This paper proves the rationality of Poincaré series whose coefficients are defined using the number of conjugacy classes in a system of finite groups defined as finite images of linear groups over $\Bbb{Z}_{p}$ modulo natural congruence subgroups.

We prove that each interval of the lattice [Escr ] of c.e. sets under inclusion is either a boolean algebra or has an undecidable theory. This solves an open problem of Maass and Stob [11]. We develop a method to prove undecidability by interpreting ideal lattices, which can also be applied to degree structures from complexity theory. We also answer a question left open in [7] by giving an example of a non-definable subclass of [Escr ]* which has an arithmetical index set and is invariant under automorphisms.

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.

In [1], Hirano gives a method for constructing families of curves with a large number of singularities. The idea is to consider an abelian covering of ℙ2 ramified along three lines in general position, and to take the pull-back of a curve C intersecting the lines non-generically. Similar constructions are used by Shimada in [10] and Oka in [8]. We apply this method for the case where C is a conic, constructing a family of curves with the following asymptotic behaviour (see [9]):

formula here

The goal of this paper is to calculate the fundamental group for the curves in this family as well as their Alexander polynomial.

Given a field k and a finite group G acting on the rational function field k(X1, . . ., Xn) as a group of k-automorphisms, an important Noether's problem asks whether the invariant subfieldformula here is purely transcendental over k.

Write $G^{\ast} = G^{{\cal L}{\cal U}{\cal C}}\setminus G$ where $G^{{\cal L}{\cal U}{\cal C}}$ is the largest semigroup compactification of the locally compact group $G$ . Then the set of points of $G^{\ast}$ which are right cancellable in $G^{{\cal L}{\cal U}{\cal C}}$ is large; in fact it has an interior in $G^{\ast}$ which is dense in $G^{\ast}$ . Corollaries are given about the number of left ideals in $G^{{\cal L}{\cal U}{\cal C}}$ and the size of right ideals in the algebra ${{\cal L}{\cal U}{\cal C}}(G)^{\ast}$ .

Let A be a group of automorphisms of the finite group G such that ([mid ]A[mid ], [mid ]G[mid ])=1. Then [mid ]A[mid ]<[mid ]G[mid ]2, and the exponent 2 here is best possible. If, moreover, A is nilpotent of class at most 2, then [mid ]A[mid ]<[mid ]G[mid ]. If A is abelian, then A has a regular orbit on G.

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.

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 aim of this paper is to study properties of sequences that are recursively defined by a linear equation and their applications to the truncated moment problem in connection with the problem of subnormal completion of the truncated weighted shifts. Special cases are considered and some classical results due to Stampfli, Curto and Fialkow are recovered using elementary techniques.

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

The title alludes to a similar title of the paper [3] by Grunewald and Segal, in which it is shown how to solve a quadratic equation in integers. This latter procedure seems to be quite difficult, and the algorithm outlined in [3] is rather involved, although it is completely effective in the logical sense.

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).

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.

We solve the functional equations

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

Let p(n) denote the ordinary partition function. Subbarao conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N ≡ r (mod t) for which p(N) is even, and infinitely many integers M ≡ r (mod t) for which p(M) is odd. We prove the conjecture for every arithmetic progression whose modulus is a power of 2.

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.

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 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.