Loading [MathJax]/jax/output/HTML-CSS/jax.js
 
 
Sign in to use this feature.

Years

Between: -

Subjects

remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline

Journals

Article Types

Countries / Regions

Search Results (14)

Search Parameters:
Keywords = quasi-rational approximation

Order results
Result details
Results per page
Select all
Export citation of selected articles as:
8 pages, 437 KiB  
Article
Accurate Analytical Approximation for the Bessel Function J2(x)
by Pablo Martin, Juan Pablo Ramos-Andrade, Fabián Caro-Pérez and Freddy Lastra
Math. Comput. Appl. 2024, 29(4), 63; https://doi.org/10.3390/mca29040063 - 9 Aug 2024
Cited by 1 | Viewed by 1349
Abstract
We obtain an accurate analytic approximation for the Bessel function J2(x) using an improved multipoint quasirational approximation technique (MPQA). This new approximation is valid for all real values of the variable x, with a maximum absolute error of [...] Read more.
We obtain an accurate analytic approximation for the Bessel function J2(x) using an improved multipoint quasirational approximation technique (MPQA). This new approximation is valid for all real values of the variable x, with a maximum absolute error of approximately 0.009. These errors have been analyzed in the interval from x=0 to x=1000, and we have found that the absolute errors for large x decrease logarithmically. The values of x at which the zeros of the exact function J2(x) and the approximated function ˜J2(x) occur are also provided, exhibiting very small relative errors. The largest relative error is for the second zero, with εrel=0.0004, and the relative errors continuously decrease, reaching 0.0001 for the eleventh zero. The procedure to obtain this analytic approximation involves constructing a bridge function that connects the power series with the asymptotic approximation. This is achieved by using rational functions combined with other elementary functions, such as trigonometric and fractional power functions. Full article
Show Figures

Figure 1

17 pages, 26853 KiB  
Article
On the Influence of Fractional-Order Resonant Capacitors on Zero-Voltage-Switching Quasi-Resonant Converters
by Wangzifan Cao and Xi Chen
Electronics 2024, 13(13), 2562; https://doi.org/10.3390/electronics13132562 - 29 Jun 2024
Cited by 3 | Viewed by 994
Abstract
This paper focuses on the influence of the fractional-order (FO) resonant capacitor on the zero-voltage-switching quasi-resonant converter (ZVS QRC). The FO impedance model of the capacitor is introduced to the circuit model of the ZVS QRC; hence, a piecewise smooth FO model is [...] Read more.
This paper focuses on the influence of the fractional-order (FO) resonant capacitor on the zero-voltage-switching quasi-resonant converter (ZVS QRC). The FO impedance model of the capacitor is introduced to the circuit model of the ZVS QRC; hence, a piecewise smooth FO model is developed for the converter. Numerical solutions of the converter are obtained by using both the fractional Adams–Bashforth–Moulton (F-ABM) method and Oustaloup’s rational approximation method. In addition, the analytical solution of the converter is obtained by the Grünwald–Letnikov (GL) definition, which reveals the influence of the FO resonant capacitor on the zero-crossing point (ZCP) and resonant state of the converter. An experimental platform was built to verify the results of the theoretical analysis and numerical calculation. Full article
(This article belongs to the Special Issue Advancements in Power Electronics Conversion Technologies)
Show Figures

Figure 1

17 pages, 665 KiB  
Article
A Rigorous Explicit Expression for the Mutual Inductance of Two Co-Axial Thin-Wire Coil Antennas Placed above a Layered Ground
by Mauro Parise, Giulio Antonini and Luisa Di Paola
Energies 2023, 16(22), 7586; https://doi.org/10.3390/en16227586 - 15 Nov 2023
Cited by 6 | Viewed by 1093
Abstract
This paper presents a quasi-analytical method that allows the derivation of a rigorous series-form representation for the mutual inductance of two co-axial coil antennas located above an arbitrarily layered earth structure. Starting from Biot–Savart law, which gives the integral representation for the primary [...] Read more.
This paper presents a quasi-analytical method that allows the derivation of a rigorous series-form representation for the mutual inductance of two co-axial coil antennas located above an arbitrarily layered earth structure. Starting from Biot–Savart law, which gives the integral representation for the primary vector potential generated by the source coil, the potential reflected by the layered ground is derived, and the resulting total vector potential is then integrated along the external circumference of the receiving coil to give the mutual inductance of the two antennas. The obtained representation for the flux is then evaluated analytically through the usage of the Gegenbauer addition theorem once an accurate, rational approximation is used in place of the factor of the integrand that exhibits branch cuts. It is shown how the resulting explicit solution exhibits the same degree of accuracy as purely numerical approaches like the finite-difference time-domain (FDTD) method and conventional numerical quadrature schemes, while it is less time-demanding than the latter methods. Full article
(This article belongs to the Section F: Electrical Engineering)
Show Figures

Figure 1

21 pages, 515 KiB  
Article
The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde Matrices
by Mutti-Ur Rehman, Jehad Alzabut, Nahid Fatima and Tulkin H. Rasulov
Axioms 2023, 12(9), 831; https://doi.org/10.3390/axioms12090831 - 28 Aug 2023
Viewed by 1352
Abstract
The numerical approximation of both eigenvalues and singular values corresponding to a class of totally positive Bernstein–Vandermonde matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices are well studied and investigated in the literature. We aim to present some new [...] Read more.
The numerical approximation of both eigenvalues and singular values corresponding to a class of totally positive Bernstein–Vandermonde matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices are well studied and investigated in the literature. We aim to present some new results for the numerical approximation of the largest singular values corresponding to Bernstein–Vandermonde, Bernstein–Bezoutian, Cauchy—polynomial-Vandermonde and quasi-rational Bernstein–Vandermonde structured matrices. The numerical approximation for the reciprocal of the largest singular values returns the structured singular values. The new results for the numerical approximation of bounds from below for structured singular values are accomplished by computing the largest singular values of totally positive Bernstein–Vandermonde structured matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices. Furthermore, we present the spectral properties of totally positive Bernstein–Vandermonde structured matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and structured quasi-rational Bernstein–Vandermonde matrices by computing the eigenvalues, singular values, structured singular values and its lower and upper bounds and condition numbers. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties II)
Show Figures

Figure 1

2 pages, 180 KiB  
Editorial
Special Issue Editorial “Special Functions and Polynomials”
by Paolo Emilio Ricci
Symmetry 2022, 14(8), 1503; https://doi.org/10.3390/sym14081503 - 22 Jul 2022
Viewed by 1074
Abstract
This Special Issue contains 14 articles from the MDPI journal Symmetry on the general subject area of “Special Functions and Polynomials”, written by scholars belonging to different countries of the world. A similar number of submitted articles was not accepted for publication. Several [...] Read more.
This Special Issue contains 14 articles from the MDPI journal Symmetry on the general subject area of “Special Functions and Polynomials”, written by scholars belonging to different countries of the world. A similar number of submitted articles was not accepted for publication. Several successful Special Issues on the same or closely related topics have already appeared in MDPI’s Symmetry, Mathematics and Axioms journals, in particular those edited by illustrious colleagues such as Hari Mohan Srivastava, Charles F. Dunkl, Junesang Choi, Taekyun Kim, Gradimir Milovanović, and many others, who testify to the importance of this matter for its applications in every field of mathematical, physical, chemical, engineering and statistical sciences. The subjects treated in this Special Issue include, in particular, the following Keywords. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
25 pages, 2412 KiB  
Article
Introducing a Novel Method for Smart Expansive Systems’ Operation Risk Synthesis
by Nikolay Zhigirev, Alexander Bochkov, Nataliya Kuzmina and Alexandra Ridley
Mathematics 2022, 10(3), 427; https://doi.org/10.3390/math10030427 - 28 Jan 2022
Cited by 5 | Viewed by 2203
Abstract
In different areas of human activity, the need to choose optimal (rational) options for actions from the proposed alternatives inevitably arises. In the case of retrospective statistical data, risk analysis is a convenient tool for solving the problem of choice. However, when planning [...] Read more.
In different areas of human activity, the need to choose optimal (rational) options for actions from the proposed alternatives inevitably arises. In the case of retrospective statistical data, risk analysis is a convenient tool for solving the problem of choice. However, when planning the growth and development of complex systems, a new approach to decision-making is needed. This article discusses the concept of risk synthesis when comparing alternative options for the development of a special class of complex systems, called smart expansive systems, by the authors. “Smart” in this case implies a system capable of ensuring a balance between its growth and development, considering possible external and internal risks and limitations. Smart expansive systems are considered in a quasi-linear approximation and in stationary conditions of problem-solving. In general, when the alternative to comparison is not the object itself, but some scalar way of determining risks, the task of selecting the objects most at risk is reduced to assessing the weights of factors affecting the integral risk. As a result, there is a complex task of analyzing the risks of objects, solved through the amount by which the integral risk can be minimized. Risks are considered as anti-potentials of the system development, being retarders of the reproduction rate of the system. The authors give a brief description of a smart expansive system and propose approaches to modeling the type of functional dependence of the integral risk of functioning of such a system on many risks, measured, as a rule, in synthetic scales of pairwise comparisons. The solution to the problem of reducing the dimension of influencing factors (private risks) using the vector compression method (in group and inter-scale formulations) is described. This article presents an original method for processing matrices of incomplete pairwise comparisons with indistinctly specified information, based on the idea of constructing reference-consistent solutions. Examples are provided of how the vector compression method can be applied to solve practical problems. Full article
Show Figures

Figure 1

9 pages, 314 KiB  
Article
Quasi-Rational Analytic Approximation for the Modified Bessel Function I1(x) with High Accuracy
by Pablo Martin, Eduardo Rojas, Jorge Olivares and Adrián Sotomayor
Symmetry 2021, 13(5), 741; https://doi.org/10.3390/sym13050741 - 23 Apr 2021
Cited by 3 | Viewed by 2724
Abstract
A new simple and accurate expression to approximate the modified Bessel function of the first kind I1(x) is presented in this work. This new approximation is obtained as an improvement of the multi-point quasi-rational approximation technique, MPQA. This method [...] Read more.
A new simple and accurate expression to approximate the modified Bessel function of the first kind I1(x) is presented in this work. This new approximation is obtained as an improvement of the multi-point quasi-rational approximation technique, MPQA. This method uses the power series of the Bessel function, its asymptotic expansion, and a process of optimization to fit the parameters of a fitting function. The fitting expression is formed by elementary functions combined with rational ones. In the present work, a sum of hyperbolic functions was selected as elementary functions to capture the first two terms of the asymptotic expansion of I1(x), which represents an important improvement with respect to previous research, where just the leading term of the asymptotic series was captured. The new approximation function presents a remarkable agreement with the analytical solution I1(x), decreasing the maximum relative error in more than one order of magnitude with respect to previous similar expressions. Concretely, the relative error was reduced from 102 to 4×104, opening the possibility of applying the new improved method to other Bessel functions. It is also remarkable that the new approximation is valid for all positive and negative values of the argument. Full article
Show Figures

Figure 1

16 pages, 2824 KiB  
Article
Application of the Segregation Potential Model to Freezing Soil in a Closed System
by Xiyan Zhang, Yu Sheng, Long Huang, Xubin Huang and Binbin He
Water 2020, 12(9), 2418; https://doi.org/10.3390/w12092418 - 28 Aug 2020
Cited by 7 | Viewed by 3262
Abstract
Previous studies have shown that an accurate prediction of frost heaves largely depends on the pore water pressure and hydraulic conductivity of frozen fringes, which are difficult to determine. The segregation potential model can avoid this problem; however, the conventional segregation potential is [...] Read more.
Previous studies have shown that an accurate prediction of frost heaves largely depends on the pore water pressure and hydraulic conductivity of frozen fringes, which are difficult to determine. The segregation potential model can avoid this problem; however, the conventional segregation potential is considered to be approximately unchanged at a steady state and only valid in an open system without dehydration in the unfrozen zone. Based on Darcy’s law and the conventional segregation potential, the segregation potential was expressed as a function of the pore water pressure at the base of the ice lens, the pore water pressure at the freezing front, the freezing temperature, the segregation freezing temperature and the hydraulic conductivity of the frozen fringe. This expression indicates that the segregation potential under quasi-steady-state conditions is not a constant in a closed system, since the pore water pressure at the freezing front varies with the freezing time owing to the dehydration of the unfrozen zone, and that when the pore water pressure at the freezing front is equal to that at the base of the ice lens, the water migration and frost heave will be terminated. To analyze the possibility of applying the segregation potential model in a closed system, a series of one-sided frost heave tests under external pressure in a closed system were carried out in a laboratory, and the existing frost heaving test data from the literature were also analyzed. The results indicate that the calculated frost heave was close to the tested data, which shows the applicability of the model in a closed system. In addition, the results show the rationality of calculating the segregation potential from the frost heaving test by comparing the potential with that calculated from the numerical simulation results. This study attempted to extend the segregation potential model to freezing soil in a closed system and is significant to the study of frost heaves. Full article
(This article belongs to the Section Hydraulics and Hydrodynamics)
Show Figures

Figure 1

11 pages, 3250 KiB  
Article
Rationalization of Lattice Thermal Expansion for Beta-Blocker Organic Crystals
by Paola Paoli, Stella Milazzo, Patrizia Rossi and Andrea Ienco
Crystals 2020, 10(5), 350; https://doi.org/10.3390/cryst10050350 - 29 Apr 2020
Cited by 7 | Viewed by 2611
Abstract
Anisotropic lattice expansion could be a source of misunderstanding in powder pattern recognitions, especially in the case of organic crystals where for the interpretation of room temperature patterns single crystal data at low temperature are usually used. Trying to rationalize the thermal lattice [...] Read more.
Anisotropic lattice expansion could be a source of misunderstanding in powder pattern recognitions, especially in the case of organic crystals where for the interpretation of room temperature patterns single crystal data at low temperature are usually used. Trying to rationalize the thermal lattice expansion, we studied two close related β-blocker molecules with similar packing in the solid state but with different thermal behavior. Solid state calculations, using the fast and accurate HF-3c method and the quasi harmonic approximation for the simulation of the lattice expansion, were able to reproduce the experimental trends with good accuracy. The complete analysis of the calculated thermal expansion of the two structures, as well as of other structures with similar packing found in a database survey, revealed the primary role of the hydrogen bonds. Secondary non-covalent interactions in the plane perpendicular to the hydrogen bond system could also play a role. Full article
Show Figures

Graphical abstract

9 pages, 265 KiB  
Article
An Approximation Theorem for Vector Equilibrium Problems under Bounded Rationality
by Wensheng Jia, Xiaoling Qiu and Dingtao Peng
Mathematics 2020, 8(1), 45; https://doi.org/10.3390/math8010045 - 1 Jan 2020
Cited by 4 | Viewed by 2070
Abstract
In this paper, our purpose is to investigate the vector equilibrium problem of whether the approximate solution representing bounded rationality can converge to the exact solution representing complete rationality. An approximation theorem is proved for vector equilibrium problems under some general assumptions. It [...] Read more.
In this paper, our purpose is to investigate the vector equilibrium problem of whether the approximate solution representing bounded rationality can converge to the exact solution representing complete rationality. An approximation theorem is proved for vector equilibrium problems under some general assumptions. It is also shown that the bounded rationality is an approximate way to achieve the full rationality. As a special case, we obtain some corollaries for scalar equilibrium problems. Moreover, we obtain a generic convergence theorem of the solutions of strictly-quasi-monotone vector equilibrium problems according to Baire’s theorems. As applications, we investigate vector variational inequality problems, vector optimization problems and Nash equilibrium problems of multi-objective games as special cases. Full article
8 pages, 1142 KiB  
Communication
Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation
by Pavel Praks and Dejan Brkić
Mathematics 2020, 8(1), 26; https://doi.org/10.3390/math8010026 - 20 Dec 2019
Cited by 4 | Viewed by 2746
Abstract
The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use [...] Read more.
The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use of transcendental functions, such as logarithm or non-integer power, which require execution of the additional number of floating-point operations in computer processor units. Instead of these, we use only rational expressions that are executed directly in the processor unit. The rational approximation was found using a combination of a Padé approximant and artificial intelligence (symbolic regression). Numerical experiments in Matlab using 2 million quasi-Monte Carlo samples indicate that the relative error of this new rational approximation does not exceed 0.866%. Moreover, these numerical experiments show that the novel rational approximation is approximately two times faster than the exact solution given by the Wright omega function. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications)
Show Figures

Figure 1

16 pages, 4173 KiB  
Article
Sentinel-2 Satellites Provide Near-Real Time Evaluation of Catastrophic Floods in the West Mediterranean
by Isabel Caballero, Javier Ruiz and Gabriel Navarro
Water 2019, 11(12), 2499; https://doi.org/10.3390/w11122499 - 27 Nov 2019
Cited by 62 | Viewed by 8513
Abstract
Flooding is among the most common natural disasters in our planet and one of the main causes of economic and human life loss worldwide. Evidence suggests the increase of floods at European scale with the Mediterranean coast being critically vulnerable to this risk. [...] Read more.
Flooding is among the most common natural disasters in our planet and one of the main causes of economic and human life loss worldwide. Evidence suggests the increase of floods at European scale with the Mediterranean coast being critically vulnerable to this risk. The devastating event in the West Mediterranean during the second week of September 2019 is a clear case of this risk crystallization, when a record-breaking flood (locally called the “Cold Drop” (Gota Fría)) has swollen into a catastrophe to the southeast of Spain surpassing previous all-time records. By using a straightforward approach with the Sentinel-2 twin satellites from the Copernicus Programme and the ACOLITE atmospheric correction processor, an initial approximation of the delineated flooded zones, including agriculture and urban areas, was accomplished in quasi-real time. The robust and flexible approach requires no ancillary data for rapid implementation. A composite of pre- and post-flood images was obtained to identify change detection and mask water pixels. Sentinel-2 identifies not only impacts on land but also on water ecosystem and its services, providing information on water quality deterioration and concentration of suspended matter in highly sensitive environments. Subsequent water quality deterioration occurred in large portions of Mar Menor, the largest coastal lagoon in the Mediterranean. The present study demonstrates the potentials brought by the free and open-data policy of Sentinel-2, a valuable source of rapid synoptic spatio-temporal information at the local or regional scale to support scientists, managers, stakeholders, and society in general during and after the emergency. Full article
(This article belongs to the Special Issue Monitoring, Modelling and Management of Water Quality)
Show Figures

Graphical abstract

12 pages, 1762 KiB  
Article
Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression
by Dejan Brkić and Pavel Praks
Computation 2019, 7(3), 48; https://doi.org/10.3390/computation7030048 - 3 Sep 2019
Cited by 7 | Viewed by 4750
Abstract
The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω [...] Read more.
The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Padé approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence. Full article
(This article belongs to the Section Computational Engineering)
Show Figures

Figure 1

9 pages, 2851 KiB  
Article
Transient Powder Melting in SLM Using an Analytical Model with Phase Change and Spherical Symmetry in a Semi-Infinite Medium
by Marios M. Fyrillas and Loucas Papadakis
J. Manuf. Mater. Process. 2019, 3(2), 50; https://doi.org/10.3390/jmmp3020050 - 20 Jun 2019
Cited by 3 | Viewed by 3078
Abstract
In this work, we introduce an analytical expression for approximating the transient melting radius during powder melting in Selective Laser Melting (SLM) assumed with a stationary laser heat source. The purpose of this work is to evaluate the suggested analytical approach in determining [...] Read more.
In this work, we introduce an analytical expression for approximating the transient melting radius during powder melting in Selective Laser Melting (SLM) assumed with a stationary laser heat source. The purpose of this work is to evaluate the suggested analytical approach in determining the melt pool geometry during laser processing, by considering heat transfer and phase change effects. This will allow for the rendering of the first findings on the way to a quasi-real time calculation of the melt pool during laser melting, which will contribute significantly to the process design and control, especially when new powders are applied. Initially, we consider the heat transfer process associated with a point heat source, releasing a continuous and constant power (in a semi-infinite powder bed. On the point of the heat source the temperature is infinite, and the material starts to melt spherically outwards, creating an interface that separates the solid from the molten material; we assume different properties between the two phases. Unlike the cases of the cartesian and cylindrical coordinates, (in a cartesian coordinate the heat source is over a plane, i.e., W/m2, and in cylindrical along a line, i.e., W/m), where the melting process is proportional to the square root of time, in spherical coordinates the melting stops at a finite radius, i.e., a maximum radius, which depends only on the heat source, the conductivity of the solid and the difference between the far-field temperature and the melting temperature of the material. Here we should also point out that to achieve continuous melting in spherical coordinates the power of the source must increase with the square root of the time. The obtained analytical expression for the maximum melting radius and the approximate expression for its dependence on the time compare well with the numerical results obtained by a finite element analysis. Full article
Show Figures

Figure 1

Back to TopTop