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A general reciprocity theorem for all orders which covered all other known reciprocity theorems when proved by E. Artin in 1927. If R is a number field and R^' a finite ...

For a real number x, the mantissa is defined as the positive fractional part x-|_x_|=frac(x), where |_x_| denotes the floor function. For example, for x=3.14159, the mantissa ...

The term "quotient" is most commonly used to refer to the ratio q=r/s of two quantities r and s, where s!=0. Less commonly, the term quotient is also used to mean the integer ...

Let K be a number field and let O be an order in K. Then the set of equivalence classes of invertible fractional ideals of O forms a multiplicative Abelian group called the ...

Consider the problem of comparing two real numbers x and y based on their continued fraction representations. Then the mean number of iterations needed to determine if x<y or ...

The nearest integer function, also called nint or the round function, is defined such that nint(x) is the integer closest to x. While the notation |_x] is sometimes used to ...

The Mittag-Leffler function (Mittag-Leffler 1903, 1905) is an entire function defined by the series E_alpha(z)=sum_(k=0)^infty(z^k)/(Gamma(alphak+1)) (1) for alpha>0. It is ...

The floor function |_x_|, also called the greatest integer function or integer value (Spanier and Oldham 1987), gives the largest integer less than or equal to x. The name ...

In the fields of functional and harmonic analysis, the Littlewood-Paley decomposition is a particular way of decomposing the phase plane which takes a single function and ...

The sawtooth wave, called the "castle rim function" by Trott (2004, p. 228), is the periodic function given by S(x)=Afrac(x/T+phi), (1) where frac(x) is the fractional part ...