In this note some results concerning the equidistant set E(−x, x) and the kernel Mθ of the metric projection PM, where M is a Chebyshev subspace of a normed linear space X, have been obtained. In particular, when X = lp (1 < p < ∞), it has been proved that every equidistant set is closed in the bw-topology of the space. In c0 no equidistant set has this property.

We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

There is a familiar construction with two finite, transitive permutation groups as input and a finite, transitive permutation group, called their wreath product, as output. The corresponding ‘imprimitive wreath decomposition’ concept is the first subject of this paper. A formal definition is adopted and an overview obtained for all such decompositions of any given finite, transitive group. The result may be heuristically expressed as follows, exploiting the associative nature of the construction. Each finite transitive permutation group may be written, essentially uniquely, as the wreath product of a sequence of wreath-indecomposable groups, amid the two-factor wreath decompositions of the group are precisely those which one obtains by bracketing this many-factor decomposition.

If both input groups are nontrivial, the output above is always imprimitive. A similar construction gives a primitive output, called the wreath product in product action, provided the first input group is primitive and not regular. The second subject of the paper is the ‘product action wreath decomposition’ concept dual to this. An analogue of the result stated above is established for primitive groups with nonabelian socle.

Given a primitive subgroup G with non-regular socle in some symmetric group S, how many subgroups W of S which contain G and have the same socle, are wreath products in product action? The third part of the paper outlines an algorithm which reduces this count to questions about permutation groups whose degrees are very much smaller than that of G.

We show the existence of a convex representation of a maximal monotone operator by a convex function which is invariant with respect to the Fenchel conjugacy (up to an interchange of variables). We use the framework of generalized convexity.

We prove that (a) if R is a commutative coherent ring, the weak global dimension of R equals the supremum of the flat (or (FP–)injective) dimensions of the simple R-modules; (b) if R is right semi-artinian, the weak (respectively, the right) global dimension of R equals the supremum of the flat (respectively, projective) dimensions of the simple right R-modules; (c) if R is right semi-artinian and right coherent, the weak global dimension of R equals the supremum of the FP-injective dimensions of the simple right R-modules.

Suppose that $f(x)=x^{n}+A(Bx+C)^{m}\in \mathbb{Z}[x]$, with $n\geq 3$ and $1\leq m<n$, is irreducible over $\mathbb{Q}$. By explicitly calculating the discriminant of $f(x)$, we prove that, when $\gcd (n,mB)=C=1$, there exist infinitely many values of $A$ such that the set $\{1,\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{2},\ldots ,\unicode[STIX]{x1D703}^{n-1}\}$ is an integral basis for the ring of integers of $\mathbb{Q}(\unicode[STIX]{x1D703})$, where $f(\unicode[STIX]{x1D703})=0$.

We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence where α,β ∈ ℝ, and α is irrational. In particular, our bounds imply that for every fixed ε > 0, if p is sufficiently large and p½+ε ≤ N ≤ p, then among the first N elements of ℬα,β, there are N/2+o(N) quadratic non-residues modulo p. When more information is available about the Diophantine properties of α, then the error term o(N) admits a sharper estimate.

The class of ordered topological spaces which are projective limits of finite partially ordered sets (equipped with the restriction of the product of the discrete topologies) is shown to coincide with the class of compact totally order-disconnected ordered topological spaces. Hence this is another category of spaces equivalent to the category of distributive lattices with zero and unit.

It is well-known that if W is a set of costant width λ then the circumsphere and the insphere of W are concentric and their radii sum to λ. Here this fact is generalized to sets of constant relative width and it is shown that the result does not depend no W being of constant width, but rather on W satisfying the spherical intersection property; that is, W = ∩{ω + λB:ω ɛ W}.

The lattice of all tolerance relations (that is, reflexive, symmetric compatible relations) on a lattice is investigated. For modular lattices some examples are given which show that such relations do naturally occur.

In this work we demonstrate that the lower radical class construction on a homomorphically closed class of associative rings generates a radical class for any class of associative rings. We also give a new description of the upper radical class using the construction on an appropriate generating class.

Bernardi has proved that if f if starlike univalent in the uni disc E, then so is the function g given by

In the first part of the paper, we extend Bernardi's theorem to a certain class of p-valent starlike functions in E. We prove that if then g, defined by

also belongs to . In the second part of the paper we examine the converse problem for functions with negative coefficients, satisfying certain conditions.

The length of every pair {A,B} of 6×6 complex matrices is shown to be at most 10, that is, the words in A,B of length at most 10, including the empty word, span the unital algebra generated by A,B. This supports the conjecture that the length of every pair of n×n complex matrices is at most 2n−2, known to be true for n<6.

We characterise solutions f,g:ℝ→ℝ of the functional equation f(x+g(x)y)=f(x)f(y) under the assumption that f is continuous. Our considerations refer mainly to a paper by Chudziak [‘Semigroup-valued solutions of the Goła̧b–Schinzel functional equation’, Abh. Math. Semin. Univ. Hambg. 76, (2006), 91–98], in which the author studied the same equation assuming that g is continuous.

We present in this note a useful extension of the criteria given in a recent paper [Advances in Appl. Probability 8 (1976), 737–771] for the finiteness of hitting times and mean hitting times of a Markov chain on sets in its (general) state space. We illustrate our results by giving conditions for the finiteness of the mean number of customers in the busy period of a queue in which both the service-times and the arrival process may depend on the waiting time in the queue. Such conditions also suffice for the embedded waiting time chain to have a unique stationary distribution.

In this paper, we study the oscillation of nth-order differential equations. Recently, Atkinson and the present authors studied (separately) the comparison properties of differential inequalities. Kartsatos treated the nth-order ordinary case and proposed several open problems.

The purpose of this paper is to answer one of them in the affirmative concerning more general functional differential equations. The result is that if under several conditions, the equation

is oscillatory for n even or a solution x(t) of (1) is oscillatory or for n odd, then this is also the case for the equation

We complete the classification of the finite special linear groups $\text{SL}_{n}(q)$ which are $(2,3)$ -generated, that is, which are generated by an involution and an element of order $3$ . This also gives the classification of the finite simple groups $\text{PSL}_{n}(q)$ which are $(2,3)$ -generated.

We prove weighted Lp-Lq estimates for the maximal operators ℳα, given by , where μt denotes the normalised surface measure on the sphere of centre 0 and radius t in Rd. The techniques used involve interpolation and the Mellin transform. To do this, we also prove weighted Lp-Lq estimates for the operators of convolution with the kernels |·|−α−iη.

If au Asplund space is weakly countably determined, then it admits an equivalent Fréchet differentiable norm and is weakly compactly generated. If on an Asplund space there exists an equivalent Gâteaux differentiable norm, then its dual has a projectional resolution of identity.

In the present paper we consider the approximation of |x|α on [−1, 1] by interpolating polynomials and we establish upper bounds for the approximation error. It turns out that these bounds, apart from the constants, are of the best possible order. We compare our results with estimates for the best approximation, established by Bernstein.