We show that the edge graph of a locally finite tiling in the plane satisfies a strong isoperimetric inequality if one among the lower averages of what we call the characteristic numbers of edges, vertices and tiles is positive.

Let P1, …, Pn be polynomials in one or several real or complex variables. Several authors, working with a variety of norms, have given estimates for a constant M depending only on the degrees of P1, …, Pn such that

∥P1∥…∥Pn∥ [les ]M∥P1 … Pn∥.

In this paper we show that inequalities of this type are valid for polynomials on any complex Banach space. Our method provides optimal constants.

We also derive analogous inequalities for polynomials on real Banach spaces, but the constants we obtain are generally not optimal. The search for optimal constants does however lead to an interesting open problem in Hilbert space geometry.

When we restrict attention to products of linear functionals, we find new characterizations of complex [lscr ]n1 (n[ges ]2), real [lscr ]21, and real Hilbert space.

It is known that for p-perfect groups G of finite virtual cohomological dimension and finite type mod-p cohomology, the p-completed classifying space BGandp has the property that ΩBGandp is a retract of the loop space on a simply-connected, [ ]p-finite, p-complete space. In this note we consider a particular example where this theorem applies, namely we study the homotopy type of BSL3(ℤ)and2. It particular we analyse ΩBSt3(ℤ)and2, a double cover of ΩBSL3(ℤ)and2, and obtain a splitting theorem for it in terms of 2-primary Moore spaces and fibres of degree 2r maps on spheres. We also give a formula for the Poincaré series of H∗(ΩBΓandp; [ ]p) for a general group Γ, as above, in terms of possibly simpler components. This formula is used to calculate the mod-2 homology of ΩBΓand2 for Γ=SL3(ℤ) or St3(ℤ) as modules over a certain tensor subalgebra.

The success of the methods of [24] and [4] in investigating the structure of the weakly almost periodic compactification wℕ of the semigroup (ℕ, +) of positive integers has prompted us to see how successful they would be in another context. We consider wG, where G=[oplus ]i∈ωGi is the direct sum of a sequence of finite groups with its discrete topology. We discover a large class of weakly almost periodic functions on G, and we use them to prove the existence of a large number of long chains of idempotents in wG. However, the closure of any singly generated subsemigroup of wG contains only one idempotent. We also prove that the set of idempotents in wG is not closed. The minimal idempotent of wG can be written as the sum of two others, with the consequence that the minimal ideal of wG is not prime. Pursuing possible parallels with the structure of βℕ, we find subsemigroups SF of wG which are ‘carried’ by closed subsets F of βω. wG contains the free abelian product of the semigroups SF corresponding to families of disjoint subsets F. Usually SF is a very large semigroup, but for some points p∈βω, Sp can be quite small.

In this paper we study the algebra L([sum ]) generated by links in the manifold [sum ]×[0, 1] where [sum ] is an oriented surface. This algebra has a filtration and the associated graded algebra LGr([sum ]) is naturally a Poisson algebra. There is a Poisson algebra homomorphism from the algebra ch ([sum ]) of chord diagrams on [sum ] to LGr([sum ]).

We show that multiplication in L([sum ]) provides a geometric way to define a deformation quantization of the algebra of chord diagrams on [sum ], provided there is a universal Vassiliev invariant for links in [sum ]×[0, 1]. If [sum ] is compact with free fundamental group we construct a universal Vassiliev invariant. The quantization descends to a quantization of the moduli space of flat connections on [sum ] and it is natural with respect to group homomorphisms.

In this note, we classify the projective flat manifolds whose holonomy group is either a cyclic group Cp or dihedral D2p, with p prime. We give an estimate for the size of the set of their positive line bundles and construct explicitly and directly in terms of the representation space of the holonomy group some interesting imbeddings.

In this paper we will study the Brill–Noether theory of vector bundles on a smooth projective curve X. As usual in papers on this topic we are mainly interested in stable or at least semistable bundles. Let Wkr, d(X) be the scheme of all stable vector bundles E on X with rank (E)=r, deg (E)=d and h0(X, E)[ges ]k+1. For a survey of the main known results, see the introduction of [6]. The referee has pointed out that the results in [6] were improved by V. Mercat in [14]; he proved that Wkr, d(X) is non-empty for d<2r if and only if k+1[les ]r+(d−r)/g. If X has general moduli the more interesting existence theorem was proved in [19]. However, in this paper we are mainly interested in very special curves X, e.g. the hyperelliptic or the bielliptic curves. We work over an algebraically closed base field K. In Section 5 we will assume char (K)=0. In Section 1 we will give some theorems of Clifford's type. In Section 2 we will construct several stable bundles with certain properties. Here the main tool is an operation (the +elementary transformation) which sends a vector bundle E on X to another vector bundle E′ with rank (E′)=rank (E) and deg (E′)=deg (E)+1 (see Section 2 for its definition and its elementary properties). Using the +elementary transformations in Section 3 we will prove the following existence theorem which covers the case of a ‘small’ number of sections.

We show that the non-existence of elements in the p-stem πSp of Hopf invariant one implies that: there exists no smooth map f[ratio ]M→N with only fold singularities when M is a closed n-dimensional manifold with odd Euler characteristic and N is an almost parallelizable p-dimensional manifold (n[ges ]p), provided that p≠1, 3, 7. In fact, the result itself is originally due to Kikuchi and Saeki [25, 34]. Our proof clarifies the relationship between the two problems and gives a new insight to the problem of the global singularity theory. Furthermore we generalize the above result to maps with only Morin singularities of types Ak with k[les ]3 when p≠1, 2, 3, 4, 7, 8.

For a wide class of categories of Banach spaces, we show that the existence of an (almost) transitive element implies the existence of a separable almost transitive element. We give some applications to C0(L) spaces and abstract M-spaces. Next, we prove that Wood's conjecture on almost transitivity of the norm in C0(L) can be reduced to the case in which the one-point compactification of L is metrizable.

We construct a simple example of transitive M-space and we show the existence of almost transitive separable M-spaces that are isomorphic to C[0, 1]. These M-spaces have a rich M-structure and they are counterexamples for several questions about centralizers.

We give a simple inequality relating the elliptic Mahler measure of a polynomial to the traditional Mahler measure (via the length of the polynomial). These bounds are essentially sharp. We also give the corresponding result for polynomials in several variables.

We study self-similar sets with overlaps, on the line and in the plane. It is shown that there exist self-similar sets that have non-integral Hausdorff dimension equal to the similarity dimension, but with zero Hausdorff measure. In many cases the Hausdorff dimension is computed for a typical parameter value. We also explore conditions for the validity of Falconer's formula for the Hausdorff dimension of self- affine sets, and study the dimension of some fractal graphs.

In two recent papers [1, 2], we have developed an algorithm for the solution of linear differential systems of the type

formula here

on an interval [X, ∞). Here Z is an n-component vector, ρ is a scalar factor, D is a constant diagonal matrix and R is a perturbation matrix such that R(x)=O(x−δ) as x→∞, for some δ>0.The algorithm implements a repeated transformation process by means of which (1·1) is transformed into other systems whose perturbation matrices are of successively smaller orders of magnitude as x→∞. When the perturbation reaches a prescribed accuracy, the Levinson asymptotic theorem [4, 5] provides the solution of the final system in the process, the solution also possessing that accuracy. The corresponding solution of (1·1) is then obtained by transforming back.

The algorithm has two main aspects. The first is the algebraic one of generating symbolically the matrix terms which appear in the transformation process. In [1] and [2], the algorithm was set up in such a way that this aspect is implemented in the symbolic algebra system Mathematica. The other aspect is the numerical one of deriving from the algebra the values of the solutions of (1·1), given ρ, D and R. Included here are the determination of explicit error-bounds and procedures for handling large numbers of matrix terms.

In this paper we develop these ideas in two different but not unrelated directions. First, we replace the constant matrix D by one which has entries of differing orders of magnitude as x→∞. Our methods are sufficiently indicated by two orders of magnitude, and thus we consider the system

formula here

where formula here

with constant and diagonal D˜ and D, while λ is a scalar factor such that λ(x)→∞ as x→∞. The amplification of our previous algorithm to cover this new situation is dealt with in Sections 2–4 below. At the end of Section 3, we state the Levinson asymptotic theorem in the form that we need, with the necessary accuracy incorporated. As in [1] and [2], a typical ρ(x) is xa (α>−1). However, the value α=−1 also occurs in applications, and our second extension is to include α=−1 in the scope of the algorithm. This matter is the subject of Section 5.

For n[ges ]2, let S(Rn) be the space of Schwartz class functions. The dual space of S(Rn) is denoted by S′(Rn) and is called the tempered distributions. For f∈S(Rn), define fˆ(w)=∫Rne−iw·xf(x)dx and for T∈S′(Rn), define Tˆ∈S′(Rn) by the formula 〈Tˆ, f〉=〈T, fˆ〉 for all f∈S(Rn). For 1[les ]p[les ]∞, let FLp(Rn)= {T∈S′(Rn)[ratio ] Tˆ∈Lp(Rn)}. Let β be a multi-index of nonnegative integers, β=(β1, β2, …, βn−1) with [mid ]β[mid ]= β1+β2+…+βn−1. For a nonnegative integer k and bounded Borel measures σβ on a compact subset K of Rn, [mid ]β[mid ][les ]k, we define T∈S′(Rn) by the formula

formula here

If T is given by (1), then we know that supp(T)⊂K and the order of T is at most k. In particular, when k=0, we say that T is a bounded measure on K. Denote EK(Rn)= {T∈S′(Rn)[ratio ]T is a bounded measure on K} and DK(Rn)= {T∈S′(Rn)[ratio ] supp(T)∪K}. If in addition we assume some connectness property of K, then one can show ([5, th. 2·3·10]) that if T∈DK(Rn), then T does have the representation (1) for some k[ges ]0. The question is when we can say that k=0. From the Hausdorff–Young inequality and the proof of theorem 7·6·6 in [5], one sees that if the interior of K (in Rn) is not empty, then DK(Rn) ∩FLp(Rn) ⊂EK(Rn) if and only if 1[les ]p[les ]2. Much remains to be understood if the interior of K is empty.

In this paper, we seek to prove a similar result for compact subsets of hypersurfaces of Rn, motivated from the differential equations with constant coefficients.