scholarly journals The Characterization of Affine Symplectic Curves in ℝ4

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 110
Author(s):  
Esra Çiçek Çetin ◽  
Mehmet Bektaş
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Fourth Order ◽  
Symplectic Geometry ◽  
Classical Mechanics ◽  
Symplectic Space ◽  
Frenet Frame ◽  
The Third ◽  
Natural Geometry

Symplectic geometry arises as the natural geometry of phase-space in the equations of classical mechanics. In this study, we obtain new characterizations of regular symplectic curves with respect to the Frenet frame in four-dimensional symplectic space. We also give the characterizations of the symplectic circular helices as the third- and fourth-order differential equations involving the symplectic curvatures.

scholarly journals Conservation Laws for a Generalized Coupled Korteweg-de Vries System

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Daniel Mpho Nkwanazana ◽  
Ben Muatjetjeja ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Fourth Order ◽  
Nonlinear Partial Differential Equations ◽  
Order System ◽  
Conserved Quantities ◽  
Third Order ◽  
The Third ◽  
New System

We construct conservation laws for a generalized coupled KdV system, which is a third-order system of nonlinear partial differential equations. We employ Noether's approach to derive the conservation laws. Since the system does not have a Lagrangian, we make use of the transformationu=Ux,v=Vxand convert the system to a fourth-order system inU,V. This new system has a Lagrangian, and so the Noether approach can now be used to obtain conservation laws. Finally, the conservation laws are expressed in theu,vvariables, and they constitute the conservation laws for the third-order generalized coupled KdV system. Some local and infinitely many nonlocal conserved quantities are found.

scholarly journals Symplectic Geometry and Its Applications on Time Series Analysis

2020 ◽  
Author(s):  
Min Lei
Keyword(s):  
Time Series ◽  
Time Series Analysis ◽  
Symplectic Geometry ◽  
Principal Component ◽  
Symplectic Space ◽  
Hamiltonian Matrix ◽  
The Third ◽  
Series Analysis ◽  
Mode Decomposition ◽  
Basic Concepts

This chapter serves to introduce the symplectic geometry theory in time series analysis and its applications in various fields. The basic concepts and basic elements of mathematics relevant to the symplectic geometry are introduced in the second section. It includes the symplectic space, symplectic transformation, Hamiltonian matrix, symplectic principal component analysis (SPCA), symplectic geometry spectrum analysis (SGSA), symplectic geometry mode decomposition (SGMD), and symplectic entropy (SymEn), etc. In addition, it also briefly reviews the applications of symplectic geometry on time series analysis, such as the embedding dimension estimation, nonlinear testing, noise reduction, as well as fault diagnosis. Readers who are familiar with the mathematical preliminaries may omit the second section, i.e. the theory part, and go directly to the third section, i.e. the application part.

scholarly journals Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Navnit Jha ◽  
R. K. Mohanty ◽  
Vinod Chauhan
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Fourth Order ◽  
Difference Equations ◽  
Computational Results ◽  
Order Convergence ◽  
Third Order ◽  
Tridiagonal Matrices ◽  
The Third ◽  
Geometric Mesh

Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourth-order ordinary differential equations. The method can be easily extended to the sixth-order differential equations. Convergence analysis proves the third-order convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block Gauss-Seidel algorithm. The computational results are provided to justify the usefulness and reliability of the proposed method.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

Gaunilo Parodies Anselm

2014 ◽  
Vol 17 (1) ◽  
pp. 45-71
Author(s):  
Geo Siegwart
Keyword(s):  
Detailed Comparison ◽  
The Third ◽  
Common Structure ◽  
Logical Reconstruction ◽  
Points Of View ◽  
And Function

The main objective is an interpretation of the island parody, in particular a logical reconstruction of the parodying argument that stays close to the text. The parodied reasoning is identified as the proof in the second chapter of the Proslogion, more specifically, this proof as it is represented by Gaunilo in the first chapter of his Liber pro insipiente. The second task is a detailed comparison between parodied and parodying argument as well as an account of their common structure. The third objective is a tentative characterization of the nature and function of parodies of arguments. It seems that parodying does not add new pertinent points of view to the usual criticism of an argument.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

Vapour-liquid equilibrium in the isobutyl formate-isobutyl alcohol and n-butyl formate-isobutyl alcohol systems at atmospheric pressure

1979 ◽  
Vol 44 (12) ◽  
pp. 3501-3508 ◽  
Author(s):  
Jan Linek
Keyword(s):  
Experimental Data ◽  
Fourth Order ◽  
Atmospheric Pressure ◽  
Isobutyl Alcohol ◽  
Liquid Equilibrium ◽  
The Third

Isobaric vapour-liquid equilibria in the isobutyl formate-isobutyl alcohol and n-butyl formate-isobutyl alcohol systems have been measured at atmospheric pressure. A modified circulation still of the Gillespie type has been used for the measurements. The experimental data have been correlated by means of the third- and fourth-order Margules equations.

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