For ${\cal X}$ a stratification with suitable regularity, in particular for any Whitney stratification and, via regular embedding, for any abstract stratified set, time-dependent vector fields are used to prove an extension theorem for diffeomorphisms near the identity defined on strata of a given dimension. Then it is shown that after isotopy a stratified map $h\,{:}\,{\cal Z} \,{\longrightarrow}\, {\cal X}$ can be made transverse to a fixed stratified map $g\,{:}\, {\cal Y} \,{\longrightarrow}\, {\cal X}$.

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.

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.

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.

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).

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.

The most important object in real singularity theory is the C∞ map germ and the most important equivalence relation among them is C∞ right left equivalence. In [7], we presented a new systematic method for the classification of C∞ map germs by characterising C∞ right left equivalence. This paper is a topological version of [7].

Two C∞ map germs f, g:(Rn, 0) → (Rp, 0) are said to be topologically equivalent if there exist homeomorphism map germs s:(Rn, 0) → (Rn 0) and t:(Rp, 0) → (Rp, 0) such that f(x) = t∘g∘s(x). The notion of topological equivalence, although it seems to be unnatural, is also important since we know the existence of C∞ moduli for the classification of C∞ map germs with respect to C∞ right left equivalence. However, we had only one method to obtain topological equivalence for two given C∞ map germs, as stated in the following.

For two given C∞ map germs f, g:(Rn, 0) → (Rp, 0), take an appropriate one-parameter family F:(Rn×[0, 1], {0}×[0, 1]) → (Rp, 0) such that F(x, 0) = f(x) and F(x, 1) = g(x). Then prove that F is in fact topologically trivial.(*)

Two C∞ map germs f, g:(Rn, 0) → (Rp, 0) are said to be [Kscr ]-equivalent if there exist a C∞ diffeomorphism map germ s:(Rn, 0) → (Rn, 0) and a C∞ map germ M:(Rn, 0) → (GL(p, R), M(0)) such that f(x) = M(x)g(s(x)). The notion of [Kscr ]-equivalence was introduced by Mather [4, 5] in order to classify the C∞ stable map germs, and we know that generally in a [Kscr ]-orbit there are uncountably many C∞ right left orbits.

Hence it is significant to give an alternative systematic method for the topological classification even in a single [Kscr ]-orbit, which is the purpose of this paper. One of our results (Theorem 1.2) yields the following well-known theorem [2] as a trivial corollary.

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.

Throughout ℤp and ℚp will, respectively, denote the ring of p-adic integers and the field of p-adic numbers (for p prime). We denote by [Copf ]p the completion of the algebraic closure of ℚp with respect to the p-adic metric. Let vp denote the p-adic valuation of [Copf ]p normalised so that vp(p)=1. Put [ ]p ={ω∈[Copf ]p[mid ] ωpn=1 for some n[ges ]0} so that [ ]p is the union of cyclic (multiplicative) groups Cpn of order pn (for n[ges ]0).

Let UD(ℤp) denote the [Copf ]p-algebra of all uniformly differentiable functions f[ratio ]ℤp→[Copf ]p under pointwise addition and convolution multiplication *, where for f, g∈UD(ℤp) and z∈ℤp we have

formula here

the summation being restricted to i, j with vp(i+j−z)[ges ]n.

This situation is a starting point for p-adic Fourier analysis on ℤp, the analogy with the classical (complex) theory being substantially complicated by the absence of a p-adic valued Haar measure on ℤp (see [5, 6] for further details).

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.

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.

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].

For a wide class of infinitely divisible laws μ numbers R(μ) are evaluated satisfying the property: if R>R(μ) then μ is uniquely determined by its values on the half-line (−∞, R); if R<R(μ) then μ is not determined by its values on (−∞, R). Necessary and sufficient conditions are given for ‘half-line’ convergence to μ on (−∞, R) to be equivalent to weak convergence to μ.

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.

The second Stiefel–Whitney class of the quadratic form trA/k(ax2) is computed, where A is a central simple algebra over a perfect field k of characteristic different from 2, a ∈ A is a fixed element, and trA/k is the reduced trace. This class is related on the one hand to the class of A in the Brauer group, and on the other hand to corestrictions of quaternion algebras over certain factors arising from E[otimes ]kE, where E is a commutative étale algebra over k that depends on the semisimple part of a.

We introduce the concept of ‘geometrical spine’ for 3-manifolds with natural metrics, in particular, for lens manifolds. We show that any spine of $L_{p,q}$ that is close enough to its geometrical spine contains at least $E(p,q)-3$ vertices, which is exactly the conjectured value for the complexity $c(L_{p,q})$. As a byproduct, we find the minimal rotation distance (in the Sleator–Tarjan–Thurston sense) between a triangulation of a regular $p$-gon and its image under rotation.

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

Standard codeterminants for Donkin's symplectic Schur algebra $\bar{S}(n,r)$ are defined. It is shown that they form a basis of $\bar{S}(n,r)$. They are then used to give a purely combinatorial proof that $\bar{S}(n,r)$ is quasi-hereditary.

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.