By L. T. Fan, D. Neogi, M. Yashima (auth.)
Fractals play a big position in modeling ordinary phenomena and engineering methods. And fractals have a detailed connection to the techniques of chaotic dynamics. This monograph provides definitions, thoughts, notions and methodologies of either spatial and temporal fractals. It addresses scholars and researchers in chemistry and in chemical engineering. The authors current the options and methodologies in adequate element for uninitiated readers. They contain many straightforward examples and graphical illustrations. They define a few examples in additional element: Perimeter fractal size of char debris, floor fractal measurement of charcoal; fractal research of strain fluctuation in multiphase movement structures. Readers who grasp the recommendations during this booklet, can expectantly follow them to their fields of interest.
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Additional resources for Elementary Introduction to Spatial and Temporal Fractals
7~ -1. 1 a .... 01 1. U Figure 4-3. Positive and negative tails of the distribution of changes in cotton price [43): a. L = tn Z(t+ 1day)-tn Z(t) where Z is the daily closing price at the New York Cotton Exchange, 1900-1905; a+ = Fr[ L(t,s) > u], a- = Fr[ L(t,s) < -u] b. , 1944-1958; b+ = Fr[ L(t,s) > u], b- = Fr[ L(t,s) < -u] c. L = tn Z(t+ 1month)-tn Z(t) where Z is the closing price on the 15th of each month at the New York Cotton Exchange, 1880-1940; c+ = Fr[ L(t,s) > u], c- = Fr[ L(t,s) < -u].
Figure 3-14. Same adsorbate molecules on the various adsorbents. 38 (3-20) and the number of molecules per unit mass of the adsorbent is na: (3-21) Thus, similar to Eq. 3-10, Eq. 3-20 can be rewritten as na: (3-22) Equation 3-14 is applicable only to the adsorbate molecules with their radii between bmin and b max. Similarly, the characteristic length of the adsorbent, R, expressed in Eqs. 3-20 and 3-22 also has an applicable range. , bmin, should be selected. IT b is large, the self-similarity below this size can only be speculated .
FRACTALS IN TIME This chapter contains three sections. The first outlines Mandelbrot's epoch-making work on application of the stable distribution to the modeling of change of commodity prices, the second the concept of fractloiIal Brownian motion, and the third the concept of fractal stochastic processes leading to the notion of fractal time. In the early 1960's, Mandelbrot published a paper related to the field of economics in which the change of commodity prices is discussed . He has contended that the process of price change is discontinuous and has an infinite variance where the Gaussian distribution is inapplicable.