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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 ...
The result of the procedure for principal value is the same as the ordinary integral; since it no longer matches the definition, it is technically not a "principal value". The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function f ( z ) : z = x + i y , {\displaystyle f(z):z=x+i\,y\;,} with x , y ...
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 ...
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 ...
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 ...
that, even after controlling for industry- and company-specific characteristics as well as background and experience of directors, having female directors reduced the insolvency risk of these companies.22 Racial diversity also is found to be good for business. A study found that a racially diverse workforce was positively associated with more
Luigi Mangione, the man accused of murdering UnitedHealthcare CEO Brian Thompson, was told "don't say a word" by his lawyer in court, following an outburst outside the hearing at a Pennsylvania ...
Problem discovery is an unconscious process which depends upon knowledge whereby an idea enters one's conscious awareness, problem formulation is the discovery of a goal; problem construction involves modifying a known problem or goal to another one; problem identification represents a problem that exists in reality but needs to be discovered ...