Multifractality of the Feigenbaum attractor and fractional derivatives

It is shown that fractional derivatives of the (integrated) invariant measure of the Feigenbaum map at the onset of chaos have power-law tails in their cumulative distributions, whose exponents can be related to the spectrum of singularities $f(\alpha)$. This is a new way of characterizing multifractality in dynamical systems, so far applied only to multifractal random functions (Frisch and Matsumoto (J. Stat. Phys. 108:1181, 2002)). The relation between the thermodynamic approach (Vul, Sinai and Khanin (Russian Math. Surveys 39:1, 1984)) and that based on singularities of the invariant measures is also examined. The theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations.Comment: 20 pages, 5 figures, J.Stat.Phys. in pres

Log-periodic self-similarity: an emerging financial law?

A hypothesis that the financial log-periodicity, cascading self-similarity through various time scales, carries signatures of a law is pursued. It is shown that the most significant historical financial events can be classified amazingly well using a single and unique value of the preferred scaling factor lambda=2, which indicates that its real value should be close to this number. This applies even to a declining decelerating log-periodic phase. Crucial in this connection is identification of a "super-bubble" (bubble on bubble) phenomenon. Identifying a potential "universal" preferred scaling factor, as undertaken here, may significantly improve the predictive power of the corresponding methodology. Several more specific related results include evidence that: (i) the real end of the high technology bubble on the stock market started (with a decelerating log-periodic draw down) in the begining of September 2000; (ii) a parallel 2000-2002 decline seen in the Standard & Poor's 500 from the log-periodic perspective is already of the same significance as the one of the early 1930s and of the late 1970s; (iii) all this points to a much more serious global crash in around 2025, of course from a level much higher (at least one order of magnitude) than in 2000.Comment: Talk given by S. Drozdz at International Econophysics Conference, Bali, August 28-31, 2002; typos correcte

Singular measures in circle dynamics

Critical circle homeomorphisms have an invariant measure totally singular with respect to the Lebesgue measure. We prove that singularities of the invariant measure are of Holder type. The Hausdorff dimension of the invariant measure is less than 1 but greater than 0

Quantifying dynamics of the financial correlations

A novel application of the correlation matrix formalism to study dynamics of the financial evolution is presented. This formalism allows to quantify the memory effects as well as some potential repeatable intradaily structures in the financial time-series. The present study is based on the high-frequency Deutsche Aktienindex (DAX) data over the time-period between November 1997 and December 1999 and demonstrates a power of the method. In this way two significant new aspects of the DAX evolution are identified: (i) the memory effects turn out to be sizably shorter than what the standard autocorrelation function analysis seems to indicate and (ii) there exist short term repeatable structures in fluctuations that are governed by a distinct dynamics. The former of these results may provide an argument in favour of the market efficiency while the later one may indicate origin of the difficulty in reaching a Gaussian limit, expected from the central limit theorem, in the distribution of returns on longer time-horizons.Comment: 10 pages, 7 PostScript figures, talk presented by the first Author at the NATO ARW on Econophysics, Prague, February 8-10, 2001; to be published in proceedings (Physica A

Dynamical chaos in the problem of magnetic jet collimation

We investigate dynamics of a jet collimated by magneto-torsional oscillations. The problem is reduced to an ordinary differential equation containing a singularity and depending on a parameter. We find a parameter range for which this system has stable periodic solutions and study bifurcations of these solutions. We use Poincar\'e sections to demonstrate existence of domains of regular and chaotic motions. We investigate transition from periodic to chaotic solutions through a sequence of period doublings.Comment: 11 pages, 29 figures, 1 table, MNRAS (published online

Convergence of the critical attractor of dissipative maps: Log-periodic oscillations, fractality and nonextensivity

For a family of logistic-like maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space. We found that the phase space volume occupied by the ensemble W(t) depicts a power-law decay with log-periodic oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractor's fractal dimension governed by the inflexion of the map near its extremal point. Further, we investigate the temporal evolution of W(t) for the circle map whose critical attractor is dense. In this case, we found W(t) to exhibit a rich pattern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of nonextensive Tsallis entropies.Comment: 8 pages and 8 fig

Attention mechanisms in the CHREST cognitive architecture

In this paper, we describe the attention mechanisms in CHREST, a computational architecture of human visual expertise. CHREST organises information acquired by direct experience from the world in the form of chunks. These chunks are searched for, and verified, by a unique set of heuristics, comprising the attention mechanism. We explain how the attention mechanism combines bottom-up and top-down heuristics from internal and external sources of information. We describe some experimental evidence demonstrating the correspondence of CHREST’s perceptual mechanisms with those of human subjects. Finally, we discuss how visual attention can play an important role in actions carried out by human experts in domains such as chess

Reverse doming of the anterior mitral leaflet with severe aortic regurgitation

The normal anatomic relation of the anterior mitral leaflet to the left ventricular outflow tract suggests that significant aortic regurgitation should have a predictable hemodynamic effect on the motion and configuration of the leaflet, an effect that should be seen by two-dimensional echocardiography. Previous reports have identified an abnormality of mitral opening in the short-axis view that was quite specific but not sensitive. This study was undertaken to evaluate mitral valve motion and configuration in aortic insufficiency using two-dimensional echocardiography. A characteristic pattern of anterior leaflet motion was found in patients with moderately severe and severe aortic regurgitation. This pattern, termed “reverse doming,” was seen in the apical and long-axis views in 19 of 22 such patients. The previously described “diastolic indentation” in the short-axis view was found in 16 of these 22 patients. Only 2 of 16 patients with lesser degrees of insufficiency had reverse doming. The sign was not seen in normal subjects nor in 16 patients with cardiomyopathy. For each of the few false positive and false negative findings, there is a seemingly logical hemodynamic explanation.It is concluded that reverse doming of the anterior mitral leaflet appears to be a sensitive and specific sign for moderately severe and severe aortic regurgitation

Restricting Zap70 Expression to CD4+CD8+ Thymocytes Reveals a T Cell Receptor–dependent Proofreading Mechanism Controlling the Completion of Positive Selection

Although T cell receptor (TCR) signals are essential for intrathymic T cell–positive selection, it remains controversial whether they only serve to initiate this process, or whether they are required throughout to promote thymocyte differentiation and survival. To address this issue, we have devised a novel approach to interfere with thymocyte TCR signaling in a developmental stage-specific manner in vivo. We have reconstituted mice deficient for Zap70, a tyrosine kinase required for TCR signaling and normally expressed throughout T cell development, with a Zap70 transgene driven by the adenosine deaminase (ADA) gene enhancer, which is active in CD4+CD8+ thymocytes but inactive in CD4+ or CD8+ single-positive (SP) thymocytes. In such mice, termination of Zap70 expression impaired TCR signal transduction and arrested thymocyte development after the initiation, but before the completion, of positive selection. Arrested thymocytes had terminated Rag gene expression and up-regulated TCR and Bcl-2 expression, but failed to differentiate into mature CD4 or CD8 SP thymocytes, to be rescued from death by neglect or to sustain interleukin 7Rα expression. These observations identify a TCR-dependent proofreading mechanism that verifies thymocyte TCR specificity and differentiation choices before the completion of positive selection

Log-periodic corrections to scaling: exact results for aperiodic Ising quantum chains

Log-periodic amplitudes of the surface magnetization are calculated analytically for two Ising quantum chains with aperiodic modulations of the couplings. The oscillating behaviour is linked to the discrete scale invariance of the perturbations. For the Fredholm sequence, the aperiodic modulation is marginal and the amplitudes are obtained as functions of the deviation from the critical point. For the other sequence, the perturbation is relevant and the critical surface magnetization is studied.Comment: 12 pages, TeX file, epsf, iopppt.tex, xref.tex which are joined. 4 postcript figure