Yang proved in [10] that if f and f(k) have no fix-points for every f∈F, where F is a family of meromorphic functions in a domain G and k a fixed integer, then F is normal in G. In this paper we prove normality for families F for which every f∈F omits ψ1 and f(k) omits ψ2, where ψ1 and ψ2 are analytic functions with ψ(k)1[nequiv ]ψ2.

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.

A new estimate for the exponential sum with square-free numbers is established. This result is applied to the problem of finding the number of representations of a large integer as a sum of three square-free numbers.

A finitary orbit equivalence mapping between the binary and ternary odometers is constructed.

In this paper we give answers to some open questions concerning generation and enumeration of finite transitive permutation groups. In [1], Bryant, Kovács and Robinson proved that there is a number c′such that each soluble transitive permutation group of degree n [ges ] 2 can be generated by [c′ n/ √log n] elements, and later A. Lucchini [5] extended this result (with a different constant c′) to finite permutation groups containing a soluble transitive subgroup. We are now able to prove this theorem in full generality, and this solves the question of bounding the number of generators of a finite transitive permutation group in terms of its degree. The result obtained is the following.

The Hofer–Zehnder theorem states that almost every hypersurface in a thickening of a hypersurface $S$ in a symplectic manifold $(M,\omega)$ carries a closed characteristic, provided that $S$ bounds a compact submanifold and $(M,\omega)$ has finite capacity. It is shown that it is enough to assume that the thickening of $S$ has finite capacity.

The present paper deals with real infinite-dimensional normed spaces; some of the main concepts here make sense, and have been treated in the literature, in the general context of topological Hausdorff linear spaces over reals.

A subset of a normed space X is a body if it is different from X itself and is the closure of its non-empty interior. A covering of X by bodies is called a tiling ofX whenever any two different members of it have disjoint interiors. The elements of such a covering are called tiles. A tiling is bounded (respectively convex) whenever each tile is bounded (respectively convex).

Some generalizations of the classical Hurewicz formula are obtained for the extension dimension and $C$ -spaces. A characterization is also given of the class of metrizable spaces that are absolute neighborhood extensors for all metrizable $C$ -spaces.

We solve the functional equations

formula here

for suitable functions f, g and h subject to x + y + z = 0. These equations essentially characterise the Weierstrass [pscr ]-function and its degenerations.

A quantum deformation of the classical conjugation action of ${\rm GL}(N,{\bb C})$ on the space of $N\times N$ matrices $M(N,{\bb C})$ is defined via a coaction of the quantum general linear group ${\cal O}({\rm GL}_q(N,{\bb C}))$ on the algebra of quantum matrices ${\cal O}(M_q(N,{\bb C}))$ . The coinvariants of this coaction are calculated. In particular, interesting commutative subalgebras of ${\cal O}(M_q(N,{\bb C}))$ generated by (weighted) sums of principal quantum minors are obtained. For general Hopf algebras, co-commutative elements are characterized as coinvariants with respect to a version of the adjoint coaction.

The authors show that any $k$ -Osserman Lorentzian algebraic curvature tensor has constant sectional curvature, and give an elementary proof that any local 2-point homogeneous Lorentzian manifold has constant sectional curvature. They also show that a Szabó Lorentzian covariant derivative algebraic curvature tensor vanishes.

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.

Let $K$ be a convex body with boundary of class ${\mathcal{C}}^{\geq 4}$. We prove that, for a sufficiently small $\delta$, the floating body $K_{\delta}$ is homothetic to $K$ if and only if $K$ is an ellipsoid. The proof relies on properties of the affine curvature flow.

We characterize the extremal functions for Heisenberg type inequalities of the form

(formula here)

The method also yields an alternative and elementary derivation of the sharp constant in Nash's inequality, originally found by Carlen and Loss.

This paper concerns rigidity of the mapping class groups. It is shown that any homomorphism $\varphi\,{:}\,\mcg_g\,{\to}\,\mcg_h$ between mapping class groups of closed orientable surfaces with distinct genera $g\,{>}\,h$ is trivial if $g\,{\geq}\, 3$, and has finite cyclic image for all $g\,{\geq}\, 1$.

Some implications are drawn for more general homomorphs of these groups.

Necessary and sufficient conditions for Lefschetz fibrations over D2 and S2 to carry spin structures are presented. The conditions are given in terms of vanishing cycles and homology sections.

We establish the peak point conjecture for uniform algebras generated by smooth functions on two-manifolds: if A is a uniform algebra generated by smooth functions on a compact smooth two-manifold M, such that the maximal ideal space of A is M, and every point of M is a peak point for A, then A = C(M). We also give an alternative proof in the case when the algebra A is the uniform closure P(M) of the polynomials on a polynomially convex smooth two-manifold M lying in a strictly pseudoconvex hypersurface in Cn.

The global and local topological zeta functions are singularity invariants associated to a polynomial f and its germ at 0, respectively. By definition, these zeta functions are rational functions in one variable, and their poles are negative rational numbers. In this paper we study their poles of maximal possible order. When f is non-degenerate with respect to its Newton polyhedron, we prove that its local topological zeta function has at most one such pole, in which case it is also the largest pole; we give a similar result concerning the global zeta function. Moreover, for any f we show that poles of maximal possible order are always of the form −1/N with N a positive integer.

Throughout this paper, we shall let Σ be a subset of [0, 1] having cardinality N. We shall consider Σ to be a set of slopes, and for any s∈Σ, we shall let es be the unit vector of slope s in ℝ2. Then, following [7], we define the maximal operator on ℝ2 associated with the set Σ by

formula here

The history of the bounds obtained on [Mfr ]0Σ is quite curious. The earliest study of related operators was carried out by Cordoba [2]. He obtained a bound of C√(1+logN) on the L2 operator norm of the Kakeya maximal operator over rectangles of length 1 and eccentricity N. This operator is analogous to [Mfr ]0Σ with

formula here

However, for arbitrary sets Σ, the best known result seems to be C(1+logN). This follows from Lemma 5.1 in [1], but a point of view which produces a proof appears already in [8]. However, in this paper, we prove the following.

We describe completely the possible Gorenstein Artin quotients of the coordinate ring of a set of points in projective space. In addition, we give a sufficient condition for an ideal of a Cohen–Macaulay graded k-algebra A to be isomorphic to the canonical module of A. This leads to a description of the canonical module in the case when A is the coordinate ring of points.