An autocorrelation and a discrete spectrum for dynamical systems on metric spaces

We study dynamical systems $(X,G,m)$ with a compact metric space $X$ , a locally compact, $\unicode[STIX]{x1D70E}$ -compact, abelian group $G$ and an invariant Borel probability measure $m$ on $X$ . We show that such a system has a discrete spectrum if and only if a certain space average over the metric is a Bohr almost periodic function. In this way, this average over the metric plays, for general dynamical systems, a similar role to that of the autocorrelation measure in the study of aperiodic order for special dynamical systems based on point sets.

Distribution of Small Values of Bohr Almost Periodic Functions with Bounded Spectrum

For f a nonzero Bohr almost periodic function on R with a bounded spectrum we proved there exist Cf > 0 and integer n > 0 such that for every u > 0 the mean measure of the set f x : jf(x)j < u g is less than Cf u1=n: For trigonometric polynomials with n + 1 frequencies we showed that Cf can be chosen to depend only on n and the modulus of the largest coefficient of f: We showed this bound implies that the Mahler measure M(h); of the lift h of f to a compactification G of R; is positive and discussed the relationship of Mahler measure to the Riemann Hypothesis

δ-Almost Periodic Functions and Applications to Dynamic Equations

In this paper, by employing matched spaces for time scales, we introduce a δ -almost periodic function and obtain some related properties. Also the hull equation for homogeneous dynamic equation is introduced and results of the existence are presented. In the sense of admitting exponential dichotomy for the homogeneous equation, the expression of a δ -almost periodic solution for a type of nonhomogeneous dynamic equation is obtained and the existence of δ -almost periodic solutions for new delay dynamic equations is considered. The results in this paper are valid for delay q-difference equations and delay dynamic equations whose delays may be completely separated from the time scale T .

Global attractivity of asymptotically almost periodic Nicholson’s blowflies models with a nonlinear density-dependent mortality term

This paper focuses on the study of a class of asymptotically almost periodic Nicholson’s blowflies models with a nonlinear density-dependent mortality term. By virtue of differential inequality techniques, a set of easily verifiable sufficient conditions are established to show that every solution of the considered model is asymptotically almost periodic, and it also converges to a same almost periodic function as [Formula: see text], which improves and supplements some previously known researches. Moreover, a numerical example is given to test the feasibility and effectiveness of the obtained results.

Diophantine approximations and almost periodic functions

In this paper we investigate the asymptotic behaviour of the classical continuous and unbounded almost periodic function in the Lebesgue measure.Using diophantine approximations we show that this function can be estimated by functions of polynomial type and we give the best polynomial estimation.

THE CONNECTION BETWEEN PSEUDO ALMOST PERIODIC FUNCTIONS DEFINED ON TIME SCALES AND ON THE REAL LINE

A necessary and sufficient condition for a continuous function $g$ to be almost periodic on time scales is the existence of an almost periodic function $f$ on $\mathbb{R}$ such that $f$ is an extension of $g$. Our aim is to study this question for pseudo almost periodic functions. We prove the necessity of the condition for pseudo almost periodic functions. An example is given to show that the sufficiency of the condition does not hold for pseudo almost periodic functions. Nevertheless, the sufficiency is valid for uniformly continuous pseudo almost periodic functions. As applications, we give some results on the connection between the pseudo almost periodic (or almost periodic) solutions of dynamic equations on time scales and of the corresponding differential equations.

A Study of Predicting Ability in State-Action Pair Prediction

When a robot considers an action-decision based on a future prediction, it is necessary to know the property of disturbance signals from the outside environment. On the other hand, the properties of disturbance signals cannot be described simply, such as non-periodic function, nonlinear time-varying function nor almost-periodic function. In case of a robot control, sampling rate for control will be affected description of disturbance signals such as frequency or amplitude. If the sampling rate for acquiring a disturbance signal is not correct, the action will be taken far from its actual property. In general, future prediction using machine learning is based on the tendency obtained through past training or learning. In this case, an optimal action will be determined uniquely based on a property of disturbance. However, in this type of situation, the learning time increases in proportional to the amount of training data, either, the tendency may not be found using prediction, in the worst case. In this paper, we focus on prediction for almost-periodic disturbance. In particular, we consider the situation where almost-periodic disturbance signals occur. From this perspective, we propose a method that identifies the frequency of an almost- periodic function based on the frequency of the disturbance using Fourier transform, nearest-neighbor one-step-ahead forecasts and Nyquist-Shannon sampling theorem.

Nonparametric Band Business Cycle Clock

We discuss representation of uncertainty in the business cycle clock. We propose approach utilising description of the unconditional mean of the process, applied for modelling dynamics of macroeco-nomic time series, as a trend component and almost period function in a non-parametric setting. We capture the dynamics over the business cycle, trend component and seasonal fluctuations and possible interactions between these features. A particular values of the almost periodic function are key for representation of the business cycle in a clock, expressing the dynamics according to phase diagram. The set of frequencies interpreted as a properties of the business fluctuations are invariant with respect to filtration methods applied in the procedure.

On Hilbert dynamical systems

Returning to a classical question in harmonic analysis, we strengthen an old result of Walter Rudin. We show that there exists a weakly almost periodic function on the group of integers ℤ which is not in the norm-closure of the algebra B(ℤ) of Fourier–Stieltjes transforms of measures on the dual group $\hat {\mathbb {Z}}=\mathbb {T}$, and which is recurrent. We also show that there is a Polish monothetic group which is reflexively but not Hilbert representable.