On the relaxation time of Gauss' continued-fraction map. II. The Banach space approach (transfer operator method)

1988 ◽  
Vol 50 (1-2) ◽  
pp. 331-344 ◽  
Author(s):  
D. Mayer ◽  
G. Roepstorff
Keyword(s):  
Banach Space ◽  
Relaxation Time ◽  
Continued Fraction ◽  
Transfer Operator ◽  
Operator Method ◽  
Space Approach

THE RELAXATION TIME AND THE CORRELATION FUNCTION FOR A LOGISTIC GROWTH SYSTEM DRIVEN BY COLORED CROSS-CORRELATION NOISES

2008 ◽  
Vol 08 (02) ◽  
pp. L213-L228
Author(s):  
CAN-JUN WANG ◽  
DONG-CHENG MEI
Keyword(s):  
Correlation Function ◽  
Relaxation Time ◽  
Tumor Cell ◽  
Noise Intensity ◽  
Multiplicative Noise ◽  
Gaussian White Noise ◽  
Operator Method ◽  
Logistic Growth ◽  
Arbitrary Initial Condition ◽  
Growth System

The associated relaxation time Tc and the normalized correlation function C(s) are investigated in the logistic growth system, which is used to describe a tumor cell growth process, driven by two Gaussian white noise sources and the correlation between the additive and multiplicative noise. The expression of Tc and C(s), which is the function of noise parameters (additive noise intensity α, multiplicative noise intensity D, correlation intensity λ and correlation time τ), is obtained by using the projection operator method. After introducing noise intensity ratio, a dimensionless parameter R = α/D, and performing the numerical computations, the two case are analyzed: (1) In the growth case, λ and τ play opposite roles on the Tc and the C(s). It must emphasize that there is a minimal evolution velocity to appear and the tumor cell numbers is hard to evolve from an arbitrary initial condition to the maximum. (2) In the decay case, λ and τ play same roles on the Tc and the C(s). There is a maximal evolution velocity to appear. The noises induce different responses of tumor cells between the growth and decay case.

scholarly journals RELATIVELY COHERENT SETS AS A HIERARCHICAL PARTITION METHOD

2013 ◽  
Vol 23 (07) ◽  
pp. 1330026 ◽  
Author(s):  
TIAN MA ◽  
ERIK M. BOLLT
Keyword(s):  
Computational Complexity ◽  
Objective Function ◽  
Oil Spill ◽  
Empirical Data ◽  
Finite Time ◽  
Transfer Operator ◽  
Operator Method ◽  
Hierarchical Partition ◽  
Partition Method ◽  
Double Gyre

Finite time coherent sets [Froyland et al., 2010] have recently been defined by a measure-based objective function describing the degree that sets hold together, along with a Frobenius–Perron transfer operator method to produce optimally coherent sets. Here, we present an extension to generalize the concept to hierarchically define relatively coherent sets based on adjusting the finite time coherent sets to use relative measures restricted to sets which are developed iteratively and hierarchically in a tree of partitions. Several examples help clarify the meaning and expectation of the techniques, as they are the nonautonomous double gyre, the standard map, an idealized stratospheric flow, and empirical data from the Mexico Gulf during the 2010 oil spill. Also for the sake of analysis of computational complexity, we include an Appendix concerning the computational complexity of developing the Ulam–Galerkin matrix estimates of the Frobenius–Perron operator centrally used here.

scholarly journals Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations

2021 ◽  
pp. 1-37
Author(s):  
WAEL BAHSOUN ◽  
CARLANGELO LIVERANI
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Absolute Continuity ◽  
Hyperbolic Systems ◽  
Transfer Operator ◽  
Statistical Properties ◽  
Functions Of Bounded Variation ◽  
Expanding Maps ◽  
New Information ◽  
Piecewise Expanding Maps

Abstract Given any smooth Anosov map, we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that, in the case of expanding maps, it reduces exactly to the usual space of functions of bounded variation which has proved to be particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations, which provides new information that could prove useful in treating hyperbolic systems with singularities.

scholarly journals The Phase–Space Approach to time Evolution of Quantum States in Confined Systems: the Spectral Split–Operator Method

2019 ◽  
Vol 29 (3) ◽  
pp. 439-451
Author(s):  
Damian Kołaczek ◽  
Bartłomiej J. Spisak ◽  
Maciej Wołoszyn
Keyword(s):  
Phase Space ◽  
Time Evolution ◽  
Quantum Dynamics ◽  
Operator Method ◽  
Time Evolution Operator ◽  
Energy Profiles ◽  
Split Operator ◽  
Energy Expectation ◽  
Space Approach ◽  
Confined System

Abstract Using the phase space approach, we consider the quantum dynamics of a wave packet in an isolated confined system with three different potential energy profiles. We solve the Moyal equation of motion for the Wigner function with the highly efficient spectral split-operator method. The main aim of this study is to compare the accuracy of the employed algorithm through analysis of the total energy expectation value, in terms of deviation from its exact value. This comparison is performed for the second and fourth order factorizations of the time evolution operator.

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