3D Fourier Coefficients on the Class of Differentiable Functions and Spline Interflatation

Lytvyn, Oleg M.; Nechuiviter, Olesya P.

doi:10.1615/jautomatinfscien.v44.i3.40

Cited by 8 publications

(4 citation statements)

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“…In addition, the algorithm for the optimal choice of lines in the construction of the cubature formula for the approximate calculation of double integrals from the function of two variables is indicated. The results of these studies are based on the theory of approximate calculation of 288 integrals from rapidly oscillating functions of many variables using new information operators [4], [25] - [29]. However, these results alone are also of great value, as numerical integration is often used in mathematical modeling of processes.…”

Section: Research Resultsmentioning

confidence: 99%

Application of the Theory of New Information Operators in Conducting Research in the Field of Information Technologies

Nechuiviter

2021

ITLT

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Due to the rapid development and implementation of the latest information technologies in many fields of science and technology significant changes have taken place. In particular, new mathematical theories have emerged; they can be effectively used in building and improving existing mathematical models of various phenomena and objects. Mathematical modeling is currently one of the main methods in scientific research of technological, social, economic processes and natural phenomena. No research thesis is complete without the use of mathematical methods and algorithms. In view of this it is important to acquaint young scientists with modern trends in mathematical modeling during their training for PhD degree, in particular. The development of information technology has contributed to the emergence of new approaches to obtaining, processing and analyzing information in scientific research. In this way, the theory of new information operators, which is widely used in mathematical modeling, was created. The analysis of the existing research reveals the need to consider the theoretical basics of new information operators and their application to the numerical integration of several variables functions. The example of the problem of numerical integration of two variables functions shows the way how to create cubature formulas depending on the type of the input information about function. The paper presents new cubature formulas for the approximate calculation of double integrals, in the case when the information is given by the set of traces on lines and values of the function at points. The proposed formulas have high accuracy but use much less input information compared to classical formulas for numerical integration of double integrals. All the results are supported by numerous examples made in the computer mathematics system Mathcad. The implementation of the developed methodology of using new information operators in scientific research has shown its efficacy during seminars with Master and PhD students.

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Section: Research Resultsmentioning

confidence: 99%

Application of the Theory of New Information Operators in Conducting Research in the Field of Information Technologies

Nechuiviter

2021

ITLT

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“…Для обробки вищезазначеної інформації та побудови нових математичних моделей також можуть бути використані нові інформаційні оператори, які гарно себе зарекомендували при математичному моделюванні задач цифрової обробки сигналів і зображень [7][8][9][10].…”

Section: вступunclassified

Mathematical modeling of foreign bodies with different density in biological and non-biological models in the experiment

Khoroshun,

Nehoduiko,

Makarov

et al. 2024

ЕМ

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Background. Modeling allows investigating both existing and predicted processes and is widely used in basic science and in many industries. The aim is to develop a mathematical model for determining the size of the foreign bodies (FB) and their radiographic density in non-biological and biological models to improve the results of diagnosis for gunshot ricochet wounds. Materials and methods. In the biological non-living model (a piece of pork) and non-biological models (polystyrene, foam rubber), we place the FB made of paper, leather, rubber, plastic, and lithium-ion batteries. The number of the FB is 9 of each type. Number of models is 3 each: pork, polystyrene, foam rubber. We measure the dimensions of the FB and models with a metric ruler. For each model, we select the FB, which we label with the study number. We immerse the FB to the same depth using a Billroth general surgical medium hemostatic clamp in the following sequence: paper, leather, rubber, plastic, and lithium-ion battery. Multislice computed tomography (MSCT) of the models is performed on the Revolution EVO (2021) apparatus with measurement of the sizes and radiographic density of the FB and models. Radiographic density was measured in conventional units on the Hounsfield scale. For each study group, the ratio of the actual sizes of the removed FB and according to MSCT data was determined in the MathCad 15 computer math software, depending on the radiological density of the FB and the model. Results. According to MSCT data, the radiographic density of the models on the Hounsfield scale is as follows: polystyrene — –990.0 ± 0.3 units; foam rubber — –985.0 ± 0.2 units; pork — 62.0 ± 0.3 units; radiographic density of the foreign bodies: paper — –743.0 ± 10.3 units, leather — –258.0 ± 14.2 units, rubber — –12.0 ± 2.6 units, plastic — 183.0 ± 14.6 units, lithium-ion batteries — 3071 units. Visualization of paper in non-biological and biological models and leather in non-biological models is problematic due to the similar radiographic density of the models and the inability to measure the dimensions. When the FB (rubber, plastic, battery) is immersed in polystyrene, the coefficient of length (CL) is 1.0612, the coefficient of width (CW) is 1.928; in foam rubber: CL is 0.9926, CW is 1.9641; in pork: CL is 0.8394, CW is 1.534. Comparing the average coefficients of the ratio (CL and CW), we find that the coefficient in a biological model is closest to 1. This means that the FB from rubber, plastic, and batteries are best detected in pork. Conclusions. The actual dimensions of the FB placed in biological and non-biological models differ from those obtained by MSCT. Data correction is performed through calculated coefficients for length and width. The radiographic density of the model affects the radial visualization of the FB. The use of mathematical modeling in determining the sizes and radiographic density allows reducing the measurement error and determine the structure of the FB.

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“…В задачах цифрової обробки сигналів та зображень наближене обчислення інтегралів від швидко осцилюючих функцій з використанням нових інформаційних операторів дозволило побудувати кубатурні формули з використанням різних типів завдання інформації [7]. Побудовані кубатурні формули використовують як дані значення функції на площинах, на лініях та в точках [8][9][10][11]. Доведено, що більшість таких формул -оптимальні за порядком точності.…”

Section: наведено оптимальну за порядком точності кубатурну формулу н...unclassified

Optimal by the Order of Accuracy Cubature Formula for the Approximate Calculation of Triple Integrals from Fast Oscillating Functions in General View

Nechuiviter,

Ivanov

2023

Cybernetics and Computer Technologies

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Introduction. The rapid development of digital technologies encourages scientists to create new or improve existing mathematical models of technical processes. It is time to develop mathematical models with different types of data. In the tasks of digital signal and image processing, the approximate calculation of integrals from rapidly oscillating functions using new information operators makes it possible to build cubature formulas using different types of information: the values of functions on planes, lines and points can be used as data. The purpose is to present and investigate the optimal cubature formula for the approximate calculation of the triple integral from rapidly oscillating functions in the general form on the class of differential functions. Information about functions are traces on systems of mutually perpendicular planes. Results. The study of the problems of digital signal and image processing continued using the example of numerical integration of triple integrals from rapidly oscillating functions in the general form. The values of functions on systems of mutually perpendicular planes are using for constructed cubature formula. The main attention in the research focuses on obtaining the estimations of errors. Proposed cubature formula for the approximate calculation of the triple integral from rapidly oscillating functions in general view is optimal in order of accuracy on the class of differential functions. The conducted numerical experiment confirmed the theoretical results. Conclusions. The obtained results make it possible to build new and improve existing mathematical models of processes with different types of input information. New information operators are a powerful tool in the development of such models. Cubature formulas for the approximate calculation of integrals from rapidly oscillating functions of many variables have been created. Іn the construction of the formulas traces of the function on planes, lines, and points are used. Formulas in their construction use function traces on planes, lines, and points. In this work, a cubature formula for the approximate calculation of the triple integral from a rapidly oscillating function in the general form, which is optimal in order of accuracy, is constructed and investigated on the class of differentiable functions. A feature of the proposed formula is the use of values of functions on systems of mutually perpendicular planes as data. Keywords: integrals of rapidly oscillating functions of many variables, cubature formulas, new information operators, digital signal and image processing.

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3D Fourier Coefficients on the Class of Differentiable Functions and Spline Interflatation

Cited by 8 publications

References 0 publications

Application of the Theory of New Information Operators in Conducting Research in the Field of Information Technologies

Application of the Theory of New Information Operators in Conducting Research in the Field of Information Technologies

Mathematical modeling of foreign bodies with different density in biological and non-biological models in the experiment

Optimal by the Order of Accuracy Cubature Formula for the Approximate Calculation of Triple Integrals from Fast Oscillating Functions in General View

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