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The meanings and dimensions of notations N ca , N c and c 3 were not clearly described in our published paper, and the approximation and the split representation of logarithmic function caused Eq. (11) to be not easily understood. In this response paper, the author first explained the dimensions of N ca , N c and c 3 , and the derivation process of Eq. (11). Then, parameter L 0 with unit length was added to the related equations, the suitability and necessity of approximating Eq. (10) to get Eq. (11) and the influence of this approximation were also explained. [ABSTRACT FROM AUTHOR]

The set of smooth functions is not dense in Morrey spaces. To address the density issue in Morrey spaces, Zorko spaces are defined by utilizing the difference of a function of first order. In this paper, we propose a subspace of Morrey spaces which is defined using the difference of a function of second order. Approximation properties in the new subspace are investigated and the relation with Zorko spaces is studied via properties of smoothness spaces. [ABSTRACT FROM AUTHOR]

This research paper is devoted to including new methods of approximation in Theory Of Approximation using Chebyshev Polynomials. The research focuses on creating a potential model for approximation using Neural Networks. This work enables mathematical analysis of the Chebyshev polynomials and activation functions. One of the goals of this paper was to minimize the time spent on the learning process. However, implementing every term of Chebyshev Polynomial could only be realized in some frameworks. [ABSTRACT FROM AUTHOR]

In this paper on a basis of functional analysis methods we justified theoretically the possibility of different orders root-polynomial functions application for interpolation and approximation of the boundary trajectory of an electron beam in case of its propagation in ionized gas with compensation of the space charge of the beam electrons. It is shown, that the root-polynomial functions satisfy to the second-order differential equation, describing the boundary trajectory of the beam electrons under such physical conditions. The results of interpolation and approximation of the boundary trajectory of the electron beam by root-polynomial functions from the second to the fifth order under the following physical conditions are presented. The interpolation results are compared with the corresponded results of the differential equation solution for the boundary trajectory of the electron beam using Runge-Kutta numerical method of the fourth order. These results are considered as reference ones for the interpolation task. To solve the approximation problem, in this paper an iterative algorithm based on the calculation of both values of the function and its derivatives at reference points is proposed. The approximation task is solved for a sample of numerical data obtained by experimental electron-beam equipment for real processes of current electron-beam technologies, which led to a rather large value of the experimental measurement error due to the effect of random factors associated with thermal treatment of products with electron beam. Test calculations show that the error of interpolation and approximation of numerical data, describing the boundary trajectory of electron beam in case of its propagation in ionized gas, does not exceed a few percent. The theoretical and practical results obtained in this paper are interesting for a wide range of specialists who are engaged in the physics of electron beams, the development of electron-beam technological equipment and implementation of current electron-beam technologies into industry. [ABSTRACT FROM AUTHOR]

A theorem of É. Borel's asserts that one of any three consecutive convergents of a real number a, which we denote p q , satisfies the inequality a - p q < C q 2 with C = 1 5 . In this paper we give more precise information about the constant C. [ABSTRACT FROM AUTHOR]

In this paper, let Ω be homogeneous of degree zero which has vanishing moment of order one, A be a function on R d such that ∇ A ∈ BMO (R d) , we consider a class of nonstandard singular integral operators, T Ω , A , with rough kernel being of the form Ω (x - y) | x - y | d + 1 (A (x) - A (y) - ∇ A (y) (x - y)) . This operator is closely related to the Calderón commutator. We prove that, under the Grafakos-Stefanov minimum size condition G S β (S d - 1) with 2 < β < ∞ for Ω , T Ω , A is bounded on L p (R d) for p with 1 + 1 / (β - 1) < p < β . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the fractal nature of the local fractional Landau–Ginzburg–Higgs Equation (LFLGHE) describing nonlinear waves with weak scattering in a fractal medium. The main goal of the paper is to introduce and apply the Local Fractional Elzaki Variational Iteration Method (LFEVIM) for solution of LFLGHE. Convergence analysis of LFEVIM solution for general nonlinear local fractional partial differential equation is also provided. Two examples of the local fractional LFLGHE are considered to demonstrate the applicability of the proposed technique with numerical simulations on Cantor set. [ABSTRACT FROM AUTHOR]

We consider the linear elasticity system for a body coated with a thin elastic layer. The effect of the thin layer on that body can be described through an impedance operator linking the displacement and the traction on the interface. This operator cannot be in general explicitly computed for an arbitrary shape. In this paper, we consider the case of a linear elastic slab with Hooke's law. We compute by the Fourier transform the exact symbol of the impedance operator and prove that it is a pseudodifferential operator of order one. Then, we give a Padé-like approximation at order three (with respect to the thickness of the layer) for the impedance operator. The pattern of this approximation will be a helpful guide for approximating the impedance operator for a body whose boundary has nonzero curvature. [ABSTRACT FROM AUTHOR]

In this paper we present and prove some new results concerning the rate of approximation of T means of functions in Lp(G) and in lip (α, p), for 1 ≤ p < ∞ and α > 0. As a corollary, we obtain some new as well as known approximation inequalities. [ABSTRACT FROM AUTHOR]

In this paper we improve and complement a result by Móricz and Siddiqi [12]. In particular, we prove that their inequality of the Nörlund means with respect to the Walsh system holds also without their additional condition. Moreover, we prove some new approximation results and inequalities in Lebesgue spaces for any 1 ≤ p < ∞. [ABSTRACT FROM AUTHOR]

This paper introduces a new modification to an iterative method for solving Cauchy problems (IVPs) based on an inverse polynomial technique. The proposed method is proven to be consistent, stable, and convergent. We also demonstrate the consistency property of the One-stage scheme and the Two-stage scheme, as well as the stability property of the proposed method. To validate the accuracy of our approach, we conduct several numerical experiments and present the results graphically. Our findings show that the proposed method outperforms existing approaches in terms of accuracy and efficiency. Additionally, we discuss the implications of our results for future research in this field. Overall, this paper provides a valuable contribution to the numerical solution of IVPs and lays the groundwork for further exploration in this area. [ABSTRACT FROM AUTHOR]

To enhance the accuracy of analyzing short datasets, this paper proposes a novel ensemble learning method that utilizes a single the General Regression Neural Network (GRNN). The core idea behind this method is the synthesis of additional pairs of vectors with different signs around each current vector from the test sample. This is achieved by employing the method of random symmetric perturbations and averaging the prediction outputs for the current vector and all synthesized vectors in its vicinity. Implementing this approach leads to a significant increase in prediction accuracy for short datasets. It achieves error compensation for each pair of additional vectors with different signs and also for the overall prediction result of the current vector and all additional pairs of synthetic vectors created for it. The effectiveness of the proposed method is validated through modeling on a small real-world biomedical dataset, and the optimal parameters have been selected. Comparative analysis with existing GRNN-based methods demonstrates a substantial improvement in accuracy. [ABSTRACT FROM AUTHOR]

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]

The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier transforms. It is also the main part of the theory of singular integral equations on the real line. Therefore, approximations of Hilbert transform are of great interest. Many papers have dealt with the numerical approximation of singular integrals in case of bounded intervals. On the other hand, the literature concerning the numerical integration on unbounded intervals is much sparser than the one on bounded intervals. There is very little literature concerning the case of Hilbert transform. This article is dedicated to the approximation of Hilbert transform in Hölder spaces by the operators introduced by V.R.Kress and E.Mortensen to approximate the Hilbert transform of analytic functions in a strip. [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, an economical unconditionally stable method of variable directions with extrapolation of numerical solutions of problems for parabolic equations containing mixed derivatives is proposed and justified by approximation and stability, which, in comparative analysis with other numerical methods, showed the highest accuracy and a unique margin of stability when changing grid characteristics. [ABSTRACT FROM AUTHOR]

In this paper, we consider norm convergence issues for matrix-based de la Vallée Poussin-like means of Fourier series for the Walsh system. In the main theorem of the paper, we state a proposition that estimates the difference between the named means above and the corresponding function in norm. The upper estimation is given by and as a function of the modulus of continuity of the function. [ABSTRACT FROM AUTHOR]

Given an edge-colored graph G c , a set of p pairs of vertices (a i , b i) together with p numbers k 1 , k 2 , … k p associated with the pairs, can we find a set of alternating paths linking the pairs (a 1 , b 1) , (a 2 , b 2) , … , in their respective numbers k 1 , k 2 , … k p ? Such is the question addressed in this paper. The problem being highly intractable, we consider a restricted version of it to edge-colored complete graphs. Even so restricted, the problem remains intractable if the paths/trails must be edge-disjoint, but it ceases to be so if the paths/trails are to be vertex-disjoint, as is proved in this paper. An approximation algorithm is presented in the end, with a performance ratio asymptotically close to 3/4 for a restricted version of the problem. [ABSTRACT FROM AUTHOR]

In this paper, we present a sharper version of the results in the paper [H. N. Mhaskar, Dimension independent bounds for general shallow networks, Neural Netw. 123 (2020) 142–152]. Let and be compact metric spaces. We consider approximation of functions of the form x ↦ ∫ G (x , y) d τ (y) , x ∈ , by G -networks of the form x ↦ ∑ k = 1 n a k G (x , y k) , y 1 , ... , y n ∈ , a 1 , ... , a n ∈ ℝ. Defining the dimensions of and in terms of covering numbers, we obtain dimension independent bounds on the degree of approximation in terms of n , where also the constants involved are all dependent at most polynomially on the dimensions. Applications include approximation by power rectified linear unit networks, zonal function networks, certain radial basis function networks as well as the important problem of function extension to higher-dimensional spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study a two-point boundary value problem involving p-Laplacian and its relationship with the associated discrete ones. Specifically, by using discrete variational methods, we first obtain a sequence of positive solutions for the associated discrete two-point boundary value problems corresponding to different step lengths. Then, utilizing the sequence of positive solutions, we construct a sequence of continuous functions which can be shown to be precompact. Finally, we prove that the limit function of the convergent subsequence is the desired positive solution. In particular, our result covers the cases when the nonlinear function in the equation is (p - 1) -superlinear and asymptotically (p - 1) -linear. [ABSTRACT FROM AUTHOR]

In this paper, we introduce, graph of lower and upper approximation of a non-empty subset of a nearring with respect to an ideal. We relate the properties of these graphs with properties of ideals. We study the relationship between the connectivity of the graph and properties of the approximation set. We obtain the properties of these graphs under nearring homomorphism. [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]

It is well known that the Kolmogorov complexity function (the minimal length of a program producing a given string, when an optimal programming language is used) is not computable and, moreover, does not have computable lower bounds. In this paper we investigate a more general question: can this function be approximated? By approximation we mean two things: firstly, some (small) difference between the values of the complexity function and its approximation is allowed; secondly, at some (rare) points the values of the approximating function may be arbitrary. For some values of the parameters such approximation is trivial (e.g., the length function is an approximation with error d except for a O (2 − d) fraction of inputs). However, if we require a significantly better approximation, the approximation problem becomes hard, and we prove it in several settings. Firstly, we show that a finite table that provides good approximations for Kolmogorov complexities of n-bit strings, necessarily has high complexity. Secondly, we show that there is no good computable approximation for Kolmogorov complexity of all strings. In particular, Kolmogorov complexity function is neither generically nor coarsely computable, as well as its approximations, and the time-bounded Kolmogorov complexity (for any computable time bound) deviates significantly from the unbounded complexity function. We also prove hardness of Kolmogorov complexity approximation in another setting: the mass problem whose solutions are good approximations for Kolmogorov complexity function is above the halting problem in the Medvedev lattice. Finally, we mention some proof-theoretic counterparts of these results. A preliminary version of this paper was presented at CiE 2019 conference (In Computing with Foresight and Industry – 15th Conference on Computability in Europe, CiE 2019, Durham, UK, July 15–19, 2019, Proceedings (2019) 230–239 Springer). [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]

Copyright of Journal of English as a Lingua Franca is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper proposes a numerical approach based on the fractional complex transform and the homotopy perturbation method to solving the space-time fractional Benjamin-Bona-Mahony (mBBM) equation with Caputo fractional derivative. Approximated solutions with high accuracy are provided without linearization or complicated computation. Numerical examples are given to illustrate the efficiency of this method. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mathematics, Mechanics & Computer Science is the property of Al-Farabi Kazakh National University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In the paper, the estimations of the approximate calculation of double integrals with variable upper limits are suggested. The theorem on estimation for the main error term of approximate integration using the approach of substituting the integrand with an approximating function is formulated and proved, as well as the theorem on the estimation of the main error term in the case of calculating double integrals using the sequential integration approach. Estimations of integration errors are obtained when dividing the variable region of integration into rectangular grids, in particular, four possible options for adapting the grid to the change of the integration region are considered. For each of these options the estimations are found for rectangular and curvilinear elements of the grid as well as total estimation of double integration with variable upper limits. [ABSTRACT FROM AUTHOR]

Because of the processing of continuous unstructured large streams of data, mining real-time streaming data is a more challenging research issue than mining static data. The privacy issue persists when sensitive data is included in streaming data. In recent years, there has been significant progress in research on the anonymization of static data. For the anonymization of quasi-identifiers, two typical strategies are generalization and suppression. However, the high dynamicity and potential infinite properties of the streaming data make it a challenging task. To end this, we propose a novel Efficient Approximation and Privacy Preservation Algorithms (EAPPA) framework in this paper to achieve efficient data pre-processing from the live streaming and its privacy preservation with minimum Information Loss (IL) and computational requirements. As the existing privacy preservation solutions for streaming data suffer from the challenges of redundant data, we first propose the efficient technique of data approximation with data pre-processing. We design the Flajolet Martin (FM) algorithm for robust and efficient approximation of unique elements in the data stream with a data cleaning mechanism. We fed the periodically approximated and pre-processed streaming data to the anonymization algorithm. Using adaptive clustering, we propose innovative k-anonymization and l-diversity privacy principles for data streams. The proposed approach scans a stream to detect and reuse clusters that fulfill the k-anonymity and l-diversity criteria for reducing anonymization time and IL. The experimental results reveal the efficiency of the EAPPA framework compared to state-of-art methods. [ABSTRACT FROM AUTHOR]

Recently, the accuracy of spike neural network (SNN) has been significantly improved by deploying convolutional neural networks (CNN) and their parameters to SNN. The deep convolutional SNNs, however, suffer from large amounts of computations, which is the major bottleneck for energy efficient SNN processor design. In this paper, we present an input-dependent computation reduction approach, where relatively unimportant neurons are identified and pruned without seriously sacrificing the accuracies. Specifically, a neuron pruning in temporal domain is proposed that prunes less important neurons and skips its future operations based on the layer-wise pruning thresholds of membrane voltages. To find the pruning thresholds, two pruning threshold search algorithms are presented that can efficiently trade-off accuracy and computational complexity with a given computation reduction ratio. The proposed neuron pruning scheme has been implemented using 65 nm CMOS process. The SNN processor achieves a 57% energy reduction and a 2.68× speed up, with up to 0.82% accuracy loss and 7.3% area overhead for CIFAR-10 dataset. [ABSTRACT FROM AUTHOR]

In this paper, we propose two new approximation methods on a general mesh for the generalized Caputo fractional derivative of order α ∈ (0 , 1). The accuracy of these two methods is shown to be of order (3 - α) which improves some previous work done to date. To demonstrate the accuracy and usefulness of the proposed approximations, we carry out experiment on test examples and apply these approximations to solve generalized fractional sub-diffusion equations. The numerical results indicate that the proposed methods perform well in practice. Our contributions lie in two aspects: (i) we propose high order approximations that work on a general mesh; (ii) we establish the well-posedness of generalized fractional sub-diffusion equations and develop numerical schemes using the new high order approximations. [ABSTRACT FROM AUTHOR]