Group Equivariant Singularity Theory in the Bifurcation of Interfacial Wilton Ripples

2015 ◽  
Vol 25 (11) ◽  
pp. 1550147
Author(s):  
Mark Jones
Keyword(s):  
Nonlinear Operator ◽  
Singularity Theory ◽  
Nonlinear Operator Equation ◽  
Infinite Dimensional ◽  
Two Fluids ◽  
Near Resonance ◽  
Classical Procedure ◽  
Symmetry Properties ◽  
System Of Polynomial Equations ◽  
Secondary Bifurcations

An investigation is made into the capillary-gravity waves which arise on the interface of two fluids and which are formed by the interaction of the first two harmonics of the motion. The problem is transformed into a nonlinear operator equation between suitable function spaces which is shown to be invariant under certain group actions. The infinite dimensional problem is reduced, by the classical procedure of Lyapunov–Schmidt, to a finite system of polynomial equations, known as the bifurcation equations. Because these equations inherit the symmetry properties of the original operator, it is possible to make quite specific statements concerning their structure, thus rendering their analysis easier. Solutions to the equations are sought, both in the cases of exact and of near-resonance. A wide variety of solutions is found depending on the values of the parameters: both simple, multiple and secondary bifurcations may occur, and in addition, there may exist isolated solution curves.

The Existence of Solutions for Nonlinear Operator Equations

2008 ◽  
Vol 15 (1) ◽  
pp. 45-52
Author(s):  
Marek Galewski
Keyword(s):  
Nonlinear Operator ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Nonlinear Operator Equation ◽  
Global Minima ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
Dual Variational Method ◽  
Dual Action ◽  
Primal And Dual

Abstract We provide the existence results for a nonlinear operator equation Λ*Φ′ (Λ𝑥) = 𝐹′(𝑥), in case 𝐹 – Φ is not necessarily convex. We introduce the dual variational method which is based on finding global minima of primal and dual action functionals on certain nonlinear subsets of their domains and on investigating relations between the minima obtained. The solution is a limit of a minimizng sequence whose existence and convergence are proved. The application for the non-convex Dirichlet problem with P.D.E. is given.

Oscillation analysis of numerical solution of Nonlinear Operator Equation

2021 ◽  
Vol 7 (5) ◽  
pp. 2111-2126
Author(s):  
Yang Zhou ◽  
Cuimei Li
Keyword(s):  
Numerical Solution ◽  
Operator Equation ◽  
Vibration Analysis ◽  
Nonlinear Operator ◽  
Vibration Spectrum ◽  
Nonlinear Operator Equation ◽  
Analysis Method ◽  
Analysis Model ◽  
Oscillation Analysis ◽  
Analysis Equation

There is a problem of low accuracy in the analysis of the vibration of the numerical solution of the nonlinear operator equation. In this work, the vibration analysis equation is constructed by the step-by-step search method, and the vibration quadrant of the equation is divided by the dichotomy method. The vibration spectrum is determined by the iteration method, and the vibration analysis model of the numerical solution of the nonlinear operator equation is constructed. The vibration analysis of the numerical solution of the nonlinear operator equation is completed based on the solution of the model and the numerical calculation and display of the step-by-step Fourier. The experimental results show that the proposed method has higher accuracy than the traditional vibration analysis method, which meets the requirements of the vibration analysis of the numerical solution of nonlinear operator equation.

Elements of Group Theory

2021 ◽  
pp. 51-110
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Group Theory ◽  
Lie Groups ◽  
Lorentz Group ◽  
Unitary Representations ◽  
Compact Groups ◽  
Mathematical Language ◽  
Group Representations ◽  
Infinite Groups ◽  
Infinite Dimensional ◽  
Symmetry Properties

The mathematical language which encodes the symmetry properties in physics is group theory. In this chapter we recall the main results. We introduce the concepts of finite and infinite groups, that of group representations and the Clebsch–Gordan decomposition. We study, in particular, Lie groups and Lie algebras and give the Cartan classification. Some simple examples include the groups U(1), SU(2) – and its connection to O(3) – and SU(3). We use the method of Young tableaux in order to find the properties of products of irreducible representations. Among the non-compact groups we focus on the Lorentz group, its relation with O(4) and SL(2,C), and its representations. We construct the space of physical states using the infinite-dimensional unitary representations of the Poincaré group.

scholarly journals TEMPERATURE MODELLING WITHIN A THIN MATERIAL SHEET INVOLVED IN CONDUCTIVE‐RADIATIVE HEAT TRANSFER

2005 ◽  
Vol 10 (2) ◽  
pp. 141-154
Author(s):  
K. Birgelis
Keyword(s):  
Heat Transfer ◽  
Operator Equation ◽  
Radiative Heat Transfer ◽  
Radiative Heat ◽  
Nonlinear Operator ◽  
Nonlinear Operator Equation

In this paper we consider a problem about finding of temperature approximation within a thin material sheet involved in conductive‐radiative heat transfer. As result, we found that temperature within the sheet can be approximated in L 2 norm by solution of a simple nonlinear operator equation. Straipsnyje modeliuojamas temperatūros pasiskirstymas tarp plonu medžiagos lakštu atsižvelgiant i radiacijai laidžios šlumos pernešima. Nustatyta, kad temperatūra tarp lakštu gali būti aproksimuojama L 2 normoje paprastos netiesines operatorines lygties sprendiniais.

scholarly journals Semilocal Convergence Theorem for the Inverse-Free Jarratt Method under New Hölder Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Yueqing Zhao ◽  
Rongfei Lin ◽  
Zdenek Šmarda ◽  
Yasir Khan ◽  
Jinbiao Chen ◽  
...  
Keyword(s):  
Banach Space ◽  
Error Estimate ◽  
Convergence Analysis ◽  
Operator Equation ◽  
Convergence Theorem ◽  
Nonlinear Operator ◽  
Semilocal Convergence ◽  
Nonlinear Operator Equation ◽  
Jarratt Method ◽  
Semilocal Convergence Theorem

Under the new Hölder conditions, we consider the convergence analysis of the inverse-free Jarratt method in Banach space which is used to solve the nonlinear operator equation. We establish a new semilocal convergence theorem for the inverse-free Jarratt method and present an error estimate. Finally, three examples are provided to show the application of the theorem.

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