In this paper, we consider the finitely 2-generated groups $K(s,l)$ and $G_m$ as follows:$$K(s,l)=langle a,b|ab^s=b^la, ba^s=a^lbrangle,\G_m=langle a,b|a^m=b^m=1, {[a,b]}^a=[a,b], {[a,b]}^b=[a,b]rangle$$ and find the explicit formulas for the probability of having nth-roots for them. Also, we investigate integers n for which, these groups are n-central.

Given a list of N numbers, the maximum can be computed in N iterations. During these N iterations, the maximum gets updated on average as many times as the Nth harmonic number. We first use this fact to approximate the Nth harmonic number as a side effect. Further, using the fact the Nth harmonic number is equal to the natural logarithm of N plus a constant that goes to zero with N, we approxim...

We study the Fredholm minors associated with a Fredholm equation of the second type. We present a couple of new linear recursion relations involving the nth and n−1th minors, whose solution is a representation of the nth minor as an n × n determinant of resolvents. The latter is given a simple interpretation in terms of a path integral over non-interacting fermions. We also provide an explicit ...

Links between the Calkin-Wilif tree and the Stern-Brocot tree are discussed answering the questions: What is the jth vertex in the nth level of the Calkin-Wilf tree? A simple mechanism is described for converting the jth vertex in the nth level of the Calkin-Wilf tree into the jth entry in the nth level of the Stern-Brocot tree. We also provide a simple method for evaluating terms in the hyperb...

We study the Fredholm minors associated with a Fredholm equation of the second type. We present a couple of new linear recursion relations involving the nth and n−1th minors, whose solution is a representation of the nth minor as an n × n determinant of resolvents. The latter is given a simple interpretation in terms of a path integral over non-interacting fermions. We also provide an explicit ...

A method for dividing a polynomial of degree (2n-l) by a precomputed nth degree polynomial in 0(n log n) arithmetic operations is given. This is used to prove that the evaluation of an nth degree polynomial at n+1 arbitrary points can be done in 0(n log^ n) arithmetic operations, and consequently, its dual problem, interpolation of an nth degree polynomial from 2 n+1 arbitrary points can be per...

Low dimensional sp3 carbon nanostructures have attracted increasing attention recently, due to their unique properties and appealing applications. Based on in silico studies, this work exploits the impacts from functional groups tensile of nanothreads (NTH) – a new nanostructure. It is found that will alter local bond configuration induce initial stress concentration, which significantly reduce...

In this paper the usefulness of the nth power law MOSFET model under Hot Carrier Injection (HCI) wearout has been experimentally demonstrated. In order to do that, three types of nFET transistors have been analyzed under different HCI conditions and the nth power law MOSFET model has been extracted for each sample. The results show that the model can reproduce the MOSFET behavior under HCI wear...

Let denote by Fk,n the nth k-Fibonacci number where Fk,n = kFk,n−1+Fk,n−2 for n 2 with initial conditions Fk,0 = 0, Fk,1 = 1, we may derive a functionalequation f(k, x) = kf(k, x − 1) + f(k, x − 2). In this paper, we solve thisequation and prove its Hyere-Ulam stability in the class of functions f : N×R ! X,where X is a real Banach space.