Sentence examples for the aforementioned theorem from inspiring English sources

The most famous consequence of the bar theorem is the fan theorem, which suffices to prove the aforementioned theorem on uniform continuity, and which will be treated first.

The following example shows that the aforementioned theorem is not correct.

The aforementioned Theorem 1 provides also an upper bound for the difference biggl| int_{a}^{b}f,mathrm{d}g - f(a bigl[g(b -g(a) b -g] biggr|.

Subjectivists have traditionally justified the three axioms of probability by appeal to one of the aforementioned theorems: the Dutch book theorem or some form of representation theorem.

The complicated contractive conditions of the aforementioned theorems can be simplified considerably by means of the following notations.

Applying the same method as that used in the proof of the aforementioned theorems and Lemma 1, we have the following sufficient conditions on analytic functions to be starlike in one direction.

We leave open the question of proving a stronger fixed point result that encompasses the existence of and the two aforementioned theorems.

However, it raises a conceptual issue about any universe sharing the capacity of producing and hosting universal computers: because of the aforementioned Halting Theorem, there cannot be any general algorithm to decide whether, given an initial configuration as input, Life will eventually die out or halt.

An alternative characterization of the aforementioned median voter theorem is given in Gans and Smart (1996) using the single-crossing property on preference profiles.

In [1], Kannan obtained the following extension of the aforementioned fixed point theorem of Banach to a larger class of mappings, now known as Kannan mappings.

A proof of the theorem can be found in [[2], Theorem 5.2]; see also Theorem 5.1 in the aforementioned reference.