We consider the fractional generalizations of Liouville equation. The normalization condition, phase volume, and average values are generalized for fractional case. The interpretation of fractional analog of phase space as a space with fractal dimension and as a space with fractional measure are discussed. The fractional analogs of the Hamiltonian systems are considered as a special class of non-Hamiltonian systems. The fractional generalization of the reduced distribution functions are suggested. The fractional analogs of the BBGKI equations are derived from the fractional Liouville equation. 1

Wavelet Packet Analysis was used to measure the global scaling behaviour of homogeneous fractal signals from the slope of decay for discrete wavelet coefficients belonging to the adapted wavelet best basis. A new scaling function for the size distribution correlation between wavelet coefficient energy magnitude and position in a sorted vector listing is described in terms of a power-law to estimate the Hurst exponent. Profile irregularity and long-range correlations in self-affine systems can be identified and indexed with the Hurst exponent, and synthetic one-dimensional fractional Brownian motion (fBm) type profiles are used to illustrate and test the proposed wavelet packet expansion. We also demonstrate an initial application to a biological problem concerning the spatial distribution of local enzyme concentration in fungal colonies which can be modelled as a self-affine trace or an 'Enzyme Walk'. The robustness of the wavelet approach applied to this stochastic system is presented...

The Lorenz manifold is an intriguing two-dimensional surface that illustrates chaotic dynamics in the well-known Lorenz system. While it is not possible to find the Lorenz manifold as an explicit analytic solution, we have developed a method for calculating a numerical approximation that builds the surface up as successive geodesic level sets. The resulting mesh approximation can be read as crochet instructions, which means that we are able to generate a three-dimensional model of the Lorenz manifold. We mount the crocheted Lorenz manifold using a stiff rod as the z-axis, and bendable wires at the outer rim and the two solutions that are perpendicular to the z-axis. The crocheted model inspired us to consider the geometrical properties of the Lorenz manifold. Specifically, we introduce a simple method to determine and visualize local curvature of a smooth surface. The colour coding according to curvature reveals a striking pattern of regions of positive and negative curvature on the Lorenz manifold. 1

Abstract. Crumpled surfaces (CS) obtained from random and irreversible compactification of aluminium foils are low-density fractal structures which become increasingly easy to deform as their size increases. In this work we study the deformation of these objects when submitted to mechanical forces. In particular, we describe the behaviour of eight scaling relations which connect quantities as stress, strain, surface roughness and geometrical variables for these CS. The critical exponents obtained from the scaling relations and some clues concerning the existence of universal behaviour in these processes are reported. 1.

The RR series extracted from human electrocardiogram signal (ECG) is considered as a fractal stochastic process. The manifestation of long-range dependencies is the presence of power laws in scale dependent process characteristics. Exponents of these laws: β- describing power spectrum decay, α- responsible for decay of detrended fluctuations or H related to, so-called, roughness of a signal, are known to differentiate hearts of healthy people from hearts with congestive heart failure. There is a strong expectation that resolution spectrum of exponents, so-called, local exponents in place of global exponents allows to study differences between hearts in details. The arguments are given that local exponents obtained in multifractal analysis by the two methods: wavelet transform modulus maxima (WTMM) and multifractal detrended fluctuation analysis (MDFA), allow to recognize the following four stages of the heart: healthy and young, healthy and advance in years, subjects with left ventricle systolic dysfunction (NYHA I–III class) and characterized by severe congestive heart failure (NYHA III-IV class). 1 Figure 1: Electrocardiogram analysis: (a)a single P-QRS-T cycle (b) identification of RR-intervals (c)a heartbeat time series from a healthy individual in a daily activity: a normal RR signal. 1

A generalized framework is developed which uses a set description instead of wavefunction to emphasize the role of the observer. Such a framework is found to be very effective in the study of the measurement problem and time’s arrow. Measurement in classical and quantum theory is given a unified treatment. With the introduction of the concept of uncertainty quantum which is the basic unit of measurement, we show that the time’s arrow within the uncertainty quantum is just opposite to the time’s arrow in the observable reality. A special constant is discussed which explains our sensation of time and provides a permanent substrate for all change. It is shown that the whole spacetime connects together in a delicate structure. 1 1

We propose a straightforward extension of our previously proposed log-periodic power law model of the “anti-bubble ” regime of the USA market since the summer of 2000, in terms of the renormalization group framework to model critical points. Using a previous work by Gluzman and Sornette (2002) on the classification of the class of Weierstrass-like functions, we show that the five crashes that occurred since August 2000 can be accurately modelled by this approach, in a fully consistent way with no additional parameters. Our theory suggests an overall consistent organization of the investors forming a collective network which interact to form the pessimistic bearish “anti-bubble ” regime with intermittent acceleration of the positive feedbacks of pessimistic sentiment leading to these crashes. We develop retrospective predictions, that confirm the existence of significant arbitrage opportunities for a trader using our model. Finally, we offer a prediction for the unknown future of the US S&P500 index extending over 2003 and 2004, that refines the previous prediction of Sornette and Zhou (2002).

Renormalization group analysis of the 2000-2002 anti-bubble in the US S&P 500 index: Explanation of the hierarchy of 5 crashes and prediction