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He obtained, for example, that the maximum (for positive x) of the cubic ax 2 – x 3 occurs when x = 2a / 3, and concluded therefrom that the equation ax 2 = x 3 + c has exactly one positive solution when c = 4a 3 / 27, and two positive solutions whenever 0 < c < 4a 3 / 27. [8] [page needed] The historian of science, Roshdi Rashed, [8] [page ...
It follows that the adjoint representation of a Lie algebra is a derivation on that algebra. The Pincherle derivative is an example of a derivation in abstract algebra. If the algebra A is noncommutative, then the commutator with respect to an element of the algebra A defines a linear endomorphism of A to itself, which is a derivation over K ...
Lebesgue measure on Euclidean space is locally finite. By definition, any Radon measure is locally finite. The counting measure is sometimes locally finite and sometimes not: the counting measure on the integers with their usual discrete topology is locally finite, but the counting measure on the real line with its usual Borel topology is not.
In functional analysis and measure theory, a differentiable measure is a measure that has a notion of a derivative.The theory of differentiable measure was introduced by Russian mathematician Sergei Fomin and proposed at the International Congress of Mathematicians in 1966 in Moscow as an infinite-dimensional analog of the theory of distributions. [1]
Properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology; an example is differential algebra. Here, it consists of the derivation of some topics in abstract algebra, such as rings, ideals, field, and so on. [52] The discrete equivalent of differentiation is finite differences.
A time scale (or measure chain) is a closed subset of the real line. The common notation for a general time scale is T {\displaystyle \mathbb {T} } . The two most commonly encountered examples of time scales are the real numbers R {\displaystyle \mathbb {R} } and the discrete time scale h Z {\displaystyle h\mathbb {Z} } .
For example, the Dirac delta function is a singular measure. Example. A discrete measure. ... as its distributional derivative. This is a measure on the real line, ...
A consequence of the Hahn decomposition theorem is the Jordan decomposition theorem, which states that every signed measure defined on has a unique decomposition into a difference = + of two positive measures, + and , at least one of which is finite, such that + = for every -measurable subset and () = for every -measurable subset , for any Hahn decomposition (,) of .