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If D(a, b) < 0 then (a, b) is a saddle point of f. If D(a, b) = 0 then the point (a, b) could be any of a minimum, maximum, or saddle point (that is, the test is inconclusive). Sometimes other equivalent versions of the test are used. In cases 1 and 2, the requirement that f xx f yy − f xy 2 is positive at (x, y) implies that f xx and f yy ...
The inhomogeneous Fredholm integral equation = + (,) ()may be written formally as = which has the formal solution =. A solution of this form is referred to as the resolvent formalism, where the resolvent is defined as the operator
In probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a noncentral generalization of the (ordinary) F-distribution. It describes the distribution of the quotient ( X / n 1 )/( Y / n 2 ), where the numerator X has a noncentral chi-squared distribution with n 1 degrees of freedom and ...
In probability theory and statistics, the F-distribution or F-ratio, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA) and other F-tests.
Let f ∈ F q [x] of degree n be the polynomial to be factored. Algorithm Distinct-degree factorization(DDF) Input: A monic square-free polynomial f ∈ F q [x] Output: The set of all pairs (g, d), such that f has an irreducible factor of degree d and g is the product of all monic irreducible factors of f of degree d.
The fundamental lemma of the calculus of variations is typically used to transform this weak formulation into the strong formulation (differential equation), free of the integration with arbitrary function. The proof usually exploits the possibility to choose δf concentrated on an interval on which f keeps sign (positive or negative). Several ...
The type–token distinction separates types (abstract descriptive concepts) from tokens (objects that instantiate concepts). For example, in the sentence "the bicycle is becoming more popular" the word bicycle represents the abstract concept of bicycles and this abstract concept is a type, whereas in the sentence "the bicycle is in the garage", it represents a particular object and this ...
The classical Pade scheme for the first derivative at a cell with index (′) reads; ′ + ′ + + ′ = +. Where is the spacing between points with index , & +.The equation yields a fourth-order accurate solution for ′ when supplemented with suitable boundary conditions (typically periodic).