Let R be a prime ring of characteristic different from 2, d a nonzero derivation of R, and I a nonzero right ideal of R such that [[d(x), x], [d(y), y]] = 0, for all x, y ∈ I. We prove that if [I, I]I ≠ 0, then d(I)I = 0. 1. Introduction. Let R be a prime ring and d a nonzero derivation of R. Define [x, y] 1 = [x, y] = xy − yx, then an Engel condition is a polynomial [x, y] k = [[x, y] k−1 ,y]

This work studies the existence of positive prime periodic solutions of higher order for rational recursive equations of the form yn = A + yn−1 yn−m , n = 0, 1, 2, . . . , with y−m , y−m+1, . . . , y−1 ∈ (0,∞) and m ∈ {2, 3, 4, . . .}. In particular, we show that for sufficiently small A > 0, there exist periodic solutions with prime period 2m +Um + 1, for almost all m, where Um = max{i ∈ N : i...

In two previous papers we computed cohomology groups H(Γ0(N);C) for a range of levels N , where Γ0(N) is the congruence subgroup of SL4(Z) consisting of all matrices with bottom row congruent to (0, 0, 0, ∗) mod N . In this note we update this earlier work by carrying it out for prime levels up to N = 211. This requires new methods in sparse matrix reduction, which are the main focus of the pap...

Theorem 1: Let un, n > 0, be the general term of a given sequence of integers and define the transformation T^yk)(un) as T^xyJc){un) = xun+lc +yun for every n > 0, A: being a positive integer. Then, if x mdy are nonzero integers and there exists a positive prime number/? which divides T(x,y,k)(n) f° every n>0 and is relatively prime to x, the distribution of the residues of (un) modulo p is eit...

The purpose of this paper is to investigate identities satisfied by centralizers on prime and semiprime rings. We prove the following result: Let R be a noncommutative prime ring of characteristic different from two and let S and T be left centralizers on R. Suppose that [S(x), T (x)]S(x) + S(x)[S(x), T (x)] = 0 is fulfilled for all x ∈ R. If S 6= 0 (T 6= 0) then there exists λ from the extende...

Let m > 0 and q > 1 be relatively prime integers. We find an explicit period ν m (q) such that for any integers n 0 and r we have n + ν m (q) r m (a) ≡ n r m (a) (mod q), provided that a = −1 and n = 0, or a is an integer with 1 − (−a) m relatively prime to q, where n r m (a) = k≡r (mod m) n k a k. This is a further extension of a congruence of Glaisher.

Proof. Using that (Z/p)× is a cyclic group of order p − 1 (i.e. the existence of primitive roots), we see that there is a square root of −1 (that is, a non-trivial fourth root of 1) in (Z/p)× if and only if p ≡ 1 mod 4. Suppose now that p ≡ −1 mod 4, and suppose that α and β are two elements of Z[i] such that p|αβ. Then p = N(p)|N(α)N(β), and so (after relabelling if necessary) we may assume th...

We claim that the prime/irreducible elements of R are the associates of primes in Z different from p. Indeed, suppose that a/p, with a ∈ Z is an irreducible. Since powers of p are units, we may assume k = 0 and p a. Since −1 is also unit, we may also assume a > 0. If a is not prime, it factors in Z as a = bc, with b, c 6= ±1. Since p b, c [as p a and p is prime], they are not units of R and hen...

Abstract The TiO B 3 Π − X Δ electronic transition ( γ ′ system) is an important opacity source in the atmospheres of M dwarfs and hot Jupiter exoplanets. 0–0, 1–0, 2–1 bands band s...

"This work studies the initial boundary value problem for Petrovsky equation with nonlinear damping \begin{equation*} \frac{\partial ^{2}u}{\partial t^{2}}+\Delta ^{2}u-\Delta u^{\prime} +\left\vert u\right\vert ^{p-2}u+\alpha g\left( u^{\prime }\right) =\beta f\left( u\right) \text{ in }\Omega \times \left[ 0,+\infty \right[, \end{equation*} where $\Omega $ is open and bounded domain $\mathbb{...