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Journal articles on the topic 'R-function method'

Author: Grafiati

Published: 30 May 2022

Last updated: 19 July 2025

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1

Proskurin, A. V., and A. M. Sagalakov. "Modeling duct flow by the R-function method." Journal of Applied and Industrial Mathematics 10, no. 3 (2016): 429–34. http://dx.doi.org/10.1134/s1990478916030133.

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2

Dmitriev, V. I., and J. G. Ingtem. "The Regularized Spline (R-Spline) Method for Function Approximation." Computational Mathematics and Modeling 30, no. 3 (2019): 198–206. http://dx.doi.org/10.1007/s10598-019-09447-w.

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3

Veretel'nik, V. V., V. F. Kravchenko, and V. L. Rvachev. "Design of groove waveguides by the R-function method." Measurement Techniques 36, no. 2 (1993): 216–19. http://dx.doi.org/10.1007/bf00977862.

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4

Hong-wei, Lu, Sun Ying, Lin Min, Li Pi-ding, and Zheng Zheng. "A probability density function method for detecting atrial fibrillation using R–R intervals." Medical Engineering & Physics 31, no. 1 (2009): 116–23. http://dx.doi.org/10.1016/j.medengphy.2008.04.013.

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5

Chrapoński, Jacek, Marian Maliński, Janusz Szala, and Jan Cwajna. "FGM Structure Characterization by Distance Functions and Systematic Scanning Method." Materials Science Forum 567-568 (December 2007): 153–56. http://dx.doi.org/10.4028/www.scientific.net/msf.567-568.153.

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The distance functions such as G(r), K(r), H(r), g(r) and systematic scanning method connected with variance analysis have been used for quantitative characterization of functionally gradient materials structures. The methods have been tested on computer generated 3D model polycrystalline structures with diverse type of gradient function. Usefulness of the presented methods for quantitative description of FGM structures has been evaluated.

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6

Pál, Ambrus, and Endre Szabó. "The fibration method over real function fields." Mathematische Annalen 378, no. 3-4 (2020): 993–1019. http://dx.doi.org/10.1007/s00208-020-02053-x.

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Abstract Let $$\mathbb R(C)$$ R ( C ) be the function field of a smooth, irreducible projective curve over $$\mathbb R$$ R . Let X be a smooth, projective, geometrically irreducible variety equipped with a dominant morphism f onto a smooth projective rational variety with a smooth generic fibre over $$\mathbb R(C)$$ R ( C ) . Assume that the cohomological obstruction introduced by Colliot-Thélène is the only one to the local-global principle for rational points for the smooth fibres of f over $$\mathbb R(C)$$ R ( C ) -valued points. Then we show that the same holds for X, too, by adopting the fibration method similarly to Harpaz–Wittenberg.

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7

Xia, Fengfei, and Shanqing Li. "R-Function and variation method for bending problem of clamped thin plate with complex shape." Advances in Mechanical Engineering 13, no. 7 (2021): 168781402110348. http://dx.doi.org/10.1177/16878140211034832.

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Solving ordinary thin plate bending problem in engineering, only a few analytical solutions with simple boundary shapes have been proposed. When using numerical methods (e.g. the variational method) to solve the problem, the trial functions can be found only it exhibits a simple boundary shape. The R-functions can be applied to solve the problem with complex boundary shapes. In the paper, the R-function theory is combined with the variational method to study the thin plate bending problem with the complex boundary shape. The paper employs the R-function theory to express the complex area as the implicit function, so it is easily to build the trial function of the complex shape thin plate, which satisfies with the complex boundary conditions. The variational principle and the R-function theory are introduced, and the variational equation of thin plate bending problem is derived. The feasibility and correctness of this method are verified by five numerical examples of rectangular, I-shaped, T-shaped, U-shaped, and L-shaped thin plates, and the results of this method are compared with that of other literatures and ANSYS finite element method (FEM). The results of the method show a good agreement with the calculation results of literatures and FEM.

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8

Diógenes, A. N., L. O. E. dos Santos, and C. P. Fernandes. "PARTICLE SIZE DISTRIBUTION CORRECTION METHOD USING A SIMULATED ANNEALING TECHNIQUE." Revista de Engenharia Térmica 10, no. 1-2 (2011): 39. http://dx.doi.org/10.5380/reterm.v10i1-2.61950.

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The procedure for obtaining the particle size distribution by visual inspection of a sample involves stereological errors, given the cut of the sample. A cut particle, supposedly spherical, with radius R, will be counted as a circular particle with radius r, r≤R. The difference between r and R depends on how far from the center of the sphere the cut was performed. This introduces errors when the extrapolation of the properties from two to three dimensions during the analysis of a sample. The usual method is to correct the distribution by probabilistic functions, which have large errors. This paper presents a method to reduce the error inherent to this problem. The method is to compute a simulation of the preparation process in a sample whose structure can be described by non-penetrating spheres of various diameters which meet a known probability distribution function, for example, a log-logistic function, or even a constant function. For each distribution radius, a number of spheres is generated and virtually cut, generating a bi-dimensional (2D) distribution. The 2D curves of the spheres distribution obtained in this simulation are compared with that obtained by the experimental procedure and then the parameters of the threedimensional distribution function are adjusted until the 2D curves are similar to the experimental one using the optimization method Simulated Annealing for the curve-fitting. In future this method will be applied to the analysis of the oil reservoir rocks.

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9

BOUGAEV, ANTON, ALEKSEY URMANOV, LEFTERI TSOUKALAS, and KENNY GROSS. "METHOD OF KEY VECTORS EXTRACTION USING R-CLOUD CLASSIFIERS." New Mathematics and Natural Computation 03, no. 03 (2007): 419–26. http://dx.doi.org/10.1142/s1793005707000884.

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A novel method for reducing a training data set in the context of nonparametric classification is proposed. The new method is based on the method of R-clouds. The advantages of the R-cloud classification method introduced recently are being investigated. The separating boundary of the R-cloud classifier is represented using Rvachev functions. The method of key vectors extraction uses the value of the R-cloud function to quantify the disturbance of the separating boundary, which is caused by removal of one data vector from the design dataset. The R-cloud method was found instructive and practical in a number of engineering problems related to pattern classification.

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10

Rvachev, V. L., and L. V. Kurpa. "Solving stability problems of plates by the R-function method." Strength of Materials 19, no. 6 (1987): 843–46. http://dx.doi.org/10.1007/bf01522846.

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11

Uvarova, E. A. "R -function method as a supportive extension for the ritz method and least squares method." Journal of Mathematical Sciences 171, no. 4 (2010): 466–73. http://dx.doi.org/10.1007/s10958-010-0150-4.

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12

Ababneh, O., and N. Zomot. "New Iterative Method with Application." JOURNAL OF ADVANCES IN MATHEMATICS 20 (October 15, 2021): 424–30. http://dx.doi.org/10.24297/jam.v20i.8289.

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13

Sizova, Natal'ya Dmitrievna, and Victor Filippovich Kravchenko. "Investigation of Some Heat Conductivity Problems by the R-Function Method." Telecommunications and Radio Engineering 58, no. 5-6 (2002): 46. http://dx.doi.org/10.1615/telecomradeng.v58.i5-6.30.

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14

Wolfenden, A., JD Landes, Z. Zhou, K. Lee, and R. Herrera. "Normalization Method for Developing J-R Curves with the LMN Function." Journal of Testing and Evaluation 19, no. 4 (1991): 305. http://dx.doi.org/10.1520/jte12574j.

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15

Breslavsky, I. D., and K. V. Avramov. "Nonlinear modes of cylindrical panels with complex boundaries. R-function method." Meccanica 46, no. 4 (2010): 817–32. http://dx.doi.org/10.1007/s11012-010-9340-x.

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16

Rvachev, V. L., L. V. Kurpa, Kh F. Nasriddinov, and A. N. Shevchenko. "The r-function method in problems of nonlinear deformation of plates." Soviet Applied Mechanics 23, no. 9 (1987): 861–66. http://dx.doi.org/10.1007/bf00887791.

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17

Gao, Liuyang, Nae Zheng, Yinghua Tian, and Jingzhi Zhang. "Target Signal Extraction Method Based on Enhanced ICA with Reference." Mathematical Problems in Engineering 2019 (August 21, 2019): 1–12. http://dx.doi.org/10.1155/2019/4128438.

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Target signal extraction has a great potential for applications. To solve the problem of error extraction of target signals in the current constrained independent component analysis (cICA) method, an enhanced independent component analysis with reference (EICA-R) method is proposed. The new algorithm establishes a unified cost function, which combines the negative entropy contrast function and the distance metric function. The EICA-R method transforms the constrained optimization problem into unconstrained optimization problem to overcome the problem of threshold setting of distance metric function in constrained optimization problem. The theoretical analysis and simulation experiment show that the proposed EICA-R algorithm overcomes the problem of the error extraction of the existing algorithm and improves the reliability of the target signal extraction.

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18

Lahav, Ofer, Donald Lynden-Bell, and Steve F. Gull. "From 2-D to 3-D by Maximum Entropy Method." Symposium - International Astronomical Union 130 (1988): 559. http://dx.doi.org/10.1017/s0074180900136885.

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We present a method of estimating distances to clusters of galaxies from twodimensional catalogues. The angular diameters (or magnitudes) of galaxies are used as distance indicators. The mapping from 2-D to 3-D is done by using a ‘diameter function’ (analogous to a luminosity function), which is based on a redshift survey from a section of the sky. The problem is formulated as follows. The number of galaxies with a metric diameter D in a volume element d3r is: where n(r) is the ‘true’ number density of galaxies at position r, n& is the mean number density of galaxies in the universe and ϕ(D)dD is the diameter function. We assume that within a narrow cone n(r) = n(r) and then express N(> θ), the number of galaxies greater than a certain angular diameter θ. In a discrete form we write the relation as: where ni is the density at the i – th distance bin and Pik is our ‘point spread function’, which is a function of the diameter function and Galactic obscuration. We express (2) in terms of χ2 statistics over the measurements, and require it to be less than a certain value. The entropy of the image is expressed as:

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19

Petersen, DR, RE Link, CJFO Fortes, and FL Bastian. "A Modified Normalization Method for Developing J-R and CTOD-R Curves with the LMN Function." Journal of Testing and Evaluation 25, no. 3 (1997): 302. http://dx.doi.org/10.1520/jte11342j.

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20

Stetsyuk, Petro, Aleksandr Pylypovskyi, and Olha Khomiak. "GNU Octave and Python Implementation of Shor's r-Algorithm with Adaptive Step Control." Cybernetics and Computer Technologies, no. 3 (November 29, 2022): 98–112. http://dx.doi.org/10.34229/2707-451x.22.3.10.

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r-algorithms, or subgradient methods with dilation of space in the direction of the difference of two sequential subgradients, were proposed by N.Z.Shor in 1970 in his doctoral thesis. Respective software implementations proved to be competitive with the most effective methods for smooth ill-conditioned problems, both in terms of reliability and calculation time and accuracy of results. The article is devoted to the description of two software implementations of Shor's r-algorithm modification with a constant coefficient of space dilation and adaptive step control. The first program is implemented using the GNU Octave, and the second program is implemented using Python. Material of the paper is presented in three sections. In the first section, we describe a modification of the r-algorithm with a constant coefficient of space dilation in the direction of the difference of two sequential subgradients and an adaptive method for step size adjustment in the direction of the antisubgradient in the transformed space of variables. The software implementation of this modification is presented in the form of octave-function ralgb5a, which allows to find either approximation of the minimum point of a convex function, or approximation of the maximum point of the concave function. The code of the ralgb5a function is given with a brief description of its input and output parameters. The second section describes test experiments to investigate efficiency of the octave-function ralgb5a to maximizing the piecewise linear concave function, which is obtained using the method of non-smooth penalty functions for the linear programming problem. Another example represents minimization of the piecewise linear convex function, which corresponds to the method of least modules. Results of these computational experiments for test problems with 200, 500, 1000, 1500 and 2000 variables are presented to demonstrate the effective operation of the octave-function ralgb5a. The third section describes the python function ralgb5a and provides its code with a description of the input and output parameters. It is show, how the ralgb5a function can be accelerated by setting two parameters. The results of computational experiments to solve the test problem using the method of least modules for 5,000 variables and 10,000 observations are presented. Keywords: r-algorithm, linear programming problem, nonsmooth penalty function, least modulus method, GNU Octave, python.

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21

Jiang, Wen Wen, and Zhi Bin Xie. "Research of Design Method of Product Function Analysis and Combination." Applied Mechanics and Materials 201-202 (October 2012): 886–89. http://dx.doi.org/10.4028/www.scientific.net/amm.201-202.886.

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This paper adopts methods of longitudinal comparison, classification research, expand competition field to research for design method of product function analysis and combination based on user research. It found initiative combination of product function and technique could share technology platform and spare module organization to form different varieties of product in functional detail, market orientation, and functional emphasis, avoid to carrying out fierce competition with competitor in narrow market area. It is benefit for company pursuing different strategy. R&D team should maintain validity and direction of design, analysis, and manufacturing work. High efficient pursuing situation from competitor, sale situation from user market would increase urgency and psychological pressure of R&D team and manufacturer, thus improve efficiency of research and development. The results show that fundamental goal of R&d team carries on product function analysis is tightly grasp functional use essence of future products to allocate various of resources to achieve product function index, satisfy user demand by best way.

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22

Christoffersen, Berit, Ulla Ribel, Kirsten Raun, Valeria Golozoubova, and Giovanni Pacini. "Evaluation of different methods for assessment of insulin sensitivity in Göttingen minipigs: introduction of a new, simpler method." American Journal of Physiology-Regulatory, Integrative and Comparative Physiology 297, no. 4 (2009): R1195—R1201. http://dx.doi.org/10.1152/ajpregu.90851.2008.

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The use of animal models in diabetes research requires reliable tests for evaluation of insulin sensitivity and β-cell function. Minipigs are being increasingly used in metabolic research, and the aim of this study was to compare different tests and indexes for evaluation of insulin sensitivity and β-cell function in Göttingen minipigs. Hyperinsulinemic, isoglycemic clamp, intravenous (IVGTT) and oral glucose tolerance tests (OGTT), and a modified insulin tolerance test were performed in minipigs fed either low- or high-energy diet. Furthermore, the reproducibility of IVGTT-derived parameters was assessed. Previously described insulin sensitivity indexes [steady-state glucose infusion rate/glucose concentration/insulin concentration from clamp (M/G/I); oral glucose insulin sensitivity (OGIS) and ISIcomp from OGTT; SI from minimal model analysis of IVGTT; and quantitative insulin sensitivity check index from fasting values] were calculated together with an insulin sensitivity index from the modified insulin tolerance test (ISIITT) and a new simple index (S2) derived from the first 30 min of the IVGTT. β-Cell function was assessed from the IVGTT and the OGTT. Reproducibility of the IVGTT-derived parameters was calculated as median intraindividual coefficient of variation (CV%).M/G/I correlated significantly only with S2 ( P < 0.05, r = 0.54). S2 furthermore correlated with SI ( P < 0.001, r = 0.81), ISIITT ( P < 0.001, r = 0.57), and the two indexes from OGTT, ISIcomp ( P < 0.001, r = 0.78) and OGIS ( p < 0.05, r = 0.48). No correlation was found between β-cell function indexes from OGTT and IVGTT. The median CV% of the new S2 index was 13. In conclusion, the new simple index of insulin sensitivity, S2, was revealed to be useful for evaluation of insulin sensitivity in pigs.

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23

Langarshoev, Mukhtor R. "The best approximation of analytic in a unit circle functions in the Bergman weight space B_(2,μ)". Russian Universities Reports. Mathematics, № 145 (2024): 65–76. http://dx.doi.org/10.20310/2686-9667-2024-29-145-65-76.

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The paper studies the issues of the best approximation of analytical functions in the Bergman weight space B_(2,μ). In this space, for best approximations of functions analytic in the circle by algebraic complex polynomials we obtain the exact inequalities by means of generalized modules of continuity of higher order derivatives Ω_m (z^r f^((r) ),t),m∈N,r∈Z. For classes of functions analytic in the unit circle defined by the characteristic Ω_m (z^r f^((r) ),t), and the majorant Φ, the exact values of some n-widths are calculated. When proving the main results of this work, we use methods for solving extremal problems in normalized spaces of functions analytic in the circle, N.P. Korneichuk’s method for estimating upper bounds for the best approximations of classes of functions by a subspace of fixed dimension, and a method for estimating from below the n-widths of function classes in various Banach spaces.

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24

Ngoc Huy, Dinh Tran, Pham Anh Dung, Le Ngoc Nuong, Le Thi Han, Dinh Tran Ngoc Hien, and Pham Hung Nhan. "Analysis of Relation Between Economic Innovation Models and R&D Function." Mediterranean Journal of Basic and Applied Sciences 08, no. 03 (2024): 143–51. http://dx.doi.org/10.46382/mjbas.2024.8315.

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The goal of this article focuses on Analysis of Relation between Economic Innovation Models and R&D Function in recent years. Method: Methodology includes Qualitative research method: synthesis and inductive methods. George, M., et al. (2009) mentioned some variables can be considered for R&D such as: R&D expenses, R&D personnel and patents of new tech. We also expand SWOT model by adding Value part analysis on it. Results: Authors stated opportunities for values adding including but not limited to: R&D will emphasize more the vital roles of technological change in improving productivity and growth; levels of R&D will influence benefits/values of global commerce among countries; as well as R&D might consist of: basic- applying- empirical.

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25

Altiparmakov, D. "Analytical Representation Of ZED-2 Reactor Geometry by Means of The R-Function Method." AECL Nuclear Review 2, no. 1 (2013): 27–37. http://dx.doi.org/10.12943/anr.2013.00004.

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The R-function method is applied to model the geometry of the critical facility ZED-2 as an illustrative example. Each material region is represented by a real, continuous and differentiable function of spatial coordinates that is referred to as the domain function. Instead of using particular functions for individual elements, a single function is constructed to represent all spatial domains that contain the same material, for instance all fuel pins, etc. Owing to continuity and differentiability, the domain functions can be used to construct a basis for approximate solution of the related boundary-value problems. A study of the approximation ability of the domain functions is carried out using the two-group spatial neutron flux distribution in the ZED-2 facility as a model problem. The least squares method is used to determine the unknown coefficients by minimizing the discrepancy between the reference MCNP solution and a power series of domain functions. The results show that a modest number of terms in the series is able to produce a good approximation of the neutron flux distribution specified on a mesh grid of 500 x 500 points.

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26

Merchela, Wassim. "One method for investigating the solvability of boundary value problems for an implicit differential equation." Russian Universities Reports. Mathematics, no. 136 (2021): 404–13. http://dx.doi.org/10.20310/2686-9667-2021-26-136-404-413.

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The article concernes a boundary value problem with linear boundary conditions of general form for the scalar differential equation f(t,x(t),x ̇(t))=y ̂(t), not resolved with respect to the derivative x ̇ of the required function. It is assumed that the function f satisfies the Caratheodory conditions, and the function y ̂ is measurable. The method proposed for studying such a boundary value problem is based on the results about operator equation with a mapping acting from a metric space to a set with distance (this distance satisfies only one axiom of a metric: it is equal to zero if and only if the elements coincide). In terms of the covering set of the function f(t,x_1,•): R→R and the Lipschitz set of the function f(t,•,x_2): R →R, conditions for the existence of solutions and their stability to perturbations of the function f generating the differential equation, as well as to perturbations of the right-hand sides of the boundary value problem: the function y ̂ and the value of the boundary condition, are obtained.

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27

González-Gaxiola, O. "Optical soliton solutions for Triki–Biswas equation by Kudryashov’s R function method." Optik 249 (January 2022): 168230. http://dx.doi.org/10.1016/j.ijleo.2021.168230.

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28

Bogolyubov, A. N., and M. I. Svetkin. "Application of V.L. Rvachev R-function method to solving mathematical modeling problems." Физические основы приборостроения 10, no. 3 (2021): 93–104. http://dx.doi.org/10.25210/jfop-2103-093104.

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29

Kravchenko, V. F., and M. A. Basarab. "Generalized method of eigenmodes and R function in problems of wave diffraction." Doklady Physics 49, no. 5 (2004): 270–74. http://dx.doi.org/10.1134/1.1763613.

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30

Cao, Weiming, Weizhang Huang, and Robert D. Russell. "An Error Indicator Monitor Function for an r-Adaptive Finite-Element Method." Journal of Computational Physics 170, no. 2 (2001): 871–92. http://dx.doi.org/10.1006/jcph.2001.6770.

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31

Li, Shan Qing, and Hong Yuan. "Application of the R-Function Theory and Variational Method for Torsion Problem with Complicated Cross-Section." Applied Mechanics and Materials 353-356 (August 2013): 3320–23. http://dx.doi.org/10.4028/www.scientific.net/amm.353-356.3320.

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The R-function theory and variational method are employed to solve the torsion problem of the bar with a complicated cross-section. When the variational method is used to solve the elastic torsion problem alone, the stress function can be set to meet the boundary condition, only with the simple cross-section such as the rectangle and ellipse. For the complicated cross-section, it is hard to find a stress function to meet the boundary condition. The R-function theory can solve the problem, and it can be used to describe the complicated cross-section by implicit function form. Introducing the R-function theory can be easy to construct the stress function that satisfied the boundary of the complicated cross-section. A numerical example demonstrates the feasibility and efficiency of the present method.

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32

Liu, Chein-Shan, Chih-Wen Chang, and Chung-Lun Kuo. "Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem." Symmetry 16, no. 1 (2024): 120. http://dx.doi.org/10.3390/sym16010120.

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In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values f′(r), f″(r), and f‴(r) of a nonlinear equation f(x)=0 with r being its simple root. We can achieve high values of the efficiency index (E.I.) over the bound 22/3=1.587 with three function evaluations and over the bound 21/2=1.414 with two function evaluations. The third-degree Newton interpolatory polynomial is derived to update these critical values per iteration. We introduce relaxation factors into the Dzˇunic´ method and its variant, which are updated to render fourth-order convergence by the memory-accelerating technique. We developed six types optimal one-step iterative schemes with the memory-accelerating method, rendering a fourth-order convergence or even more, whose original ones are a second-order convergence without memory and without using specific optimal values of the parameters. We evaluated the performance of these one-step iterative schemes by the computed order of convergence (COC) and the E.I. with numerical tests. A Lie symmetry method to solve a second-order nonlinear boundary-value problem with high efficiency and high accuracy was developed.

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33

Wang, Tianji, and Qingdao Huang. "Research on Three-Dimensional Extension of Barzilai-Borwein-like Method." Mathematics 13, no. 2 (2025): 215. https://doi.org/10.3390/math13020215.

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The Barzilai-Borwein (BB) method usually uses BB stepsize for iteration so as to eliminate the line search step in the steepest descent method. In this paper, we modify the BB stepsize and extend it to solve the optimization problems of three-dimensional quadratic functions. The discussion is divided into two cases. Firstly, we study the case where the coefficient matrix of the quadratic term of quadratic function is a special third-order diagonal matrix and prove that using the new modified stepsize, this case is R-superlinearly convergent. In addition to that, we extend it to n-dimensional case and prove the rate of convergence is R-linear. Secondly, we analyze that the coefficient matrix of the quadratic term of quadratic function is a third-order asymmetric matrix, that is, when the matrix has a double characteristic root and prove the global convergence of this case. The results of numerical experiments show that the modified method is effective for the above two cases.

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34

Graef, John, Lingju Kong, Feliz Minhós, and João Fialho. "On the lower and upper solution method for higher order functional boundary value problems." Applicable Analysis and Discrete Mathematics 5, no. 1 (2011): 133–46. http://dx.doi.org/10.2298/aadm110221010g.

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The authors consider the nth-order differential equation ?(?(u(n?1)(x)))?= f(x, u(x), ..., u(n?1)(x)), for 2?(0, 1), where ?: R? R is an increasing homeomorphism such that ?(0) = 0, n?2, I:= [0,1], and f : I ?Rn ? R is a L1-Carath?odory function, together with the boundary conditions gi(u, u?, ..., u(n?2), u(i)(1)) = 0, i = 0, ..., n? 3, gn?2 (u, u?, ..., u(n?2), u(n?2)(0), u(n?1)(0)) = 0, gn?1 (u, u?, ..., u(n?2), u(n?2)(1), u(n?1)(1)) = 0, where gi : (C(I))n?1?R ? R, i = 0, ..., n?3, and gn?2, gn?1 : (C(I))n?1?R2 ? R are continuous functions satisfying certain monotonicity assumptions. The main result establishes sufficient conditions for the existence of solutions and some location sets for the solution and its derivatives up to order (n?1). Moreover, it is shown how the monotone properties of the nonlinearity and the boundary functions depend on n and upon the relation between lower and upper solutions and their derivatives.

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35

Headd, Jeffrey J., Nathaniel Echols, Pavel V. Afonine, et al. "Use of knowledge-based restraints in phenix.refine to improve macromolecular refinement at low resolution." Acta Crystallographica Section D Biological Crystallography 68, no. 4 (2012): 381–90. http://dx.doi.org/10.1107/s0907444911047834.

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Traditional methods for macromolecular refinement often have limited success at low resolution (3.0–3.5 Å or worse), producing models that score poorly on crystallographic and geometric validation criteria. To improve low-resolution refinement, knowledge from macromolecular chemistry and homology was used to add three new coordinate-restraint functions to the refinement program phenix.refine. Firstly, a `reference-model' method uses an identical or homologous higher resolution model to add restraints on torsion angles to the geometric target function. Secondly, automatic restraints for common secondary-structure elements in proteins and nucleic acids were implemented that can help to preserve the secondary-structure geometry, which is often distorted at low resolution. Lastly, we have implemented Ramachandran-based restraints on the backbone torsion angles. In this method, a φ,ψ term is added to the geometric target function to minimize a modified Ramachandran landscape that smoothly combines favorable peaks identified from nonredundant high-quality data with unfavorable peaks calculated using a clash-based pseudo-energy function. All three methods show improved MolProbity validation statistics, typically complemented by a lowered R free and a decreased gap between R work and R free.

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Li, Ping Ping, Hong Yuan, and Shan Qing Li. "Application of the R-Function Theory and Least Square Method for Torsion Problem with H-Shaped Cross-Section." Applied Mechanics and Materials 217-219 (November 2012): 2313–16. http://dx.doi.org/10.4028/www.scientific.net/amm.217-219.2313.

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The R-function theory and least square method are employed to solve the torsion problem of the bar with H-shaped cross-section. When the least square method is used to solve the elastic torsion problem alone, the stress function can be set to meet the boundary condition, only with the simple cross-section such as the rectangle and ellipse. For the H-shaped cross-section, it is hard to find a stress function to meet the boundary condition. The R-function theory can solve the problem, and it can be used to describe H-shaped cross-section by implicit function form. Introducing the R-function theory can be easy to construct the stress function that satisfied the boundary of H-shaped cross-section. A numerical example demonstrates the feasibility and efficiency of the present method.

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Yang, Zhipeng, Wei Zhang, and Fukun Zhao. "Existence and concentration results for fractional Schrodinger-Poisson system via penalization method." Electronic Journal of Differential Equations 2021, no. 01-104 (2021): 14. http://dx.doi.org/10.58997/ejde.2021.14.

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This article concerns the positive solutions for the fractional Schrodinger-Poisson system $$\displaylines{ \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=f(u)\quad \text{in }\mathbb{R}^3,\cr \varepsilon^{2t}(-\Delta)^t\phi=u^2\quad \text{in }\mathbb{R}^3, }$$ where \(\varepsilon>0\) is a small parameter, \((-\Delta)^\alpha\) denotes the fractional Laplacian of orders \(\alpha=s,t\in(3/4,1)\), \(V\in C(\mathbb{R}^3,\mathbb{R})\) is the potential function and \(f:\mathbb{R}\to\mathbb{R}\) is continuous and subcritical. Under a local condition imposed on the potential function, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values. Moreover, we considered some properties of these positive solutions, such as concentration behavior and decay estimate. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory. For more information see https://ejde.math.txstate.edu/Volumes/2021/14/abstr.html

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Poltorak, K. "Cross-Approximation Method for Solving Dynamics Problems of Arbitrarily Shaped Plates." Journal of Applied Mechanics 57, no. 2 (1990): 370–75. http://dx.doi.org/10.1115/1.2891999.

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This paper deals with a new method for solving dynamics problems of arbitrarily shaped plates. The method consists in cross approximation of functions satisfying solely an equation of motion and functions satisfying solely boundary conditions on irregular plate edges. This approach is applied in the paper to the case of free and forced vibration of homogeneous and sandwich plates with edges of various shapes. Functions satisfying boundary conditions are determined by the Rvachev’s R-function method. The validity of the method is discussed.

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Esfandiari, Aakbar, and Masoud Sanayei. "Discussion on: Function-weighted frequency response function sensitivity method for analytical model updating, by R. M. Lin." Journal of Sound and Vibration 432 (October 2018): 699–705. http://dx.doi.org/10.1016/j.jsv.2018.06.038.

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Bhalgamiya, Bhavika, and Mark A. Novotny. "Exact solutions for the inverse problem of the time-independent Schrödinger equation." American Journal of Physics 92, no. 12 (2024): 975–79. http://dx.doi.org/10.1119/5.0172824.

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We introduce an “inverse method” for solving the time-independent Schrödinger equation. Rather than derive wave functions that are solutions for a given external potential V(r→), we ask the inverse question of which V(r→) will have a given probability density function P(r→). Several examples of ground states in one, two, and three dimensions are presented for both well-known and more exotic probability density functions in position space.

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BUNDSCHUH, PETER, and KEIJO VÄÄNÄNEN. "ALGEBRAIC INDEPENDENCE OF CERTAIN MAHLER FUNCTIONS AND OF THEIR VALUES." Journal of the Australian Mathematical Society 98, no. 3 (2014): 289–310. http://dx.doi.org/10.1017/s1446788714000524.

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This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations $af(z^{r})=f(z)+R(z)$, where $a$ is a nonzero complex number, $r$ an integer greater than 1, and $R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained.

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Sangiorgi, Davide. "From enhanced coinduction towards enhanced induction." Proceedings of the ACM on Programming Languages 6, POPL (2022): 1–29. http://dx.doi.org/10.1145/3498679.

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There exist a rich and well-developed theory of enhancements of the coinduction proof method, widely used on behavioural relations such as bisimilarity. We study how to develop an analogous theory for inductive behaviour relations, i.e., relations defined from inductive observables. Similarly to the coinductive setting, our theory makes use of (semi)-progressions of the form R->F(R), where R is a relation on processes and F is a function on relations, meaning that there is an appropriate match on the transitions that the processes in R can perform in which the process derivatives are in F(R). For a given preorder, an enhancement corresponds to a sound function, i.e., one for which R->F(R) implies that R is contained in the preorder; and similarly for equivalences. We introduce weights on the observables of an inductive relation, and a weight-preserving condition on functions that guarantees soundness. We show that the class of functions contains non-trivial functions and enjoys closure properties with respect to desirable function constructors, so to be able to derive sophisticated sound functions (and hence sophisticated proof techniques) from simpler ones. We consider both strong semantics (in which all actions are treated equally) and weak semantics (in which one abstracts from internal transitions). We test our enhancements on a few non-trivial examples.

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Wang, G., T. L. Rhodes, and W. A. Peebles. "Analysis method for calculating radial correlation length of electron temperature turbulence from correlation electron cyclotron emission radiometer." Review of Scientific Instruments 93, no. 11 (2022): 113511. http://dx.doi.org/10.1063/5.0101680.

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The radial correlation length ( L r) is one of the essential quantities to measure in order to more fully characterize and understand turbulence and anomalous transport in magnetic fusion plasmas. The analysis method for calculating L r of electron temperature (Te) turbulence from correlation electron cyclotron emission (correlation ECE or CECE) radiometer measurements has not been fully developed partly due to the fact that the turbulent electron temperature fluctuations are generally imbedded in much larger amplitude thermal noise, which leads to a greatly reduced cross correlation coefficient ( ϱ) between two spatially separated ECE signals. This work finds that this ϱ reduction factor due to thermal noise is a function of the local relative temperature fluctuation power and CECE system bandwidths of intermediate and video frequencies, independent of radial separations. This indicates that under the approximation of constant relative temperature fluctuation power for a small radial range of local CECE measurements, the original shape of ϱ as a function of radial separation without thermal noise is preserved in the CECE data with thermal noise present. For Te turbulence with a Gaussian radial wavenumber spectrum, a fit function using the product of Gaussian and sinusoidal functions is derived for calculating L r. This analysis method has been numerically tested using simulated ECE radiometer data over a range of parameters. Using this method, the experimental temperature turbulence correlation length L r in a DIII-D L-mode plasma is found to be ∼10 times the local ion gyroradius.

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Rvachev, Vladimir Logvinovich, Vadim Valerievich Torchinov, and Victor Filippovich Kravchenko. "The R-Function Method in Electrodynamic's Boundary Value Problems with Arbitrary Boundary Conditions." Telecommunications and Radio Engineering 54, no. 11-12 (2000): 3–22. http://dx.doi.org/10.1615/telecomradeng.v54.i11-12.10.

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Kurpa, L. V., and A. V. Arkhipov. "R-function method in problems of elastoplastic bending of plates of arbitrary shape." International Applied Mechanics 35, no. 1 (1999): 76–80. http://dx.doi.org/10.1007/bf02682066.

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Kurpa, L. V., and A. V. Chistilina. "R-function method in studying the vibrations of orthotropic shells with varying thickness." International Applied Mechanics 45, no. 7 (2009): 742–53. http://dx.doi.org/10.1007/s10778-009-0222-7.

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Rinki, Hada. "A Study Of Correlation Of Pulmonary Functions And Percentage Of Body Fat In Normal And Overweight Medical Students." International Journal of Basic and Applied Physiology 1, no. 1 (2021): 61–95. https://doi.org/10.5281/zenodo.4468641.

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Background & Objectives: Clinically less severe obesity like overweight is not generally thought to have significant effect on ventilator function. The purpose of our study is to explore effects of body fat percentage and its distribution on ventilatory functions. We hypothesized that there is significant difference in Pulmonary Function among normal and overweight young healthy adults. Objectives: To assess & compare the difference in Pulmonary Function Tests among normal and overweight young healthy adults and to detect the correlation between Pulmonary Function Tests and Body fat percentage if any. Method: The study underwent in one forty eight young healthy medical students (74 male & 74 female) of 18-25 years of SMS Medical College, Jaipur categorized into 4 groups - Normal weight (37 males and 37 females) & over weight (37males and 37 females) on the basis of BMI. After measuring skin fat thickness with the help of Lange calliper, body fat percentage was calculated by Durnin and Womersley method. Body fat distribution was also estimated by measuring waist hip ratio and then pulmonary function tests were recorded by spirometer(Medspiror) in all groups. Results: Analysis done by using SPSS window. Simple regression was used to determine correlation and ‘r’ is coefficient of correlation. In males and females overweight groups forced vital capacity (FVC), and FEV1 were decreased significantly (P<0.05). In males group body fat % showed significant negative correlation with FVC (r = –0.2), and FEV1 (r = –0.20). It is seen that body fat % in females also had negative correlation with FVC (r = –0.2) and FEV1(r= –0.24). Whereas WHR with FVC in males shows(r= –7.3) and with FEV1(r= –7.4). In case of females subject WHR also shows negative correlation with FVC(r= –11.6) and FEV1 (r= –10.5). Interpretation & Conclusion: Our study shows that increase in body fat percentage itself and the way of fat distribution have sovereign effects on pulmonary functions.

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Zaprawa, Paweł, Anna Futa, and Magdalena Jastrzębska. "On Coefficient Functionals for Functions with Coefficients Bounded by 1." Mathematics 8, no. 4 (2020): 491. http://dx.doi.org/10.3390/math8040491.

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In this paper, we discuss two well-known coefficient functionals a 2 a 4 - a 3 2 and a 4 - a 2 a 3 . The first one is called the Hankel determinant of order 2. The second one is a special case of Zalcman functional. We consider them for functions in the class Q R ( 1 2 ) of analytic functions with real coefficients which satisfy the condition ( ) f ( z ) z > 1 2 for z in the unit disk Δ . It is known that all coefficients of f ∈ Q R ( 1 2 ) are bounded by 1. We find the upper bound of a 2 a 4 - a 3 2 and the bound of | a 4 - a 2 a 3 | . We also consider a few subclasses of Q R ( 1 2 ) and we estimate the above mentioned functionals. In our research two different methods are applied. The first method connects the coefficients of a function in a given class with coefficients of a corresponding Schwarz function or a function with positive real part. The second method is based on the theorem of formulated by Szapiel. According to this theorem, we can point out the extremal functions in this problem, that is, functions for which equalities in the estimates hold. The obtained estimates significantly extend the results previously established for the discussed classes. They allow to compare the behavior of the coefficient functionals considered in the case of real coefficients and arbitrary coefficients.

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Zhang, Guangxun. "The additional proof and application of Cauchy-Riemann Equation." Highlights in Science, Engineering and Technology 38 (March 16, 2023): 158–67. http://dx.doi.org/10.54097/hset.v38i.5801.

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As far back as the seventeenth century, Newton and Leibniz introduced and refined the idea of calculus, and the concepts of derivative and differentiation gradually entered the mathematical world as powerful tools. However, after the appearance of the theory of complex functions in the 18th century, the conventional methods used to analyze the properties of complex functions were somewhat limited. Thus, mathematicians Cauchy and Riemann came up with the idea of the “C-R equation” and provided the necessary and sufficient conditions in derivable complex functions based on the research done by d'Alembert. This equation laid the foundation for the analytic function. This paper introduces the idea of the limit of two variables function and shows both the concept of and graphical meaning partial derivative, paving way for the concept of complex function and that of C-R equation. In addition, the paper specifically proves the C-R equation by a much more rigorous method with the routes from the linear equation and applies this important equation to determine the number of and the locations of the differentiable points in a complex plane. It is also one of the factors to determine the analyticity of complex functions. At the end of the paper, an example will be provided to illustrate the application of the C-R equation.

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Podhornyj, O. R., and M. V. Sidorov. "Mathematical modeling of fluid flows through the piecewise homogeneous porous medium by R-function method." Mathematical Modeling and Computing 8, no. 3 (2021): 499–508. http://dx.doi.org/10.23939/mmc2021.03.499.

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The stationary fluid flow through a piecewise homogeneous porous medium is considered under the assumption that Darcy's law holds. The mathematical model of this problem is defined as an elliptic equation for the stream function, supplemented by the second-type boundary conditions at the water boundaries and the first-type boundary conditions at the impervious to liquid boundaries. The problem statement also includes the conditions of conjugation at the separation line between two soils and the unknown value of fluid discharge, which can be established from the additional integral ratio. It is proposed to use the structure-variational method of R-functions in order to numerically analyze and solve the current problem. The complete solution structure for the boundary value problem of stream function regarding the R-functions method is established, moreover, the application of the Ritz method for approximating an unspecified structural formula component is substantiated. Then, the approximate value of the fluid discharge and the approximate solution of the original problem are found from the additional integral ratio. The computational experiment was carried out with different coefficients of permeability within the area, which has the shape of the lower half ring. It is established that as the number of coordinate functions increases, the value of fluid discharge becomes constant, indicating the convergence of the proposed method.

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