Self-similar Series Research Articles

Random self-similar series over a rotation

We study the law of random self-similar series defined above an irrational rotation on the Circle. This provides a natural class of continuous singular non-Rajchman measures.

Synthesis and recovery of bulk Fe4O5 from magnetite, Fe3O4. A member of a self-similar series of structures for the lower mantle and transition zone

AbstractA multi-anvil press was combined with monochromatic synchrotron X-ray radiation to investigate the synthesis, at high-pressure high-temperature conditions, of a recoverable bulk sample of Fe4O5, from an initial magnetite, Fe3O4, sample. Angle-dispersive diffraction patterns show that magnetite firstly breaks down, into an assemblage of hematite (Fe2O3) + Fe4O5. By increasing temperature at constant load, hematite disappears progressively by either reduction or by melting, or a combination of both. In the final product only Fe4O5 remains and, in the absence of hematite, can be kept stable and be recovered at ambient conditions. Refinement of the diffraction patterns at standard conditions demonstrate that Fe4O5 has the Sr2Tl2O5-type-structure with space group Cmcm and a = 2.8964(2) Å, b = 9.8225(6) Åand c = 12.5808(7) Å. This structure-type and related members of a homologous series, offer the possibility that the general sequence of AB(2+n)X(4+n) chemistries could, under certain conditions, be extended to accommodate prevalent oxygen fugacity or, indeed, other ordered stoichiometries through extension of the c axis by the addition of FeO6 octahedral blocks. This structural series, as in other systems, offers possibilities of hosting charge-transfer, Jahn-Teller and other electronic phenomena − as well as supporting metric distortions. Each of these possibilities is highlighted through illustration and extension to related structure-types, most notably from those of the spinels, post-spinels and post-perovskites.

Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups

One of the main features of analysis on post-critically finite self-similar (pcfss) sets is that it is possible to understand the behavior of the Laplacian and its inverse, the Green operator, in terms of the self-similar structure of the set. Indeed, a major step in the approach to analysis on self-similar fractals via Dirichlet forms was Kigami’s proof [8, 10] that for a self-similar Dirichlet form the Green kernel can be written explicitly as a series in which each term is a rescaling of a single expression via the self-similar structure. In [6] this result was extended to show that the resolvent kernel of the Laplacian, meaning the kernel of (z − ∆)−1, can also be written as a self-similar series for suitable values of z ∈ C. Part of the motivation for that work was that it gives a new understanding of functions of the Laplacian (such as the heat operator et∆) by writing them as integrals of the resolvent. The purpose of the present work is to establish estimates that permit the above approach to be carried out. We study the functions occurring in the series decomposition from [6] (see Theorem 3.3 below for this decomposition) and give estimates on their decay. From this we determine estimates on the resolvent kernel and on kernels of operators defined as integrals of the resolvent kernel. In particular we recover the sharp upper estimates for the heat kernel (see Theorem 10.2) that were proved for pcfss sets by Hambly and Kumagai [4] by probabilistic methods (see also [1, 13, 3] for earlier results of this type on less general classes of sets). It is important to note that the preceding authors were able to prove not just upper estimates but also lower bounds for the heat kernel, and therefore were able to prove sharpness of their bounds. Our methods permit sharp bounds for the resolvent kernel on the positive real axis, but we do not know how to obtain these globally in the complex plane or how to obtain lower estimates for the heat kernel from them. Therefore in this direction our results are not as strong as those obtained in [4]. However in other directions we obtain more information than that known from heat kernel estimates, and we hope that our approach will complement the existing probabilistic methods. In particular we are able to obtain resolvent bounds on any ray in C other than the negative real axis (where the spectrum lies), while standard calculations from heat kernel bounds only give these estimates in a half-plane. A further consequence of our approach is that we extend (in Theorem 9.7) the decomposition from [6] to the case of blowups, which are non-compact sets with local structure equivalent to that of the underlying self-similar sets. The blowup of a pcfss set bears the same relation to the original set as the real line bears to the unit interval, see [17] for details. The structure of the paper is as follows. In Section 3 we recall some basic features of analysis on pcfss sets, as well as the main result of [6], which is the decomposition of the resolvent as a weighted sum of piecewise eigenfunctions. Section 4 then discusses

Internet Traffic Prediction Using Artificial Neural Network

In this paper we employed neural network on real network traffic measurements to predict Internet traffic. The traffic data consists of hourly traces from freestats.com collected over periods of 7 and 82 days respectively. First, self-similarity tests were carried out on the data series to ascertain its self-similar nature. The data were aggregated at various frequencies to smooth out burstiness. A feed-forward multi-layer perceptions (MLP) neural network was constructed and trained to predict the traffic. The results show that, using neural network, a reliable prediction can be made on packet data despite its self-similar profile. The results also showed that a neural network trained on a self-similar series can reliably predict the aggregated forms of the series. It was concluded that the results obtained can be used to moderate congestion using limited network resources and sharing the bandwidth allocation. Keywords: Self-similarity, neural networks, Hurst parameter, Internet traffic, aggregation, long range dependence (LRD).

The design and evaluation of the Simple Self-Similar Sequences Generator

This paper describes a new algorithm for the generation of pseudo random numbers with approximate self-similar structure. The Simple Self-Similar Sequences Generator ( 4SG) elaborates on an intuitive approach to obtain a fast and accurate procedure, capable of reproducing series of points exhibiting the property of persistence and anti-persistence. 4SG has a computational complexity of O ( n ) and memory requirements of the order of log 2 ( N ) , where N is the number of points to be generated. The accuracy of the algorithm is evaluated by means of computer-based simulations, recurring to several Hurst parameter estimators, namely Variance Time (VT) and the Wavelets-based estimator. The Hosking and the Wavelets-based methods for the generation of self-similar series were submitted to the same tests the 4SG was analysed with, providing for a basis for comparison of several performance aspects of the algorithm. Results show that the proposal embodies a good candidate not only for on-demand emulation of arbitrarily long self-similar sequences, but also for fast and efficient online simulations.

Generation of Gaussian Self-similar series via Wavelets for Use in Traffic Simulations

Measurements have shown that network traffic has fractal properties such as self-similarity and long memory or long-range dependence. Long memory is characterized by the existence of a pole at the origin of the power spectrum density function (1/f shape). It was also noticed that traffic may present short-range dependence at some time scales. The use of a “realistic” aggregated network traffic generator, one that synthesizes fractal time series, is fundamental to the validation of traffic control algorithms. In this article, the synthesis of approximate realizations of a kind of self-similar random process named fractional Gaussian noise is done via wavelet transform. The proposed method is also capable of synthesizing Gaussian time series with more generic spectra than 1/f, that is, time series that also have short-range dependence. The generation is done in two stages. The first one generates an approximate realization of fractional Gaussian noise via discrete Wavelet transform. The second one introduces short-range dependence through IIR (Infinite Impulse Response) filtering at the output of the first stage. A detailed characterization of the resulting series was done, using statistical moments of first, second, third and fourth orders, as well as specific statistical tests for self-similar series. It was verified that the Whittle estimator of the Hurst parameter is more robust than the periodogram method for series that simultaneously present short-range and long-range dependence.

Fracton dimension of a mutually self-similar series of base-catalyzed aerogels

Mutual self-similarity of a series of base-catalyzed aerogels was established by comparison of small-angle neutron scattering data and numerical simulations. Investigation of phonon-fracton crossover in these samples by Brillouin scattering allowed determination of the fracton dispersion curve. A fracton dimension d ∼ = 1.1 ± 0.1 was obtained. The correlation between the sharpness of the phonon-fracton crossover and the well-defined structure of the materials is discussed.

SANS and SAXS investigations of silica aerogels: crossover from fractal structure to short-range packing

The structure of silica aerogels prepared by destabilization of colloidal sols was investigated by transmission electron microscopy (TEM), small-angle neutron (SANS) and X-ray (SAXS) scatterings. The size of the elementary particles was directly measured in real space from the micrographs. This allowed one to relate quantitatively the crossovers observed in the wavevector dependence of the SANS and SAXS intensities to the particle dimensions. By changing the silica concentration in the initial solution, mutually self-similar series of aerogels were obtained.

Fractal graphs and wavelet series

The potential usefulness of wavelets as basic building blocks for constructing fractal objects by self-similar series is illustrated by two examples.