In this paper, some new sufficient conditions for oscillation of first order neutral delay differential equations with several variable coefficients are obtained. These sufficient conditions include and are in many cases weaker than those known.

The main results proved in this note are the following:

(i) Any finitely generated group can be expressed as a quotient of a finitely presented, centreless group which is simultaneously Hopfian and co-Hopfian.

(ii) There is no functorial imbedding of groups (respectively finitely generated groups) into Hopfian groups.

(iii) We prove a result which implies in particular that if the double orientable cover N of a closed non-orientable aspherical manifold M has a co-Hopfian fundamental group then π1(M) itself is co-Hopfian.

Usual duality results are proved for Wolfe and Mond-Weir type multiobjective symmetric dual problems without nonnegativity constraints under invexity/generalised invexity assumptions. Moreover, assuming the kernel function to be skew symmetric, the multiobjective problems are exhibited to be self duals.

We classify unit speed curves contained in a real space form of arbitrary dimension Nm(c), whose mean curvature vector is proper for the Laplacian. Then we use these results to classify Hopf cylinders of S3 and semi-Riemannian Hopf cylinders of with proper mean curvature function.

We consider normed and Banach algebras satisfying a condition topologically analogous to bounded index for rings. We investigate stability properties, prove a topological version of a theorem of Jacobson, and find in many cases co-incidence with well-known finiteness properties.

It is shown that a complete simply connected negatively curved manifold supports nontrivial bounded harmonic functions if the singular set of the ideal boundary is disconnected.

Suppose kn denotes either φ(n) or φ(rn) (n = 1, 2, …) where the polynomial φ maps the natural numbers to themselves and rk denotes the kth rational prime. Let denote the sequence of convergents to a real numbers x for the optimal continued fraction expansion. Define the sequence of approximation constants by

In this paper we study the behaviour of the sequence for all most all x with respect to Lebesgue measure. In the special case where kn = n (n = 1, 2, …) these results are due to Bosma and Kraaikamp.

Recently the authors considered Newman-type rational interpolation to |x| induced by arbitrary sets of interpolation nodes and showed that under mild restrictions on the location of the interpolation nodes, the corresponding sequence of rational interpolants converges to |x|. In the present paper we consider the special case of the Chebyshev nodes which are known to be very efficient for polynomial interpolation. It is shown that, in contrast to the polynomial case, the approximation of |x| induced by rational interpolation at the Chebyshev nodes has the same order as rational interpolation at equidistant points.

Let C be a cube in Rn+1 and let F = (f1, …, fn+1) be a polynomial vector field. In this note we propose a recursive algorithm for the computation of the degree of F on C. The main idea of the algorithm is that the degree of F is equal to the algebraic sum of the degrees of the map (f1, f2, …, fi−1, fi, fi+1, …, fn+1) over all sides of C, thereby reducing an (n + 1)–dimensional problem to an n–dimensional one.

For a self-adjoint rational matrix function, not necessarily analytic at infinity, the existence of a right (symmetric) spectral factorisation is described in terms of a given left spectral factorisation. The formula for the right spectral factor is given in terms of the formula for the given left spectral factor. All formulas are based on a special realisation of a rational matrix function, which is different from ones that have been used before.

The equality between the closures of the sets of periodic and of recurrent points (called the periodic-recurrent property) is extended from mappings of a tree to mappings defined on a λ-dendroid obtained as a compactification of the complement of a finite subset of a tree provided that the components of the remainder have the same finite depth and each has the periodic-recurrent property.

We study the qualitative properties of the one dimensional analogue of the Helmholtz vorticity advection equation. The second order hyperbolic equation has the unusual characteristic of disturbances propagating at infinite speed. The global solution for Goursat data is given in closed form. We also obtain qualitative results on the nodal curve where the solution is zero. A related perturbation problem is considered and solutions for small data are obtained. The forced vorticity equation admits a class of soliton solutions.

Explicit automorphisms of the irrational rotation algebra are constructed which are associated with the two 2 × 2 diagonal integer matrices of determinant −1. The fixed point algebra of the product of these two automorphisms is shown to be isomorphic to the fixed point algebra of the flip.

In this paper we prove the existence theorem for the implicit Darboux problem on the quarterplane x ≥ 0, y ≥ 0. Moreover, we study the topological structure of the solution set of this problem.