2,383 research outputs found
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
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