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Journal articles on the topic 'R-functions method'

Author: Grafiati

Published: 30 May 2022

Last updated: 31 May 2022

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1

Basarab, Mikhail, Alain Giani, and Philippe Combette. "Thermal Accelerometer Simulation by the R‑Functions Method." Applied Sciences 10, no. 23 (November 25, 2020): 8373. http://dx.doi.org/10.3390/app10238373.

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As well as many modern devices, thermal accelerometers (TAs) need a sophisticated mathematical simulation to find the ways for their performance optimization. In the paper, a novel approach for solving computational fluid dynamics (CFD) problems in the TA’s cavity is proposed (MQ-RFM), which is based on the combined use of Rvachev’s R-functions method (RFM) and the Galerkin technique with multiquadric (MQ) radial basis functions (RBFs). The semi-analytical RFM takes an intermediate position between traditional analytical approaches and numerical methods, such as the finite-element method (FEM), belonging to the family of the so-called meshless techniques which became popular in the last decades in solving various CFD problems in complex-shaped cavities. Mathematical simulation of TA by using the MQ-RFM was carried out with the purpose to simulate the temperature response of the device and to study and improve its performance. The results of numerical experiments were compared with well-known analytical and numerical benchmark solutions for the circular annulus geometry and it demonstrated the effectiveness of the MQ-RFM for solving the convective heat-transfer problem in the TA’s cavity. The use of solution structures allows one to take a relatively small number of expansion terms to achieve an appropriate accuracy of the approximate solution satisfying at the same time the given boundary conditions exactly. The application of the MQ-RFM gives the possibility to obtain semi-analytical solutions to the diffusion-convection problems and to identify the main thermal characteristics of the TA, that allows one to improve the device performance.

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2

Ranenko, A. A., and T. I. Sheiko. "Method of R-functions in problems with natural boundary conditions." Journal of Soviet Mathematics 66, no. 1 (August 1993): 2097–102. http://dx.doi.org/10.1007/bf01097401.

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3

Miyajima, Keiichi, Takahiro Kato, and Yasunari Shidama. "Riemann Integral of Functions R into C." Formalized Mathematics 18, no. 4 (January 1, 2010): 201–6. http://dx.doi.org/10.2478/v10037-010-0024-8.

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Riemann Integral of Functions R into C In this article, we define the Riemann Integral on functions R into C and proof the linearity of this operator. Especially, the Riemann integral of complex functions is constituted by the redefinition about the Riemann sum of complex numbers. Our method refers to the [19].

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4

Rvachev, V. L., and T. I. Sheiko. "R-Functions in Boundary Value Problems in Mechanics." Applied Mechanics Reviews 48, no. 4 (April 1, 1995): 151–88. http://dx.doi.org/10.1115/1.3005099.

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Described are the concepts and applications of the R-functions theory in continuum mechanics boundary value problems which model fields of different physical natures. With R-functions there appears the possibility of creating a constructive mathematical tool which incorporates the capabilities of classical continuous analysis and logic algebra. This allows one to overcome the main obstacle which hinders the use of variational methods when solving boundary value problems in domains of complex shape with complex boundary conditions, this obstacle being connected with the construction of so-called coordinate sequences. In contrast to widely used methods of the network type (finite difference, finite and boundary elements), in the R-functions method all the geometric information present in the boundary value problem statement is reduced to analytical form, which allows one to search for a solution in the form of formulae called solution structures containing some indefinite functional components. A method of constructing solution structures satisfying the required conditions of completeness has been developed. The structural formulae include the left-hand sides of the normalized equations of the boundaries of the domains or their regions being considered, thus allowing one to change the solution structure expeditiously when changing the geometric shape. Given in the work is a definition of the basic class of R-functions, solution with their help of the inverse problem of analytical geometry (construction of equations of specified configurations); generalization of the Taylor-Hermite formulae for functional spaces in which points are represented by lines and surfaces; and construction of solution structures of some types of boundary value problems. Shown are the solutions of a number of concrete problems in these application fields with the use of the RL language and POLYE system.

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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

Torres-Hernandez, A., F. Brambila-Paz, U. Iturrarán-Viveros, and R. Caballero-Cruz. "Fractional Newton–Raphson Method Accelerated with Aitken’s Method." Axioms 10, no. 2 (March 31, 2021): 47. http://dx.doi.org/10.3390/axioms10020047.

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In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order α of the derivative is different from one. A simplified way of constructing the Riemann–Liouville (R–L) fractional operators, fractional integral and fractional derivative is presented along with examples of its application on different functions. Furthermore, an introduction to Aitken’s method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken’s method in the F N–R method, where it is shown that F N–R with Aitken’s method converges faster than the simple F N–R.

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7

Gudmundsson, Sigmundur, and Marko Sobak. "Proper r-harmonic functions from Riemannian manifolds." Annals of Global Analysis and Geometry 57, no. 1 (December 10, 2019): 217–23. http://dx.doi.org/10.1007/s10455-019-09696-3.

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AbstractWe introduce a new method for constructing complex-valued r-harmonic functions on Riemannian manifolds. We then apply this for the important semisimple Lie groups $$\mathbf{SO }(n)$$SO(n), $$\mathbf{SU }(n)$$SU(n), $$\mathbf{Sp }(n)$$Sp(n), $$\mathbf{SL }_{n}({\mathbb {R}})$$SLn(R), $$\mathbf{Sp }(n,{\mathbb {R}})$$Sp(n,R), $$\mathbf{SU }(p,q)$$SU(p,q), $$\mathbf{SO }(p,q)$$SO(p,q), $$\mathbf{Sp }(p,q)$$Sp(p,q), $$\mathbf{SO }^*(2n)$$SO∗(2n) and $$\mathbf{SU }^*(2n)$$SU∗(2n).

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8

Avramov, K. V. "Nonlinear vibrations of circular plates with notches. Method of R-functions." Journal of Mathematical Sciences 170, no. 6 (October 14, 2010): 708–20. http://dx.doi.org/10.1007/s10958-010-0115-7.

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9

Stetsyuk, P., V. Stovba, and A. Suprun. "B-FORM OF THE DAVIDON–FLETCHER–POWELL METHOD." Journal of Numerical and Applied Mathematics, no. 2 (136) (2021): 93–110. http://dx.doi.org/10.17721/2706-9699.2021.2.08.

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A special form (B-form) of methods of Quasi-Newton type is discussed, which makes it easy to interpret these methods as gradient in appropriately transformed argument space. B-form of the Davidon–Fletcher–Powell method is given and compared with r-algorithms. To minimize smooth convex functions, a gradient method with space transformation is built, combining properties of both quasi-Newtonian methods and r-algorithms. Possible schemes of this type of methods for minimizing non-smooth convex functions are discussed.

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10

Bray, William O. "TRANSPLANTATION FORMULAE AND HADAMARD'S METHOD OF DESCENT." Proceedings of the Edinburgh Mathematical Society 50, no. 2 (May 17, 2007): 277–92. http://dx.doi.org/10.1017/s0013091505000933.

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AbstractWe define a partial Radon transform mapping functions on $\mathbb{R}^{n+l}$ to functions on $\mathbb{R}^{n}$ which intertwines the Laplace operator on the two spaces. As a consequence, transplantation formulae relating the radial eigenfunctions of the Laplacian on Euclidean spaces of different dimensions are obtained. Our formulae provide a geometric interpretation of integral formulae for Bessel functions of Abel type, which are found useful in potential theory. The formulae portray a view of Hadamard's method of descent within the realm of harmonic analysis, allowing the transplant of local problems from even dimensions to odd dimensions and unifying the techniques of several authors.

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11

Miyajima, Keiichi, and Yasunari Shidama. "Riemann Integral of Functions from R into Rn." Formalized Mathematics 17, no. 2 (January 1, 2009): 179–85. http://dx.doi.org/10.2478/v10037-009-0021-y.

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Riemann Integral of Functions from R into Rn In this article, we define the Riemann Integral of functions from R into Rn, and prove the linearity of this operator. The presented method is based on [21].

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12

Kravchenko, V. F., and M. A. Basarab. "A new method of multidimensional-signal processing with the use of R functions and atomic functions." Doklady Physics 47, no. 3 (March 2002): 195–200. http://dx.doi.org/10.1134/1.1467861.

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13

Chen, Chunlin, Yulong Li, and Fuh-Gwo Yuan. "Development of Time-Reversal Method for Impact Source Identification on Plate Structures." Shock and Vibration 20, no. 3 (2013): 561–73. http://dx.doi.org/10.1155/2013/312169.

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This paper presents a detailed study on the impact source identification of a plate structure using time-reversal (T-R) method. Prior to impact monitoring, the plate is calibrated (or characterized) by transfer functions at discrete locations on the plate surface. Both impact location and impact loading time-history are identified using T-R technique and associated signal processing algorithms. Numerical verification for finite-size isotropic plates under low velocity impacts is performed to demonstrate the versatility of T-R method for impact source identification. The tradeoff between accuracy of the impact location detection and calibration spacing is studied in detail. In particular, the effect of plate thickness on calibration spacing has been examined. A number of parameters selected for determining the impact location, approximated transfer functions and steps taken for reconstructing the impact loading time-history are also examined. Two types of noise with various intensities contaminated in strain response and/or transfer functions are investigated for demonstrating the stability and reliability of the T-R method. The results show that T-R method is robust against noise in impact location detection and force reconstruction in circumventing the inherent ill-conditioned inverse problem. Only transfer functions are needed to be calibrated and four sensors are requested in T-R method for impact identification.

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14

Artiukh, A. V., S. I. Kozyrenko, O. O. Shabalin, and I. R. Lytvyn. "APPLICATION OF R-FUNCTIONS METHOD AND SMOOTHED PARTICLE HYDRODYNAMICS FOR FLUID SIMULATION." Radio Electronics, Computer Science, Control, no. 3 (October 1, 2019): 25–33. http://dx.doi.org/10.15588/1607-3274-2019-3-3.

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15

Orkisz, J., W. Rachowicz, and M. Wnuk. "Computerized R-functions method and its application in linear elastic fracture mechanics." Mathematical Modelling 8 (1987): 493–99. http://dx.doi.org/10.1016/0270-0255(87)90631-2.

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16

Pokras, V. D., and M. A. Rvachev. "Application of the R-functions method to visioplastic analysis in metal forming." Journal of Materials Processing Technology 60, no. 1-4 (June 1996): 493–500. http://dx.doi.org/10.1016/0924-0136(96)02376-x.

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17

Kim, Henry H. "Langlands-Shahidi Method and Poles of Automorphic L-Functions: Application to Exterior Square L-Functions." Canadian Journal of Mathematics 51, no. 4 (August 1, 1999): 835–49. http://dx.doi.org/10.4153/cjm-1999-036-0.

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AbstractIn this paper we use Langlands-Shahidimethod and the result of Langlands which says that non selfconjugatemaximal parabolic subgroups do not contribute to the residual spectrum, to prove the holomorphy of several completed automorphic L-functions on the whole complex plane which appear in constant terms of the Eisenstein series. They include the exterior square L-functions of GLn, n odd, the Rankin-Selberg L-functions of GLn × GLm, n ≠ m, and L-functions L(s, σ, r), where σ is a generic cuspidal representation of SO10 and r is the half-spin representation of GSpin(10, ). The main part is proving the holomorphy and non-vanishing of the local normalized intertwining operators by reducing them to natural conjectures in harmonic analysis, such as standard module conjecture.

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18

Zippo, Antonio, Giovanni Iarriccio, Francesco Pellicano, and Tetyana Shmatko. "Vibrations of Plates with Complex Shape: Experimental Modal Analysis, Finite Element Method, and R-Functions Method." Shock and Vibration 2020 (December 3, 2020): 1–23. http://dx.doi.org/10.1155/2020/8882867.

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In this paper, the dynamic behavior of 3D-printed plates with different shapes and boundary conditions is investigated. The natural frequencies and mode shapes were determined using three different methods: the experimental analysis, the finite element method, using Nastran, and the R-functions method. The experimental and theoretical results are compared. The specimens tested included four cases. The test procedure is deeply described, and the material properties of the plates are given. The fixed-fixed configuration shows a better agreement both in the rectangular plate and in the plate with rectangular cuts, and the R-functions method gives better convergence with respect to the experimental and finite element analysis. The simply supported arrangement indicates some uncertainty in the boundary realization of the specimen.

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19

Gudmundsson, Sigmundur, and Marko Sobak. "r-Harmonic and Complex Isoparametric Functions on the Lie Groups $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$." Annals of Global Analysis and Geometry 58, no. 4 (September 21, 2020): 477–96. http://dx.doi.org/10.1007/s10455-020-09736-3.

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Abstract In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ R m ⋉ R n and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$ R m ⋉ H 2 n + 1 , where $$\mathrm {H}^{2n+1}$$ H 2 n + 1 denotes the classical $$(2n+1)$$ ( 2 n + 1 ) -dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.

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20

Podhornyi, O. R. "Numerical Analysis of Filtration Flows in Inhomogeneous Media Using the R-Functions Method." Mathematical and computer modelling. Series: Physical and mathematical sciences, no. 18 (December 18, 2018): 113–25. http://dx.doi.org/10.32626/2308-5878.2018-18.113-125.

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21

Kurpa, Lidia, Galina Pilgun, and Marco Amabili. "Nonlinear vibrations of shallow shells with complex boundary: R-functions method and experiments." Journal of Sound and Vibration 306, no. 3-5 (October 2007): 580–600. http://dx.doi.org/10.1016/j.jsv.2007.05.045.

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22

Zhao, T. C., and S. Y. Tong. "R-matrix method for calculating wave functions in reflection high-energy electron diffraction." Physical Review B 47, no. 7 (February 15, 1993): 3923–28. http://dx.doi.org/10.1103/physrevb.47.3923.

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23

Maksimenko-Sheiko, K. V., A. M. Matsevityi, and T. I. Sheiko. "Automation of constructing equations of geometric objects in the method of R-functions." Cybernetics and Systems Analysis 42, no. 2 (March 2006): 284–90. http://dx.doi.org/10.1007/s10559-006-0064-9.

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24

Antczak, Tadeusz. "A new method of solving nonlinear mathematical programming problems involving r-invex functions." Journal of Mathematical Analysis and Applications 311, no. 1 (November 2005): 313–23. http://dx.doi.org/10.1016/j.jmaa.2005.02.049.

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25

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 (November 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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26

DRUET, T., D. BAYE, P. DESCOUVEMONT, and J. M. SPARENBERG. "APPLICATION OF THE R-MATRIX METHOD TO CDCC CALCULATIONS." Modern Physics Letters A 25, no. 21n23 (July 30, 2010): 1745–49. http://dx.doi.org/10.1142/s021773231000023x.

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The R-matrix method is applied to the Continuum Discretized Coupled Channel (CDCC) approximation. The variational basis is chosen as Lagrange functions, which are shown to be efficient and accurate. We apply the general formalism to the d + 58 Ni elastic scattering at Ed = 80 MeV . Future developments are briefly discussed.

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27

Semerich, Yuriy. "ANALYTICAL MODELLING OF PERFORATED GEOMETRICAL DOMAINS BY THE R-FUNCTIONS." Mathematical Modelling and Analysis 25, no. 3 (July 8, 2020): 490–504. http://dx.doi.org/10.3846/mma.2020.11174.

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This paper deals with the construction of boundary equations for geometric domains with perforation. Different types of perforated geometric domains are considered. The R-functions method for analytical modelling of perforated geometrical domains is used. For all constructed equations, function plots are obtained.

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28

Miyajima, Keiichi, Takahiro Kato, and Yasunari Shidama. "Riemann Integral of Functions from R into Real Normed Space." Formalized Mathematics 19, no. 1 (January 1, 2011): 17–22. http://dx.doi.org/10.2478/v10037-011-0003-8.

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Riemann Integral of Functions from R into Real Normed Space In this article, we define the Riemann integral on functions from R into real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to a wider range of functions. The proof method follows the [16].

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29

Scott, W. F. "The Z-Method for Assurances." Transactions of the Faculty of Actuaries 43 (1992): 376–79. http://dx.doi.org/10.1017/s0071368600010107.

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AbstractThe use of the Z-method for finding the probability of survival of at least r lives, and the corresponding pure endowment and annuity functions, is well known in acturial circles. (See, for example, Neill [1; section 7.4].) The purpose of this note is to show that similar formulae apply to assurances (including varying, temporary and deferred assurances).

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30

Semerich, Yu S. "THE R-FUNCTIONS METHOD IN THE BOUNDARY VALUE PROBLEM FOR A COMPLEX DOMAIN POSSESSING THE SYMMETRY." Mathematical Modelling and Analysis 8, no. 1 (March 31, 2003): 77–86. http://dx.doi.org/10.3846/13926292.2003.9637212.

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This paper considers a case of the punctual symmetry of cyclic type and application of the R‐functions method in building equations of the boundaries for symmetric objects. The R‐functions method combined with the Ritz method is used for solving the Laplace equation in the complex symmetric domain. The problem is described for calculating an electrostatic field in the system with crossed fields (the cylindrical magnetron of reversed construction).

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Babolian, Esmail, Jamshid Saeidian, and Mahmood Paripour. "Computing the Fourier Transform via Homotopy Perturbation Method." Zeitschrift für Naturforschung A 64, no. 11 (November 1, 2009): 671–75. http://dx.doi.org/10.1515/zna-2009-1101.

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In this paper, the homotopy perturbation method (HPM) is applied to compute the Fourier transform (FT) of functions on R. The basic properties of the Fourier transform are again obtained by HPM and examples assuring the applicability of HPM are presented.

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32

Poltorak, K. "Cross-Approximation Method for Solving Dynamics Problems of Arbitrarily Shaped Plates." Journal of Applied Mechanics 57, no. 2 (June 1, 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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33

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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Kurpa, L., G. Timchenko, and A. Osetrov. "Research of Nonlinear Vibrations of Laminated Shallow Shells with Cutouts by R-functions Method." Bulletin of the National Technical University «KhPI» Series: Dynamics and Strength of Machines 1, no. 46 (January 1, 2016): 86–89. http://dx.doi.org/10.20998/2078-9130.2016.46.88056.

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35

Barbarics, T., H. Igarashi, A. Iványi, and T. Honma. "Electrostatic field calculation using R-functions and the method of characteristics in electrostatic precipitator." Journal of Electrostatics 38, no. 4 (December 1996): 269–82. http://dx.doi.org/10.1016/s0304-3886(96)00031-9.

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36

Gladkov, Svyatoslav, and Bob Svendsen. "Application of the R-functions method to the solution of elliptic PDEs (Poster Presentation)." PAMM 7, no. 1 (December 2007): 2010033–34. http://dx.doi.org/10.1002/pamm.200701055.

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37

Semerich, Yu S. "THE CONSTRUCTION OF LOCI WITH A CYCLICAL SYMMETRY BY THE R-FUNCTIONS." Mathematical Modelling and Analysis 12, no. 1 (March 31, 2005): 73–82. http://dx.doi.org/10.3846/13926292.2005.9637272.

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This paper deals with the development of the R‐functions method for construction of equations describing a boundary of locus in the case of cyclical symmetry. The proposed method is based on transformations of coordinates and canonical locus. Darbe pateiktas būdas, kaip pritaikyti R‐funkciju metoda taip vadinamu ciklines simetrijos sričiu kontūru lygtims sudaryti. Pasiūlytasis metodas pagristas loordinačiu transformavimu, pasinaudojant kanoniniu hodografu.

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Plankovskyy, Sergiy, Olga Shypul, Yevgen Tsegelnyk, Oleg Tryfonov, and Ivan Golovin. "SIMULATION OF SURFACE HEATING FOR ARBITRARY SHAPE’S MOVING BODIES/SOURCES BY USING R-FUNCTIONS." Acta Polytechnica 56, no. 6 (December 31, 2016): 472–77. http://dx.doi.org/10.14311/ap.2016.56.0472.

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The purpose of this article is to propose an efficient algorithm for determining the place of an action of a heat source with a given motion law for a body of an arbitrary shape using methods of analytical geometry. The solution to this problem is an important part of a modeling of a laser, plasma, ion beam treatment. In addition, it can also be used for mass transfer problems, such as simulation of coating, sputtering, painting etc. The problem is solved by the method of R-functions to define the shape of the test body and the heat source and the analytical determination zone shadowing. As an example, we consider the problem of using the method of ion cleaning parameters optimization considering temperature limitations. Application of the R-functions can significantly reduce the amount of computation with usage of the ray tracing algorithm. The numerical realization of the proposed method requires an accurate creation of a numerical mesh. The best results in terms of accuracy of determination the scope of the source can be expected when applying adaptive tunable meshes. In case of integration of the R-functions into the CAD system, the use of the proposed method would be simple enough. The proposed method allows to determine the range of the source by the expression, which is constructed only once for the body and the source of arbitrary geometric shapes moving in any law. This distinguishes the proposed approach against all known algorithms for ray tracing. The proposed method can also be used for time-dependent multisource with arbitrary shapes, which move in different directions.

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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 (November 11, 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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40

Mayzel, O. V. "The Method of Increasing the Cognitive Potential of a Person (R. Feuerstein and R. Guzman)." Клиническая и специальная психология 5, no. 3 (2016): 130–44. http://dx.doi.org/10.17759/cpse.2016050309.

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The article is devoted to the description and analysis of method of development of intelligence Reuven of Feierstein, a distinguished Israeli scholar and a follower of J. Piaget and L.S. Vygotsky. Based on theoretical experience of his great predecessors R. Feuerstein has developed practical methods to develop cognitive functions in children. A significant role is given to training in adult (mediator), whose tasks include the direction and organization of the learning process by using specially developed techniques. Later, Israeli psychologist R. Guzman has adapted these methods for children suffering from Attention Deficient Hyperactivity Disorder, as well as for born blind people. The exception of the visual perception channel in the proposed set of original methods allows children suffering from Attention Deficient Hyperactivity Disorder, to avoid extra stimulation of brain activity and to focus on completing the tasks given by the mediator. Meanwhile, using the tactile methods when working with born blind people, on the contrary, promotes creating “the library of visual images”. Carrying out consistent lessons based on tactile methodology lead to the behavior correction of the children who suffer from Attention Deficient Hyperactivity Disorder and of born blind people of all age groups, as well as it caused the development of immature or absent cognitive functions.

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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 (July 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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Bellavia, Stefania, Nataša Krejić, and Nataša Krklec Jerinkić. "Subsampled inexact Newton methods for minimizing large sums of convex functions." IMA Journal of Numerical Analysis 40, no. 4 (July 4, 2019): 2309–41. http://dx.doi.org/10.1093/imanum/drz027.

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Abstract This paper deals with the minimization of a large sum of convex functions by inexact Newton (IN) methods employing subsampled functions, gradients and Hessian approximations. The conjugate gradient method is used to compute the IN step and global convergence is enforced by a nonmonotone line-search procedure. The aim is to obtain methods with affordable costs and fast convergence. Assuming strongly convex functions, R-linear convergence and worst-case iteration complexity of the procedure are investigated when functions and gradients are approximated with increasing accuracy. A set of rules for the forcing parameters and subsample Hessian sizes are derived that ensure local q-linear/q-superlinear convergence of the proposed method. The random choice of the Hessian subsample is also considered and convergence in the mean square, both for finite and infinite sums of functions, is proved. Finally, the analysis of global convergence with asymptotic R-linear rate is extended to the case of the sum of convex functions and strongly convex objective function. Numerical results on well-known binary classification problems are also given. Adaptive strategies for selecting forcing terms and Hessian subsample size, streaming out of the theoretical analysis, are employed and the numerical results show that they yield effective IN methods.

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A. Nazirov, Sh, F. M. Nuraliev, and Sh A. Anorova. "Study of Numeric Convergence of the Method of R − functions in Problems of Constraint Torsion." American Journal of Computational and Applied Mathematics 2, no. 4 (August 9, 2012): 189–96. http://dx.doi.org/10.5923/j.ajcam.20120204.07.

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Mulak, J. "Non-integral expansion method of f(R)Y(Θ,φ) functions about a displaced centre." Journal of Alloys and Compounds 219, no. 1-2 (March 1995): 316–21. http://dx.doi.org/10.1016/0925-8388(94)05031-7.

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Kurpa, Lidiya, Tatiana Shmatko, and Galina Timchenko. "Free vibration analysis of laminated shallow shells with complex shape using the R-functions method." Composite Structures 93, no. 1 (December 2010): 225–33. http://dx.doi.org/10.1016/j.compstruct.2010.05.016.

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Rvachev, V. L., T. I. Sheiko, and V. Shapiro. "Application of the method of r-functions to integration of differential equations with partial derivatives." Cybernetics and Systems Analysis 35, no. 1 (January 1999): 1–18. http://dx.doi.org/10.1007/bf02667909.

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Kurpa, Lidiya, Galina Timchenko, Andrey Osetrov, and Tetyana Shmatko. "Nonlinear vibration analysis of laminated shallow shells with clamped cutouts by the R-functions method." Nonlinear Dynamics 93, no. 1 (November 22, 2017): 133–47. http://dx.doi.org/10.1007/s11071-017-3930-2.

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Sklepus, A. N., and N. G. Sklepus. "Investigation of the Creep of Plates of Complex Shape by the Method of R-Functions." Strength of Materials 37, no. 3 (May 2005): 304–9. http://dx.doi.org/10.1007/s11223-005-0043-9.

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Hansevi, Daniel. "The Perron method for $$\varvec{p}$$ p -harmonic functions in unbounded sets in $$\mathbf {R}^n$$ R n and metric spaces." Mathematische Zeitschrift 288, no. 1-2 (March 30, 2017): 55–74. http://dx.doi.org/10.1007/s00209-017-1877-0.

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Barile, Sara, and Addolorata Salvatore. "Some multiplicity results of homoclinic solutions for second order Hamiltonian systems." Opuscula Mathematica 40, no. 1 (2020): 21–36. http://dx.doi.org/10.7494/opmath.2020.40.1.21.

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We look for homoclinic solutions \(q:\mathbb{R} \rightarrow \mathbb{R}^N\) to the class of second order Hamiltonian systems \[-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}\] where \(L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}\) and \(a,b: \mathbb{R}\rightarrow \mathbb{R}\) are positive bounded functions, \(G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}\) are positive homogeneous functions and \(f:\mathbb{R}\rightarrow\mathbb{R}^N\). Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if \(f\equiv 0\) and the existence of at least three solutions if \(f\) is not trivial but small enough.

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