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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 ...
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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 second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
A polynomial f of degree n greater than one, which is irreducible over F q, defines a field extension of degree n which is isomorphic to the field with q n elements: the elements of this extension are the polynomials of degree lower than n; addition, subtraction and multiplication by an element of F q are those of the polynomials; the product ...
In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum ( functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf .
The category of such F-isocrystals is abelian and semisimple, so every F-isocrystal is a direct sum of simple F-isocrystals. The simple F-isocrystals are the modules E s/r where r and s are coprime integers with r>0. The F-isocrystal E s/r has a basis over K of the form v, Fv, F 2 v,...,F r−1 v for some element v, and F r v = p s v.
PDF 2.0 defines 256-bit AES encryption as the standard for PDF 2.0 files. The PDF Reference also defines ways that third parties can define their own encryption systems for PDF. PDF files may be digitally signed, to provide secure authentication; complete details on implementing digital signatures in PDF are provided in ISO 32000-2.