What are self affine fractals?
In mathematics, self-affinity is a feature of a fractal whose pieces are scaled by different amounts in the x- and y-directions. This means that to appreciate the self similarity of these fractal objects, they have to be rescaled using an anisotropic affine transformation.
What is self affine surface?
The surface is statistically self-affine if the height distribution at magnification ζ corresponds to the one at ζ = 1 but with all heights rescaled by ζ−H, where H is the Hurst exponent (1).
Is the Mandelbrot self-similar?
The Mandelbrot set is highly complex. It is self-similar – that is, the set contains mini-Mandelbrot sets, each with the same shape as the whole. of the Mandelbrot set is more complicated than the whole,’ says Shishikura.
What is difference between self-similar and strictly self-similar?
If parts of a figure contain small replicas of the whole, then the figure is called self-similar. If the figure can be decomposed into parts which are exact replicas of the whole, then the figure is called strictly self-similar.
What is an example of self-similarity?
The property of having a substructure analogous or identical to an overall structure. For example, a part of a line segment is itself a line segment, and thus a line segment exhibits self-similarity. By contrast, no part of a circle is a circle, and thus a circle does not exhibit self-similarity.
What is self-similar flow?
From Wikipedia, the free encyclopedia. In the study of partial differential equations, particularly in fluid dynamics, a self-similar solution is a form of solution which is similar to itself if the independent and dependent variables are appropriately scaled.
Is any figure that is self-similar a fractal?
Simply put, a fractal is a geometric object that is similar to itself on all scales. If you zoom in on a fractal object it will look similar or exactly like the original shape. This property is called self-similarity. On all scales the Sierpenski triangle is an exactly self-similar object.
What self similarity means?
Self similarity can be defined as the phenomenon where a certain property of an object (e.g. a natural image or a mathematical time series) is preserved with respect to scaling in space and/or time [19].
What is critical Reynold number?
A critical Reynolds number is determined as a limit where the laminar flow changes to turbulent flow. If the calculated NRe is greater than the critical Reynolds number NRec, the flow regime is turbulent; otherwise the flow regime is laminar.
What are PI groups?
The Pi groups are formulated by multiplying each of the remaining variables (those that were not chosen as repeating variables) in turn by the repeating variables, each in turn raised to some unknown exponent. Also, the Pi group can be multiplied by any dimensionless constant without altering its dimensions.
Is Reynolds number dimensionless?
The Reynolds number is a dimensionless number. High values of the parameter (on the order of 10 million) indicate that viscous forces are small and the flow is essentially inviscid. The Euler equations can then be used to model the flow.
Which is the best definition of the word affine?
affine (comparative more affine, superlative most affine) (mathematics) Assigning finite values to finite quantities.
How is the Hausdorff dimension of a self-affine set built?
Unlike self-similar sets, the Hausdorff dimension of self-affine sets depends on the position of the iterated elements and there is no formula, so far, for the general case. Built by exchanging iteratively each pentagon by a flake of 6 pentagons. ( golden ratio ).
When is an affine transformation a semiaffine transformation?
If the dimension of the affine space (X, V, k) is at least two, then an affine transformation is a semiaffine transformation f that satisfies the condition: If x ≠ y and p ≠ q are points of X such that the line segments xy and pq are parallel, then ‖ p q ¯ ‖ ‖ x y ¯ ‖ = ‖ f ( p ) f ( q ) ¯ ‖ ‖ f ( x ) f ( y ) ¯ ‖ .
How is affine arithmetic used in everyday life?
Affine arithmetic is potentially useful in every numeric problem where one needs guaranteed enclosures to smooth functions, such as solving systems of non-linear equations, analyzing dynamical systems, integrating functions differential equations, etc. Applications include ray tracing, plotting curves,…