An extension is obtained to certain maps into full flag manifolds of compact simple Lie groups of the classical Plücker formulae for holomorphic curves in complex projective space.

A complete characterization of the categorical quotients of $({\mathbb P}^1)^n$ by the diagonal action of $\SL(2, {\mathbb C})$ with respect to any polarization is given by M. Polito, in ‘$\SL(2, {\mathbb C})$-quotients de $({\mathbb P}^1)^n$’, C. R. Acad. Sci. Paris Sér. I 321 (1995) 1577–1582. In this paper, these categorical quotients are obtained by certain linear systems on ${\mathbb P}^{n-3}$?>, depending on the given polarization.

Let $X$ be a non-singular real algebraic curve in ${\rm C\!\!\!I}P^2$ of even degree. In this paper a refinement is proved of a theorem of Kharlamov about ($M - 2$)-curves that are invariants under the projective involution. In particular, if the ($M - 2$)-symmetric curve $X$ satisfies the Arnold congruence, then either $X$ or its twin is a separating curve.

A Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold is proved, under the condition that $E^* \otimes F$ is Griffiths $k$-positive. This result is applied to find some homotopy groups of the Brill–Noether loci for a generic curve.

The Hall–Paige conjecture deals with conditions under which a finite group $G$ will possess a complete mapping, or equivalently a Latin square based on the Cayley table of $G$ will possess a transversal. Two necessary conditions are known to be: (i) that the Sylow 2-subgroups of $G$ are trivial or non-cyclic, and (ii) that there is some ordering of the elements of $G$ which yields a trivial product. These two conditions are known to be equivalent, but the first direct, elementary proof that (i) implies (ii) is given here.

It is also shown that the Hall–Paige conjecture implies the existence of a duplex in every group table, thereby proving a special case of Rodney's conjecture that every Latin square contains a duplex. A duplex is a ‘double transversal’, that is, a set of $2n$ entries in a Latin square of order $n$ such that each row, column and symbol is represented exactly twice.

Let ${\cal D}$ be a complex of non-trivial $p$-subgroups of a finite group $G$, closed under $G$-conjugation. In this paper, a $p$-local geometry $\Delta_p(G; {\cal D})$ is introduced for $G$ associated with the complex ${\cal D}$. The homotopy equivalence between $\Delta_p(G; {\cal D})$ and ${\cal D}$ is also studied.

If $G$ is a finite group and $V$ is a finite-dimensional ${\rm Q\!\!\!I}\,[G]$-module, $V$ is isomorphic to its contragredient module $V^*$. In general, $V$ need not contain any ${\bb Z}[G]$-lattice which is locally isomorphic to its contragredient lattice. Nevertheless, it turns out that for every $V$ there exists another ${\rm Q\!\!\!I}\,[G]$-module $V^{\prime}$; such that both $V^{\prime}$; and $V \oplus V^{\prime}$; contain ${\bb Z}[G]$-lattices which are locally isomorphic to their contragredient lattices.

Given a coaction $\alpha$ of a Hopf ${\rm C}^*$-algebra $A$ on a ${\rm C}^*$-algebra $B$ with an $\alpha$-invariant ${\rm C}^*$-subalgebra $C$, and a conditional expectation $E:B \rightarrow C$ commuting with $\alpha$, it is shown that if $(\pi, u)$ is a covariant representation of the system ($C, A, \alpha\mid_C$), then there is an associated covariant representation ($\tilde{\pi}, \tilde{u}$) of the system ($B, A, \alpha$), where $\tilde{\pi}$ is the representation induced from $\pi$ up to $B$ via $E$, and $\tilde{u}$ is a unitary corepresentation of $A$ naturally associated with $u$. Some applications are also discussed, including a lifting of ergodic coactions to von Neumann algebras, and a characterization of the amenability of multiplicative unitary operators via infinite tensor product covariant representations.

An element u of a norm-unital Banach algebra A is said to be unitary if u is invertible in A and satisfies $\Vert u\Vert =\Vert u^{-1}\Vert =1$. The norm-unital Banach algebra A is called unitary if the convex hull of the set of its unitary elements is norm-dense in the closed unit ball of A. If X is a complex Hilbert space, then the algebra $\BL(X)$ of all bounded linear operators on X is unitary by the Russo–Dye theorem. The question of whether this property characterizes complex Hilbert spaces among complex Banach spaces seems to be open. Some partial affirmative answers to this question are proved here. In particular, a complex Banach space X is a Hilbert space if (and only if) $\BL(X)$ is unitary and, for Y equal to $X,$ $X^*$ or $X^{**},$ there exists a biholomorphic automorphism of the open unit ball of Y that cannot be extended to a surjective linear isometry on Y.

Suppose that $R$ goes to infinity through a second-order lacunary set. Let $S_R$ denote the $R$th spherical partial inverse Fourier integral on ${\rm I\!R}^d$. Then $S_R f$ converges almost everywhere to $f$, provided that $f$ satisfies\[\int \widehat{f}(\xi)\log\log(8+|\xi|)^2\,d\xi < \infty.\]

The paper presents an improvement of the author's earlier estimate for the average number of limit cycles of a planar polynomial vector field situated in a neighbourhood of the origin, provided that the field is close enough to a linear centre in a larger neighbourhood. The result follows from a new distributional inequality for the number of zeros of a family of univariate holomorphic functions depending holomorphically on a parameter.

In this note a proof is given that in the case of finitely generated amenable groups, the classical zero divisor conjecture implies the analytic zero divisor conjecture of Linnell.

This paper is concerned with the oscillatory behaviour of first-order delay differential equations of the form\begin{equation}x^{\prime}(t) + p(t)x(\tau(t)) = 0, \quad t \geqslant t_0,\end{equation}where $p, \tau \in C([t_0,\infty),{\rm I\!R}^+),{\rm I\!R}^+ = [0,\infty),\tau(t)$ is non-decreasing, $\tau(t) < t$ for $t \geqslant t_0$ and $\lim_{t\rightarrow\infty}\tau(t) = \infty$. Let the numbers $k$ and $L$ be defined by\[k = \mathop{{\lim\inf}}_{{t\rightarrow\infty}} \int\nolimits^t_{\tau(t)} p(s)ds\quad\hbox{and}\quad L = \mathop{\lim\sup}_{{t\rightarrow\infty}} \int\nolimits^t_{\tau(t)} p(s)ds.\]

It is proved here that when $L < 1$ and $0 < k \leqslant 1/e$ all solutions of equation (1) oscillate in several cases in which the condition\[L > \frac{\ln\lambda_1 - 1 + \sqrt{5 - 2\lambda_1 + 2k\lambda_1}}{\lambda_1}\]holds, where $\lambda_1$ is the smaller root of the equation $\lambda = e^{k\lambda}$.

Suppose that $T(S_0)$ is the Teichmüller space of a compact Riemann surface $S_0$ of genus $g>1$. Let $d_T(\cdot,\cdot)$ be the Teichmüller metric of $T(S_0)$ and let $d_S(\cdot,\cdot)$ be a metric of $T(S_0)$ defined by the length spectrums of Riemann surfaces. The author showed in a previous paper that $d_T$ and $d_S$ are topologically equivalent and $d_S(\tau_1,\tau_2)\leq d_T(\tau_1,\tau_2) \leq 2d_S(\tau_1,\tau_2)+C(\tau_1)$, where $C(\tau_1)$ is a constant depending on $\tau_1$. In this paper, it is shown that $d_T$ and $d_S$ are not metrically equivalent; that is, there is no constant $C>0$ such that $d_S(\tau_1,\tau_2)\leq d_T(\tau_1,\tau_2)\leq Cd_S(\tau_1,\tau_2)$ for all $\tau_1 {\rm and} \tau_2 {\rm in} T(S_0)$.

Let $D\subset H^n(-k^2)$ be a convex compact subset of the hyperbolic space $H^n(-k^2)$ with non-empty interior and smooth boundary. It is shown that the volume of D can be estimated by the total curvature of $\partial D$. More precisely, $(n-1)k^n{\rm Vol}(D)+ {\rm Vol}(S^{n-1})\leq \int_{\partial D}K$, where K denotes the Gauss–Kronecker curvature of $\partial D$ and Vol$(S^{n-1})$?> denotes the Euclidean volume of the sphere.

Let $\phi\,{:}\,G \to G$ be a group endomorphism where G is a finitely generated group of exponential growth, and denote by $R(\phi)$ the number of twisted ϕ-conjugacy classes. Fel'shtyn and Hill (K-theory 8 (1994) 367–393) conjectured that if ϕ is injective, then R(ϕ) is infinite. This paper shows that this conjecture does not hold in general. In fact, R(ϕ) can be finite for some automorphism ϕ. Furthermore, for a certain family of polycyclic groups, there is no injective endomorphism ϕ with $R({\phi}^n)\,{<}\,\infty$ for all n.

It is proved here that if $f : {\rm C\!\!\!I} \rightarrow \bar{{\rm C\!\!\!I}}$ is an elliptic function and $q$ is the maximal multiplicity of all poles of $f$, then the Hausdorff dimension of the Julia set of $f$ is greater than $2q/(q+1)$, and the Hausdorff dimension of the set of points that escape to infinity is less than or equal to $2q/(q+1)$. In particular, the area of this latter set is equal to 0.