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Before PDF version 1.5, the table would always be in a special ASCII format, be marked with the xref keyword, and follow the main body composed of indirect objects. Version 1.5 introduced optional cross-reference streams, which have the form of a standard stream object, possibly with filters applied. Such a stream may be used instead of the ...
In mathematics, the support of a real-valued function is the subset of the function domain of elements that are not mapped to zero. If the domain of is a topological space, then the support of is instead defined as the smallest closed set containing all points not mapped to zero.
File size is a measure of how much data a computer file contains or how much storage space it is allocated. Typically, file size is expressed in units based on byte . A large value is often expressed with a metric prefix (as in megabyte and gigabyte ) or a binary prefix (as in mebibyte and gibibyte ).
If −1 < x < 1 there are two possible values of y, one positive and one negative. For x = ± 1, these two values become both equal to 0. Otherwise, there is no possible value of y. This means that the equation defines two implicit functions with domain [−1, 1] and respective codomains [0, +∞) and (−∞, 0].
As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false.
In many branches of mathematics, the term map is used to mean a function, [5] [6] [7] sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous function" in topology, a "linear transformation" in linear algebra, etc.
The diameter may be called the length, width, measure, range, or size of the interval. The size of unbounded intervals is usually defined as +∞, and the size of the empty interval may be defined as 0 (or left undefined). The centre of a bounded interval with endpoints a and b is (a + b)/2, and its radius is the half-length | a − b |/2
The same idea underlies the solution of a first order equation as an integral of the Monge cone. [5] The Monge cone is a cone field in the R n+1 of the (x,u) variables cut out by the envelope of the tangent spaces to the first order PDE at each point. A solution of the PDE is then an envelope of the cone field.