Euler's totient function φ is the function defined on the positive natural numbers N * in the following way: if n ∈ N * , then φ(n) is the cardinal of the set {x ∈ N * : 1 ≤ x ≤ n, (x, n) = 1}, where (x, n) is the pgcd of x and n. Thus φ(1) = 1, φ(2) = 1, φ(3) = 2, φ(4) = 2, and so on. The principle aim of this article is to study certain aspects of the image of the function φ. 1 Elementary pro...

Let J n+p−1 (α) (p ∈ N, n > −p, 0 α < p) denote the class of functions of the form f (z) = 1 z p + a 0 z p−1 + a 1 z p−2 + ··· which are regular and p-valent in the punctured unit disc U * = {z ∈ C : 0 < |z| < 1} and satisfy the condition Re (D n+p f (z)) (D n+p−1 f (z)) − (p + 1) < − p (n + p − 1) + α n + p , where D n+p−1 f (z) = 1 z p (1 − z) n+p * f (z). It is proved that J n+p (α) ⊂ J n+p−...

at each point x in M, then M is called a QR-submanifold of r QR-dimension, where ] denotes the complementary orthogonal distribution to ] in TM (cf. [1–3]). Real hypersurfaces, which are typical examples of QR-submanifold with r = 0, have been investigated by many authors (cf. [2–9]) in connection with the shape operator and the induced almost contact 3-structure (for definition, see [10–13]). ...

with an additional restriction q = p − 1. In the limit ε → 0, we use formal asymptotics to construct a solution whose activator component a concentrates on a circle. Under additional constraints p > 1, m > 0, and 1 < m − s < 3, we find that such a solution exists and is unique. The radius of the circle of concentration is given explicitly in terms of certain integrals. Full numerical computatio...