than 100 years ago and are gradually becoming an important tool for the study of differential equations systems. Not only do they allow you to qualitatively investigate the equation, but, as the authors of this article show, they also enable you to find exact solutions to the boundary value problems. For the equations of the iso-tropic theory of elasticity, the conservation laws were first calculated by P. Olver. For the equations of the theory of plasticity in the two-dimensional case, the conservation laws were found by one of the authors of this article and used to solve the main boundary value problems of the plasticity equations. Later it turned out that the conservation laws can also be used to find the boundaries between elastic and plastic zones in twisted rods, bent beams, and deformable plates. The proposed work found conservation laws for equations describing the orthotropic elastic state of the twisted straight-line rod. It is assumed that the remaining current depends linearly on the voltage tensor component. In the workit was also found an endless series of laws of preservation, which allows you to find an elastic-plastic boundary, which arises when twisting the orthotropic rod.
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Using of the contour method to solving the problem of optimal traffic distribution in the network
Статья научная
The purpose of this work is to create a method for solving the problem of optimal traffic distribution in a network using the contour data analysis method. In the first section of the work, the principle of converting any available network to a contour form is explained, and the case is considered both for networks without loss and for networks with losses. The second section shows in a general way the method of bringing the network in contour form to a system of non-linear inequalities, by solving which one can obtain a certain distribution of traffic in the system. In the final section, using the M/M/1/N queuing system as an example, the solution of the problem of optimal traffic distribution according to the loss minimization criterion is shown. The initial data for the task were the incidence matrix, service intensity and buffer dimension for communication channels. A feature of the proposed algorithm is the search for a contour matrix, for the compilation of which it is proposed to use loss edges as elements of the spanning tree of the graph, which allows you to immediately determine the contour matrix using the concept of a fundamental cycle of a graph. This approach to optimal traffic distribution reduces the number of variables used compared to the known methods based on loopless routes, and also does not require their preliminary search, since they are determined from the dimension of the incidence matrix of the simulated network graph.
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Using signals of navigation satellites in the monitoring of the Earth covers
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The features and capabilities of remote sensing of earth covers by means of signals of navigation satellites are presented. The methods of reflectometry of the surfaces of earth covers and radioscopy of forest canopy are described. The options for using the signals of GLONASS, GPS systems are considered. Test measurements of interference diagrams were carried out on 5 test platforms from heterogeneous soil surfaces: salt marshes and asphalt; water surfaces of saline and freshwater bodies in summer, including ice cover of small thickness in the period of autumn freeze-up. The method of radioscopy helped obtain the data on the spatial and temporal characteristics of attenuated signals of the GLONASS and GPS satellites in the pine forest. Estimates of the linear attenuation coefficients of the signals passing through a forest canopy with a coordinate reference were made. The results obtained are the basis for the development of methods and technologies for continuous monitoring of the characteristics and state of earth covers by means of signals of navigation satellites for solving a wide range of applied tasks.
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Volume temperature control at automated high-frequency processing of polymer and composite materials
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The purpose of this work is to develop and justify a method for volumetric temperature control of a polymer and composite material in automated high-frequency processing. The developed method is implemented by in-troducing thermocouples into the prism-shaped or cube-shaped sample body according to a certain pattern of their location throughout the volume. This technique is cost-effective and easy to implement compared to expen-sive and specialized equipment with complex design, as well as to the cost of thermocouples having a simple de-sign. Methods to achieve the purpose of the research of the contact method of volumetric temperature of a polymer or composite sample control are development and outlining thermocouples throughout the volume so as to iden-tify the most accurate temperature spectrum of the polymer or composite sample during automated high-frequency processing. Another method to achieve this purpose is the method of finding out how it will affect the measurements accuracy of the heating sample temperature from the introduction of thermocouples by making holes in it for installation. For this, a finite-difference mathematical calculation of the dependence of the sample temperature on the number of holes for thermocouples in it was performed in the MSC Patran Sinda software package. The calculation results were summarized and presented on graphic data. Further, a general mathemat-ical calculation was performed according to the formulas for the process of heat and mass conductivity calcula-tion, the results of which were table and graphic data. At the end of the finite-difference and general mathematical calculation, a comparative analysis of the ob-tained error of temperature measurement from the introduction of thermocouples into the body of the sample was performed. Based on this analysis, the developed method is applicable for further research on automated high-frequency processing of polymer and composite materials, since the errors obtained do not exceed the permissi-ble 3 %.
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m-aperiodic words on three-letter alphabet
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The work is devoted to the study of sets of aperiodic words over a finite alphabet. The set of aperiodic words can be considered as a dictionary of some finite formal language. The existence of infinite words in two-letter or three-letter alphabets that do not contain subwords that are third powers or, respectively, squares of other words was first discovered more than a hundred years ago. S.I. Adyan in 2010 constructed an example of an infinite sequence of irreducible words, each of which is the beginning of the next and does not contain word squares in a two-letter alphabet. S.E. Arshon established the existence of an n-digit asymmetric repetition-free sequence for an alphabet of at least three letters. In the monograph by S.I. Adyan proved that in an alphabet of two symbols there exist infinite 3-aperiodic sequences. In the works of other authors, generalizations of aperiodicity were considered, when not only the powers of some subwords were excluded. In the monograph by A.Yu. Olshansky proved the infinity of the set of 6-aperiodic words in a two-letter alphabet and obtained an estimate for the number of such words of any given length. The author previously considered the case of a three-letter alphabet only in the case of 6-aperiodic words. In this article, we prove the infinity of the set of m-aperiodic words in the three-letter alphabet at m 4 and obtain an estimate for the set of such words. The results can be applied when encoding information in space communications.
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