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.

We give a complete characterisation of univalent logharmonic mappings from the domain D of such that has countable many components onto where pj is a singleton in C.

The definition of bent is redefined for any finite field. Our main result is a complete description of the relationship between bent polynomials and perfect non-linear functions over finite fields: we show they are equivalent. This result shows that bent polynomials can also be viewed as the generalisation to several variables of the class of polynomials known as planar polynomials. An explicit method for obtaining large sets of not necessarily distinct maximal orthogonal systems using bent polynomials is given and we end with a short discussion on the existence of bent polynomials over finite fields.

The paper gives a detailed description of all those finite A-groups of nilpotent length three that satisfy the cyclic subnormal separation condition. It is shown that every monolithic group under discussion is an extension of its Fitting subgroup P, which is a homocyclic p-group, by a p′ metabelian subgroup H, where p is a prime. The centraliser of P in H is trivial while the monolith W is equal to ω1(P) and the action of H on W is faithful and irreducible. H is further shown to have non trivial centre and is an extension of its derived subgroup M by a subgroup L such that

for all primes q where Mq and Lq are the respective Sylow q-subgroups of M and L. The Fitting subgroup of F of H is shown to be M × Z(H), while Z(H) = F ∩ L and every element of L of prime order is in Z(H). Finally it is shown that if ql(q) is the exponent of Mq then every element of order dividing ql(q) in L belongs to Z(H).

A solution of the equation must be C2 smooth, need not be C3 smooth, but at the same time the ratio zxy: zxx is C1 (provided zxx ≠ 0). An analytical proof of this fact is given; the underlying geometrical interpretation is discussed.

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.

A finite group G is said to have the m-CI property if, for any two Cayley graphs Cay(G, S) and Cay(G, T) of valency m, Cay(G, S) ≅ Cay(G, T) implies Sσ = T for some automorphism σ of G. In this paper, we investigate finite groups with the m-CI property. We first construct groups with the 3-CI property but not with the 2-CI property, and then prove that a nonabelian simple group has the 3-CI property if and only if it is A5. Finally, for infinitely many values of m, we construct Frobenius groups with the m-CI property but not with the nontrivial k-CI property for any k < m.

We show that if a continuous bump function on a Banach space X locally depends on finitely many elements of a set F in X*, then the norm closed linear span of F equals to X*. Some corollaries for Markuševič bases and Asplund spaces are derived.

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.

The periodic-point or cycle structure of a continuous map of a topological space has been a subject of great interest since A.N. Sharkovsky completely explained the hierarchy of periodic orders of a continuous map f: R → R, where R is the real line. In this paper the topological idea of “stirring” is invoked in an effort to obtain a transparent proof of a generalisation of Sharkovsky's Theorem to continuous functions f: I → I where I is any interval. The stirring approach avoids all graph-theoretical and symbolic abstraction of the problem in favour of a more concrete intermediate-value-theorem-oriented analysis of cycles inside an interval.

We characterise Banach spaces not containing l1 by a differentiability property of each equivalent norm and show that a slightly stronger differentiability property characterises Asplund spaces.

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.

We determine the complexity of torsion-freeness of finitely presented groups in Kleene's arithmetical hierarchy as -complete. This implies in particular that there is no effective listing of all torsion-free finitely presented groups, or of all non-torsion-free finitely presented groups.

We introduce a definition of generalised solutions of the Hessian equation Sm(D2u) = f in a convex set ω ⊂ ℝn, where Sm(D2u) denotes the m-th symmetric function of the eigenvalues of D2u, f ∈ Lp(ω), p ≥ 1, and m ∈ {1, …, n}. Such a definition is given in the class of semi-convex functions, and it extends the definition of convex generalised solutions for the Monge-Ampère equation. We prove that semiconvex weak solutions are solutions in the sense of the present paper.

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.

Let End (T) be the number of ends of a tree T and f: T → T be continuous. We show that f has a non divisible periodic orbit if and only if there are some x ∈ T and n > 1 with (n, m) = 1 for each 2 ≤ m ≤ End(T) such that x ∈ (f(x), fn(x)). Consequently the property of a tree map with a non divisible periodic orbit is preserved under small perturbation.

We show that the left and the right uniformities on a locally connected topological group G coincide if and only if every left uniformly continuous real-valued function on G is right uniformly continuous.

We generalise to partially ordered vector spaces, with a new technique, Arendt's approach to Kim's characterisation of Riesz homomorphisms.