Construction of Known and New Cubature Formulas of Degree Five for Three-Dimensional Symmetric Regions, Using Orthogonal Polynomials

1982 ◽  
pp. 119-127 ◽  
Author(s):  
Ann Haegemans
Keyword(s):  
Orthogonal Polynomials ◽  
Three Dimensional ◽  
Cubature Formulas

scholarly journals On the Modeling of Five-Layer Thin Prismatic Bodies

2019 ◽  
Vol 24 (3) ◽  
pp. 69 ◽  
Author(s):  
Mikhail U. Nikabadze ◽  
Armine R. Ulukhanyan ◽  
Tamar Moseshvili ◽  
Ketevan Tskhakaia ◽  
Nodar Mardaleishvili ◽  
...  
Keyword(s):  
Orthogonal Polynomials ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Micropolar Theory ◽  
Prismatic Bodies ◽  
Initial Boundary Value

Proceeding from three-dimensional formulations of initial boundary value problems of the three-dimensional linear micropolar theory of thermoelasticity, similar formulations of initial boundary value problems for the theory of multilayer thermoelastic thin bodies are obtained. The initial boundary value problems for thin bodies are also obtained in the moments with respect to systems of orthogonal polynomials. We consider some particular cases of formulations of initial boundary value problems. In particular, the statements of the initial-boundary value problems of the micropolar theory of K-layer thin prismatic bodies are considered. From here, we can easily get the statements of the initial-boundary value problems for the five-layer thin prismatic bodies.

scholarly journals Non-Equilibrium Liouville and Wigner Equations: Classical Statistical Mechanics and Chemical Reactions for Long Times

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 179 ◽  
Author(s):  
Ramon Álvarez-Estrada
Keyword(s):  
Orthogonal Polynomials ◽  
Chemical Reactions ◽  
Thermal Equilibrium ◽  
Three Dimensional ◽  
Liapunov Function ◽  
Heat Bath ◽  
Wigner Functions ◽  
Statistical Systems ◽  
Non Equilibrium

We review and improve previous work on non-equilibrium classical and quantum statistical systems, subject to potentials, without ab initio dissipation. We treat classical closed three-dimensional many-particle interacting systems without any “heat bath” ( h b ), evolving through the Liouville equation for the non-equilibrium classical distribution W c , with initial states describing thermal equilibrium at large distances but non-equilibrium at finite distances. We use Boltzmann’s Gaussian classical equilibrium distribution W c , e q , as weight function to generate orthogonal polynomials ( H n ’s) in momenta. The moments of W c , implied by the H n ’s, fulfill a non-equilibrium hierarchy. Under long-term approximations, the lowest moment dominates the evolution towards thermal equilibrium. A non-increasing Liapunov function characterizes the long-term evolution towards equilibrium. Non-equilibrium chemical reactions involving two and three particles in a h b are studied classically and quantum-mechanically (by using Wigner functions W). Difficulties related to the non-positivity of W are bypassed. Equilibrium Wigner functions W e q generate orthogonal polynomials, which yield non-equilibrium moments of W and hierarchies. In regimes typical of chemical reactions (short thermal wavelength and long times), non-equilibrium hierarchies yield approximate Smoluchowski-like equations displaying dissipation and quantum effects. The study of three-particle chemical reactions is new.

scholarly journals Boundary Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation

2012 ◽  
Vol 19 (3) ◽  
pp. 349-364 ◽  
Author(s):  
R. Lal ◽  
Yajuvindra Kumar
Keyword(s):  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Cartesian Product ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Rectangular Plates ◽  
Transverse Vibrations ◽  
Modes Of Vibration ◽  
Boundary Characteristic

The free transverse vibrations of thin nonhomogeneous rectangular plates of variable thickness have been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. Gram-Schmidt process has been used to generate these orthogonal polynomials in two variables. The thickness variation is bidirectional and is the cartesian product of linear variations along two concurrent edges of the plate. The nonhomogeneity of the plate is assumed to arise due to linear variations in Young's modulus and density of the plate material with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of the nonhomogeneity and thickness variation with varying values of aspect ratio on the natural frequencies of vibration is illustrated for the first three modes of vibration. Three dimensional mode shapes for all the four boundary conditions have been presented. A comparison of results with those available in the literature has been made.

scholarly journals Algorithm for Calculating the Global Minimum of a Smooth Function of Several Variables

2021 ◽  
Vol 8 (4) ◽  
pp. 591-596
Author(s):  
Zhetkerbay Kaidassov ◽  
Zhailan S. Tutkusheva
Keyword(s):  
Global Minimum ◽  
Dimensional Space ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Computational Experiments ◽  
Quadrature Formulas ◽  
Cubature Formulas ◽  
Integration Formulas ◽  
C Programming ◽  
Approximate Integration

Every year the interest of theorists and practitioners in optimisation problems is growing, and extreme problems are found in all branches of science. Local optimisation problems are well studied and there are constructive methods for their solution. However, global optimisation problems do not meet the requirements in practice; therefore, the search for the global minimum remains one of the major challenges for computational and applied mathematics. This study discusses the search for the global minimum of multidimensional and multiextremal problems with high precision. Mechanical quadrature formulas, that is, the formulas for approximate integration were applied to calculate the integrals. Of all the approximate integration formulas, the Sobolev lattice cubature formulas with a regular boundary layer were chosen. In multidimensional examples, the Sobolev formulas are optimal. Computational experiments were carried out in the most popular C++ programming language. Based on the computational experiments, a new algorithm was proposed. In three-dimensional space, the calculations of the global minimum have been described using specific examples. Computational experiments show that the proposed algorithm works for multiextremal problems with the same amount of time as for convex ones.

scholarly journals Discrete Transforms and Orthogonal Polynomials of (Anti)symmetric Multivariate Sine Functions

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 938 ◽  
Author(s):  
Adam Brus ◽  
Jiří Hrivnák ◽  
Lenka Motlochová
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Three Dimensional ◽  
Weight Functions ◽  
Recursive Algorithms ◽  
Interpolation Methods ◽  
Orthogonality Relations ◽  
Discrete Transforms ◽  
Unitary Transform ◽  
Sine Functions

Sixteen types of the discrete multivariate transforms, induced by the multivariate antisymmetric and symmetric sine functions, are explicitly developed. Provided by the discrete transforms, inherent interpolation methods are formulated. The four generated classes of the corresponding orthogonal polynomials generalize the formation of the Chebyshev polynomials of the second and fourth kinds. Continuous orthogonality relations of the polynomials together with the inherent weight functions are deduced. Sixteen cubature rules, including the four Gaussian, are produced by the related discrete transforms. For the three-dimensional case, interpolation tests, unitary transform matrices and recursive algorithms for calculation of the polynomials are presented.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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