Let p denote a prime, G a. finite soluble p'-group and A an elementary abelian p-group of operators on G. Suppose that A has order p4 and that if ω ∈ A# then CG(ω) has nilpotent derived group. Then G has nilpotent derived group.

We present a generalisation of the quasilinearisation method to obtain a monotone sequence of approximate solutions that converges quadratically to a solution of a nonlinear ordinary differential equation of order n ≥ 1.

We compute commutativity degrees of wreath products of finite Abelian groups A and B. When B is fixed of order n the asymptotic commutativity degree of such wreath products is 1/n2. This answers a generalized version of a question posed by P. Lescot. As byproducts of our formula we compute the number of conjugacy classes in such wreath products, and obtain an interesting elementary number-theoretic result.

Using an old result of Von Staudt on sums of consecutive integer powers, we shall show by an elementary method that the Diophantine equation 1k + 2k + … + (x − l)k = axk has no solutions (a, x, k) with k > 1, . For a = 1 this equation reduces to the Erdös-Moser equation and the result to a result of Moser. Our method can also be used to deal with variants of the equation of the title, and two examples will be given. For one of them there are no integer solutions with

An example is given of a Banach space with normal structure which does not satisfy the geometrical conditions commonly expected to be related to normal structure.

The family ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ of orientation-preserving harmonic functions $f=h+\overline{g}$ in the unit disc $\mathbb{D}$ (normalised in the standard way) satisfying

$$\begin{eqnarray}h^{\prime }(z)+g^{\prime }(z)=\frac{1}{(1+\unicode[STIX]{x1D706}z)(1+\overline{\unicode[STIX]{x1D706}}z)},\quad z\in \mathbb{D},\end{eqnarray}$$

$\unicode[STIX]{x1D706}\in \unicode[STIX]{x2202}\mathbb{D}$

${\mathcal{F}}_{\unicode[STIX]{x1D706}}$

Sufficient conditions are provided in order that some classes of finite soluble groups, defined by properties of the Sylow normalisers, are saturated formations.

The paper considers a system of differential equations with impulse perturbations at fixed moments in time of the form

where x ∈ Rn, ε is a small parameter,

Sufficient conditions are found for the existence of the periodic solution of the given system in the critical and non-critical cases.

In this paper we consider the situation that a group G has a subgroup H which is a dihedral 2-group and with connected transversals A and B in G. We show that G is then solvable and moreover, if G is generated by the set A ∪ B, then H is subnormal in G. We apply these results to loop theory and it follows that if the inner mapping group of a loop Q is a dihedral 2-group then Q is centrally nilpotent.

Weighted sums of products of independent normal random variables arise naturally as distributional limits for various statistics. This note investigates the rate at which the tail probability of these sums approaches zero.

The elementary nature and simplicity of the theory of continued fractions is mostly well disguised in the literature. This makes one reluctant to quote sources when making a remark on the subject and seems to necessitate redeveloping the theory ab initio. That had best be done succinctly. That is done here and allows the retrieval of some amusing results on pattern in the period of the continued fraction expansion of quadratic integers.

A convex function f is said to be pointed if its epigraph has a recession cone which is pointed. Partial pointedness of f refers to the case in which such a recession cone is only partially pointed. In this note we show that the degree of pointedness of f is related to the “thickness” of the effective domain of the conjugate function f*.

We give a class C of continuous functions from [0, 1] onto itself which are chaotic in the sense of Li and Yorke, but with the property that almost all (in the sense of Lebesgue) points of [0, 1] are eventually fixed. For some continuous functions from [0, 1] onto itself which are not in C, We also show that their non-wandering sets are all equal to the interval [0, 1].

Let H be a subgroup of a finite group G. In this paper the Brauer correspondence between blocks of H and blocks of G is characterized in terms of a relationship among the block idempotents.

An unsolved problem in discrete analytic function theory has been to find a suitable analogue of the function . An analogue z(α), of the function zα, is found here for discrete analytic functions of the first kind (or monodiffric functions). This function resolves a conjecture of Isaacs in the negative, and at the same time it introduces multi-valued functions into the discrete analytic theory.

This paper looks at simple rings which have right ideals satisfying various types of injectivity conditions. We characterise when a simple regular ring is right self-injective and show that if R is a simple ring in which every right ideal is the direct sum of quasi-continuous right ideals then R is either Artinian or a non-selfinjective right Goldie ring in which every right ideal is a direct sum of uniform right ideals.

Let G be the group of isometries of a homogeneous tree . In the first part of this paper we decompose G in terms of certain subgroups N, ℤ, and K to obtain the related integral formula

Then, by using ideas of A. Ionescu and the formula above, we prove that and that a related maximal operator on  is bounded from L2, 1() to L2,∞(). We finally show that Lp, 1(K\G/K) is a commutative Banach algebra of convolutors for Lp(G) and give an explicit description of its Gelfand spectrum.

We find the spectrum of the Berezin operator T on L∞(Bn), then we show that if f ∈ L∞(Bn) satisfies Sf = rf for some r in the unit circle, where S is any convex combination of the iterations of T, then f is M-harmonic.

Finally we decompose the subspace of L∞(Bn) where lim Tkf exists into the direct sum of two subspaces of L∞(Bn).

The indecomposable representations in characteristic two of the groups PSL(2, q) where q is congruent to 3 or 5 modulo 8 are classified. For q = 3 or 5 the classification is obtained by explicit construction of modules, using the Green correspondence to prove completeness. For larger q, the classification is obtained using equivalences between appropriate categories of modules.

A survey is made of the way in which a variety of physical scalar and vector quantities are diffused, and convected out of fixed regions in moving fluids. In particular, viscous flow itself is viewed as the diffusion and convection of circulation which is generated at the boundaries of the fluid by the no-slip condition. The non-linearity of the problem arises from the fact that the convection field is in part self-generated by the diffused circulation. Ways of overcoming these difficulties are reviewed in the light of the above point of view. New kinematic interpretations are given for the equations governing axisymmetric viscous flow and these are used to determine the flow vector for ring circulation, angular momentum and other relevant physical quantities.