Some of Ramanujan's original discoveries about hypergeometric functions and their relation to modular integrals, especially Eisenstein series of negative weight, are still not very well understood. These discoveries take the form of identities that he recorded, without proof, as entries in his notebooks. In the following sections I will introduce some of these entries, discuss their status, give new proofs of several of them and also provide new results of a similar nature.

We show that in quasi-logarithmic additive number systems all partition sets have asymptotic density, and we obtain a corresponding monadic second-order limit law for adequate classes of relational structures. Our conditions on the local counting function p(n) of the set of irreducible elements of allow situations which are not covered by the density theorems of Compton [6] and Woods [15]. We also give conditions on p(n) which are sufficient to show the assumptions of Compton's result are satisfied, but which are not necessarily implied by those of Bell and Burris [2], Granovsky and Stark [8] or Stark [14].

Suppose that A and A′ are subsets of ℤ/Nℤ. We write A+A′ for the set {a+a′:a ∈ A and a′ ∈ A′} and call it the sumset of A and A′. In this paper we address the following question. Suppose that A1,. . .,Am are subsets of ℤ/Nℤ. Does A1+· · ·+Am contain a long arithmetic progression?

The situation for m=2 is rather different from that for m ≥ 3. In the former case we provide a new proof of a result due to Green. He proved that A1+A2 contains an arithmetic progression of length roughly where α1 and α2 are the respective densities of A1 and A2. In the latter case we improve the existing estimates. For example we show that if A ⊂ ℤ/Nℤ has density then A+A+A contains an arithmetic progression of length Ncα. This compares with the previous best of Ncα2+ϵ.

Two main ingredients have gone into the paper. The first is the observation that one can apply the iterative method to these problems using some machinery of Bourgain. The second is that we can localize a result due to Chang regarding the large spectrum of L2-functions. This localization seems to be of interest in its own right and has already found one application elsewhere.

Let θ be a Jordan homomorphism from an algebra A into an algebra B. We find various conditions under which the restriction of θ to the commutator ideal of A is the sum of a homomorphism and an antihomomorphism. Algebraic results, obtained in the first part of the paper, are applied to the second part dealing with the case where A and B are C*-algebras.

Mislin and Talelli showed that a torsion-free group in with periodic cohomology after some steps has finite cohomological dimension. In this note we look at similar questions for groups with torsion by considering Bredon cohomology. In particular we show that every elementary amenable group acting freely and properly on some × Sm admits a finite dimensional model for G.

MacLane cohomology is an algebraic version of the topological Hochschild cohomology. Based on the computation of the third author (see Appendix) we obtain an interpretation of the third Mac Lane cohomology of rings using certain kind of crossed extensions of rings in the quadratic world. Actually we obtain two such interpretations corresponding to the two monoidal structures on the category of square groups.

A closed formula is obtained for the integral of tautological classes over the locus of hyperelliptic Weier points in the moduli space of curves. As a corollary, a relation between Hodge integrals is obtained.

The calculation utilizes the homeomorphism between the moduli space of curves and the combinatorial moduli space , a PL-orbifold whose cells are enumerated by fatgraphs. This cell decomposition can be used to naturally construct combinatorial PL-cycles whose homology classes are essentially the Poin duals of the Mumford–Morita–Miller classes κa. In this paper we construct another PL-cycle representing the locus of hyperelliptic Weier points and explicitly describe the chain level intersection of this cycle with W1. Using this description of , the duality between Witten cycles Wa and the κa classes, and the Kontsevich--Penner method of integration, scheme of integrating ε classes, the integral is reduced to a weighted sum over graphs and is evaluated by the enumeration of trees.

We show that the minimal dimension of a faithful action of a metacyclic group , for primes p and q, on a homology sphere coincides with the minimal dimension of a faithful linear action on a sphere; as a consequence, we obtain the analogous result for various finite simple groups.

If the Bing double of a knot K is slice, then K is algebraically slice. In addition the Heegaard–Floer concordance invariants τ, developed by Ozsváth–Szabó, and δ, developed by Manolescu and Owens, vanish on K.

The first author conjectured certain relations for Morita–Mumford classes and Newton classes in the integral cohomology of mapping class groups (integral Riemann–Roch formulae). In this paper, the conjecture is verified for cyclic subgroups of mapping class groups.

In this paper, for a possibly singular complex variety X, generating functions of total orbifold Chern homology classes of the symmetric products SnX are given. These are very natural “class versions” of known generating function formulae of (generalized) orbifold Euler characteristics of SnX. Our Chern classes work covariantly for proper morphisms. We state the result more generally. Let G be a finite group and Gn the wreath product G ∼ Sn. For a G-variety X and a group A, we show a “Dey–Wohlfahrt type formula“ for equivariant Chern–Schwartz–MacPherson classes associated to Gn-representations of A (Theorem 1ċ1 and 1ċ2). When X is a point, our formula is just the classical one in group theory generating numbers |Hom(A, Gn)|.

X. Buff and A. Chéritat proved that there are quadratic polynomials having Siegel disks with smooth boundaries. Based on a simplification of Avila, we give yet another simplification of their proof. The main tool used is a harmonic function introduced by Yoccoz whose boundary values are the sizes of the Siegel disks. The proof also applies to some other families of polynomials, entire and meromorphic functions.

We give a version of Shimizu's lemma for groups of complex hyperbolic isometries one of whose generators is a parabolic screw motion. Suppose that G is a discrete group containing a parabolic screw motion A and let B be any element of G not fixing the fixed point of A. Our result gives a bound on the radius of the isometric spheres of B and B−1 in terms of the translation lengths of A at their centres. We use this result to give a sub-horospherical region precisely invariant under the stabiliser of the fixed point of A in G.

We give an interpretation of the Chern–Heinz inequalities for graphs in order to extend them to transversally oriented codimension one C2-foliations of Riemannian manifolds. It contains Salavessa's work on mean curvature of graphs and fully generalizes results of Barbosa–Kenmotsu–Oshikiri [3] and Barbosa–Gomes–Silveira [2] about foliations of 3-dimensional Riemannian manifolds by constant mean curvature surfaces. This point of view of the Chern–Heinz inequalities can be applied to prove a Haymann–Makai–Osserman inequality (lower bounds of the fundamental tones of bounded open subsets Ω ⊂ ℝ2 in terms of its inradius) for embedded tubular neighbourhoods of simple curves of ℝn.

Let Sj: ℝd → ℝd for j = 1, . . ., N be contracting similarities. Also, let (p1,. . ., pN, p) be a probability vector and let ν be a probability measure on ℝd with compact support. We show that there exists a unique probability measure μ such thatThe measure μ is called an in-homogenous self-similar measure. In this paper we study the asymptotic behaviour of the Fourier transforms of in-homogenous self-similar measures. Finally, we present a number of applications of our results. In particular, non-linear self-similar measures introduced and investigated by Glickenstein and Strichartz are special cases of in-homogenous self-similar measures, and as an application of our main results we obtain simple proofs of generalizations of Glickenstein and Strichartz's results on the asymptotic behaviour of the Fourier transforms of non-linear self-similar measures.

We identify the support of a tempered distribution by evaluation of a sequence of test functions against the Fourier transform of the distribution. This improves previous results by removing the restriction that the distribution's Fourier transform be in and be of polynomial growth. We use an apparently new technical lemma that implies that certain bounded approximate identities for are also topological approximate identities for elements of the space of Schwartz functions.

We show that in the Haar wavelet basis is not equivalent to any permutation with any signs of the Strω wavelet basis. We also construct a Haar-type system in L1[0,1] which is not equivalent to any subsequence with signs of the classical Haar basis.