scholarly journals Purely Iterative Algorithms for Newton’s Maps and General Convergence

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1158
Author(s):  
Sergio Amat ◽  
Rodrigo Castro ◽  
Gerardo Honorato ◽  
Á. A. Magreñán
Keyword(s):  
Fixed Points ◽  
Large Class ◽  
Parameter Space ◽  
Critical Points ◽  
Broad Class ◽  
Iterative Algorithms ◽  
Dynamical Behaviour ◽  
Root Finding ◽  
Cubic Polynomials ◽  
General Convergence

The aim of this paper is to study the local dynamical behaviour of a broad class of purely iterative algorithms for Newton’s maps. In particular, we describe the nature and stability of fixed points and provide a type of scaling theorem. Based on those results, we apply a rigidity theorem in order to study the parameter space of cubic polynomials, for a large class of new root finding algorithms. Finally, we study the relations between critical points and the parameter space.

scholarly journals ETHOS – an effective parametrization and classification for structure formation: the non-linear regime at z ≳ 5

2020 ◽  
Vol 498 (3) ◽  
pp. 3403-3419
Author(s):  
Sebastian Bohr ◽  
Jesús Zavala ◽  
Francis-Yan Cyr-Racine ◽  
Mark Vogelsberger ◽  
Torsten Bringmann ◽  
...  
Keyword(s):  
Power Spectrum ◽  
Structure Formation ◽  
Parameter Space ◽  
Broad Class ◽  
High Redshift ◽  
Acoustic Oscillations ◽  
Effective Parameters ◽  
Matter Power Spectrum ◽  
Wide Range ◽  
Non Linear

ABSTRACT We propose two effective parameters that fully characterize galactic-scale structure formation at high redshifts (z ≳ 5) for a variety of dark matter (DM) models that have a primordial cutoff in the matter power spectrum. Our description is within the recently proposed ETHOS framework and includes standard thermal warm DM (WDM) and models with dark acoustic oscillations (DAOs). To define and explore this parameter space, we use high-redshift zoom-in simulations that cover a wide range of non-linear scales from those where DM should behave as CDM (k ∼ 10 h Mpc−1), down to those characterized by the onset of galaxy formation (k ∼ 500 h Mpc−1). We show that the two physically motivated parameters hpeak and kpeak, the amplitude and scale of the first DAO peak, respectively, are sufficient to parametrize the linear matter power spectrum and classify the DM models as belonging to effective non-linear structure formation regions. These are defined by their relative departure from cold DM (kpeak → ∞) and WDM (hpeak = 0) according to the non-linear matter power spectrum and halo mass function. We identify a region where the DAOs still leave a distinct signature from WDM down to z = 5, while a large part of the DAO parameter space is shown to be degenerate with WDM. Our framework can then be used to seamlessly connect a broad class of particle DM models to their structure formation properties at high redshift without the need of additional N-body simulations.

scholarly journals Ishikawa's iterations of real Lipschitz functions

1992 ◽  
Vol 46 (1) ◽  
pp. 107-113 ◽  
Author(s):  
Lei Deng ◽  
Xie Ping Ding
Keyword(s):  
Fixed Points ◽  
Iteration Scheme ◽  
Lipschitz Functions ◽  
Convergence Theorems ◽  
General Convergence

In this paper, we consider Ishikawa's iteration scheme to compute fixed points of real Lipschitz functions. Two general convergence theorems are obtained. Our results generalise the result of Hillam.

scholarly journals Groups generated by elements with rational fixed points

1997 ◽  
Vol 40 (1) ◽  
pp. 19-30 ◽  
Author(s):  
A. W. Mason
Keyword(s):  
Normal Subgroup ◽  
Fixed Points ◽  
Large Class ◽  
Integral Domain ◽  
Structure Theorem ◽  
Quotient Field ◽  
Linear Fractional Transformations ◽  
Krull Domain ◽  
Dedekind Domains ◽  
Precise Version

Let R be a commutative integral domain and let S be its quotient field. The group GL2(R) acts on Ŝ = S ∪ {∞} as a group of linear fractional transformations in the usual way. Let F2(R, z) be the stabilizer of z ∈ Ŝ in GL2(R) and let F2(R) be the subgroup generated by all F2(R, z). Among the subgroups contained in F2(R) are U2(R), the subgroup generated by all unipotent matrices, and NE2(R), the normal subgroup generated by all elementary matrices.We prove a structure theorem for F2(R, z), when R is a Krull domain. A more precise version holds when R is a Dedekind domain. For a large class of arithmetic Dedekind domains it is known that the groups NE2(R),U2(R) and SL2(R) coincide. An example is given for which all these subgroups are distinct.

Cubic Polynomials with Rational Roots and Critical Points

2010 ◽  
Vol 41 (5) ◽  
pp. 365-369 ◽  
Author(s):  
Shiv K. Gupta ◽  
Waclaw Szymanski
Keyword(s):  
Critical Points ◽  
Cubic Polynomials

scholarly journals The iteration of cubic polynomials Part I: The global topology of parameter space

1988 ◽  
Vol 160 (0) ◽  
pp. 143-206 ◽  
Author(s):  
Bodil Branner ◽  
John H. Hubbard
Keyword(s):  
Parameter Space ◽  
Global Topology ◽  
Cubic Polynomials

ONE-DIMENSIONAL CHAOS IN IMPULSED LINEAR OSCILLATING CIRCUITS

1993 ◽  
Vol 03 (04) ◽  
pp. 921-941 ◽  
Author(s):  
LAURA GARDINI ◽  
RENZO LUPINI
Keyword(s):  
Asymptotic Behavior ◽  
Fixed Points ◽  
Critical Point ◽  
Critical Points ◽  
Phase Plane ◽  
Global Bifurcation ◽  
Homoclinic Bifurcation ◽  
Combined Effect ◽  
Chaotic Oscillations ◽  
One Dimensional

The dynamics of a damped linear oscillating circuit subject to impulses is represented by a one-dimensional endomorphism (or noninvertible map) π: ℝ → ℝ. The asymptotic behavior of orbits in the phase-plane is characterized in terms of critical points and point singularities of π (fixed points or cycles). Their combined effect, that is, the merging of a critical point into a repelling cycle, causes a global bifurcation or a homoclinic bifurcation, with transition to chaotic oscillations.

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