Let K be a bounded, closed convex set in the euclidean plane. We denote the diameter, width, perimeter, area, inradius, and circumradius of K by d, w, p, A, r, and R respectively. We establish a number of best possible upper bounds for (w−2r)d, (w−2r)R,(w−2r)p, (w−2r)A in terms of w and r. Examples are:

A new coincidence point theorem is proved for a pair of multivalued mappings operating between G-convex spaces. From this theorem, a generalisation of the classical Fan-Glicksberg fixed point theorem is established.

An asymptotic formula is derived for the number of partitions of a large positive integer n into r unequal positive integer parts and maximal summand k. The number of parts has a normal distribution about its maximum, the largest summand an extreme-value distribution. For unrestricted partitions the two distributions coincide and both are extreme-valued. The problem of joint distribution of unrestricted partitions with r parts and largest summand k remains unsolved.

It is proved that the product of two non-zero derivations of a prime C*-algebra is bounded only if both of them are bounded.

We construct two families of few-weight codes for the Lee weight over the ring $R_{k}$ based on two different defining sets. For the first defining set, taking the Gray map, we obtain an infinite family of binary two-weight codes which are in fact $2^{k}$ -fold replicated MacDonald codes. For the second defining set, we obtain two infinite families of few-weight codes. These few-weight codes can be used to implement secret-sharing schemes.

For the twice-punctured unit disc Ωp = {z: |z| < 1, z ≠ ±p}, where 0 < p < 1, we obtain precise descriptions for p near 0 of various parameters associated with the uniformisation of Ωp by the upper half-plane U = {τ: Im τ > 0}. These parameters include the hyperbolic length of the geodesic surrounding ±p, the so-called “accessory parameters”, and the “proximity parameter” which determines the behaviour of the hyperbolic density near the punctures of Ωp.

We show that every topological grading of a $C^{\ast }$-algebra by a discrete abelian group is implemented by an action of the compact dual group.

Let S be the family of functions f(z) = z + a2z2 + … which are analytic and univalent in |z| < 1. We find the value

as a function of r 0 < r < 1. The known lower estimate of

is improved. Relations with the growth theorem are considered and the radius of univalence of f(z)/z is discussed.

The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The object of this paper is to compute the commutativity degree of a class of finite groups obtained by semidirect product of two finite abelian groups. As a byproduct of our result, we provide an affirmative answer to an open question posed by Lescot.

Let G be a group with a normal subgroup H whose index is a power of a prime p, and which is nilpotent with exponent a power of p. Gilbert Baumslag (Proc. Cambridge Philos. Soc. 55 (1959), 224–231) has shown that such a group is nilpotent; the main result of this paper is an upper bound on its nilpotency class in terms of parameters of H and G/H. It is shown that this bound is attained whenever G is a wreath product and H its base group.

A descending central series, here called the cpp-series, is involved in these calculations more closely than is the lower central series, and the class of the wreath product in terms of this series is also found.

Two tools used to obtain the main result, namely a useful basis for a finite p-group and a result about the augmentation ideal of the integer group ring of a finite p-group, may have some independent interest. The main result is applied to the construction of some two-generator groups of large nilpotency class with exponents 8, 9, and 25.

The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.

Dedicated to George Szekeres on the occasion of his 90th birthday

Necessary and sufficient conditions for equality over the pseudovariety CR of all finite completely regular semigroups are obtained. They are inspired by the solution of the word problem for free completely regular semigroups and clarify the role played by groups in the structure of such semigroups. A strengthened version of Ash's inevitability theorem (κ-reducibility of the pseudovariety G of all finite groups) is proposed as an open problem and it is shown that, if this stronger version holds, then CR is also κ-reducible and, therefore, hyperdecidable.

We calculate the rank and idempotent rank of the semigroup ${\mathcal{E}}(X,{\mathcal{P}})$ generated by the idempotents of the semigroup ${\mathcal{T}}(X,{\mathcal{P}})$ which consists of all transformations of the finite set $X$ preserving a nonuniform partition ${\mathcal{P}}$. We also classify and enumerate the idempotent generating sets of minimal possible size. This extends results of the first two authors in the uniform case.

Let A be a finite set. A partial clone on A is a composition closed set of operations containing all projections. It is well known that the partial clones on A, ordered by inclusion, form a lattice. We show that the minimal partial clones on A are:

(a) minimal clones of full operations or

(b) generated by partial projections defined on a totally reflexive and totally symmetric domain.

Let G be a group and M a G-module; then d(G) denotes the minimal number of generators of G and dG(M) the minimal number of generators over ZG of M. For G a finite nilpotent group let G = F/R, F free, be a presentation for G; then it is shown that d(R/[F, R]) = dG(R/[R, R]), that is d(G) + d(M(G)) = dG(R/R′), where M(G) denotes the Schur multiplicator of G.

Let G be a graph of order n, and let a, b, k be nonnegative integers with 1 ≤ a < b. An [a, b]-factor of graph G is defined as a spanning subgraph F of G such that a ≤ dF(x) ≤ b for each x ϵ V (F). Then a graph G is called an (a, b, k)-critical graph if after deleting any k vertices of G the remaining graph of G has an [a, b]-factor. In this paper, it is proved that G is an (a, b, k)-critical graph if the binding number and

Furthermore, it is showed that the result in this paper is best possible in some sense.

For a ring R with identity we show that the existence of certain nilpotent elements forces R to be a matrix ring of size ≥ 2.

Kaplansky's observation, namely, a commutative ring R is (von Neumann) regular if and only if each simple R-module is injective, is generalized to projective modules over a commutative ring.

Let $G$ be a general weighted graph (with possible self-loops) on $n$ vertices and $\lambda _1,\lambda _2,\ldots ,\lambda _n$ be its eigenvalues. The Estrada index of $G$ is a graph invariant defined as $EE=\sum _{i=1}^ne^{\lambda _i}$. We present a generic expression for $EE$ based on weights of short closed walks in $G$. We establish lower and upper bounds for $EE$in terms of low-order spectral moments involving the weights of closed walks. A concrete example of calculation is provided.

In this paper the result of Sobczyk about complemented copies of c0 is extended to a class of Banach spaces X such that the unit ball of their dual endowed with the weak* topology has a certain topological property satisfied by every Corson-compact space. By means of a simple example it is shown that if Corson-compact is replaced by Rosenthal-compact, this extension does not hold. This example gives an easy proof of a result of Phillips and an easy solution to a question of Sobczyk about the existence of a Banach space E, c0 ⊂ E ⊂ l∞, such that E is not complemented in l∞ and c0 is not complemented in E. Assuming the continuum hypothesis, it is proved that there exists a Rosenthal-compact space K such that C(K) has no projectional resolution of the identity.