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The identity element of this algebra is the condensed interval [1, 1]. If interval [x, y] is not in one of the ideals, then it has multiplicative inverse [1/x, 1/y]. Endowed with the usual topology, the algebra of intervals forms a topological ring.
The term domain is also commonly used in a different sense in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis , a domain is a non-empty connected open subset of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} or the complex coordinate space C n ...
A comprehensive paper on interval algebra in numerical analysis was published by Teruo Sunaga (1958). [11] The birth of modern interval arithmetic was marked by the appearance of the book Interval Analysis by Ramon E. Moore in 1966. [12] [13] He had the idea in spring 1958, and a year later he published an article about computer interval ...
In this case, an element x of the domain is represented by an interval of the x-axis, and the corresponding value of the function, f(x), is represented by a rectangle whose base is the interval corresponding to x and whose height is f(x) (possibly negative, in which case the bar extends below the x-axis).
The notation is also used to denote the characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function. A related concept in statistics is that of a dummy variable .
When no confusion is possible, notation f(S) is commonly used. [ , ] 1. Closed interval: if a and b are real numbers such that , then [,] denotes the closed interval defined by them. 2. Commutator (group theory): if a and b belong to a group, then [,] =. 3.
In complex analysis, a complex domain (or simply domain) is any connected open subset of the complex plane C. For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function.
The factorial notation X! may be used for permutations of a single set X. In functional analysis, the same is seen for continuous linear transformations, including topologies on the vector spaces in the above, and many of the major examples are function spaces carrying a topology; the best known examples include Hilbert spaces and Banach spaces.