Sentence examples for distinct proofs from inspiring English sources

Since Euclid, a host of professional and amateur mathematicians (even U.S. President James Garfield) have found more than 300 distinct proofs of the Pythagorean theorem.

In his Meditations (1641), for example, Descartes offered two distinct proofs of the existence of God and asserted that no one who does not have a rationally well-founded belief in God can have knowledge in the proper sense of the term.

At least five distinct proofs of (1.1) could be found in [3] and [4].

Consequently the observation cannot be regarded as a distinct proof for an inherited aetiology of the brachyspina syndrome and the sire cannot at present be regarded as a carrier of the defect.

This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used.

Here, we demonstrated six distinct NAMs as proof-of-principle and verified all the NAMs described in Figure 1A are present in cancer.

Both results are remarkable as clinical narratives have specific idiosyncrasies (telegram-style, numerous ad-hoc abbreviations, spelling and typing errors), and are thus quite distinct from the proof-read content of textbooks and scientific papers [ 50].

"The formation of different languages and of distinct species, and the proofs that both have been developed through a gradual process, are curiously parallel" (Darwin 1871).

Moreover, we develop a proof of Theorem 1.1 which is an alternative proof of (1.1) for all positive real numbers a, b, and r ∈ [ 0, e ], which is distinct from the existing proofs given in [14, 29].

After all, more and more people seem to be drawn to alternative medical therapies -- half of all Americans say they have tried them -- despite a distinct lack of empirical proof for their effectiveness.

In the previous study, the sets of NISE, for which the existence of two or more distinct folds (the ultimate proof of the lack of homology), could be demonstrated directly was a small minority but this fraction changed to majority in the present study thanks to the advances of structural biology in general and structural genomics in particular.