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For example, using single-precision IEEE arithmetic, if x = −2 −149, then x/2 underflows to −0, and dividing 1 by this result produces 1/(x/2) = −∞. The exact result −2 150 is too large to represent as a single-precision number, so an infinity of the same sign is used instead to indicate overflow.
j=1 a j X j has a (univariate) normal distribution. The variance of X is a k×k symmetric positive-definite matrix V. The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso-density loci in the k = 2 case are ellipses and in the case of arbitrary k are ellipsoids.
This is an accepted version of this page This is the latest accepted revision, reviewed on 24 December 2024. See also: Orders of magnitude (numbers) and Long and short scales Natural number 1000000000 List of numbers Integers ← 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 Cardinal One billion (short scale) One thousand million, or one milliard (long scale) Ordinal One billionth (short ...
[163] [164] [165] In two dimensions, the critical Sobolev inequality is ‖ ‖ ‖ ‖ for f a smooth function with compact support in R 2, is the gradient of f, and ‖ ‖ and ‖ ‖ refer respectively to the L 2 and L 1-norm. The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants.
The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. [1] Angles in polar notation are generally expressed in either degrees or radians (2 π rad being equal to 360°).
Lucas numbers have L 1 = 1, L 2 = 3, and L n = L n−1 + L n−2. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite. Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have P n = 2P n−1 + P n−2.
Morey et al. [27] point out that several of these confidence procedures, including the one for ω 2, have the property that as the F statistic becomes increasingly small—indicating misfit with all possible values of ω 2 —the confidence interval shrinks and can even contain only the single value ω 2 = 0; that is, the CI is infinitesimally ...