Iterative Roots Research Articles

On Iteration of Bijective Functions with Discontinuities

Abstract We present three different types of bijective functions f : I → I on a compact interval I with finitely many discontinuities where certain iterates of these functions will be continuous. All these examples are strongly related to permutations, in particular to derangements in the first case, and permutations with a certain number of successions (or small ascents) in the second case. All functions of type III form a direct product of a symmetric group with a wreath product. It will be shown that any iterative root F : J → J of the identity of order k on a compact interval J with finitely many discontinuities is conjugate to a function f of type III, i.e., F = φ− 1 ∘ f ∘ φ where φ is a continuous, bijective, and increasing mapping between J and [0, n] for some integer n.

A Newton method for harmonic mappings in the plane

Abstract We present an iterative root finding method for harmonic mappings in the complex plane, which is a generalization of Newton’s method for analytic functions. The complex formulation of the method allows an analysis in a complex variables spirit. For zeros close to poles of $f = h + \overline{g}$ we construct initial points for which the harmonic Newton iteration is guaranteed to converge. Moreover, we study the number of solutions of $f(z) = \eta $ close to the critical set of $f$ for certain $\eta \in \mathbb{C}$. We provide a MATLAB implementation of the method, and illustrate our results with several examples and numerical experiments, including phase plots and plots of the basins of attraction.

Subcommuting and comparable iterative roots of order preserving homeomorphisms

It is known that the iterative roots of continuous functions are not necessarily unique, if it exist. In this note, by introducing the set of points of coincidence, we study the iterative roots of order preserving homeomorphisms. In particular, we prove a characterization of identical iterative roots of an order preserving homeomorphism using the points of coincidence of functions.

Embeddability of pairs of weighted quasi-arithmetic means into a semiflow

We determine the form of all semiflows of pairs of weighted quasi-arithmetic means, those over positive dyadic numbers as well as the continuous ones. Then the iterability of such pairs is characterized: necessary and sufficient conditions for a given pair of weighted quasi-arithmetic means to be embeddable into a continuous semiflow are given. In particular, it turns out that surprisingly the existence of a square iterative root in the class of such pairs implies the embeddability.

Decreasing case on characteristic endpoints question for iterative roots of PM functions

ABSTRACTFor PM functions of height 1, the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition. A natural open question about iterative roots without that condition was raised. This question was answered partially in the case that the function is increasing on its characteristic interval. In this paper, to the opposite, we consider the decreasing case and give the existence and nonexistence results for their iterative roots.

Coherency structure exploitation in DOA estimation for non‐circular sources

This study addresses the problem of direction-of-arrival (DOA) estimation for non-circular (NC) sources in the presence of multipath propagation. Recent research reveals that the use of the association of incident signals, i.e. source association (SA), can provide a significant gain in reducing DOA estimation error, especially in the presence of multiple sources. Here, in order to make full use of both the properties of NC sources and the SA information, the multipath model with NC sources is studied, and then a simple combination algorithm is proposed to estimate the SA, based on which, the authors derive an iterative root estimator to refine a given coarse DOA estimation. Meanwhile, the stochastic Cramer-Rao bound taking these two factors into account is derived as well. The proposed method is search-free and maintains the array aperture which is up to twice of the physical array aperture. Numerical simulations have been provided to verify the effectiveness of the proposed method.

A characterization of additive iterative roots of identity with respect to invariant subspaces

Let mathbb {K} be an extension of the field mathbb {Q} of rational numbers. We prove an algebraic characterization of the additive iterative roots of identity defined on a vector space X over the field mathbb {K} which save a Hamel basis of X (a basis of X over mathbb {Q}). The method is based on a canonical decomposition theorem.

Phantom Ray-Hair Intersector

We present a new approach to ray tracing swept volumes along trajectories defined by cubic Bézier curves. It performs at two-thirds of the speed of ray-triangle intersection, allowing essentially even treatment of such primitives in ray tracing applications that require hair, fur, or yarn rendering. At each iteration, we approximate a radially symmetric swept volume with a tangential cone. A distance from the ray-cone intersection to the cone's base is then used to compute the next curve parameter t. When this distance is zero, the ray intersects the swept volume and the cone at the same point and we stop the iterations. To enforce continuity of the iterative root finding, we introduce "phantom" intersection, padding the cone until it touches the ray if the ray-cone intersection does not exist.

On iterative roots of order $n$ of some multifunctions with a unique set-value point

On iterative roots of order $n$ of some multifunctions with a unique set-value point

Iterative roots of some functions

The iterative equation $f^{q}(x)=g(x)$, $x\in X$ for a given function $g$ and a positive integer $q$ is solved in the following two main cases: (i) $X=\mathbb{Z}$, $g(x)=ax+b$, ($a,b\in\mathbb{Z}$; $a\neq0,1$); (ii) $X=\mathbb{N}\cup\left\{ 0\right\} $, $g$ is increasing with no fixed point.

Conjugation of Rational Functions to Power Functions and Applications to Iteration

In this paper we investigate normalization of rational functions, reducing in the sense of conjugation to monomials or more general power functions. We give conditions for the normalization by computing minimal irreducible decomposition of algebraic varieties. We use those conditions to compute the general n-th order iterates and iterative roots for those rational functions.

Open question on lower order iterative roots for PM functions

It is known that any PM function of nonmonotonicity height has no continuous iterative roots of an order n being greater than the number of forts of the function, from which there was known the open question on lower order roots: Does such a function have an iterative root of lower order? This question was answered partially for order being equal to the number of forts. In this paper we discuss the question for lower orders.

Iterative Roots of Non-PM Functions and Denseness

The characteristic interval plays a vital role on the existence of iterative roots of PM functions with height less than or equal to one. In this paper, we define the characteristic interval for continuous functions and prove theorems on extension and nonexistence of iterative roots for a class of continuous non-PM functions on a closed and bounded interval I. Also, we prove that a class of continuous non-PM functions, which do not possess any iterative roots, is dense in C(I, I).

Iterative roots of upper semicontinuous multifunctions

The square iterative roots for strictly monotonic and upper semicontinuous functions with one set-valued point were fully described in (Li et al. in Publ. Math. (Debr.) 75:203-220, 2009). As a continuation, we study both strictly monotonic and nonmonotonic multifunctions. We present sufficient and necessary conditions under which those multifunctions have nth iterative roots. This equivalent condition and the construction method of nth iterative roots extend the previous results.

The question on characteristic endpoints for iterative roots of PM functions

Iterative roots of mappings are of special interests because it defines fractional iteration, displays middle procedure of evolution and proposes a weak version of the embedding flow problem. For PM functions of height 1, the class of 1-dimensional mappings having the simplest nonmonotonicity, the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition and the condition was proved to be necessary for those orders greater than the number of forts plus 1. This suggests an open question about iterative roots without that condition, called the question on characteristic endpoints. In this paper, the question is answered completely in the case that the number of forts is equal to the order. Although results of nonexistence are also obtained for the case that the number of forts is greater than the order, a full description is still open.

On iterative convexity of diffeomorphism

A function f is said to be iteratively convex if f possesses convex iterative roots of all orders. We give several constructions of iteratively convex diffeomorphisms and explain the phenomenon that the non-existence of convex iterative roots is a typical property of convex functions. We show also that a slight perturbation of iteratively convex functions causes loss of iterative convexity. However, every convex function can be approximate by iteratively convex functions.

Iterative roots of continuous functions with non-isolated forts

Iterative root problem is one of the classical problem in analysis and is described as follows: given a topological space X, a continuous self map F on X and a fixed positive integer n, to find another self map f on X such that $$f^n=F$$. This problem is solved only for a particular class of continuous functions and is still unsolved for any continuous function. In this paper, we present results on non-existence of iterative roots of continuous functions having finitely many non-isolated forts.

A square iterative roots of some mean-type mappings

We determine all pairs of weighted quasi-arithmetic means being square iterative roots of another pair , that is we find all continuous strictly monotonic functions and parameters such that the equationis satisfied.

On convex iterative roots of non-monotonic mappings

Let I be an interval. We consider the non-monotonic convex self-mappings f:Irightarrow I such that f^2 is convex. They have the property that all iterates f^n are convex. In the class of these mappings we study three families of functions possessing convex iterative roots. A function f is said to be iteratively convex if f possesses convex iterative roots of all orders. A mapping f is said to be dyadically convex if for every nge 2 there exists a convex iterative root f^{1/2^n} of order 2^n and the sequence {f^{1/2^n}} satisfies the condition of compatibility, that is f^{1/2^n}circ f^{1/2^n}= f^{1/2^{n-1}}. A function f is said to be flowly convex if it possesses a convex semi-flow of f, that is a family of convex functions {f^t,t>0} such that f^tcirc f^s=f^{t+s}, t,s >0 and f^1=f. We show the relations among these three types of convexity and we determine all convex iterative roots of non-monotonic functions.

A topological classification for piecewise monotone iterative roots

It is proved that every continuous iterative root of a piecewise monotone function with finite nonmonotonicity height, is an extension from a root on the characteristic interval. In this paper, a topological classification for those iterative roots is investigated. We first characterize all conjugacies between those piecewise monotone functions with finite nonmonotonicity height, and then determine a topological relation for their continuous iterative roots.