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Throughout this article, boldfaced unsubscripted and are used to refer to random vectors, and Roman subscripted and are used to refer to scalar random variables.. If the entries in the column vector = (,, …,) are random variables, each with finite variance and expected value, then the covariance matrix is the matrix whose (,) entry is the covariance [1]: 177 ...
In the Y-factor technique, P out is measured for two different, known values of T R. P out is then converted to an effective temperature T out (in units of kelvin) by dividing by k B and the measurement bandwidth B. The two values of T out are then plotted as a function of T R (also in units of kelvin), and a line is fit to these points (see ...
Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid. It approximates the value of a function at an intermediate point ( x , y , z ) {\displaystyle (x,y,z)} within the local axial rectangular prism linearly, using function data on the lattice points.
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...
Degenerate states are also obtained when the sum of squares of quantum numbers corresponding to different energy levels are the same. For example, the three states (n x = 7, n y = 1), (n x = 1, n y = 7) and (n x = n y = 5) all have = and constitute a degenerate set.
Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform.
corresponds to a rotation of approximately −74° around the axis (− 1 / 2 ,1,1) in three-dimensional space. The 3 × 3 permutation matrix = [] is a rotation matrix, as is the matrix of any even permutation, and rotates through 120° about the axis x = y = z. The 3 × 3 matrix
It graphically depicts the ratios of the three variables as positions in an equilateral triangle. It is used in physical chemistry, petrology, mineralogy, metallurgy, and other physical sciences to show the compositions of systems composed of three species. Ternary plots are tools for analyzing compositional data in the three-dimensional case.