A coordinate-system called $\lambda$-lengths is constructed for an SL$(2,{\Bbb C})$ representation space of punctured surface groups. These $\lambda$-lengths can be considered as complexification of R. C. Penner's $\lambda$-lengths for decorated Teichmüller spaces of punctured surfaces. Via the coordinates the mapping class group acts on the representation space as a group of rational transformations. This fact is applied to find hyperbolic 3-manifolds which fibre over the circle.

The paper considers stationary critical points of the heat flow in sphere SN and in hyperbolic space ℍN, and proves several results corresponding to those in Euclidean space ℝN which have been proved by Magnanini and Sakaguchi. To be precise, it is shown that a solution u of the heat equation has a stationary critical point, if and only if u satisfies some balance law with respect to the point for any time. In Cauchy problems for the heat equation, it is shown that the solution u has a stationary critical point if and only if the initial data satisfies the balance law with respect to the point. Furthermore, one point, say x0, is fixed and initial-boundary value problems are considered for the heat equation on bounded domains containing x0. It is shown that for any initial data satisfying the balance law with respect to x0 (or being centrosymmetric with respect to x0) the corresponding solution always has x0 as a stationary critical point, if and only if the domain is a geodesic ball centred at x0 (or is centrosymmetric with respect to x0, respectively).

Let [Bscr ]2 denote the family of all circular discs in the plane. It is proved that the discrepancy for the family {B1 × B2 [ratio ] B1, B2 ∈ [Bscr ]2} in R4 is O(n1/4+ε) for an arbitrarily small constant ε > 0, that is, it is essentially the same as that for the family [Bscr ]2 itself. The result is established for the combinatorial discrepancy, and consequently it holds for the discrepancy with respect to the Lebesgue measure as well. This answers a question of Beck and Chen. More generally, we prove an upper bound for the discrepancy for a family {Πki=1Ai[ratio ] Ai∈Ai, i = 1, 2, …, k}, where each Ai is a family in Rdi, each of whose sets is described by a bounded number of polynomial inequalities of bounded degree. The resulting discrepancy bound is determined by the ‘worst’ of the families Ai , and it depends on the existence of certain decompositions into constant-complexity cells for arrangements of surfaces bounding the sets of Ai. The proof uses Beck's partial coloring method and decomposition techniques developed for the range-searching problem in computational geometry.

The paper shows that, if the operator T[ratio ]A(γ)→B(γ) is compact for almost every γ∈Γ, then T[ratio ]F(Ā)→F([Bmacron]) is compact when F([Xmacron])=(X)SK or F([Xmacron])=(X)SJ is the interpolation functor constructed for infinite families of Banach spaces and S satisfies certain conditions.

For a linear and bounded operator T from a Banach space X into a Banach space Y, let ϱ(T[mid ][Iscr ]n, [Rscr ]n) and ϱ(T[mid ][Iscr ]n, [Gscr ]n) denote the Rademacher and Gaussian cotype 2 norm of T computed with n vectors, respectively. It is shown that the sequence ϱ(T[mid ][Iscr ]n, [Rscr ]n) has submaximal behaviour if and only if ϱ(T[mid ][Iscr ]n, [Gscr ]n) has. This means that

[formula here]

Moreover, the class of these operators coincides with the class of operators preserving copies of ln∞ uniformly. The tool connecting these concepts is the equal norm Rademacher cotype of operators.

Let

formula here

be a formal power series. In 1913, G. Pólya [7] proved that if, for all sufficiently large n, the sections

formula here

have real negative zeros only, then the series (0.1) converges in the whole complex plane C, and its sum f(z) is an entire function of order 0. Since then, formal power series with restrictions on zeros of their sections have been deeply investigated by several mathematicians. We cannot present an exhaustive bibliography here, and restrict ourselves to the references [1, 2, 3], where the reader can find detailed information.

In this paper, we propose a different kind of generalisation of Pólya's theorem. It is based on the concept of multiple positivity introduced by M. Fekete in 1912, and it has been treated in detail by S. Karlin [4].

In this paper we define three elements of a certain generalised cohomology ring BP〈m, n〉* BVk. Here m, n and k are non-negative integers with k+m[les ]n+1, there is a fixed prime p not exhibited in the notation, and Vk is an elementary Abelian p-group of rank k. We show that these elements are equal; this is striking, because the three definitions are very different. The significance of our equation is not yet entirely clear, but it makes contact with other work in the literature in a number of fascinating ways.

The selection problem for strong M-bases is shown to have a negative solution in Hilbert space. The constructions are used to demonstrate that spectral synthesis for compact operators is not preserved by inflation. The results show that a number of statements in the literature are incorrect.

A family [Fscr ] of functions from ℝn to ℝ is k-point separating if, for every k-subset S of ℝn, there is a function f∈[Fscr ] such that f is one-to-one on S. The paper shows that, if the functions are required to be linear (or smooth), then a minimum k-point separating family [Fscr ] has cardinality n(k−1). In the linear case, this result is extended to a larger class of fields including all infinite fields as well as some finite fields (depending on k and n). Also, some partial results are obtained for continuous functions on ℝn, including the case when k is infinite. The proof of the main result is based on graph theoretic results that have some interest in their own right. Say that a graph is an n-tree if it is a union of n edge-disjoint spanning trees. It is shown that every graph with k[ges ]2 vertices and n(k−1) edges has a non-trivial subgraph which is an n-tree. A determinantal criterion is also established for a graph with k vertices and n(k−1) edges to be an n-tree.

We investigate the closability of those derivations D defined on a (non necessarily closed) subalgebra B of a complex Banach algebra A for which the conditions

BAB⊂B and dim[Bk∩Rad(A)]<∞

hold for some k∈ℕ, where Rad(A) stands for the Jacobson radical of A. In this situation we show that the separating subspace [Sscr ](D) for D satisfies the property

B[B∩[Sscr ](D)]B⊂Rad(A).

Furthermore, we demonstrate several specially relevant situations in which we get a ‘closability property’ which is more precise than the former one.

In [6] S. Shelah showed that in the endomorphism semi-group of an infinitely generated algebra which is free in a variety one can interpret some set theory. It follows from his results that, for an algebra Fℵ which is free of infinite rank ℵ in a variety of algebras in a language L, if ℵ > |L|, then the first-order theory of the endomorphism semi-group of Fℵ, Th(End(Fℵ)), syntactically interprets Th(ℵ,L2), the second-order theory of the cardinal ℵ. This means that for any second-order sentence χ of empty language there exists χ*, a first-order sentence of semi-group language, such that for any infinite cardinal ℵ > |L|,

formula here

In his paper Shelah notes that it is natural to study a similar problem for automorphism groups instead of endomorphism semi-groups; a priori the expressive power of the first-order logic for automorphism groups is less than the one for endomorphism semi-groups. For instance, according to Shelah's results on permutation groups [4, 5], one cannot interpret set theory by means of first-order logic in the permutation group of an infinite set, the automorphism group of an algebra in empty language. On the other hand, one can do this in the endomorphism semi-group of such an algebra.

In [7, 8] the author found a solution for the case of the variety of vector spaces over a fixed field. If V is a vector space of an infinite dimension ℵ over a division ring D, then the theory Th(ℵ, L2) is interpretable in the first-order theory of GL(V), the automorphism group of V. When a field D is countable and definable up to isomorphism by a second-order sentence, then the theories Th(GL(V)) and Th(ℵ, L2) are mutually syntactically interpretable. In the general case, the formulation is a bit more complicated.

The main result of this paper states that a similar result holds for the variety of all groups.

Functionals of higher derivative type are linear combinations of functionals of the form f→f(n)(ζ), where n[ges ]2 and 0<[mid ]ζ[mid ]<1. The paper shows that, if L is a functional of higher derivative type and f is a function in the class S of univalent functions that maximises Re{L} over S, then L(f)≠0. In addition, if the function f is a rational function, then it must be a rotation of the Koebe function k(z)=z(1−z)−2. These results are applied to establish several cases of the two-functional conjecture for functionals of higher derivative type.

An integral of the generalized Riemann type is developed which inverts the Schwarz derivative of a continuous function.

We study genus 5 curves C with a fixed point free involution. We give a geometrical (embedded) characterisation of these curves among all genus 5 curves: the points of the Prym variety associated to the involution give embeddings of the curve C in ${{\mathbb P}}_3$ so that C has infinitely many quadrisecant lines. Conversely, any genus 5 curve having such an embedding is endowed with a fixed point free involution and the embedding is given by a point of the Prym variety.

Let T = {T(t)}t[ges ]0 be a C0-semigroup on a Banach space X. The following results are proved.

(i) If X is separable, there exist separable Hilbert spaces X0 and X1, continuous dense embeddings j0[ratio ]X0 → X and j1[ratio ]X → X1, and C0-semigroups T0 and T1 on X0 and X1 respectively, such that j0 ∘ T0(t) = T(t) ∘ j0 and T1(t) ∘ j1 = j1 ∘ T(t) for all t [ges ] 0.

(ii) If T is [odot ]-reflexive, there exist reflexive Banach spaces X0 and X1 , continuous dense embeddings j[ratio ]D(A2) → X0, j0[ratio ]X0 → X, j1[ratio ]X → X1, and C0-semigroups T0 and T1 on X0 and X1 respectively, such that T0(t) ∘ j = j ∘ T(t), j0 ∘ T0(t) = T(t) ∘ j0 and T(t) ∘ j1 = j1 ∘ T(t) for all t [ges ] 0, and such that σ(A0) = σ(A) = σ(A1), where Ak is the generator of Tk, k = 0, [emptyv ], 1.

All finite groups G are determined that admit a subgroup K of index pa, p a prime, under the assumption that G has an irreducible and faithful GF(q)-module in characteristic p whose dimension over GF(q) is at most a. As an application to the theory of permutation groups, the maximal transitive subgroups of the primitive affine permutation groups are determined. The above-mentioned classification is generalized by dropping the assumption that p|q. In both cases surprisingly nice results are obtained.

Let Y be a topological space; and suppose that the circle group, S1, which will be denoted by G throughout this paper, is acting on Y. Then the equivariant cohomology H*G(Y) of Y is defined to be H*(YG), where YG=(EG×Y)/G, the Borel construction. Because of the bundle map YG→BG, the cohomology H*G(Y) is an H*(BG)-module. With rational coefficients H*(BG; ℚ)=ℚ[u], where deg(u)=2. And, when Y is a reasonably nice finite-dimensional space, for example, a finite-dimensional G-CW-complex, then the localized module H*G(Y; ℚ)[u−1] is isomorphic to H*G (YG; ℚ)[u−1] via the restriction homomorphism, where YG⊆Y is the fixed point set. (See, for example, [2].) That this very useful Localization Theorem does not generalize to infinite dimensional spaces can be seen by taking Y=EG, for example.

The Beurling algebras $l^1({\cal D},\omega)\;({\cal D}={\bb N},{\bb Z})$ that are semi-simple, with compact Gelfand transform, are considered. The paper gives a necessary and sufficient condition (on $\omega$ ) such that $l^1({\cal D},\omega)$ possesses a uniform quantitative version of Wiener's theorem in the sense that there exists a function $\phi:]0,+\infty[\longrightarrow ]0,+\infty[$ such that, for every invertible element $x$ in the unit ball of $l^1({\cal D},\omega)$ , we have \[ \|x^{-1}\|\le \phi(r(x^{-1}))\quad r(x^{-1})\hbox{ is the spectral radius of }x^{-1}. \]

The first example of a non-finitely based variety of centre-by-abelian-by-nilpotent groups was found in 1995. This example has an infinite exponent but it can be seen that the method used to construct it also gives a variety that has an exponent of 128. Later it was proved that there exists a variety with similar properties but with an exponent of 32. The paper constructs a non-finitely based variety of centre-by-abelian-by-nilpotent groups which has an exponent of 8.

It is shown that many hyperbolic monopoles can be distinguished from each other via their asymptotic values in contrast to the case of Euclidean monopoles.