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Negative correlation can be seen geometrically when two normalized random vectors are viewed as points on a sphere, and the correlation between them is the cosine of the circular arc of separation of the points on a great circle of the sphere. [1] When this arc is more than a quarter-circle (θ > π/2), then the cosine is negative.
The Bingham distribution is a distribution over axes in N dimensions, or equivalently, over points on the (N − 1)-dimensional sphere with the antipodes identified. [9] For example, if N = 2, the axes are undirected lines through the origin in the plane. In this case, each axis cuts the unit circle in the plane (which is the one-dimensional ...
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
As a result, the sum of these spaces is also dense in the space L 2 (S n−1) of square-integrable functions on the sphere. Thus every square-integrable function on the sphere decomposes uniquely into a series of spherical harmonics, where the series converges in the L 2 sense.
Whilst the sphere is rigid and can not be bent using an isometry, if a small region removed, or even a cut along a small segment, then the resulting surface can be bent. Such bending preserves Gaussian curvature so any such bending of a sphere with a region removed will also have constant Gaussian curvature. [5]
calculation of () Radial distribution function for the Lennard-Jones model fluid at =, =.. In statistical mechanics, the radial distribution function, (or pair correlation function) () in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle.
Under many-body theory, the term is also used in physics, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics, seismology and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition.
A correlation coefficient is a numerical measure of some type of linear correlation, meaning a statistical relationship between two variables. [ a ] The variables may be two columns of a given data set of observations, often called a sample , or two components of a multivariate random variable with a known distribution .