We show deterministic polynomial time algorithms over some family of finite fields for solving univariate polynomial equations and some related problems such as taking nth roots, constructing nth nonresidues, constructing primitive elements and computing elliptic curve “nth roots”. In additional, we present a deterministic polynomial time primality test for some family of integers. All algorith...

in this paper, we consider the finitely 2-generated groups k(s,l) and g_m as follows:k(s,l) = ;g_m = and find the explicit formulas for the probability of having nth-roots for them. also weinvestigate integers n for which, these groups are n-central.

here we consider all finite non-abelian 2-generator $p$-groups ($p$ an odd prime) of nilpotency class two and study the probability of having $n^{th}$-roots of them. also we find integers $n$ for which, these groups are $n$-central.

Based on the generalized continued-fraction method for finding the nth roots of real numbers, this paper presents a fast computation method for finding the principal nth roots of complex matrices. Computation algorithms with high convergence rates are developed, and their global convergence properties are investigated from the viewpoint of systems theory.

In this set of lecture notes we focus on the point-value representation obtained by looking at a particular set of points, the nth roots of unity. In the field of complex numbers C, there are exactly n different solutions to the equation x = 1. We call these solutions the n-th roots of unity. One of these roots of unity is ωn = cos(2π/n)+ i sin(2π/n), and this is called the principal nth root o...

In this set of lecture notes we focus on the point-value representation obtained by looking at a particular set of points, the nth roots of unity. In the field of complex numbers C, there are exactly n different solutions to the equation x = 1. We call these solutions the n-th roots of unity. One of these roots of unity is ωn = cos(2π/n)+ i sin(2π/n), and this is called the principal nth root o...

*Correspondence: [email protected] Department of Mathematics, Binzhou University, Shandong, 256603, P.R. China Abstract The square iterative roots for strictly monotonic and upper semicontinuous functions with one set-valued point were fully described in (Li et al. in Publ. Math. (Debr.) 75:203-220, 2009). As a continuation, we study both strictly monotonic and nonmonotonic multifunctio...

We develop the nth order Fourier-Bessel series expansion of 1-D functions in the interval (0,α). Hence we establish the sampling theorem for a function with α-bandlimited nth order Hankel transform. The latter statement implies that the function is also Fourier transform αbandlimited. The samples’ locations are given by the roots of nth order Bessel functions. In addition, the sampling distance...