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Non-monotonicity height is an important index to describe the complexity of dynamics for piecewise monotone functions. Although it is used extensively in the theory of iterative roots, its calculation is still difficult especially in the infinite case. In this paper, by introducing the concept of spanning interval, we first present a sufficient condition for piecewise monotone functions to have height infinity and then an algorithm for finding the spanning intervals is given. We further investigate the density of all piecewise monotone functions with infinite and finite height, respectively, and the results indicate the instability of height. At the end of this paper, the variance of height under composition, especially for functions of height 1 and infinity, are also discussed.

It is known that every piecewise monotone function with height finity has a characteristic interval after finite times iteration, and then the study of dynamics for such functions is able to be restricted to their characteristic intervals, which becomes monotone case. To the opposite, the description for piecewise monotone functions with height being infinity is much more complicated since the theory of characteristic interval does not work anymore. In this paper, we consider the problem of topological conjugacy for piecewise monotone functions with height being infinity. Some necessary and sufficient conditions are given for the existence of conjugacies between these functions. Moreover, the height of infinity under composition is also discussed. The fact shows a kind of symmetry for the height.

In this paper we consider iteration of single-plateau functions, an important class of continuous functions with infinitely many forts, and investigate changes of number and length of plateaux under iteration. We use the indices flatness, plateau limit and limit length to formulate those changes. Furthermore, we compute flatness, plateau limit and limit length for all nine types of single-plateau functions.

In this paper we study a functional equation of linear combination of nonautonomous iterations, which can be reduced from invariance of an interval homeomorphism under nonautonomous iteration. We discuss existence, uniqueness and continuous dependence for increasing Lipschitzian solutions on a compact interval, and also discuss for bounded increasing Lipschitzian solutions and unbounded ones on the whole \begin{document}$ \mathbb{R} $\end{document} .

In this paper we study a general n -dimensional mixed type differential equation with state dependence. Many known works give the existence of solutions with the so-called Return Condition. In this paper, without requiring the Return Condition, we prove a non-local existence of solutions by using a technique of compactification and the dependence of the maximum of functions on the size of their domains. We apply our result to differential equations with iterates.

Concerning iterative roots, a main difficulty comes from forts of continuous functions. A case of infinitely many forts is the class of functions with plateaus. In this paper we consider functions in this class and their iterative roots. We discuss on changes of plateaus under iteration and give a criterion to determine the clenched class. Then we find all iterative roots for clenched single-plateau functions.

A crisis in the smoothness of iterative roots appears at fixed points. The local C r smoothness of iterative roots near a hyperbolic fixed point is well known, but its global C r extension to both fixed points is generically impossible because of their violent oscillation near the next hyperbolic fixed point. In this study, we consider how violently these smooth iterative roots oscillate. We prove that their first order derivatives are bounded but their r-th ( r ≥ 2 ) order derivatives are generically unbounded. In addition, we prove that the global C 1 smoothness is of vital significance for the global higher order derivatives. As a corollary, we show that iterative roots on the maximal interval oscillate with C 1 boundedness but C r ( r ≥ 2 ) unboundedness.

Stability of iterative roots is important in the numerical computation of iterative roots. Known results show that under some conditions iterative roots of strictly monotonic self-mappings are C 0 stable in both the local sense and the global sense. In this paper we discuss the C 1 stability for iterative roots of strictly increasing self-mappings on a compact interval between two fixed points. We prove that those iterative roots are locally C 1 stable but globally C 1 unstable.

It is an open problem proposed by N. Brillou¨et-Belluot to solve the equation f2(x) = f(x+a)..x. Although some related results have been obtained, the problem has remained open. In this paper we prove that it has no continuous real solutions, finally answering Brillou¨et-Belluot�s problem. Furthermore, we give existence of continuous real solutions for the general equation f2(x) = f(x + a) + x on the whole R in some cases which neither include the equation f2(x) = f(x + a) .. x nor are considered in [J. Difference Equ. Appl. 16(11) (2010), 1237�1255].

We present a Condensation Nucleus Clustering (CNC) method based on our study of SVC algorithm. In CNC, data points are mapped by means of a Gaussian kernel to a high dimensional feature space, where we search for the minimal enclosing sphere. We consider the inner data images as the rough estimate of each cluster's core, and they can be easily clustered. We name the groups of cluster result as Condensation Nucleus. Then assign the remaining data points into each cluster by linear discriminant analysis. Furthermore, we improve the CNC method to GCNC (Gradational Condensation Nucleus Clustering). In GCNC, the remaining data are assigned to each cluster gradationally. With the Condensation Nucleus bigger and the remaining data less, the Condensation Nucleus grow up to the final cluster results. We compare our methods with other similar clustering algorithm to demonstrate the performance of the proposed method on several datasets.

Global $C^{0}$ and local $C^{1}$ stability of iterative roots for monotonic functions defined on a compact interval, as well as global $C^{1}$ instability under some assumptions, are well-known facts. In this paper, we investigate the stability of iterative roots for piecewise monotonic functions with nonmonotonicity height equal to 1. We prove the roots are $C^{1}$ locally stable and $C^{0}$ global stable with the same extension.