Let $q\geq 1$ be any integer and let $\unicode[STIX]{x1D716}\in [\frac{1}{11},\frac{1}{2})$ be a given real number. We prove that for all primes $p$ satisfying

$$\begin{eqnarray}p\equiv 1\!\!\!\!\hspace{0.6em}({\rm mod}\hspace{0.2em}q),\quad \log \log p>\frac{2\log 6.83}{1-2\unicode[STIX]{x1D716}}\quad \text{and}\quad \frac{\unicode[STIX]{x1D719}(p-1)}{p-1}\leq \frac{1}{2}-\unicode[STIX]{x1D716},\end{eqnarray}$$

$g$

$p$

$\text{gcd}(g,(p-1)/q)=1$

In this paper, we investigate properties of finite-order transcendental meromorphic solutions, rational solutions and polynomial solutions of the difference Painlevé I equation where a, b and c are constants, ∣a∣+∣b∣≠0.

Using some techniques of perturbation theory for Banach space complexes, we obtain necessary and sufficient conditions for the stability of the topological index of an open linear relation under small (with respect to the gap topology) perturbations with linear relations.

Majorant functions for contractors can be defined in a natural war. Such a case is considered here in order to find iterative solutions of general equations in Banach spaces by means of contractors. A class of majorant functions is defined which contains in particular the linear majorant ones. Local and global existence and convergence theorems are proved.

In this paper, we prove that, if $N$ is a positive odd number with $r$ distinct prime factors such that $N\mid \sigma (N)$, then $N\lt {2}^{{4}^{r} - {2}^{r} } $ and $N{\mathop{\prod }\nolimits}_{p\mid N} p\lt {2}^{{4}^{r} } $, where $\sigma (N)$ is the sum of all positive divisors of $N$. In particular, these bounds hold if $N$ is an odd perfect number.

In this paper, a generalisation of the Andersen, Hilton, Rodger Theorem for embedding partial idempotent latin squares is proved. This result is then used to prove that a partial directed m-cycle system of order n can be embedded in a directed m-cycle system of order (2n + 1)m if m is odd, of order 2nm if m ≥ 8 is even, 12n + 1 if m = 6 and approximately 2n + √2n if m = 4.

In this paper, we consider an optimal control problem involving second-order hyperbolic systems with boundary controls. Necessary and sufficient conditions are derived and a result on the existence of optimal controls is obtained. Also, a computational algorithm which generated minimizing sequences of controls is devised and the convergence properties of the algorithm are investigated.

We introduce ‘generalised higher-rank k-graphs’ as a class of categories equipped with a notion of size. They extend not only higher-rank k-graphs, but also the Levi categories introduced by the first author as a categorical setting for graphs of groups. We prove that examples of generalised higher-rank k-graphs can be constructed using Zappa–Szép products of groupoids and higher-rank graphs.

We prove a divisibility result on sums involving the Apéry-like polynomials

$$ \begin{align*} V_n(x)=\sum_{k=0}^n {n\choose k}{n+k\choose k}{x\choose k}{x+k\choose k}, \end{align*} $$

which confirms a conjectural congruence of Z.-H. Sun. Our proof relies on some combinatorial identities and transformation formulae.

A symmetry of an ordered Banach space is an order and norm isomorphism which commutes with its ideal centre. A class of ordered Banach spaces is introduced to show that, for a space in this class, the group of symmetries is trivial if and only if the space is lattice-ordered. When this group becomes larger, the space approaches an antilattice. This phenomenon is also investigated.

Given a positive integer $m$, a finite $p$-group $G$ is called a $BC(p^{m})$-group if $|H_{G}|\leq p^{m}$ for every nonnormal subgroup $H$ of $G$, where $H_{G}$ is the normal core of $H$ in $G$. We show that $m+2$ is an upper bound for the nilpotent class of a finite $BC(p^{m})$-group and obtain a necessary and sufficient condition for a $p$-group to be of maximal class. We also classify the $BC(p)$-groups.

It is shown that expanding self-maps of the interval with a finite number of turning points must have periodic points whose periods are not integral power of 2 and therefore are chaotic.

A unified theory of continuous and certain non-continuous functions, initiated in an earlier paper, is further elaborated. The proposed theory provides a common platform for dealing simultaneously with continuous functions and a host of non-continuous functions including lower (upper) semicontinuous functions, almost continuous functions, weakly continuous functions (encountered in functional analysis), c-continuous functions, δ-continuous functions, semiconnected functions, H-continuous functions s-continuous functions, ε-continuous functions of Klee and several other variants of continuity.

We announce results of Landesman-Lazer type for boundary-value problems for ordinary differential equations. Details will appear elsewhere.

I.N. Herstein has shown that an associative ring in which the nilpotent elements are “well-behaved”, and such that every element satisfies a certain polynomial identity, is commutative. This result is generalized here. Specifically, it is shown that an alternative ring R which satisfies the following three properties is commutative:

(i) for x ∈ R, there exists an integer n(x) and a polynomial px (t) with integer coefficients such that xn+1p(x) = xn;

(ii) for a fixed positive integer m, a a nilpotent and b an arbitrary element of R, a - am commutes with b - bm;

(iii) for the same m, a and b, (ab+b)m = (ba+b)m and (ab)m = ambm.

Examples are given to show that all three properties are essential, and it is shown that for associative rings certain modified versions of these properties are individually enough to assure that the commutator ideal of the ring is nil.

The following is a correct proof of the main theorem of [1]. It should be substituted for the published Section 3, which, as pointed out by Professor B.L.S. Prakasa Rao, contains an error in the equations following (15) on page 49.

It is proved here that for two distribution functions with equal moments up to order n, the number of crossings is at least n, and if exactly n, the remaining odd or even moments (for n even or odd respectively) do not cross again. This both generalises and extends a number of previous results.

This paper considers a special class of stochastic optimal parameter selection problem subject to probability constraints on the state. The system dynamics are governed by a linear Ito stochastic differential equation with controllable parameters appearing nonlinearly in the dynamics. The problem seeks to optimise a cost functional which is quadratic in the state with weighting matrices being time invariant but depending nonlinearly on the parameters. Although the inclusion of the probability state constraints renders the problem insolvable by the conventional LQG theory, we show that the problem can in fact be transformed into an equivalent deterministic optimal parameter selection problem solvable by an existing software MISER. Numerical examples are presented to demonstrate the feasibility and efficiency of the proposed approach.

It is proved that every metrizable topological space without isolated points is the union of a continuum or fewer nowhere dense subsets.

The propogation of disturbance when a shear flow with a free surface, in a channel of infinite horizontal extent and finite depth, is disturbed by the application of time-oscillatory pressure, is studied. The initial value problem is solved by using transform techniques and the steady state solution is obtained therefrom in the limit t → ∞. The effect of the initial shear on the development of the wave system is investigated.