The geometry of fractal percolation

A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions: {itemize} the statements work for all directions, not almost all, the statements are true for more general projections, for example radial projections onto a circle, in the case $\dim_H >1$, each projection has not only positive Lebesgue measure but also has nonempty interior. {itemize}Comment: Survey submitted for AFRT2012 conferenc

Slicing Sets and Measures, and the Dimension of Exceptional Parameters

We consider the problem of slicing a compact metric space \Omega with sets of the form \pi_{\lambda}^{-1}\{t\}, where the mappings \pi_{\lambda} \colon \Omega \to \R, \lambda \in \R, are \emph{generalized projections}, introduced by Yuval Peres and Wilhelm Schlag in 2000. The basic question is: assuming that \Omega has Hausdorff dimension strictly greater than one, what is the dimension of the 'typical' slice \pi_{\lambda}^{-1}{t}, as the parameters \lambda and t vary. In the special case of the mappings \pi_{\lambda} being orthogonal projections restricted to a compact set \Omega \subset \R^{2}, the problem dates back to a 1954 paper by Marstrand: he proved that for almost every \lambda there exist positively many $t \in \R$ such that \dim \pi_{\lambda}^{-1}{t} = \dim \Omega - 1. For generalized projections, the same result was obtained 50 years later by J\"arvenp\"a\"a, J\"arvenp\"a\"a and Niemel\"a. In this paper, we improve the previously existing estimates by replacing the phrase 'almost all \lambda' with a sharp bound for the dimension of the exceptional parameters.Comment: 31 pages, three figures; several typos corrected and large parts of the third section rewritten in v3; to appear in J. Geom. Ana

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and \al, how large Hausdorff dimension can a compact set A\su\Rn have if $A$ does not contain three points that form an angle \al? (2) Given \al and \de, how large Hausdorff dimension can a %compact subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the \de-neighborhood of \al? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles 0 and $180^\circ$, the angles $60^\circ,90^\circ$ and $120^\circ$ also play special role in problem (2): the maximal dimension is smaller for these special angles than for the other angles. In problem (1) the angle $90^\circ$ seems to behave differently from other angles

The Lagrange and Markov spectra from the dynamical point of view

This text grew out of my lecture notes for a 4-hours minicourse delivered on October 17 \& 19, 2016 during the research school "Applications of Ergodic Theory in Number Theory" -- an activity related to the Jean-Molet Chair project of Mariusz Lema\'nczyk and S\'ebastien Ferenczi -- realized at CIRM, Marseille, France. The subject of this text is the same of my minicourse, namely, the structure of the so-called Lagrange and Markov spectra (with an special emphasis on a recent theorem of C. G. Moreira).Comment: 27 pages, 6 figures. Survey articl

Self-similar sets: projections, sections and percolation

We survey some recent results on the dimension of orthogonal projections of self-similar sets and of random subsets obtained by percolation on self-similar sets. In particular we highlight conditions when the dimension of the projections takes the generic value for all, or very nearly all, projections. We then describe a method for deriving dimensional properties of sections of deterministic self-similar sets by utilising projection properties of random percolation subsets.Postprin

Left Ventricular Systolic Dysfunction in Patients Diagnosed With Hypertrophic Cardiomyopathy During Childhood: Insights From the SHaRe Registry.

BACKGROUND: The development of left ventricular systolic dysfunction (LVSD) in hypertrophic cardiomyopathy (HCM) is rare but serious and associated with poor outcomes in adults. Little is known about the prevalence, predictors, and prognosis of LVSD in patients diagnosed with HCM as children. METHODS: Data from patients with HCM in the international, multicenter SHaRe (Sarcomeric Human Cardiomyopathy Registry) were analyzed. LVSD was defined as left ventricular ejection fraction <50% on echocardiographic reports. Prognosis was assessed by a composite of death, cardiac transplantation, and left ventricular assist device implantation. Predictors of developing incident LVSD and subsequent prognosis with LVSD were assessed using Cox proportional hazards models. RESULTS: We studied 1010 patients diagnosed with HCM during childhood (<18 years of age) and compared them with 6741 patients with HCM diagnosed as adults. In the pediatric HCM cohort, median age at HCM diagnosis was 12.7 years (interquartile range, 8.0-15.3), and 393 (36%) patients were female. At initial SHaRe site evaluation, 56 (5.5%) patients with childhood-diagnosed HCM had prevalent LVSD, and 92 (9.1%) developed incident LVSD during a median follow-up of 5.5 years. Overall LVSD prevalence was 14.7% compared with 8.7% in patients with adult-diagnosed HCM. Median age at incident LVSD was 32.6 years (interquartile range, 21.3-41.6) for the pediatric cohort and 57.2 years (interquartile range, 47.3-66.5) for the adult cohort. Predictors of developing incident LVSD in childhood-diagnosed HCM included age <12 years at HCM diagnosis (hazard ratio [HR], 1.72 [CI, 1.13-2.62), male sex (HR, 3.1 [CI, 1.88-5.2), carrying a pathogenic sarcomere variant (HR, 2.19 [CI, 1.08-4.4]), previous septal reduction therapy (HR, 2.34 [CI, 1.42-3.9]), and lower initial left ventricular ejection fraction (HR, 1.53 [CI, 1.38-1.69] per 5% decrease). Forty percent of patients with LVSD and HCM diagnosed during childhood met the composite outcome, with higher rates in female participants (HR, 2.60 [CI, 1.41-4.78]) and patients with a left ventricular ejection fraction <35% (HR, 3.76 [2.16-6.52]). CONCLUSIONS: Patients with childhood-diagnosed HCM have a significantly higher lifetime risk of developing LVSD, and LVSD emerges earlier than for patients with adult-diagnosed HCM. Regardless of age at diagnosis with HCM or LVSD, the prognosis with LVSD is poor, warranting careful surveillance for LVSD, especially as children with HCM transition to adult care

Hypertrophic Cardiomyopathy with Left Ventricular Systolic Dysfunction: Insights from the SHaRe Registry

Background: The term "end stage" has been used to describe hypertrophic cardiomyopathy (HCM) with left ventricular systolic dysfunction (LVSD), defined as occurring when left ventricular ejection fraction is <50%. The prognosis of HCM-LVSD has reportedly been poor, but because of its relative rarity, the natural history remains incompletely characterized. Methods: Data from 11 high-volume HCM specialty centers making up the international SHaRe Registry (Sarcomeric Human Cardiomyopathy Registry) were used to describe the natural history of patients with HCM-LVSD. Cox proportional hazards models were used to identify predictors of prognosis and incident development. Results: From a cohort of 6793 patients with HCM, 553 (8%) met the criteria for HCM-LVSD. Overall, 75% of patients with HCM-LVSD experienced clinically relevant events, and 35% met the composite outcome (all-cause death [n=128], cardiac transplantation [n=55], or left ventricular assist device implantation [n=9]). After recognition of HCM-LVSD, the median time to composite outcome was 8.4 years. However, there was substantial individual variation in natural history. Significant predictors of the composite outcome included the presence of multiple pathogenic/likely pathogenic sarcomeric variants (hazard ratio [HR], 5.6 [95% CI, 2.3-13.5]), atrial fibrillation (HR, 2.6 [95% CI, 1.7-3.5]), and left ventricular ejection fraction <35% (HR, 2.0 [95% CI, 1.3-2.8]). The incidence of new HCM-LVSD was ≈7.5% over 15 years. Significant predictors of developing incident HCM-LVSD included greater left ventricular cavity size (HR, 1.1 [95% CI, 1.0-1.3] and wall thickness (HR, 1.3 [95% CI, 1.1-1.4]), left ventricular ejection fraction of 50% to 60% (HR, 1.8 [95% CI, 1.2, 2.8]-2.8 [95% CI, 1.8-4.2]) at baseline evaluation, the presence of late gadolinium enhancement on cardiac magnetic resonance imaging (HR, 2.3 [95% CI, 1.0-4.9]), and the presence of a pathogenic/likely pathogenic sarcomeric variant, particularly in thin filament genes (HR, 1.5 [95% CI, 1.0-2.1] and 2.5 [95% CI, 1.2-5.1], respectively). Conclusions: HCM-LVSD affects ≈8% of patients with HCM. Although the natural history of HCM-LVSD was variable, 75% of patients experienced adverse events, including 35% experiencing a death equivalent an estimated median time of 8.4 years after developing systolic dysfunction. In addition to clinical features, genetic substrate appears to play a role in both prognosis (multiple sarcomeric variants) and the risk for incident development of HCM-LVSD (thin filament variants)

A computational evaluation of over-representation of regulatory motifs in the promoter regions of differentially expressed genes

BACKGROUND: Observed co-expression of a group of genes is frequently attributed to co-regulation by shared transcription factors. This assumption has led to the hypothesis that promoters of co-expressed genes should share common regulatory motifs, which forms the basis for numerous computational tools that search for these motifs. While frequently explored for yeast, the validity of the underlying hypothesis has not been assessed systematically in mammals. This demonstrates the need for a systematic and quantitative evaluation to what degree co-expressed genes share over-represented motifs for mammals. RESULTS: We identified 33 experiments for human and mouse in the ArrayExpress Database where transcription factors were manipulated and which exhibited a significant number of differentially expressed genes. We checked for over-representation of transcription factor binding sites in up- or down-regulated genes using the over-representation analysis tool oPOSSUM. In 25 out of 33 experiments, this procedure identified the binding matrices of the affected transcription factors. We also carried out de novo prediction of regulatory motifs shared by differentially expressed genes. Again, the detected motifs shared significant similarity with the matrices of the affected transcription factors. CONCLUSIONS: Our results support the claim that functional regulatory motifs are over-represented in sets of differentially expressed genes and that they can be detected with computational methods

Distance sets, orthogonal projections, and passing to weak tangents

The author is supported by a Leverhulme Trust Research Fellowship (RF-2016-500).We consider the Assouad dimension analogues of two important problems in geometric measure theory. These problems are tied together by the common theme of ‘passing to weak tangents’. First, we solve the analogue of Falconer’s distance set problem for Assouad dimension in the plane: if a planar set has Assouad dimension greater than 1, then its distance set has Assouad dimension 1. We also obtain partial results in higher dimensions. Second, we consider how Assouad dimension behaves under orthogonal projection. We extend the planar projection theorem of Fraser and Orponen to higher dimensions, provide estimates on the (Hausdorff) dimension of the exceptional set of projections, and provide a recipe for obtaining results about restricted families of projections. We provide several illustrative examples throughout.PostprintPeer reviewe

Isoform Diversity and Regulation in Peripheral and Central Neurons Revealed through RNA-Seq

To fully understand cell type identity and function in the nervous system there is a need to understand neuronal gene expression at the level of isoform diversity. Here we applied Next Generation Sequencing of the transcriptome (RNA-Seq) to purified sensory neurons and cerebellar granular neurons (CGNs) grown on an axonal growth permissive substrate. The goal of the analysis was to uncover neuronal type specific isoforms as a prelude to understanding patterns of gene expression underlying their intrinsic growth abilities. Global gene expression patterns were comparable to those found for other cell types, in that a vast majority of genes were expressed at low abundance. Nearly 18% of gene loci produced more than one transcript. More than 8000 isoforms were differentially expressed, either to different degrees in different neuronal types or uniquely expressed in one or the other. Sensory neurons expressed a larger number of genes and gene isoforms than did CGNs. To begin to understand the mechanisms responsible for the differential gene/isoform expression we identified transcription factor binding sites present specifically in the upstream genomic sequences of differentially expressed isoforms, and analyzed the 3′ untranslated regions (3′ UTRs) for microRNA (miRNA) target sites. Our analysis defines isoform diversity for two neuronal types with diverse axon growth capabilities and begins to elucidate the complex transcriptional landscape in two neuronal populations