Floquet theory for doubly-periodic differential equations and a number theory conjecture

1983 ◽  
pp. 449-501
Author(s):  
BD Sleeman ◽  
PD Smith
Keyword(s):  
Number Theory ◽  
Differential Equations ◽  
Floquet Theory

Titchmarsh's theorem of Hankel transform

2019 ◽  
pp. 11-24
Author(s):  
Mohamed-Ahmed Boudref
Keyword(s):  
Number Theory ◽  
Data Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Analytic Number Theory ◽  
Hankel Transform ◽  
Partial Differential ◽  
Analytic Number ◽  
Bessel Transform

Hankel transform (or Fourier-Bessel transform) is a fundamental tool in many areas of mathematics and engineering, including analysis, partial differential equations, probability, analytic number theory, data analysis, etc. In this article, we prove an analog of Titchmarsh's theorem for the Hankel transform of functions satisfying the Hankel-Lipschitz condition.

Instability and Unbalance Response of Dissymmetric Rotor-Bearing Systems

1988 ◽  
Vol 110 (3) ◽  
pp. 288-294 ◽  
Author(s):  
P. M. Guilhen ◽  
P. Berthier ◽  
G. Ferraris ◽  
M. Lalanne
Keyword(s):  
Differential Equations ◽  
Coordinate System ◽  
Transfer Matrix ◽  
Differential System ◽  
Industrial Application ◽  
Floquet Theory ◽  
Unbalance Response ◽  
Stationary Coordinate System ◽  
The Matrix ◽  
The Stability

The study deals with the instability and unbalance response of dissymmetric rotors, when periodic differential equations are impossible to avoid. The method which yields motion instability is based on an extension of the well-known Floquet theory. A transfer matrix over one period of the motion is obtained, and the stability of the system can be tested with the eigenvalues of the matrix. To find the instability and the unbalance response, the Newmark formulation is used. Here, the dissymmetry comes either from the rotor or from the bearings in such a way that it is possible to solve a regular differential system without periodic coefficients, either in the stationary coordinate system or in the rotating one. Three examples are given, including an industrial application. The results show that the method proposed is satisfactory.

Studies in multiplicative solutions to linear differential equations

1987 ◽  
Vol 106 (3-4) ◽  
pp. 277-305 ◽  
Author(s):  
F. M. Arscott
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Floquet Theory ◽  
Special Functions ◽  
Linear Differential Equations ◽  
Ordinary Linear Differential Equation ◽  
Closed Path ◽  
Starting Point ◽  
Linear Differential

SynopsisGiven an ordinary linear differential equation whose singularities are isolated, a solution is called multiplicative for a closed path C if, when continued analytically along C, it returns to its starting-point merely multiplied by a constant. This paper first classifies such paths into three types, then investigates combinations of two such paths, in which a number of qualitatively different situations can arise. A key result is also given relating to a three-path combination. There are applications to special functions and Floquet theory for periodic equations.

Bifurcation and Modal Interaction in a Simplified Model of Bending-Torsion Vibrations of the Thin Elastica

1995 ◽  
Vol 117 (1) ◽  
pp. 30-42 ◽  
Author(s):  
J. P. Cusumano ◽  
D.-C. Lin
Keyword(s):  
Differential Equations ◽  
Floquet Theory ◽  
Numerical Study ◽  
Normal Modes ◽  
Elastic Beam ◽  
Pitchfork Bifurcation ◽  
Statistical Technique ◽  
Simplified Model ◽  
Modal Interaction ◽  
Amplitude Parameter

This paper presents a numerical study of bifurcation and modal interaction in a system of partial differential equations first proposed as a simplified model for bending-torsion vibrations of a thin elastic beam. A system of seven ordinary differential equations obtained using the first six bending and first torsional normal modes is studied, and Floquet theory is used to locate regions in the forcing frequency, forcing amplitude parameter plane where “planar” (i.e., zero torsion) motions are unstable. Numerical branch following and symmetry considerations show that the initial instability arises from a subcritical pitchfork bifurcation. The subsequent nonplanar chaotic attractor is part of a branch of 2-frequency quasiperiodic orbits which undergoes torus-doubling bifurcations. A new statistical technique which identifies interacting modes and the average stability properties of the associated subspaces is presented. The technique employs the Lyapunov vectors used in the calculation of the Lyapunov exponents. We show how this method can be used to split the modes into active and passive sets: active modes interact to contain the attractor, whereas passive modes behave like isolated driven oscillators. In particular, large amplitude modes may simply serve as conduits through which energy is supplied to the active modes.

scholarly journals Floquet Theory for Parabolic Differential Equations

1994 ◽  
Vol 109 (1) ◽  
pp. 147-200 ◽  
Author(s):  
S.N. Chow ◽  
K.N. Lu ◽  
J. Malletparet
Keyword(s):  
Differential Equations ◽  
Floquet Theory ◽  
Parabolic Differential Equations

scholarly journals Studying in Lee groups and most important examples of it (Heisenberg group): دراسة في زمر لي وأهم الأمثلة عنها (زمرة هايزنبرغ)

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Number Theory ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Representation Theory ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
General Definition ◽  
Nilpotent Lie Groups ◽  
Basic Facts

In this research, we present some basic facts about Lie algebra and Lie groups. We shall require only elementary facts about the general definition and knowledge of a few of the more basic groups, such as Euclidean groups. Then we introduce the Heisenberg group which is the most well-known example from the realm of nilpotent Lie groups and plays an important role in several branches of mathematics, such as representation theory, partial differential equations and number theory... It also offers the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis.

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