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Each iteration of the Sierpinski triangle contains triangles related to the next iteration by a scale factor of 1/2. In affine geometry, uniform scaling (or isotropic scaling [1]) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions (isotropically).
The compressed exponential function (with β > 1) has less practical importance, with the notable exceptions of β = 2, which gives the normal distribution, and of compressed exponential relaxation in the dynamics of amorphous solids. [1] In mathematics, the stretched exponential is also known as the complementary cumulative Weibull distribution.
Top: The action of M, indicated by its effect on the unit disc D and the two canonical unit vectors e 1 and e 2. Left: The action of V ⁎, a rotation, on D, e 1, and e 2. Bottom: The action of Σ, a scaling by the singular values σ 1 horizontally and σ 2 vertically. Right: The action of U, another rotation.
A flow is a process in which the points of a space continuously change their locations or properties over time. More specifically, in a one-dimensional geometric flow such as the curve-shortening flow, the points undergoing the flow belong to a curve, and what changes is the shape of the curve, its embedding into the Euclidean plane determined by the locations of each of its points. [2]
Plot of the inverse Prandtl–Glauert factor / as a function of freestream Mach number. Notice the infinite limit at Mach 1. Notice the infinite limit at Mach 1. Inviscid compressible flow over slender bodies is governed by linearized compressible small-disturbance potential equation: [ 1 ]
Liouville's equation appears in the study of isothermal coordinates in differential geometry: the independent variables x,y are the coordinates, while f can be described as the conformal factor with respect to the flat metric. Occasionally it is the square f 2 that is referred to as the conformal factor, instead of f itself.
Pauli's equation is derived by requiring minimal coupling, which provides a g-factor g=2. Most elementary particles have anomalous g -factors, different from 2. In the domain of relativistic quantum field theory , one defines a non-minimal coupling, sometimes called Pauli coupling, in order to add an anomalous factor
For instance, the Pythagorean theorem has been illustrated with shear mapping [3] as well as the related geometric mean theorem. Shear matrices are often used in computer graphics. [4] [5] [6] An algorithm due to Alan W. Paeth uses a sequence of three shear mappings (horizontal, vertical, then horizontal again) to rotate a digital image by an ...