Let f (x) = x6 + Ax2k + B E Z[x], with A =6 0 and k E {1, 2}. We say that f (x) is monogenic if f (x) is irreducible over Q and {1, 9, B2, B3, B4, B5} is a basis for the ring of integers of Q(9), where f(9) = 0. For each value of k and each possible Galois group G of f (x) over Q, we use a theorem of Jakhar, Khanduja and Sangwan to give explicit descriptions of all monogenic trinomials f (x) having Galois group G. We also determine when these descriptions provide infinitely many such trinomials, and we investigate when these trinomials generate distinct sextic fields. These results extend recent work on monogenic power-compositional sex-tic trinomials of the form g(x3) to the situation g(x2), and thereby complete the characterization, in terms of their Galois groups, of monogenic power-compositional sextic trinomials.
Consider the following higher order difference equation x(n + 1) = a(n)x(n) + b(n)f (x(n)) + c(n )f (x(n - k)), n = 0, 1, . . . , where f : [0, infinity) -> [0, infinity) is a continuous function with f (x) > 0 for x > 0, {a(n)} is a sequence in (0, 1), {b(n)} and {c(n)} are sequences in [0, 1) with a(n) + b(n) + c(n) = 1 and a(n), b(n) and c(n) are convergent, and k is a positive integer. Our aim in this paper is to study the global attractivity of positive solutions of this equation and its applications.
An example of a quartic extension of the rational number field that does not have quadratic subfields and there exists a set of rational prime numbers p-3 (mod 4) of positive Dirichlet density such that either p or 31p is a sum of two squares of integers of the extension is given.
The aim of this paper is to explore the relations between some geometric invariants associated with surfaces immersed in the Euclidean four-space, by investigating the extrinsic and intrinsic geometries of our surfaces from a global point of view, as well as considering the curvature ellipses of a given surface and their associated isoptic curves. We establish an elegant formula which shows how the isoptic curves of the curvature ellipses of a given surface are related to the asymptotic directions on the surface, and derive a Wintgen type inequality which provides both a simple relationship between the main intrinsic and extrinsic invariants, and a natural and geometric characterization of the hyperbolic points, on the given surface. To indicate an application of our Wintgen type inequality to the theory of Mo & uml;bius invariant Euclidean submanifolds, we conclude the paper with a novel geometric result on a remarkable family of surfaces known as Wintgen ideal surfaces.
