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.

A classical problem in analytic number theory is the distribution in short intervals of integers with a prescribed multiplicative structure, such as primes, almost-primes, k-free numbers and others. Recently, partly due to applications to cryptology, much attention has been received by the problem of the distribution in short intervals of integers without large prime factors, see Lenstra–Pila–Pomerance [3] and section 5 of the excellent survey by Hildebrand–Tenenbaum [1].

In this paper we deal with the distribution in short intervals of numbers representable as a product of a prime and integers from a given set [Sscr ], defined in terms of cardinality properties. Our results can be regarded as an extension of the above quoted results, and we will provide a comparison with such results by a specialization of the set [Sscr ].

The main aim of this paper is to provide two new proofs of Ramanujan's cubic transformation formula for 2F1(1/3, 2/3; 1; z) (see (1·8) below). For our first proof, we have to develop Ramanujan's elliptic functions in the theory of signature 3 using a different approach from that given in a recent paper by Berndt, Bhargava and Garvan. For our second proof, we use two of Goursat's formulas.

A classical reciprocity formula for Gauss sums due to Cauchy, Dirichlet, and Kronecker states that

formula here

where a, b are nonzero integers and w∈ℚ such that ab+2aw∈2ℤ. For a detailed proof and historical background, see [4, chapter IX]. Various versions of formula (1) have been extensively studied in connection with transformation properties of theta functions. A version of (1) for multivariate Gauss sums was first obtained by A. Krazer [10] in 1912, see also [2, 5, 11, 14]. Krazer's formula generalizes the case w=0 of (1) via replacing one of the numbers a, b by an integer quadratic form of several variables.

Recently, Florian Deloup [6] found a new and most beautiful reciprocity formula for multivariate Gauss sums. His formula is a far reaching generalization of Krazer's result. Roughly speaking, Deloup replaces both numbers a, b with quadratic forms. Deloup involves Wu classes of quadratic forms which allows him to remove the evenness condition appearing in Krazer's formulation. However, Deloup's formula covers only the cases w=0 and w=b/2 of (1).

In this paper we establish a more general reciprocity for Gauss sums including the Krazer and Deloup formulas and formula (1) in its full generality. Our reciprocity law involves two quadratic forms and a so-called rational Wu class, as defined below.

The original proofs of Cauchy, Dirichlet, Kronecker and Krazer are analytical and involve a study of a limit of a transformation formula for theta-functions. Deloup's proof goes by a reduction to the case w=b/2 of (1) based on a careful study of Witt groups of quadratic forms. Our proof is more direct and uses only (a generalization of) the van der Blij computation of Gauss sums via signatures of integer quadratic forms. In particular, our argument provides a new proof of (1).

Schur algebras are certain finite-dimensional algebras that completely control the polynomial representation theory of the general linear groups over an infinite field. Infinitesimal Schur algebras are truncated versions of the classical Schur algebras which control the polynomial representation theory of the Frobenius kernels of general linear groups. In this paper we use some elementary results on symmetric powers to classify the semisimple Schur algebras. We then classify the semisimple infinitesimal Schur algebras as well.

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.

Let A be a commutative ring. We denote by a standard A-algebra a commutative graded A-algebra U=[oplus ]n[ges ]0Un with U0=A and such that U is generated as an A-algebra by the elements of U1. Take x a (possibly infinite) set of generators of the A-module U1. Let V=A[t] be the polynomial ring with as many variables t (of degree one) as x has elements and let f[ratio ]V→U be the graded free presentation of U induced by the x. For n[ges ]2, we will call the module of effective n-relations the A-module E(U)n= ker fn/V1· ker fn. The minimum positive integer r[ges ]1 such that the effective n-relations are zero for all n[ges ]r+1 is known to be an invariant of U. It is called the relation type of U and is denoted by rt(U). For an ideal I of A, we define E(I)n= E([Rscr ](I))n and rt(I)=rt([Rscr ](I)), where [Rscr ](I)= [oplus ]n[ges ]0Intn⊂A [t] is the Rees algebra of I.

In this paper we give two descriptions of the A-module of effective n-relations. In terms of André–Quillen homology we have that E(U)n= H1(A, U, A)n (see 2·3). It turns out that this module does not depend on the chosen [x]. In terms of Koszul homology we prove that E(U)n= H1([x], U)n (see 2·4). Using these characterizations, we show later some properties on the module of effective n-relations and the relation type of a graded algebra. Our approach has connections with several earlier works on the subject (see [2, 5–7, 9, 10, 13, 14]).

For a fixed finite group G, the numbers Ng of equivalence classes of orientation-preserving actions of G on closed orientable surfaces Σg of genus g can be encoded by a generating function [sum ]Ngzg. When equivalence is determined by the isomorphism class of the quotient orbifold Σg/G, we show that the generating function is rational. When equivalence is topological conjugacy, we examine the cases where G is abelian and show that the generating function is again rational in the cases where G is cyclic.

In this paper a new discreteness criterion for subgroups in SL(2, C) is established, which implies all known discreteness criteria by two-generator subgroup and is the best possible. And it is proved that every non-elementary and non-discrete subgroup of SL(2, C) has a non-elementary and non-discrete subgroup generated by two loxodromic elements.

In a recent paper [7], Erdmann has calculated Ext1G between Weyl modules for SL2. In this paper we generalize this result to solve the corresponding problem for quantum GL2 as defined by Dipper and Donkin in [2]. We also show how our result also holds for the Manin quantization. To apply the methods of [7], it is necessary to determine the block structure of quantum GL2, so the first main result of this paper is a description of this, derived from the analysis of the subcomodule structure of the symmetric powers in [10].

After an initial section of generalities, the next section consists of the determination of the block structure. We also need a quantum analogue of two short exact sequences from [11], which we give in the following section. With these results, the argument now follows much as in [7]; we consider the infinitesimal case, and then use the Lyndon–Hochschild–Serre spectral sequence to obtain the desired result. Finally we show how the result also holds for the Manin quantization.

It should be noted that the result here uses the classical case, so is not independent of that in [7]. The only real difference in the arguments used occurs in Lemma 4·8 where the original methods do not generalize, so we use a more direct argument. There is also an unfortunate typographical error in the statement of the main result in [7].

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.

Let Γ be a discrete group and p be a prime. One of the fundamental results in group cohomology is that H*(Γ, [ ]p) is a finitely generated [ ]p-algebra if Γ is a finite group [8, 24]. The purpose of this paper is to study the analogous question when Γ is no longer finite.

Recall that Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torion-free subgroup Γ′ of Γ such that Γ′ has finite cohomological dimension over ℤ [4]. By definition vcd Γ is the cohomological dimension of Γ′. It is easy to see that the mod p cohomology ring of a finite vcd-group does not have to be a finitely generated [ ]p-algebra in general. For instance, if Γ is a countably infinite free product of ℤ's, then H1(Γ, [ ]p) is not finite dimensional over [ ]p. The three most important classes of examples of finite vcd-groups in which the mod p cohomology ring is a finitely generated [ ]p-algebra are arithmetic groups [2], mapping class groups [9, 10] and outer automorphism groups of free groups [5]. In each of these examples, the proof of finite generation involves the construction of a specific Γ-complex with appropriate finiteness conditions. These constructions should be regarded as utilizing the geometry underlying these special classes of groups. In contrast, the result we prove will depend only on the algebraic structure of the group Γ.

We attack the conjecture that the only spherical classes in the homology of Q0S0 are Hopf invariant one and Kervaire invariant one elements. We do this by computing products in the E2-term of the unstable Adams spectral sequence converging to π∗(Q0S0), using results about the Dickson algebra and by studying the Lannes–Zarati homomorphism.

Let [Mfr ]g be the moduli space of smooth curves C of genus g[ges ]4 over the complex field [Copf ] and, as in [4], denote by [Mfr ](n)g⊂[Mfr ]g the locus of points representing curves having n vanishing theta-null, i.e. n points of order two in Sing (Θ).

I. V. Dolgachev proved the rationality of [Mfr ](1)4 (see [4, p. 13]) and of [Mfr ](2)5 (see [4, p. 11]). The aim of this paper is to prove the

Theorem. [Mfr ](1)tis rational.

For G a classical group of type Bn, Cn or Dn defined over an algebraically closed field (of characteristic other than two if G is a special orthogonal group), we show that the centralizer algebra of G on the space of endomorphisms of the module of traceless tensors Wr is naturally isomorphic to the group algebra of the symmetric group Sr if r[les ]n (for G of type Bn or Cn) or if r<n (for G of type Dn). This extends a characteristic zero result of Brauer and Weyl and recovers some of the related characteristic free results that De Concini and Strickland obtained for G of type Cn.

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.

The usefulness of the Koszul complex in handling in an algebraic setting the two geometric notions of multiplicity and depth first became apparent with the work of Auslander and Buchsbaum [1] following a suggestion of Serre. Regarding the generators a1, …, an of the complex as a 1×n matrix first Eagon and Northcott [4] extended this work to a complex associated with an m×n matrix, then shortly afterwards a different extension was given by Buchsbaum and Rim [2, 3]. These two complexes are two of an infinite family [6] some of which inherit the depth sensitive property of the Koszul complex and all of which under a certain finiteness condition provide the same multiplicity as Euler–Poincaré characteristic [7].

These two properties prove useful in geometric applications, see for example Lago and Rodicio [7] for depth sensitivity and Kirby [8] for the characteristic as multiplicity of intersection. During the course of these developments it became clear that it was most appropriate to regard the complexes as being generated by linear forms. From this point of view it is natural to ask if the linearity of the forms is necessary. In the present note we begin a response to this question by extending the work of [6] to complexes associated with forms of arbitrary positive degree. In a sequel we shall similarly extend the results of multiplicity in [7].