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It is a theory of testing based on the relationship between individuals' performances on a test item and the test takers' levels of performance on an overall measure of the ability that item was designed to measure. Several different statistical models are used to represent both item and test taker characteristics. [1]
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients [1]: ch. 17 [2]: ch. 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence.
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. [1] [2] The order of the equation is the maximum time gap between any two indicated values of the variable vector. For ...
A contrast is defined as the sum of each group mean multiplied by a coefficient for each group (i.e., a signed number, c j). [10] In equation form, = ¯ + ¯ + + ¯ ¯, where L is the weighted sum of group means, the c j coefficients represent the assigned weights of the means (these must sum to 0 for orthogonal contrasts), and ¯ j represents the group means. [8]
where σ 1 and σ 2 are the population standard deviations of the Y i1 and Y i2 data, respectively. Thus the variance of D is lower if there is positive correlation within each pair. Such correlation is very common in the repeated measures setting, since many factors influencing the value being compared are unaffected by the treatment.
A general 1-form is a linear combination of these differentials at every point on the manifold: + +, where the f k = f k (x 1, ... , x n) are functions of all the coordinates. A differential 1-form is integrated along an oriented curve as a line integral.
Any scalar product on a real vector space V is a pairing (set M = N = V, R = R in the above definitions).. The determinant map (2 × 2 matrices over k) → k can be seen as a pairing .
Definition: [7] The midpoint of two elements x and y in a vector space is the vector 1 / 2 (x + y). Theorem [ 7 ] [ 8 ] — Let A : X → Y be a surjective isometry between normed spaces that maps 0 to 0 ( Stefan Banach called such maps rotations ) where note that A is not assumed to be a linear isometry.