Algebraic solutions of certain Painlevé VI equations are produced by solving linear systems with monodromy contained in the octahedral subgroup of $\SO(3)$. The method uses the algebraic geometry of special plane curves, and makes contact with some classical geometrical problems.

Ramanujan presented (without proof) the following double inequality for the gamma function:\begin{equation}\sqrt{\pi}\Bigl(\frac{x}{e}\Bigr)^x\Bigl[8x^3+4x^2+x+\frac{1}{100}\Bigr]^{1/6} < \Gamma(x+1)< \sqrt{\pi}\Bigl(\frac{x}{e}\Bigr)^x\Bigl[8x^3+4x^2+x+\frac{1}{30}\Bigr]^{1/6} \quad{(x\geq 0)}.\end{equation}Recently, Karatsuba established that these inequalities hold for $x\geq 1$. We show that this can be slightly improved: the inequalities hold for all $x\geq 0$, even if we replace $1/100$ by \[\alpha= \min_{0.6\leq x\leq 0.7}f(x)=0.0100450\dots,\]where $f(x)=(1/\pi)^3[\Gamma(x+1)(e/x)^x]^6-8x^3-4x^2-x$. Moreover, $\alpha$ and $1/30$ are the best possible constant terms.

Barbasch and Vogan showed that the Kazhdan–Lusztig cells of a finite Weyl group are compatible with parabolic subgroups. Their proof uses the known bridge between the theory of cells and the theory of primitive ideals. In this paper, an elementary, self-contained proof of this result is provided, which works for arbitrary Coxeter groups and Lusztig's general definition of cells (involving Iwahori–Hecke algebras with unequal parameters). The argument is based on a recent paper by Howlett and Yin.

It is shown that the mod $3$ cohomology of a $1$-connected, homotopy associative mod $3$$H$-space that is rationally equivalent to the Lie group $E_6$ is isomorphic to that of $E_6$ as an algebra. Moreover, it is shown that the mod $3$ cohomology of a nilpotent, homotopy-associative mod $3$$H$-space that is rationally equivalent to $E_6$, and whose fundamental group localized at $3$ is non-trivial, is isomorphic to that of the Lie group $\Ad E_6$ as a Hopf algebra over the mod $3$ Steenrod algebra. It is also shown that the mod $3$ cohomology of the universal cover of such an $H$-space is isomorphic to that of $E_6$ as a Hopf algebra over the mod $3$ Steenrod algebra.

Necessary and sufficient conditions are given for a Polish topological group to be ‘almost free’. It is deduced that the existence of one free subgroup of a Polish group can lead to the existence of many free subgroups of maximal rank. Applications are given to permutation groups, profinite groups, Lie groups and unitary groups.

The paper gives lists of all the primitive permutation groups of squarefree degree. All such groups are either solvable and act on a prime number of points, or are almost simple. Among the almost simple examples, the groups of Lie type have rank at most $2$, or the point stabilizer is a parabolic subgroup.

This paper demonstrates that, in dimensions $n\ge 3$, the metric boundary of a conformal deformation of the unit ball is pathwise connected, and even of bounded turning, provided that the conformal scaling factor satisfies a Harnack inequality and the volume growth of the deformed space is at most euclidean.

Let $\sigma$ be a non-degenerate positive $M_n$-valued measure on a locally compact group $G$ with $\|\sigma\|\,{=}\,1$. An $M_n$-valued Borel function $f$ on $G$ is called $\sigma$-harmonic if $f(x) = \int_G f(xy^{-1})\,d\sigma(y)$ for all $x\,{\in}\,G$. Given such a function $f$ which is bounded and left uniformly continuous on $G$, it is shown that every central element in $G$ is a period of $f$. Further, it is shown that $f$ is constant if $G$ is nilpotent or central.

It is proved that if $P_G(s)$ has an Euler product expansion with all factors of the form $1-c_i/q_i^s$ where each $q_i$ is a prime power, then $G$ is soluble.

Let $M$ be a compact complex Kähler manifold of dimension $n\geq 3$, and suppose that $N\hookrightarrow M$ is a closed complex submanifold of dimension $n-1$. Denote by $\mathcal{KC}(1)$ the space of classes of Kähler forms $[\omega]\in H^{1,1}(M,\mathbb R)$ that define Kähler metrics of volume $1$ on $M$, and define $\mathbf f:\mathcal{KC}(1)\longrightarrow\mathbb R$ by $\mathbf f([\omega])=({1}/{(n-1)!})\int _N\omega^{n-1}$, equal to the volume of $N$ in the metric induced by $\omega$. In this paper, the critical points of $\mathbf f$ are studied. The Riemann–Hodge bilinear relations of $M$ and $N$ are shown to imply that any critical point of $\mathbf f$ is the strict global minimum; also, the hard Lefschetz theorem determines the critical point $[\omega]$. A positive multiple of $[\omega]\in H^2(M,\mathbb R)$ is the Poincaré dual of the homology class of $N$. If this is applied to the Theta divisor of the Jacobian $J(C)$ of a compact Riemann surface $C$, the Theta metric minimizes the volume of the Theta divisor within all Kähler metrics of volume one on $J(C)$.

A generalization of norm type cone-compression and -expansion results due to Krasnosel'skiì (see Dokl. Akad. Nauk SSSR NS 135 (1960) 527–530) is presented here for single-valued completely continuous maps defined on a Fréchet space. Applications to second-order differential equations on the half line are presented, and the existence of nontrivial solutions is established.

Global convergence is established in this paper for monotone and subhomogeneous discrete dynamical systems on product Banach spaces. This result is then used to obtain the asymptotic periodicity of solutions to a class of periodic and cooperative reaction-diffusion systems.

For arbitrary infinite directed graphs $E$, the characterisation of the (not necessarily simple) Cuntz–Krieger algebras $C^*(E)$ which are purely infinite in the sense of Kirchberg–Rørdam is given. It is also shown that $C^*(E)$ has real rank zero if and only if the graph $E$ satisfies Condition (K).

If $\Gamma$ is a discrete group, conditions are given that ensure that the full group C*-algebra $C^*(\Gamma)$ is primitive, thereby partially answering a question posed by P. de la Harpe.

Let $A_n=K\langle x_1,\ldots,x_n \rangle$ be a free associative algebra over a field $K$. In this paper, examples are given of elements $u \in A_n$, $n \ge 3$, such that the factor algebra of $A_n$ over the ideal generated by $u$ is isomorphic to $A_{n-1}$, and yet $u$ is not a primitive element of $A_n$ (that is, it cannot be taken to $x_1$ by an automorphism of $A_n$). If the characteristic of the ground field $K$ is $0$, this yields a negative answer to a question of G. Bergman. On the other hand, by a result of Drensky and Yu, there is no such example for $n=2$. It should be noted that a similar question for polynomial algebras, known as the embedding conjecture of Abhyankar and Sathaye, is a major open problem in affine algebraic geometry.