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1.25 ≤ SI <1.50: twisty; 1.50 ≤ SI: meandering; It has been claimed that river shapes are governed by a self-organizing system that causes their average sinuosity (measured in terms of the source-to-mouth distance, not channel length) to be π, [3] but this has not been borne out by later studies, which found an average value less than 2. [4]
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). [1]
The two dimensional Manhattan distance has "circles" i.e. level sets in the form of squares, with sides of length √ 2 r, oriented at an angle of π/4 (45°) to the coordinate axes, so the planar Chebyshev distance can be viewed as equivalent by rotation and scaling to (i.e. a linear transformation of) the planar Manhattan distance.
Also, let Q = (x 1, y 1) be any point on this line and n the vector (a, b) starting at point Q. The vector n is perpendicular to the line, and the distance d from point P to the line is equal to the length of the orthogonal projection of on n. The length of this projection is given by:
where denotes the vector (x 1, x 2). In this example, the first line defines the function to be minimized (called the objective function , loss function, or cost function). The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint.
The coordinate-independent definition of the square of the line element ds in an n-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement [2] (in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length: = = (,) where g is the metric tensor ...
Using the convention that ignores taking the (n+1)st root, this turns into the maximization of the following product: (1 − e −2λ) · (e −2λ − e −4λ) · (e −4λ). Letting μ = e −2λ, the problem becomes finding the maximum of μ 5 −2μ 4 +μ 3. Differentiating, the μ has to satisfy 5μ 4 −8μ 3 +3μ 2 = 0. This equation ...
A maximum length sequence (MLS) is a type of pseudorandom binary sequence.. They are bit sequences generated using maximal linear-feedback shift registers and are so called because they are periodic and reproduce every binary sequence (except the zero vector) that can be represented by the shift registers (i.e., for length-m registers they produce a sequence of length 2 m − 1).