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Questions regarding the well-definedness of a function often arise when the defining equation of a function refers not only to the arguments themselves, but also to elements of the arguments, serving as representatives. This is sometimes unavoidable when the arguments are cosets and when the equation refers to coset representatives. The result ...
Hence the functional calculus is well-defined. Consequently, if f 1 and f 2 are two holomorphic functions defined on neighborhoods D 1 and D 2 of σ(T) and they are equal on an open set containing σ(T), then f 1 (T) = f 2 (T). Moreover, even though the D 1 may not be D 2, the operator (f 1 + f 2) (T) is well-defined. Same holds for the ...
The well-definedness condition corresponds to the requirement that every infinite path must eventually pass through a sufficiently long node: the same requirement that is needed to invoke a bar induction. The principles of bar induction and bar recursion are the intuitionistic equivalents of the axiom of dependent choices. [3]
Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules: . For a singularity at a finite number b + [() + + ()] with < < and where b is the difficult point, at which the behavior of the function f is such that = for any < and = for any >.
Today, any formal statement or calculation that exhibits this quality of well-definedness is termed computable, while the statement or calculation itself is referred to as a computation. Turing's definition apportioned "well-definedness" to a very large class of mathematical statements, including all well-formed algebraic statements , and all ...
A U.S. Postal Service worker from Compton was arrested on suspicion of swiping more than 20 checks from the mail and depositing $281,000 into various bank accounts under her name, authorities said.
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This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces for < < (Adams & Fournier 2003). The space L p {\displaystyle L^{p}} for 0 < p < 1 {\displaystyle 0<p<1} is an F-space : it admits a complete translation-invariant metric with respect to which the vector space ...