Non-uniqueness of ergodic measures with full Hausdorff dimension on a Gatzouras-Lalley carpet

In this note, we show that on certain Gatzouras-Lalley carpet, there exist more than one ergodic measures with full Hausdorff dimension. This gives a negative answer to a conjecture of Gatzouras and Peres

Pseudo-Automorphisms of positive entropy on the blowups of products of projective spaces

We use a concise method to construct pseudo-automorphisms f_n of the first dynamical degree d_1(f_n) > 1 on the blowups of the projective n-space for all n > 1 and more generally on the blowups of products of projective spaces. These f_n, for n = 3 have positive entropy, and for n > 3 seem to be the first examples of pseudo-automorphisms with d_1(f_n) > 1 (and of non-product type) on rational varieties of higher dimensions.Comment: Mathematische Annalen (to appear

Toddlers’ fine motor milestone achievement is associated with early touchscreen scrolling

Touchscreen technologies provide an intuitive and attractive source of sensory/cognitive stimulation for young children. Despite fears that usage may have a negative impact on toddlers’ cognitive development, empirical evidence is lacking. The current study presents results from the UK Toddler Attentional Behaviours and LEarning with Touchscreens (TABLET) project, examining the association between toddlers’ touchscreen use and the attainment of developmental milestones. Data were gathered in an online survey of 715 parents of 6- to 36-month-olds to address two research questions: (1) How does touchscreen use change from 6 to 36 months? (2) In toddlers (19–36 months, i.e., above the median age, n = 366), how does retrospectively reported age of first touchscreen usage relate to gross motor (i.e., walking), fine motor (i.e., stacking blocks), and language (i.e., producing two-word utterances) milestones? In our sample, the proportion of children using touchscreens, as well as the average daily usage time, increased with age (youngest quartile, 6–11 months: 51.22% users, 8.53 min per day; oldest quartile, 26–36 months: 92.05% users, average use of 43.95 min per day). In toddlers, aged 19–36 months, age of first touchscreen use was significantly associated with fine motor (stacking blocks), p = 0.03, after controlling for covariates age, sex, mother’s education (a proxy for socioeconomic status) as well as age of early fine motor milestone achievement (pincer grip). This effect was only present for active scrolling of the touchscreen p = 0.04, not for video watching. No significant relationships were found between touchscreen use and either gross motor or language milestones. Touchscreen use increases rapidly over the first 3 years of life. In the current study, we find no evidence to support a negative association between the age of first touchscreen usage and developmental milestones. Indeed, earlier touchscreen use, specifically scrolling of the screen, was associated with earlier fine motor achievement. Future longitudinal studies are required to elucidate the temporal order and mechanisms of this association, and to examine the impact of touchscreen use on other, more fine-grained, measures of behavioral, cognitive, and neural development

Unitarity Cuts: NLO Six-Gluon Amplitudes in QCD

We report on a technique for evaluating finite unitarity cut for one-loop amplitudes in gauge theories, and discuss its application to the cut-constructible part of six-gluon amplitude in QCD.Comment: talk given at Loops & Legs 2006, April 23-28, Eisenach (Germany

On the complexity of some birational transformations

Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines, and relies mainly on univariate polynomial algebra, the second approach is a singularity analysis, and the third method is more numerical, using integer arithmetics. Each method has its own domain of application, but they give corroborating results, and lead us to a conjecture on the complexity of a class of maps constructed from matrix inversions

On the cohomology of pseudoeffective line bundles

The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact K{\"a}hler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation, one can still obtain interesting information on cohomology, e.g. a Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his PhD thesis defended in Grenoble, obtained a general K{\"a}hler vanishing theorem that depends on the concept of numerical dimension of a given pseudoeffective line bundle. The proof of these results depends in a crucial way on a general approximation result for closed (1,1)-currents, based on the use of Bergman kernels, and the related intersection theory of currents. Another important ingredient is the recent proof by Guan and Zhou of the strong openness conjecture. As an application, we discuss a structure theorem for compact K{\"a}hler threefolds without nontrivial subvarieties, following a joint work with F.Campana and M.Verbitsky. We hope that these notes will serve as a useful guide to the more detailed and more technical papers in the literature; in some cases, we provide here substantially simplified proofs and unifying viewpoints.Comment: 39 pages. This survey is a written account of a lecture given at the Abel Symposium, Trondheim, July 201

Covariant Balance Laws in Continua with Microstructure

The purpose of this paper is to extend the Green-Naghdi-Rivlin balance of energy method to continua with microstructure. The key idea is to replace the group of Galilean transformations with the group of diffeomorphisms of the ambient space. A key advantage is that one obtains in a natural way all the needed balance laws on both the macro and micro levels along with two Doyle-Erickson formulas

The rational parts of one-loop QCD amplitudes I: The general formalism

A general formalism for computing only the rational parts of oneloop QCD amplitudes is developed. Starting from the Feynman integral representation of the one-loop amplitude, we use tensor reduction and recursive relations to compute the rational parts directly. Explicit formulas for the rational parts are given for all bubble and triangle integrals. Formulas are also given for box integrals up to two-masshard boxes which are the needed ingredients to compute up to 6-gluon QCD amplitudes. We use this method to compute explicitly the rational parts of the 5- and 6-gluon QCD amplitudes in two accompanying papers.Comment: 49 pages, 8 figure and LaTeX file; minor corrections, references added, to be published in Nucl. Phys.

Mixture of Fluids involving Entropy Gradients and Acceleration Waves in Interfacial Layers

Through an Hamiltonian action we write down the system of equations of motions for a mixture of thermocapillary fluids under the assumption that the internal energy is a function not only of the gradient of the densities but also of the gradient of the entropies of each component. A Lagrangian associated with the kinetic energy and the internal energy allows to obtain the equations of momentum for each component and for the barycentric motion of the mixture. We obtain also the balance of energy and we prove that the equations are compatible with the second law of thermodynamics. Though the system is of parabolic type, we prove that there exist two tangential acceleration waves that characterize the interfacial motion. The dependence of the internal energy of the entropy gradients is mandatory for the existence of this kind of waves. The differential system is non-linear but the waves propagate without distortion due to the fact that they are linearly degenerate (exceptional waves).Comment: 30 page