Let Γ⊂SL2(ℤ) be a subgroup of finite index, and let [Hscr ] denote the Hecke algebra of Γ. The aim of this note is to give some information about the action of [Hscr ] on spaces of modular forms for certain noncongruence subgroups Γ, which can be deduced from the geometric results of [9].

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.

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.

An $\varepsilon$ -entropy theoretical solution is given to an analytic function-theoretical version of the 13th problem formulated by D. Hilbert.

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).

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.

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.

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.

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 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.

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.

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.

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.

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.

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 ${\rm PG}(3,q^n)$).

Nous donnons une majoration de la somme d'exponentielles SM = [sum ]Mm=1e(f(m)) lorsque la dérivée cinquième de f est d'un ordre de grandeur constant petit, noté λ, en fonction de M et de λ, améliorant un résultat ancien de Van Der Corput. La démonstration utilise un théorème de moyenne des puissances sixièmes de sommes d'exponentielles qui fait l'objet d'un article indépendant [6].

We give a bound for the exponential sum SM = [sum ]Mm=1e(f(m)) where f is a real-valued function whose fifth derivative is of a constant small size, say λ, by means of M and λ, improving an old result of Van Der Corput. The proof relies on a mean value theorem for sixth powers of exponential sums which is treated independently in [6].

The existence of periodic solutions for the nonlinear asymmetric oscillator $ x''{+}\alpha x^+{-}\beta x^- {=} h(t),\qquad (' = d/dt) $ is discussed, where $\alpha, \beta$ are positive constants satisfying $ {1}/{\sqrt{\alpha}} + {1}/{\sqrt{\beta}} = {2}/{n} $ for some positive integer $n\in {\bf \mathbb{N}}$ and $h(t)\in L^\infty(0,2\pi)$ is $2\pi$-periodic with $x^{\pm}=\max\{\pm x, 0\}$.

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 $n=3$.

In this paper, we show first that any prime (or semiprime) element of a commutative Banach algebra must have closed range. As a corollary, we find that in a commutative radical Banach algebra, all primes are zero divisors; indeed, all semiprimes are zero divisors (see below for the definition of semiprimeness). Our result is also true of a semiprime that is in the centre of a noncommutative Banach algebra.

The proof is fairly simple and entertaining, and we obtain a result that is helpful for the ambitious classification of elements in commutative radical Banach algebras being attempted by Marc Thomas. It is also related to the unbounded Kleinecke–Shirov conjecture.