In this paper, Conway's group, $\Co_3$, is characterized among simple groups of even type with $e(G)=3$, by a restriction on the $p$-local structure for some odd prime $p$ for which $m_{2,p}(G) = 3$.

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.

It is shown that the classical result of Pringsheim on the convergence of complex continued fractions extends to all dimensions, both geometrically and as a result on continuedfractions whose coefficients lie in a Clifford algebra.

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. The asymptotics of the number of geodesics in M starting from and returning to a given cusp, and of the number of horoballs at parabolic fixed points in the universal cover of M, are studied in this paper. The case of SL(2, ℤ), and of Bianchi groups, is developed.

a general, albeit simple, approach to obtaining rearrangement inequalities of maximal operators is given. being applied to specific operators it leads not only to new proofs of well-known relations in a simpler and shorter way, but also to new, profitable results.

If s1, s2, …, st are integers such that n − 1 = s1 + s2 + ··· + st and such that for each i (1 [les ] i [les ] t), 2 [les ] si [les ] n − 1 and sin is even, then Kn can be expressed as the union G1 ∪ G2 ∪ ··· ∪ Gt of t edge-disjoint factors, where for each i, Gi is si-regular and si-connected. Moreover, whenever si = sj, Gi and Gj are isomorphic.

The author recently proved that 1-cotilting modules are pure-injective; however, the problem of determining whether n-cotilting modules are pure-injective remained open. In this paper, necessary and sufficient conditions are given for the pure-injectivity of n-cotilting modules.

Let ϕ[ratio ]G→G be an endomorphism of a finitely generated residually finite group. R. Hirshon asked if there exists n such that the restriction of ϕ to ϕn(G) is injective. We give an example to show that this is not always the case.

For each d[ges ]2 we construct a connected open set Ω⊂ℝd such that Ω=int (clos(Ω)), and for each k[ges ]1 and each p∈[1, +∞), the subset Wk, ∞(Ω) fails to be dense in the Sobolev space Wk, p(Ω), in the norm of Wk, p(Ω).

An ovoid of ${\rm PG}(3,q)$ can be defined as a set of $q^2+1$ points with the property that every three points span a plane, and at every point there is a unique tangent plane. In 2000, M. R. Brown proved that if an ovoid of ${\rm PG}(3,q)$, $q$ even, contains a conic, then the ovoid is an elliptic quadric. Generalising the definition of an ovoid to a set of $(n-1)$-spaces of ${\rm PG}(4n-1,q)$, J. A. Thas introduced the notion of pseudo-ovoids or eggs: a set of $q^{2n}+1$$(n-1)$-spaces in ${\rm PG}(4n-1,q)$, with the property that any three egg elements span a $(3n-1)$-space and at every egg element there is a unique tangent $(3n-1)$-space. In this paper, a proof is given that an egg in ${\rm PG}(4n-1,q)$, $q$ even, contains a pseudo-conic (that is, a pseudo-oval arising from a conic of ${\rm PG}(2,q^n)$) if and only if the egg is classical (that is, arising from an elliptic quadric in ${\rm PG}(3,q^n)$).

It is shown that for every compact group G, L1(G)ˆ is unique and minimal among all the closed subsets I of M(G)** such that I is a proper (≠0, ≠M(G)**) algebraic ideal, and such that I is solid with respect to absolute continuity; that is, n∈L1(G)ˆ whenever n∈M(G)** and n[Lt ]μ∈L1(G)ˆ.

In this paper, we define a Hardy–Littlewood maximal function operator M associated with the Jacobi transform, and prove that M is of weak (1, 1) and strong (p, p) (p > 1) type.

It is shown that every transcendental entire function f grows transcendentally in a sequence of cercles de remplissage. An example shows that if

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then there may be no sequence of cercles de remplissage the union of which contains infinitely many zeros of f. It is also shown that every transcendental entire function f has a Hayman direction, that is, a direction θ such that, in every open sector containing θ, either f assumes all complex values infinitely often, or else every derivative of f assumes all complex values, except possibly zero, infinitely often.

Holomorphic principal bundles over a compact Riemann surface X that admits a flat connection are considered. A holomorphic G-bundle over X, where G is a connected semisimple linear algebraic group over ${\Bbb C}$, admits a flat connection if and only if the adjoint vector bundle admits one. More generally, for a complex reductive group G, the necessary and sufficient condition on a G-bundle to admit a flat connection is described. This simplifies the criterion obtained by the authors and given in Math. Ann. 322 (2002) 333–346.

It is known that quadratic polynomials do not have real rational orbits of period four. By using a two-dimensional model for the quadratic family, this result is generalized for complex rational orbits.

Denote by f(n) the number of subgroups of the symmetric group Sym(n) of degree n, and by ftrans(n) the number of its transitive subgroups. It was conjectured by Pyber [9] that almost all subgroups of Sym(n) are not transitive, that is, ftrans(n)/f(n) tends to 0 when n tends to infinity. It is still an open question whether or not this conjecture is true. The difficulty comes from the fact that, from many points of view, transitivity is not a really strong restriction on permutation groups, and there are too many transitive groups [9, Sections 3 and 4]. In this paper we solve the problem in the particular case of permutation groups of prime power degree, proving the following result.

Let p be a prime number, and let G be a p-group which is not elementary abelian. For every mod-p cohomology class ξ of G which restricts trivially to all proper subgroups, we show that ξp = 0. This gives upper bounds for nilpotency degrees of such classes of G and of nilpotent mod-p cohomology classes of finite groups.

Using the twistorial approach and some previous results, we prove the conjecture that the dimension of the moduli space of harmonic maps of area $4 \pi d$ from the 2-sphere to the $2n$-sphere is $2 d+n^2$ for the particular case $n=3$.

The main theorem of this paper generalizes a classic Aumann's characterization of compact convex sets, via the acyclicity of their hypersections, to arbitrary weakly closed subsets of a locally convex linear space.

Let X be a separable infinite dimensional Banach space. There exist a closed set A ⊂ X which contains a translate of any compact set in the unit ball of X, and a bi-Lipschitz homeomorphism f of X onto X so that every line in X intersects f(A) in a set of Lebesgue measure zero.