We prove Davis and Garsia Inequalities for dyadic perturbations of Hardy martingales and show that those inequalities play a substantial role in the proof of Bourgain's [1] embedding L1 ↪ L1/H10. This paper continues [17] on Davis and Garsia Inequalities (DGI).

For a nonsingular projective threefold of general type X over the field of complex numbers, we show that the fourth pluricanonical map ϕ4 is not birational onto its image if and only if X is birationally fibred by (1,2)-surfaces, provided that vol(X) ≥ 303. We also have similar characterization of birationality of ϕ3.

Let F:${\mathbb R}$n→${\mathbb R}$n be a polynomial local diffeomorphism and let SF denote the set of not proper points of F. The Jelonek's real Jacobian Conjecture states that if codim(SF) ≥ 2, then F is bijective. In this work we prove a weak version of such Conjecture, but for more general maps than polynomial, namely: the semi-algebraic maps.

We develop a new technique for deriving asymptotic series expansions for moments of combinatorial generating functions that uses the transformation theory of Jacobi forms and “mock” Jacobi forms, as well as the Hardy-Ramanujan Circle Method. The approach builds on a suggestion of Zagier, who observed that the moments of a combinatorial statistic can be simultaneously encoded as the Taylor coefficients of a function that transforms as a Jacobi form. Our use of Jacobi transformations is a novel development in the subject, as previous results on the asymptotic behavior of the Taylor coefficients of Jacobi forms have involved the study of each such coefficient individually using the theory of quasimodular forms and quasimock modular forms.

As an application, we find asymptotic series for the moments of the partition rank and crank statistics. Although the coefficients are exponentially large, the error in the series expansions is polynomial, and have the same order as the coefficients of the residual Eisenstein series that are undetectable by the Circle Method. We also prove asymptotic series expansions for the symmetrized rank and crank moments introduced by Andrews and Garvan, respectively. Equivalently, the former gives asymptotic series for the enumeration of Andrews k-marked Durfee symbols.

This paper examines the Hausdorff dimension of the level sets f−1(y) of continuous functions of the form

\begin{equation*} f(x)=\sum_{n=0}^\infty 2^{-n}\omega_n(x)\phi(2^n x), \quad 0\leq x\leq 1, \end{equation*}

$\log ((9+\sqrt{105})/2)/\log 16\approx .8166$

The theory of twist maps is applied to prove the existence of many harmonic and sub-harmonic solutions for certain Newtonian systems of differential equations. The method of proof leads to very precise information on the oscillatory properties of these solutions.

We establish growth tightness for a class of groups acting geometrically on a geodesic metric space and containing a contracting element. As a consequence, any group with non-trivial Floyd boundary are proven to be growth tight with respect to word metrics. In particular, all non-elementary relatively hyperbolic group are growth tight. This generalizes previous works of Arzhantseva-Lysenok and Sambusetti. Another interesting consequence is that CAT(0) groups with rank-1 elements are growth tight with respect to CAT(0)-metric.

We prove local existence of complex-valued harmonic morphisms from any Riemannian homogeneous space of positive curvature, except the Berger space Sp(2)/SU(2).

In this paper we prove the generalized version of the same n-type conjecture posed by McGibbon and Møller in [15, page 287]: it is shown that the set of all the same homotopy n-types of the suspension of the smash products of the Eilenberg-MacLane spaces is the one element set consisting of a single homotopy type.

We construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus g, and determine its abelianization for g ≥ 4.

We discuss a new method to compute the canonical height of an algebraic point on a hyperelliptic jacobian over a number field. The method does not require any geometrical models, neither p-adic nor complex analytic ones. In the case of genus 2 we also present a version that requires no factorisation at all. The method is based on a recurrence relation for the ‘division polynomials’ associated to hyperelliptic jacobians, and a diophantine approximation result due to Faltings.

In this note we wish to correct several mistakes which appeared in our paper [1]. Fortunately, all the results claimed in that reference may be recovered, with small modifications to the statements and the argument. We are grateful to Nathan Ng for drawing our attention to an issue in the application of Theorem 6 to divisor sums, and to Régis de la Bretèche for helpful discussions on the problem addressed here.