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In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero. When only finitely many of the angles are ...
A uniformly recurrent word is a recurrent word in which for any given factor X in the sequence, there is some length n X (often much longer than the length of X) such that X appears in every block of length n X. [1] [6] [7] The terms minimal sequence [8] and almost periodic sequence (Muchnik, Semenov, Ushakov 2003) are also used.
The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or ...
The precise definition of fractal sequence depends on a preliminary definition: a sequence x = (x n) is an infinitive sequence if for every i, (F1) x n = i for infinitely many n. Let a(i,j) be the jth index n for which x n = i. An infinitive sequence x is a fractal sequence if two additional conditions hold:
For an infinite sequence, one is often more interested in the long-term behaviors of the sequence than the behaviors it exhibits early on. In which case, one way to formally capture this concept is to say that the sequence possesses a certain property eventually, or equivalently, that the property is satisfied by one of its subsequences (), for some .
where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.
Thomae mentioned it as an example for an integrable function with infinitely many discontinuities in an early textbook on Riemann's notion of integration. [ 4 ] Since every rational number has a unique representation with coprime (also termed relatively prime) p ∈ Z {\displaystyle p\in \mathbb {Z} } and q ∈ N {\displaystyle q\in \mathbb {N ...
Henryk Iwaniec showed that there are infinitely many numbers of the form + with at most two prime factors. [ 26 ] [ 27 ] Ankeny [ 28 ] and Kubilius [ 29 ] proved that, assuming the extended Riemann hypothesis for L -functions on Hecke characters , there are infinitely many primes of the form p = x 2 + y 2 {\displaystyle p=x^{2}+y^{2}} with y ...