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Weighted least squares (WLS), also known as weighted linear regression, [1] [2] is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (heteroscedasticity) is incorporated into the regression.
The notation convention chosen here (with W 0 and W −1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. [3]The name "product logarithm" can be understood as follows: since the inverse function of f(w) = e w is termed the logarithm, it makes sense to call the inverse "function" of the product we w the "product logarithm".
Formally, the Stanley symmetric function F w (x 1, x 2, ...) indexed by a permutation w is defined as a sum of certain fundamental quasisymmetric functions. Each summand corresponds to a reduced decomposition of w , that is, to a way of writing w as a product of a minimal possible number of adjacent transpositions .
What follows is the Richtmyer two-step Lax–Wendroff method. The first step in the Richtmyer two-step Lax–Wendroff method calculates values for f(u(x, t)) at half time steps, t n + 1/2 and half grid points, x i + 1/2. In the second step values at t n + 1 are calculated using the data for t n and t n + 1/2.
[1] [2] The order of the equation is the maximum time gap between any two indicated values of the variable vector. For example, = + is an example of a second-order matrix difference equation, in which x is an n × 1 vector of variables and A and B are n × n matrices. This equation is homogeneous because there is no vector constant term added ...
If t 1 and t 2 are terms then t 1 =t 2 is an atomic formula If R is an n -ary predicate symbol, and t 1 ,..., t n are terms, then R ( t 1 ,..., t n ) is an atomic formula Finally, the set of formulas is defined to be the smallest set containing the set of atomic formulas such that the following holds:
MD(X + c) = MD(X), MD(−X) = MD(X), and; MD(c X) = |c| MD(X). The relative mean absolute difference is invariant to positive scaling, commutes with negation, and varies under translation in proportion to the ratio of the original and translated arithmetic means. That is to say, if X is a random variable and c is a constant:
The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m 2 ⋅s −3. [ 1 ] [ 2 ] [ 3 ] It is used to quantify the rate of energy transfer .