This article has multiple issues. Text document with red question mark. Nts solved papers pdf often appears in lectures, and sometimes in print, as in
This article has multiple issues. Text document with red question mark. Nts solved papers pdf often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense.
Some phrases, like “in general”, appear below in more than one section. Grothendieck and others showed that classical problemswhich had resisted efforts of several generations of talented mathematicians, could be solved in terms ofcomplicated concepts. A reference to a standard or choice-free presentation of some mathematical object. The proof that there are infinitely many prime numbers.
The proof of the irrationality of the square root of two. A result is called “deep” if its proof requires concepts and methods that are advanced beyond the concepts needed to formulate the result. The beauty of a mathematical theory is independent of the aesthetic qualitiesof the theory’s rigorous expositions. Some beautiful theories may never be given a presentation which matches their beauty. Instances can also be found of mediocre theories of questionable beauty which are given brilliant, exciting expositions. In retrospect, one wonders what all the fuss was about. Mathematicians may say that a theorem is beautiful when they really mean to say that it is enlightening.
We acknowledge a theorem’s beauty when we see how the theorem ‘fits’ in its place. We say that a proof is beautiful when such a proof finally gives away the secret of the theorem. A result is called “folklore” if it is non-obvious, has not been published, and yet is generally known among the specialists in a field. Usually, it is unknown who first obtained the result. If the result is important, it may eventually find its way into the textbooks, whereupon it ceases to be folklore. Many of the results mentioned in this paper should be considered “folklore” in that they merely formally state ideas that are well-known to researchers in the area, but may not be obvious to beginners and to the best of my knowledge do not appear elsewhere in print. Though long used informally, this term has found a formal definition in category theory.
These can be and often are contradictory requirements. Sometimes the term is more pointed, referring to an object which is specifically and artificially exhibited as a counterexample to these properties. Since half a century we have seen arise a crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose. Nay more, from the logical point of view, it is these strange functions which are the most general. This function is continuous but not differentiable. Banach found out in 1931, differentiable functions are colloquially speaking a rare exception among the continuous ones.