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.
In this paper, we show the existence of a timelike minimal surface with an arbitrary number of weak complete ends. Then, we discuss the asymptotic behaviour of the simple ends and the topology of the singularity set of the constructed timelike minimal surface.
We study the sum & sum;(abc <= x )ohm([a, b, c]), where ohm(n) denotes the number of distinct prime divisors of n is an element of Z(>= 1 )counted with multiplicity, and [a, b, c] = lcm (a, b, c). An asymptotic formula is derived for this sum over the hyperbolic region {(a, b, c) is an element of Z(>= 1)(3), abc <= x}.
Characteristic class relations in Dolbeault cohomology follow from the existence of a holomorphic geometric structure (for example, holomorphic conformal structures, holomorphic Engel distributions, holomorphic projective connections, and holomorphic foliations). These relations can be calculated directly from the representation theory of the structure group, without selecting any metric or connection or having any knowledge of the Dolbeault cohomology groups of the manifold. This paper improves on its predecessor [17] by allowing infinite type geometric structures.
This note deals with the classical topic of interpolating bounded sequences of complex numbers by bounded analytic functions on sequences in the unit disk. We introduce linear, polynomial, and multiplicative interpolation over fixed-length consecutive blocks of points taken from these sequences, and examine the corresponding interpolating sequences.
The present investigation deals with a new subclass of alpha-convex bi-univalent functions in the unit disc E = {z : vertical bar z vertical bar< 1} defined with q-derivative operator. Bounds for the first two coefficients and Fekete-Szego inequality are established for this class. Many known results follow as consequences of the results derived here.
This note deals with interpolation of values of analytic functions belonging to a given space, on finite sets of consecutive points of sequences in the disc, performed by rational functions and polynomials. Our goal is to identify sequences and spaces whose functions provide a bound of the error at the first uninterpolated point that is as small as desired. For certain sequences, we prove that this happens for bounded functions, Lipschitz functions and those that have derivatives in the disc algebra.
The purpose of the present paper is to determine lower bounds R Skf(z) , R (Skf)m(z) R n S0 o n (Skf)' o kf(z) and R m(z) , where (Skf)m(z) Skf(z) (Skf)m (z) Skf(z) Skf is the generalized normalized error function of the form Skf (z) P infinity ((n-1)k+1)(n-1)! zn and(Skf)m its partial sum. Furthermore, we (-1)n-1 lower bounds for R I[Skf](z) n=2 and R (I[Skf])m(z) , where I [Skf] (I[Skf])m(z) I[Skf](z) Alexander transform of Skf. Several examples of the main results are considered.
We prove that there is an orbit-cone correspondence for the proalgebraic completion of normal toric varieties, which is analogous to the classical orbit-cone correspondence for toric varieties.
It is shown that oscillation of perturbed second order half-linear differential equations can be derived from oscillation of second order linear differential equations associated with modified Riccati equations. In the main result of the present paper, some of technical assumptions in the known results of this type are removed.
We construct generalized stationary discs to perturbations of decoupled real submanifolds of co dimension 2 in C4.
We study the geometry associated with the kinematics of a planar robot known as the "three-segment snake," whose velocity distribution belongs to a class of (2,3,5) distributions. We discover that, under certain assumptions on its construction parameters, the snake may be endowed with a CR structure of CR dimension 1 and real codimension 3. We solve the associated Cartan equivalence problem and find the invariants of the snake's CR structure.
In this paper we would like to introduce some new methods for studying magic type-colorings of graphs or domination of graphs, based on combinatorial spectrum on polynomial rings. We hope that this concept will be potentially useful for the graph theorists.
In this paper, we prove new rigidity results related to some generalised Ricci-Hessian equation on Riemannian manifolds.
We show the change of coordinates that maps the maximally symmetric $(2,3,5)$-distribution given by solutions to the $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy equation to the flat Cartan distribution. This establishes the local equivalence between the maximally symmetric $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy distribution and the flat Cartan or Hilbert-Cartan distribution. We give the set of vector fields parametrised by solutions to the $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy equation and the corresponding Ricci-flat conformal scale that bracket-generate to give the split real form of $\frak{g}_2$.
Motivated by our attempts to construct an analogue of the Dirac operator in the setting of Uq(sln), we write down explicitly the braided copro duct, antipode, and adjoint action for quantum algebra Uq(sl2). The braided adjoint action is seen to coincide with the ordinary quantum adjoint action, which also follows from the general results of S. Majid.
