Fractals

We study the tick dynamical behavior of the bond futures in Korean Futures Exchange (KOFEX) market. Since the survival probability in the continuous-time random walk theory is applied to the bond futures transaction, the form of the decay function in our bond futures model is discussed from two kinds of Korean Treasury Bond (KTB) transacted recently in KOFEX. The decay distributions for survival probability are particularly displayed stretched exponential forms with novel scaling exponents β = 0.82 (KTB203) and 0.90 (KTB112), respectively, for our small time intervals. We obtain the scaling exponents for survival probability ∊ = 17 and 18 decayed rapidly in large time limit, and our results are compared with recent numerical calculations.

Interest in climate change has increased over the last 30 years due largely to global predictions associated with the greenhouse effect, which appear to lead to a substantial increase in planetary temperature. Implications of such results have led many scientists to examine climatic records from different regions of the world in order to understand temperature behavior. However, many researchers have noted that changes in temperature variability are also important in determining the future temperature distributions. In this context, we have analyzed a long-term record of monthly extreme minimum temperature registered at Guanajuato, Mexico. Data set was treated as a fractal profile to estimate the fractal dimension through variography (D_v) and power-spectral (Ds) approaches under two situations: (1) complete series, from January 1895 to December 1997 with 312 missing observations, and (2) partial series, from January, 1921 to April, 1963 with no missing values. In both cases, we obtained similar values for the two types of fractal dimensions meaning there is not a significant effect of missing values. The estimated fractal dimensions for the partial series (508 observations) are near 1.5 (D_v = 1.445 ± 0.06, D_s = 1.486 ± 0.155), which means monthly extreme minimum temperature is almost equally characterized by both short- and long-range variations. Evaluating through scaling arguments did not evidence multifractality in the scale range of two to 254 months. Then interpolation can make use of the fact that monthly extreme minimum temperature has a power-law spectrum. Interpolated data generated by this way may develop greater confidence in their capability to forecast near future climate.

We present a fractal dust model of the Universe based on Mandelbrot's proposal to replace the standard Cosmological Principle by his Conditional Cosmological Principle within the framework of General Theory of Relativity. This model turns out to be free from the Hubble de-Vaucouleurs paradox and is consistent with the SNe1a observations. The expected galaxy count as a function of red-shift is obtained for this model. An interesting variation is a steady state version, which can account for an accelerating scale factor without any cosmological constant in the model.

We study the fractal dimension of the contour of the liquid-gas interface in a spray. Our images include both the linking region and the break-up region, and are obtained with a high-resolution shadowgraph technique. This means that the images can then be subjected to an intensity filtering, equivalent to a threshold analysis, that enables the establishment of the fractal range.

The complexity of the Nautilus pompilius shell is analyzed in terms of its fractal dimension and its equiangular spiral form. Our findings assert that the shell is fractal from its birth and that its growth is dictated by a self-similar criterion (we obtain the fractal dimension of the shell as a function of time). The variables that have been used for the analysis show an exponential dependence on the number of chambers/age of the cephalopod, a property inherited from its form.

While conducting a study with air in a simple closed rectangular cavity, we have discovered the formation of highly impressive, complex-structured, and symmetrical smoke density patterns. Using the box counting method the fractal characteristic of the patterns is demonstrated.

We construct indexes of the price-volume relation and calculate correlation dimensions, based on the Fractal Market Hypothesis (FMH). According to this result, we propose a new method for the detection of stock price manipulation on the secondary market in China. The result of empirical research indicates: the great change of the stock price alone is insufficient to prove that there is stock price manipulation. However, the drastic fluctuations of correlation dimensions indicate the instability of the secondary market; and especially, the sudden drop of the correlation dimensions always means that there are stock manipulations on the market.

Avalanches of various sizes occur due to instability during sand dune dynamics. A physically viable equation to model sand dune dynamics is a first order nonlinear difference equation. To visualize the size distribution of avalanches, sand dune dynamics has been numerically simulated by changing the strength of nonlinearity parameter (λ) that shows different impacts on dune dynamics. An equation to compute avalanche diameters, by considering the inter-slipface angles of a simulated sand dune under dynamics, is proposed. Using this equation, avalanche diameters have been computed from a dynamically changing simulated sand dune.

The complexity of the cardiac rhythm is demonstrated to exhibit self-affine multifractal variability. The dynamics of heartbeat interval time series was analyzed by application of the multifractal formalism based on the Cramer theory of large deviations. The continuous multifractal large deviation spectrum uncovers the nonlinear fractal properties in the dynamics of heart rate and presents a useful diagnostic framework for discrimination and classification of patients with cardiac disease, e.g. congestive heart failure. The characteristic multifractal spectral pattern in heart transplant recipients or chronic heart disease highlights the importance of neuroautonomic control mechanisms regulating the fractal dynamics of the cardiac rhythm.

The chorioallantoic membrane (CAM) of the avian chick is an often used in vivo model to investigate angiogenesis. An image processing was developed to detect the vessels and especially the capillaries of the CAM. The result is a binary image of the microvascular network. To quantify the angiogenesis, the fractal dimension was calculated by the box-, information- and correlation-dimensions. The influence of noise on the result was investigated. The stability criterion for the fractal dimension was used to evaluate the most suitable dimension. The correlation-dimension has due to its definition the best properties in regards to the box- and information-dimensions. A native CAM possesses a box-dimension of 1.8606 ± 0.0093, an information-dimension of 1.853 ± 0.018 and a correlation-dimension of 1.844 ± 0.021.