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Assessment of the current state of the near-surface part of the geological environment and understanding of its layered structure play an important role in various scientific and applied fields. The presented work is devoted to the application of phasometric modifications of geoelectric control methods to solve the problem of the detailed complex study of the underground layers of the environment in the process of drilling operations with the use of special equipment. These studies are based on the analysis of variations in phase parameters and characteristics of an artificially excited multiphase electric field to assess poorly distinguishable details and changes in the layered structure of the medium. The proposed method has increased accuracy, sensitivity and noise proofness of measurements, which allows for extracting detailed information about the heterogeneity, composition and stratification of underground geological formations not only in the zone where the drill makes contact with the medium, but also in the entire control zone. This paper considers practical mathematical models of phase images for basic scenarios of drill penetration between the layers of the near-surface part of the geological medium with different characteristics, obtained by means of approximation apparatus based on continuous piecewise linear functions, and also suggests the use of modern machine learning methods for intelligent analysis of its structure. Studying the phase shifts in electrical signals during drilling highlights their value for understanding the dynamics of soil response to the process. The observed signal changes during the drilling cycle reveal in detail the heterogeneity in soil structure and its response to changes caused by drilling. The stability of phase shifts at the last stages of the process indicates a quasi-equilibrium state. The results make a significant contribution to geotechnical science by offering an improved approach to monitoring a layered structure without the need for deep drilling. [ABSTRACT FROM AUTHOR]

Dimension reduction is a technique used to transform data from a high-dimensional space into a lower-dimensional space, aiming to retain as much of the original information as possible. This approach is crucial in many disciplines like engineering, biology, astronomy, and economics. In this paper, we consider the following dimensionality reduction instance: Given an n-dimensional probability distribution p and an integer m < n , we aim to find the m-dimensional probability distribution q that is the closest to p, using the Kullback–Leibler divergence as the measure of closeness. We prove that the problem is strongly NP-hard, and we present an approximation algorithm for it. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximation of continuous functions on a subclass of singular space—the subcartesian space. As is well known, the orbit space of the proper action of a Lie group on a smooth manifold is a subcartesian space. We prove that continuous functions on the orbit space can be approximated by smooth functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a variant of dominating set problem, i.e., the total dominating set problem. Given an undirected graph G = (V , E) , a subset of vertices T ⊆ V is called a total dominating set if every vertex in V is adjacent to at least one vertex in T. Based on LP relaxation techniques, this paper gives a distributed approximation algorithm for the total dominating set problem in general graphs. The presented algorithm obtains a fractional total dominating set that is, at most, k (1 + Δ 1 k) Δ 1 k times the size of the optimal solution to this problem, where k is a positive integer and Δ is the maximum degree of G. The running time of this algorithm is constant communication rounds under the assumption of a synchronous communication model. [ABSTRACT FROM AUTHOR]

The Internet of Things (IoT) plays a fundamental role in monitoring applications; however, existing approaches relying on cloud and edge-based IoT data analysis encounter issues such as network delays and high costs, which can adversely impact time-sensitive applications. To address these challenges, this paper proposes an IoT framework called Sazgar IoT. Unlike existing solutions, Sazgar IoT leverages only IoT devices and IoT data analysis approximation techniques to meet the time-bounds of time-sensitive IoT applications. In this framework, the computing resources onboard the IoT devices are utilised to process the data analysis tasks of each time-sensitive IoT application. This eliminates the network delays associated with transferring large volumes of high-velocity IoT data to cloud or edge computers. To ensure that each task meets its application-specific time-bound and accuracy requirements, we employ approximation techniques for the data analysis tasks of time-sensitive IoT applications. These techniques take into account the available computing resources and optimise the processing accordingly. To evaluate the effectiveness of Sazgar IoT, experimental validation has been conducted. The results demonstrate that the framework successfully meets the time-bound and accuracy requirements of the COVID-19 citizen compliance monitoring application by effectively utilising the available IoT devices. The experimental validation further confirms that Sazgar IoT is an efficient and scalable solution for IoT data processing, addressing existing network delay issues for time-sensitive applications and significantly reducing the cost related to cloud and edge computing devices procurement, deployment, and maintenance. [ABSTRACT FROM AUTHOR]

This paper analyses the possibility of damaging and destroying an identification chip of the Mifare type in a frequently used contactless identification card of size ID-1, following the standard ISO/IEC 7810 (i.e., with dimensions 85.60 × 53.98 × 0.76 mm), using the magnetic field of an adjacent conductor in which a current pulse of a defined shape and amplitude is flowing. For analysis purposes, the nonlinear current–voltage characteristic of the Mifare chip voltage limiter was measured and approximated, and the mutual inductance of the straight conductor and the rectangle coil antenna in the card was calculated. Next, a mathematical analysis was conducted based on the description of the equivalent electrical circuit by the differential equations. The results of the mathematical analysis were verified by a simulation in the free simulation software Micro-Cap 12. The peak value of the current pulse that can damage the Mifare chip was measured by a combination wave generator. Based on these measurements and the chip characteristics, the energy capable of destroying the chip was calculated. The characteristics of chip damage were determined using a comparison of the resonant characteristics of undamaged and damaged RFID cards with Mifare chips. [ABSTRACT FROM AUTHOR]

The Gaussian Q-function has considerable applications in numerous areas of science and engineering. However, the fact that a closed-form expression for this function does not exist encourages finding approximations or bounds of the Q-function. In this paper, we determine analytically two novel interval upper bound Q-function approximations and show that they could be used efficiently not only for the symbol error probability (SEP) estimation of transmission over Nakagami-m fading channels, but also for the average symbol error probability (ASEP) evaluation for two modulation formats. Specifically, we determine analytically the composition of the upper bound Q-function approximations specified at disjoint intervals of the input argument values so as to provide the highest accuracy within the intervals, by utilizing the selected one of two upper bound Q-function approximations. We show that a further increase of the accuracy, achieved in the case with two upper-bound approximations composing the interval approximation, can be obtained by forming a composite interval approximation of the Q-function that assumes another extra interval and by specifying the third form for the upper-bound Q-function approximation. The proposed analytical approach can be considered universal and widely applicable. The results presented in the paper indicate that the proposed Q-function approximations outperform in terms of accuracy other well-known approximations carefully chosen for comparison purposes. This approximation can be used in numerous theoretical communication problems based on the Q-function calculation. In this paper, we apply it to estimate the average bit error rate (ABER), when the transmission in a Nakagami-m fading channel is observed for the assumed binary phase-shift keying (BPSK) and differentially encoded quadrature phase-shift keying (DE-QPSK) modulation formats, as well as to design scalar quantization with equiprobable cells for variables from a Gaussian source. [ABSTRACT FROM AUTHOR]

In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments. [ABSTRACT FROM AUTHOR]

This paper presents a solution for the modelling, implementation and simulation of the fractional-order process of producing the enriched 13C isotope, through the chemical exchange between carbamate and carbon dioxide. To achieve the goal of implementation and simulation of the considered process, an original solution for the approximation of fractional-order systems at the variation of the system's differentiation order is proposed, based on artificial intelligence methods. The separation process has the property of being strongly non-linear and also having fractional-order behaviour. Consequently, in the implementation of the mathematical model of the process, the theory associated with the fractional-order system's domain has to be considered and applied. For learning the dynamics of the structure parameters of the fractional-order part of the model, neural networks, which are associated with the artificial intelligence domain, are used. Using these types of approximations, the simulation and the prediction of the produced 13C isotope concentration dynamics are made with high accuracy. In order to prove the efficiency of the proposed solutions, a comparation between the responses of the determined model and the experimental responses is made. The proposed model implementation is made based on using four trained neural networks. Moreover, in the final part of the paper, an original method for the online identification of the separation process model is proposed. This original method can identify the process of fractional differentiation order variation in relation to time, a phenomenon which is quite frequent in the operation of the real separation plant. In the last section of the paper, it is proven that artificial intelligence methods can successfully sustain the system model in all the scenarios, resulting in the feasible premise of designing an automatic control system for the 13C isotope concentration, a method which can be applied in the case of other industrial applications too. [ABSTRACT FROM AUTHOR]

In this paper, control analysis of a stochastic lagging discrete ecosystem is investigated. Two-dimensional stochastic hysteresis discrete ecosystem equilibrium points with symmetry are discussed, and the dynamical behavior of equilibrium points with symmetry and their control analysis is discussed. Using the orthogonal polynomial approximation theory, the stochastic lagged discrete ecosystems are approximately transformed as its equivalent deterministic ecosystem. Based on the stability and bifurcation theory of deterministic discrete systems, through mathematical analysis, asymptotic stability and Hopf bifurcation are existent in the ecosystem, constructing control functions, controlling the behavior of the system dynamics. Finally, the effects of different random strengths on the bifurcation control and asymptotic stability control are verified by numerical simulations, which validate the correctness and effectiveness of the main results of this paper. [ABSTRACT FROM AUTHOR]

This paper proposes two new encoders for data bus inversion (DBI), which conventionally uses a majority voter to pick a data representation that minimizes switching activities and thus reduces the corresponding energy consumption. The new encoders employ simpler approximate voters comprising only two gate levels, which improve latency more than twice while still achieving switching activity savings by 9% and 11%, respectively. Although the proposed voters are not always accurate, the errors in the voters do not affect the correctness of data movement. We report various metrics, including latencies, areas, and operating powers, regarding five different designs, two proposed designs along with three conventional designs, based on 65-nm process implementations. [ABSTRACT FROM AUTHOR]

The strong approximation of a function is a useful tool to analyze the convergence of its Fourier series. It is based on the summability techniques. However, unlike matrix summability methods, it uses non-linear methods to derive an auxiliary sequence using approximation errors generated by the series under analysis. In this paper, we give some direct results on the strong means of Fourier series of functions in generalized Hölder and Zygmund spaces. To elaborate its use, we deduce some corollaries. [ABSTRACT FROM AUTHOR]

In this paper we study pseudoultrametrics, which are a natural mixture of ultrametrics and pseudometrics. They satisfy a stronger form of the triangle inequality than usual pseudometrics and naturally arise in problems of classification and recognition. The text focuses on the natural partial order on the set of all pseudoultrametrics on a fixed (not necessarily finite) set. In addition to the "way below" relation induced by a partial order, we introduce its version which we call "weakly way below". It is shown that a pseudoultrametric should satisfy natural conditions closely related to compactness, for the set of all pseudoultrametric weakly way below it to be non-trivial (to consist not only of the zero pseudoultrametric). For non-triviality of the set of all pseudoultrametrics way below a given one, the latter must be compact. On the other hand, each compact pseudoultrametric is the least upper bound of the directed set of all pseudoultrametrics way below it, which are compact as well. Thus it is proved that the set C P s U (X) of all compact pseudoultrametric on a set X is a continuous poset. This shows that compactness is a crucial requirement for efficiency of approximation in methods of classification by means of ultrapseudometrics. [ABSTRACT FROM AUTHOR]

In this paper, we propose two Maple procedures and some related utilities to determine the maximum curvature of a cubic Bézier-spline curve that interpolates an ordered set of points in R 2 or R 3 . The procedures are designed from closed-form formulas for such open and closed curves. [ABSTRACT FROM AUTHOR]

Development of energy-efficient and high-performance bioreactors requires progress in methods for assessing the key parameters of the biosynthesis process. With a wide variety of approaches and methods for determining the phase contact area in gas–liquid flows, the question of obtaining its accurate quantitative estimation remains open. Particularly challenging are the issues of getting information about the mass transfer coefficients instantly, as well as the development of predictive capabilities for the implementation of effective flow control in continuous fermentation both on the laboratory and industrial scales. Motivated by the opportunity to explore the possibility of applying classical and non-classical computer vision methods to the results of high-precision video records of bubble flows obtained during the experiment in the bioreactor vessel, we obtained a number of results presented in the paper. Characteristics of the bioreactor's bubble flow were estimated first by classical computer vision (CCV) methods including an elliptic regression approach for single bubble boundaries selection and clustering, image transformation through a set of filters and developing an algorithm for separation of the overlapping bubbles. The application of the developed method for the entire video filming makes it possible to obtain parameter distributions and set dropout thresholds in order to obtain better estimates due to averaging. The developed CCV methodology was also tested and verified on a collected and labeled manual dataset. An onwards deep neural network (NN) approach was also applied, for instance the segmentation task, and has demonstrated certain advantages in terms of high segmentation resolution, while the classical one tends to be more speedy. Thus, in the current manuscript both advantages and disadvantages of the classical computer vision method (CCV) and neural network approach (NN) are discussed based on evaluation of bubbles' number and their area defined. An approach to mass transfer coefficient estimation methodology in virtue of obtained results is also represented. [ABSTRACT FROM AUTHOR]

Polynomials are common algebraic structures, which are often used to approximate functions, such as probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of systems rather than to assume polynomials for only approximating known or empirically estimated distributions. Polynomial distributions offer great modeling flexibility and mathematical tractability. However, unlike canonical distributions, polynomial functions may have non-negative values in the intervals of support for some parameter values; their parameter numbers are usually much larger than for canonical distributions, and the interval of support must be finite. Hence, polynomial distributions are defined here assuming three forms of a polynomial function. Transformations and approximations of distributions and histograms by polynomial distributions are also considered. The key properties of the polynomial distributions are derived in closed form. A piecewise polynomial distribution construction is devised to ensure that it is non-negative over the support interval. A goodness-of-fit measure is proposed to determine the best order of the approximating polynomial. Numerical examples include the estimation of parameters of the polynomial distributions and generating polynomially distributed samples. [ABSTRACT FROM AUTHOR]

This paper uses computational intelligence and machine learning methods to describe experimental modeling performed to approximate the static characteristics of one type of fluidic muscle from the manufacturer FESTO for three different muscle sizes. For the experiments, measured data from the manufacturer and data from a real system (i.e., test device) were used. The measurements, which took place on the experimental equipment, were carried out in two stages (i.e., when the muscle was pressed and when the muscle was relaxed). The resulting measured characteristics were obtained by averaging two values at a given moment. MATLAB® software was used for simulations, in which four models were created: MLP, SVM, ANFIS, and a custom model (i.e., polynomial model). Given that most articles mainly interpret their results graphically when approximating characteristics, in this article, the outputs of the models are also compared with the measured data based on the SSE, NRMSE, SBC, and AIC performance indicators, enabling a more relevant and comprehensive overview of the performance of the individual models. The outputs of the best models described in this article reach an accuracy of 89.90% to 98.74% (all from the MLP group), depending on the muscle size, compared to real measured outputs. [ABSTRACT FROM AUTHOR]

The modern development of the biomedical engineering area is accompanied by the availability of large volumes of data with a non-linear response surface. The effective analysis of such data requires the development of new, more productive machine learning methods. This paper proposes a cascade ensemble that combines the advantages of using a high-order Wiener polynomial and Stochastic Gradient Descent algorithm while eliminating their disadvantages to ensure a high accuracy of the approximation of such data with a satisfactory training time. The work presents flow charts of the learning algorithms and the application of the developed ensemble scheme, and all the steps are described in detail. The simulation was carried out based on a real-world dataset. Procedures for the proposed model tuning have been performed. The high accuracy of the approximation based on the developed ensemble scheme was established experimentally. The possibility of an implicit approximation by high orders of the Wiener polynomial with a slight increase in the number of its members is shown. It ensures a low training time for the proposed method during the analysis of large datasets, which provides the possibility of its practical use in the biomedical engineering area. [ABSTRACT FROM AUTHOR]

This paper proposes a new method to get explicit expressions of some quantities associated with performance analysis of the maximum likelihood DOA algorithm in the presence of an additive Gaussian noise on the antenna elements. The motivation of the paper is to make a quantitative analysis of the ML DOA algorithm in the case of multiple incident signals. We present a simple method to derive a closed-form expression of the MSE of the DOA estimate based on the Taylor series expansion. Based on the Taylor series expansion and approximation, we get explicit expressions of the MSEs of estimates of azimuth angles of all incident signals. The validity of the derived expressions is shown by comparing the analytic results with the simulation results. [ABSTRACT FROM AUTHOR]

Amplitude-comparison monopulse radar in tracking radar uses the tracking scheme of a monopulse radar to estimate the angle components of a target. The performance of the amplitude comparison monopulse radar under measurement uncertainty is analysed. Measurement noises are modelled as Gaussian random variables. Taylor series expansion is adopted to get analytic expression of the mean square error (MSE). Estimation accuracy, in terms of the MSEs for estimate the direction-of-arrival (DOA) estimation algorithm, is usually obtained from the Monte Carlo simulation, which can be computationally intensive especially for large number of repetitions in the Monte Carlo simulation. To get reliable MSE in the Monte Carlo simulation, the number of repetitions should be very large, which implies that there is a trade-off between reliability of the MSE and computational burden in the Monte Carlo simulation. This paper shows the performance of amplitude comparison monopulse radar by linear approximation of nonlinear equations to estimate the DOA. The performance of amplitude comparison monopulse radar is quantitatively analysed via the MSEs, and the derived expression is validated by comparing the analytic MSEs with the simulation based MSEs. In addition, it is shown in the numerical results that analytically derived MSE is much less computationally intensive in comparison with the Monte Carlo simulation-based MSE, which implies that the proposed scheme in this paper results in drastic reduction in computational complexity for evaluation of the MSE. [ABSTRACT FROM AUTHOR]

This paper proposes a new structure and substantiates the choice of a technique for controller design using fractional integration and fractional differentiation. The method of fractional integration is based on extrapolation using a series of integrating and differentiating links, the time constant of which changes symmetrically from one step to another. The new controller structure does not contain the three traditional links (proportional, integrating, and differentiating) but contains six links, three of which are incomplete integration, incomplete differentiation, and redundant (for example, one and a half) integration. The proposed technique consists of modeling a controller with the indicated six paths and using the numerical optimization method to calculate the coefficients of each such path. Alternative controllers are known PID controllers and PIλDµ controllers. Both of these species contain only three tracts. The proposed structure has never been used before, and publications with examples of calculation of such structures were not found in the literature. The effectiveness of the proposed method is confirmed by modeling and examples. [ABSTRACT FROM AUTHOR]

In connection with applications, the solution of a system of logical equations plays an important role in computational mathematics and in many other areas. As a result, many new directions and algorithms for solving systems of logical equations are being developed. One of these directions is transformation into the real continuous domain. The real continuous domain is a richer domain to work with because it features many algorithms, which are well designed. In this study, firstly, we transformed any system of logical equations in the unit n -dimensional cube K n into a system of polylinear–polynomial equations in a mathematically constructive way. Secondly, we proved that if we slightly modify the system of logical equations, namely, add no more than one special equation to the system, then the resulting system of logical equations and the corresponding system of polylinear–polynomial equations in K n + 1 is equivalent. The paper proposes an algorithm and proves its correctness. Based on these results, further research plans are developed to adapt the proposed method. [ABSTRACT FROM AUTHOR]

The properties of the material applied to the finite element (FE) simulation can be expressed by constitutive models, and simple constitutive and complex constitutive models can be used to show the actual phenomenon. The technology to improve the accuracy of the constitutive model applied to FE simulation is the inverse method. The inverse method is a method to curve fit the FE simulation result to the test data by utilizing finite element model updating (FEMU). Inverse methods are general approaches to update material properties. The inverse method can iteratively run many FE simulations for constitutive model optimization and consider metamodel-based simulation optimization (MBSO) to reduce this resource waste. With MBSO, one can obtain significant results with fewer resources. However, the MBSO algorithm has the problem in that the optimization performance deteriorates as the number of parameters increases. The typical process of the inverse method is to adjust these factor values individually. If there are many factors in the constitutive model, the optimization result may deteriorate owing to the performance limit of the MBSO when the structural method is used. This paper proposes a method of fitting a stress–strain constitutive model with a scaling factor to improve the efficiency of the inversion method using MBSO. For this purpose, a process was performed to determine the curve characteristics during the pretreatment stage. The results show that the proposed method significantly improved the prediction efficiency of the combination function. Thus, we conclude that initializing the combination function and setting the parameters of the inverse method by applying the proposed approach improves the efficiency of large deformation analyses. [ABSTRACT FROM AUTHOR]

In this paper, the dual hesitant fuzzy rough set (DHFRS) is studied from the viewpoint of assessment deviations. Firstly, according to the relationship between intuitionistic fuzzy set and vague set, the DHFRS is transferred into a fuzzy set, where the membership of any given element to it has multi-grouped values. By the idea of bootstrap sampling, a group of four sets are generated to describe the membership degree on DHFRS, where the elements of the aforementioned sets are all considered as assessment values. Secondly, the generated sets are dealt with by assessment deviation theories, and specifically, two variables are proposed to describe the systematic and random deviations of the sets. Thirdly, the true-value of the membership degree of any elements to the set is estimated by a deviation-based dual hesitant fuzzy rough weighted aggregating operator. Fourthly, a dual hesitant fuzzy rough pattern recognition approach based on assessment deviation theories is proposed. Finally, an urban traffic modes recognition example is given to illustrate the validity of the proposed theories on DHFRSs. [ABSTRACT FROM AUTHOR]

A grommet is a representative component that fixes the position of a cable. It is made from hyper-elastic materials (rubber), such as ethylene propylene diene monomer (EPDM). The grommet and cable are conventionally fixed through bonding; however, this method has numerous disadvantages that can be improved through relevant research. To apply a fixing method using the elastic force of EPDM rubber, this paper presents an empirical equation for approximating the bonding force of EPDM grommet parts with a hollow shaft geometry. First, tensile tests and the inverse method were used to approximate the basic mechanical properties. The physical properties were derived through basic tests; furthermore, bonding force tests and the inverse method were used on a grommet with a hollow shaft structure. In addition, the Box–Behnken design of experiments was used to predict the amount of change in the bonding force according to the geometry variables. Finally, this study was validated by comparing the approximation results derived through the design of experiments with the analysis and bonding force test results. [ABSTRACT FROM AUTHOR]

This paper features the study of global optimization problems and numerical methods of their solution. Such problems are computationally expensive since the objective function can be multi-extremal, nondifferentiable, and, as a rule, given in the form of a "black box". This study used a deterministic algorithm for finding the global extremum. This algorithm is based neither on the concept of multistart, nor nature-inspired algorithms. The article provides computational rules of the one-dimensional algorithm and the nested optimization scheme which could be applied for solving multidimensional problems. Please note that the solution complexity of global optimization problems essentially depends on the presence of multiple local extrema. In this paper, we apply machine learning methods to identify regions of attraction of local minima. The use of local optimization algorithms in the selected regions can significantly accelerate the convergence of global search as it could reduce the number of search trials in the vicinity of local minima. The results of computational experiments carried out on several hundred global optimization problems of different dimensionalities presented in the paper confirm the effect of accelerated convergence (in terms of the number of search trials required to solve a problem with a given accuracy). [ABSTRACT FROM AUTHOR]

This study introduces an innovative numerical approach for polylinear approximation (polylinearization) of non-self-intersecting compact sensor characteristics (transfer functions) specified either pointwise or analytically. The goal is to partition the sensor characteristic optimally, i.e., to select the vertices of the approximating polyline (approximant) along with their positions, on the sensor characteristics so that the distance (i.e., the separation) between the approximant and the characteristic is rendered below a certain problem-specific tolerance. To achieve this goal, two alternative nonlinear optimization problems are solved, which differ in the adopted quantitative measure of the separation between the transfer function and the approximant. In the first problem, which relates to absolutely integrable sensor characteristics (their energy is not necessarily finite, but they can be represented in terms of convergent Fourier series), the polylinearization is constructed by the numerical minimization of the L 1 -metric (a distance-based separation measure), concerning the number of polyline vertices and their locations. In the second problem, which covers the quadratically integrable sensor characteristics (whose energy is finite, but they do not necessarily admit a representation in terms of convergent Fourier series), the polylinearization is constructed by numerically minimizing the L 2 -metric (area- or energy-based separation measure) for the same set of optimization variables—the locations and the number of polyline vertices. [ABSTRACT FROM AUTHOR]

The well-known equations for the powder compaction process (PCP) in a rigid die published from the beginning of the last century until today were considered in this review. Most of the considered equations are converted into the dependences of densification pressure on the powder's relative density. The equations were analyzed and their ability to describe PCP was assessed by defining the coefficient of determination when approximating experimental data on the compaction of various powders. It was shown that most of the equations contain two constants the values of which are determined by fitting the mathematical dependence to the experimental curve. Such equations are able to describe PCP with high accuracy for the compaction of powders up to a relative density of 0.9–0.95. It was also shown that different equations can describe PCP in the density range from the initial density to 0.9 with the same high accuracy, but when the process of compaction is extrapolated to higher values of density, the curves diverge. This indicates the importance of equations that can unambiguously describe PCP to a relative density equal to or close to 1.0. For an adequate description of PCP for relative density greater than 0.95, equations containing three or four constants have proven useful. [ABSTRACT FROM AUTHOR]

The problem of the approximation of the coefficients of the objective function of a scheduling problem for a single machine is considered. It is necessary to minimize the total weighted completion times of jobs with unknown weight coefficients when a set of problem instances with known optimal schedules is given. It is shown that the approximation problem can be reduced to finding a solution to a system of linear inequalities for weight coefficients. For the case of simultaneous job release times, a method for solving the corresponding system of inequalities has been developed. Based on it, a polynomial algorithm for finding values of weight coefficients that satisfy the given optimal schedules was constructed. The complexity of the algorithm is O (n 2 (N + n)) operations, where n is the number of jobs and N is the number of given instances with known optimal schedules. The accuracy of the algorithm is estimated by experimentally measuring the function ε (N , n) = 1 n ∑ j = 1 n ∣ w j − w j 0 ∣ w j 0 , which is an indicator of the average modulus of the relative deviation of the found values w j from the true values w j 0 . An analysis of the results shows a high correlation between the dependence ε (N , n) and a function of the form α (n) / N , where α (n) is a decreasing function of n. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of high-order, accurate difference schemes' numerical solutions of local and non-local problems for ordinary differential equations of the fourth order. Local and non-local problems for ordinary differential equations with constant coefficients can be solved by classical integral transform methods. However, these classical methods can be used simply in the case when the differential equation has constant coefficients. We study fourth-order differential equations with dependent coefficients and their corresponding boundary value problems. Novel compact numerical solutions of high-order, accurate finite difference schemes generated by Taylor's decomposition on five points have been studied in these problems. Numerical experiments support the theoretical statements for the solution of these difference schemes. [ABSTRACT FROM AUTHOR]

There are a couple of purposes in this paper: to study a problem of approximation with exponential functions and to show its relevance for economic science. The solution of the first problem is as conclusive as it can be: working with the max-norm, we determine which datasets have best approximation by means of exponentials of the form f (t) = b + a exp (k t) , we give a necessary and sufficient condition for some a , b , k ∈ R to be the coefficients that give the best approximation, and we give a best approximation by means of limits of exponentials when the dataset cannot be best approximated by an exponential. For the usual case, we have also been able to approximate the coefficients of the best approximation. As for the second purpose, we show how to approximate the coefficients of exponential models in economic science (this is only applying the R-package nlstac) and also the use of exponential autoregressive models, another well-established model in economic science, by utilizing the same tools: a numerical algorithm for fitting exponential patterns without initial guess designed by the authors and implemented in nlstac. We check one more time the robustness of this algorithm by successfully applying it to two very distant areas of economy: demand curves and nonlinear time series. This shows the utility of TAC (Spanish for CT scan) and highlights to what extent this algorithm can be useful. [ABSTRACT FROM AUTHOR]

In this paper, the shortest paths search for all departure times (profile search) are discussed. This problem is called a time-dependent shortest path problem (TDSP) and is suitable for time-dependent travel-time analysis. Particularly, this paper deals with the ε -approximation of profile search computation. The proposed algorithms are based on a label correcting modification of Dijkstra's algorithm (LCA). The main idea of the algorithm is to simplify the arrival function after every relaxation step so that the maximum relative error is maintained. When the maximum relative error is 0.001, the proposed solution saves more than 97% of breakpoints and 80% of time compared to the exact version of LCA. Furthermore, the runtime can be improved by other 15% to 40% using heuristic splitting of the original departure time interval to several subintervals. The algorithms we developed can be used as a precomputation step in other routing algorithms or for some travel time analysis. [ABSTRACT FROM AUTHOR]

The paper introduces a new approach to enhance optimization algorithms when solving the piecewise linearization problem of a given function. Eight swarm intelligence algorithms were selected to be experimentally compared. The problem is represented by the calculation of the distance between the original function and the estimation from the piecewise linear function. Here, the piecewise linearization of 2D functions is studied. Each of the employed swarm intelligence algorithms is enhanced by a newly proposed automatic detection of the number of piecewise linear parts that determine the discretization points to calculate the distance between the original and piecewise linear function. The original algorithms and their enhanced variants are compared on several examples of piecewise linearization problems. The results show that the enhanced approach performs sufficiently better when it creates a very promising approximation of functions. Moreover, the degree of precision is slightly decreased by the focus on the speed of the optimization process. [ABSTRACT FROM AUTHOR]

High resolution Digital Surface Models (DSMs) produced from airborne laser-scanning or stereo satellite images provide a very useful source of information for automated 3D building reconstruction. In this paper an investigation is reported about extraction of 3D building models from high resolution DSMs and orthorectified images produced from Worldview-2 stereo satellite imagery. The focus is on the generation of 3D models of parametric building roofs, which is the basis for creating Level Of Detail 2 (LOD2) according to the CityGML standard. In particular the building blocks containing several connected buildings with tilted roofs are investigated and the potentials and limitations of the modeling approach are discussed. The edge information extracted from orthorectified image has been employed as additional source of information in 3D reconstruction algorithm. A model driven approach based on the analysis of the 3D points of DSMs in a 2D projection plane is proposed. Accordingly, a building block is divided into smaller parts according to the direction and number of existing ridge lines for parametric building reconstruction. The 3D model is derived for each building part, and finally, a complete parametric model is formed by merging the 3D models of the individual building parts and adjusting the nodes after the merging step. For the remaining building parts that do not contain ridge lines, a prismatic model using polygon approximation of the corresponding boundary pixels is derived and merged to the parametric models to shape the final model of the building. A qualitative and quantitative assessment of the proposed method for the automatic reconstruction of buildings with parametric roofs is then provided by comparing the final model with the existing surface model as well as some field measurements. [ABSTRACT FROM AUTHOR]

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ″ (x) + α u ′ (x) + β u (x) = r (x) , with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ″ (x) + α u ′ (x) + β u (x) = 0 . Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation. [ABSTRACT FROM AUTHOR]

Function interpolation and approximation are classical problems of vital importance in many science/engineering areas and communities. In this paper, we propose a powerful methodology for the optimal placement of centers, when approximating or interpolating a curve or surface to a data set, using a base of functions of radial type. In fact, we chose a radial basis function under tension (RBFT), depending on a positive parameter, that also provides a convenient way to control the behavior of the corresponding interpolation or approximation method. We, therefore, propose a new technique, based on multi-objective genetic algorithms, to optimize both the number of centers of the base of radial functions and their optimal placement. To achieve this goal, we use a methodology based on an appropriate modification of a non-dominated genetic classification algorithm (of type NSGA-II). In our approach, the additional goal of maintaining the number of centers as small as possible was also taken into consideration. The good behavior and efficiency of the algorithm presented were tested using different experimental results, at least for functions of one independent variable. [ABSTRACT FROM AUTHOR]

In recent years, various methods and directions for solving a system of Boolean algebraic equations have been invented, and now they are being very actively investigated. One of these directions is the method of transforming a system of Boolean algebraic equations, given over a ring of Boolean polynomials, into systems of equations over a field of real numbers, and various optimization methods can be applied to these systems. In this paper, we propose a new transformation method for Solving Systems of Boolean Algebraic Equations (SBAE). The essence of the proposed method is that firstly, SBAE written with logical operations are transformed (approximated) in a system of harmonic-polynomial equations in the unit n-dimensional cube K n   with the usual operations of addition and multiplication of numbers. Secondly, a transformed (approximated) system in K n   is solved by using the optimization method. We substantiated the correctness and the right to exist of the proposed method with reliable evidence. Based on this work, plans for further research to improve the proposed method are outlined. [ABSTRACT FROM AUTHOR]

Zadeh's extension principle is one of the elementary tools in fuzzy set theory, and among other things, it provides a natural extension of a real-valued continuous self-map to a self-map having fuzzy sets as its arguments. The purpose of this paper is to introduce a new algorithm that can be used for simulations of fuzzy dynamical systems induced by interval maps. The core of the proposed algorithm is based on calculations including piecewise linear maps, and consequently, an implementation of an optimization algorithm (in our case, particle swarm optimization) was prepared to obtain necessary piecewise linear approximations. For all parts of this algorithm, we provide detailed testing and numerous examples. [ABSTRACT FROM AUTHOR]

In this paper, it is shown how the performance of the monopulse algorithm in the presence of an additive noise can be obtained analytically. In a previous study, analytic performance analysis based on the first-order Taylor series and the second-order Taylor series was conducted. By adopting the third-order Taylor series, it is shown that the analytic performance based on the third-order Taylor series can be brought closer to the performance of the original monopulse algorithm than the analytic performance based on the first-order Taylor series and the second-order Taylor series. [ABSTRACT FROM AUTHOR]

We consider a new approach to approximate stability analysis for a tri-additive functional inequality and to obtain the optimal approximation for permuting tri-derivations and tri-homomorphisms in unital matrix algebras via the vector-valued alternative fixed-point theorem, which is a popular technique of proving the stability of functional equations. We also present a small list of aggregation functions on the classical, well-known special functions to investigate the best approximation error estimates using a different concept of perturbation stability. [ABSTRACT FROM AUTHOR]

This paper aims to consider approximation-estimation tests for decision-making by machine-learning methods, and integral-estimation tests are defined, which is a generalization for the continuous case. Approximation-estimation tests are measurable sampling functions (statistics) that estimate the approximation error of monotonically increasing number sequences in different classes of functions. These tests make it possible to determine the Markov moments of a qualitative change in the increase in such sequences, from linear to nonlinear type. If these sequences are trajectories of discrete quasi-deterministic random processes, then moments of change in the nature of their growth and qualitative change in the process match up. For example, in cluster analysis, approximation-estimation tests are a formal generalization of the "elbow method" heuristic. In solid mechanics, they can be used to determine the proportionality limit for the stress strain curve (boundaries of application of Hooke's law). In molecular biology methods, approximation-estimation tests make it possible to determine the beginning of the exponential phase and the transition to the plateau phase for the curves of fluorescence accumulation of the real-time polymerase chain reaction, etc. [ABSTRACT FROM AUTHOR]

In a series of three articles, spline approximation is presented from a geodetic point of view. In part 1, an introduction to spline approximation of 2D curves was given and the basic methodology of spline approximation was demonstrated using splines constructed from ordinary polynomials. In this article (part 2), the notion of B-spline is explained by means of the transition from a representation of a polynomial in the monomial basis (ordinary polynomial) to the Lagrangian form, and from it to the Bernstein form, which finally yields the B-spline representation. Moreover, the direct relation between the B-spline parameters and the parameters of a polynomial in the monomial basis is derived. The numerical stability of the spline approximation approaches discussed in part 1 and in this paper, as well as the potential of splines in deformation detection, will be investigated on numerical examples in the forthcoming part 3. [ABSTRACT FROM AUTHOR]

Fractional calculus has found multiple applications around the world. It is especially prevalent in the domains of control and electronics. One of the key elements of fractional applications is the fractional integral (or integrator) which is a backbone of famous PI λ D controller. It gives advantages of traditional PID with a limited phase lag. The are, however, issues with implementation, which will allow good low-frequency behavior. In this paper, we consider a diffusive realization of a fractional integrator with the use of quadratures. We implemented this method in numerical package SoftFrac, and we illustrate how different quadratures work for this purpose. We show superiority of bounded domain integration with logarithmic transformation and explain issues with behavior for extremely low frequencies. [ABSTRACT FROM AUTHOR]

In this paper, the Singular-Polynomial-Fuzzy-Model (SPFM) approach problem and impulse elimination are investigated based on sliding mode control for a class of nonlinear singular system (NSS) with impulses. Considering two numerical examples, the SPFM of the nonlinear singular system is calculated based on the compound function type and simple function type. According to the solvability and the steps of two numerical examples, the method of solving the SPFM form of the nonlinear singular system with (and without) impulse are extended to the more general case. By using the Heine–Borel finite covering theorem, it is proven that a class of nonlinear singular systems with bounded impulse-free item (BIFI) properties and separable impulse item (SII) properties can be approximated by SPFM with arbitrary accuracy. The linear switching function and sliding mode control law are designed to be applied to the impulse elimination of SPFM. Compared with some published works, a human posture inverted pendulum model example and Example 3.2 demonstrate that the approximation error is small enough and that both algorithms are effective. Example 3.3 is to illustrate that sliding mode control can effectively eliminate impulses of SPFM. [ABSTRACT FROM AUTHOR]

The transfer functions G (s) for different types of heat exchangers obtained from their partial differential equations usually contain some irrational components which reflect quite well their spatio-temporal dynamic properties. However, such a relatively complex mathematical representation is often not suitable for various practical applications, and some kind of approximation of the original model would be more preferable. In this paper we discuss approximate rational transfer functions G ^ (s) for a typical thick-walled double-pipe heat exchanger operating in the counter-flow mode. Using the semi-analytical method of lines, we transform the original partial differential equations into a set of ordinary differential equations representing N spatial sections of the exchanger, where each nth section can be described by a simple rational transfer function matrix G n (s) , n = 1 , 2 , ... , N . Their proper interconnection results in the overall approximation model expressed by a rational transfer function matrix G ^ (s) of high order. As compared to the previously analyzed approximation model for the double-pipe parallel-flow heat exchanger which took the form of a simple, cascade interconnection of the sections, here we obtain a different connection structure which requires the use of the so-called linear fractional transformation with the Redheffer star product. Based on the resulting rational transfer function matrix G ^ (s) , the frequency and the steady-state responses of the approximate model are compared here with those obtained from the original irrational transfer function model G (s) . The presented results show: (a) the advantage of the counter-flow regime over the parallel-flow one; (b) better approximation quality for the transfer function channels with dominating heat conduction effects, as compared to the channels characterized by the transport delay associated with the heat convection. [ABSTRACT FROM AUTHOR]

We establish a new fixed point theorem in the setting of convex b-metric spaces that ensures the existence of fixed point for Cirić contraction with the assumption k < 1 s 2 . Also, the fixed point is approximated by Krasnoselskij iterative procedure. Moreover, we discuss the stability of fixed point for the aforesaid contraction. As a consequence, we develop a common fixed point and coincidence point result. Finally, we provide a number of examples to illustrate the findings presented here and incorporate these findings to solve an initial value problem. [ABSTRACT FROM AUTHOR]

This article presents a harmonic analysis of the stator currents of a squirrel-cage induction motor fed by a voltage source inverter with PWM space vector control (SVPWM). The influence of PWM switching frequency and dead time (dead band) of controlled transistors on THD and electromagnetic torque ripple is shown. The aim is to determine the lowest switching frequency of the transistors for which the drive will operate correctly. Characteristics were determined as functions in the form of THD ( f P W M ), where the least square approximation was used for stator current measurements when the PWM switching frequency was changed. The approximations were realized for simulation and experimental results. To clarify the results, the operation of hardware PWM circuits in microcontrollers is analyzed. [ABSTRACT FROM AUTHOR]

The main goal of machine learning is the creation of self-learning algorithms in many areas of human activity. It allows a replacement of a person with artificial intelligence in seeking to expand production. The theory of artificial neural networks, which have already replaced humans in many problems, remains the most well-utilized branch of machine learning. Thus, one must select appropriate neural network architectures, data processing, and advanced applied mathematics tools. A common challenge for these networks is achieving the highest accuracy in a short time. This problem is solved by modifying networks and improving data pre-processing, where accuracy increases along with training time. Bt using optimization methods, one can improve the accuracy without increasing the time. In this review, we consider all existing optimization algorithms that meet in neural networks. We present modifications of optimization algorithms of the first, second, and information-geometric order, which are related to information geometry for Fisher–Rao and Bregman metrics. These optimizers have significantly influenced the development of neural networks through geometric and probabilistic tools. We present applications of all the given optimization algorithms, considering the types of neural networks. After that, we show ways to develop optimization algorithms in further research using modern neural networks. Fractional order, bilevel, and gradient-free optimizers can replace classical gradient-based optimizers. Such approaches are induced in graph, spiking, complex-valued, quantum, and wavelet neural networks. Besides pattern recognition, time series prediction, and object detection, there are many other applications in machine learning: quantum computations, partial differential, and integrodifferential equations, and stochastic processes. [ABSTRACT FROM AUTHOR]

Pressure intensifiers have their utilization in hydraulic circuits, when there is a need for pressure multiplication for some period of time. Using intensifiers provides energy savings, cost reductions or smaller dimensioning. This article is about a pressure intensifier based on rotary flow dividers. Its functioning is evaluated by its pressure-multiplying efficiency. This efficiency is always calculated from values gained from measurements. However, it would be very convenient to be able to evaluate these types of intensifiers without a need to provide specific measurements for every type of intensifier and rather calculate their efficiency using the general parameters of a hydraulic circuit. Therefore, this article is about the description of a model which calculates the pressure-multiplying efficiency only with two input values: the pressure and the speed of an intensifier. Final derived formulas are also used for the efficiency prediction at pressure levels never measured before and then verified by measurements. [ABSTRACT FROM AUTHOR]

Based on the idea of integral averaging and function extension, an extended Kantorovich-type neural network operator is constructed, and its error estimate of approximating continuous functions is obtained by using the modulus of continuity. Furthermore, by introducing the normalization factor, the approximation property of the new version of the extended Kantorovich-type neural network (normalized extended Kantorovich-type neural network) operator is obtained in L p [ − 1 , 1 ] . The numerical examples show that this newly proposed neural network operator has a better approximation performance than the classical one, especially at the endpoints of a compact interval. [ABSTRACT FROM AUTHOR]