Algebras generated by two bounded holomorphic functions

We study the closure in the Hardy space or the disk algebra of algebras generated by two bounded functions, of which one is a finite Blaschke product. We give necessary and sufficient conditions for density or finite codimension of such algebras. The conditions are expressed in terms of the inner part of a function which is explicitly derived from each pair of generators. Our results are based on identifying z-invariant subspaces included in the closure of the algebra. Versions of these results for the case of the disk algebra are given.Comment: 22 pages ; a number of minor mistakes have been corrected, and some points clarified. Conditionally accepted by Journal d'Analyse Mathematiqu

Invariant elliptic curves as attractors in the projective plane

Let f be a rational self-map of P^2 which leaves invariant an elliptic curve C with strictly negative transverse Lyapunov exponent. We show that C is an attractor, i.e. it possesses a dense orbit and its basin is of strictly positive measure

Lyapunov exponents, bifurcation currents and laminations in bifurcation loci

Bifurcation loci in the moduli space of degree $d$ rational maps are shaped by the hypersurfaces defined by the existence of a cycle of period $n$ and multiplier 0 or $e^{i\theta}$. Using potential-theoretic arguments, we establish two equidistribution properties for these hypersurfaces with respect to the bifurcation current. To this purpose we first establish approximation formulas for the Lyapunov function. In degree $d=2$, this allows us to build holomorphic motions and show that the bifurcation locus has a lamination structure in the regions where an attracting basin of fixed period exists

Young Adults With Anterior Ischemic Optic Neuropathy: A Multicenter Optic Disc Drusen Study.

PURPOSE: Optic disc drusen (ODD), present in 2% of the general population, have occasionally been reported in patients with nonarteritic anterior ischemic optic neuropathy (NA-AION). The purpose of this study was to examine the prevalence of ODD in young patients with NA-AION. DESIGN: Retrospective, cross-sectional multicenter study. METHODS: All patients with NA-AION 50 years old or younger, seen in neuro-ophthalmology clinics of the international ODDS (Optic Disc Drusen Studies) Consortium between April 1, 2017, and March 31, 2019, were identified. Patients were included if ODD were diagnosed by any method, or if ODD were excluded by enhanced-depth imaging optical coherence tomography (EDI-OCT) using ODDS Consortium guidelines. NA-AION eyes with ODD were termed "ODD-AION"; those without were termed "NODD-AION". RESULTS: A total of 65 patients (127 eyes) with NA-AION were included (mean 41 years old). Of the 74 eyes with NA-AION, 51% had ODD-AION, whereas 43% of fellow eyes without NA-AION had ODD (P = .36). No significant differences were found between ODD-AION and NODD-AION eyes in terms of Snellen best-corrected VA or perimetric mean deviation. According to EDI-OCT results, 28% of eyes with NODD-AION had peripapillary hyperreflective ovoid mass-like structures (PHOMS); 7% had hyperreflective lines, whereas 54% with ODD-AION had PHOMS; and 66% had hyperreflective lines (P = .006 and P < .001, respectively). CONCLUSIONS: Most of these young NA-AION patients had ODD. This indicates that ODD may be an independent risk factor for the development of NA-AION, at least in younger patients. This study suggests ODD-AION be recognized as a novel diagnosis

Multirater Validation of Peripapillary Hyperreflective Ovoid Mass-like Structures (PHOMS)

Peripapillary hyperreflective ovoid mass-like structures (PHOMS) are a new retinal optical coherence tomography (OCT) finding. The Optic Disc Drusen Studies Consortium had made recommendations to distinguish PHOMS from true optic disc drusen (ODD) in 2018. While publications on PHOMS have increased since then, the accuracy of the definition of PHOMS and reliability of detection is unknown. In this multi-rater study, we demonstrate that the 2018 definition of PHOMS resulted in a poor multi-rater kappa of 0.356. We performed a Delphi consensus process to develop a consistent and refined definition of PHOMS with clear principles around the nature of PHOMS and how they differ from normal anatomy. Fifty explanatory teaching slides, provided as supplementary material, allowed our expert group of raters to achieve a good level of agreement (kappa 0.701, 50 OCT scans, 21 raters). We recommend adopting the refined definition for PHOMS

Angiotensin-converting enzyme and male fertility

The angiotensin-converting enzyme (ACE; EC 3.4.15.1) gene (Ace) encodes both a somatic isozyme found in blood and several other tissues, including the epididymis, and a testis-specific isozyme (testis ACE) found only in developing spermatids and mature sperm. We recently used gene targeting to disrupt the gene coding for both ACE isozymes in mice and reported that male homozygous mutants mate normally but have reduced fertility; the mutant females are fertile. Here we explore the male fertility defect. We demonstrate that ACE is important for achieving in vivo fertilization and that sperm from mice lacking both ACE isozymes show defects in transport within the oviducts and in binding to zonae pellucidae. Males generated by gene targeting that lack somatic ACE but retain testis ACE are normally fertile, establishing that somatic ACE in males is not essential for their fertility. Furthermore, male and female mice lacking angiotensinogen have normal fertility, indicating that angiotensin I is not a necessary substrate for testis ACE. Males heterozygous for the mutation inactivating both ACE isozymes sire wild-type and heterozygous offspring at an indistinguishable frequency, indicating no selection against sperm carrying the mutation

Oka's inequality for currents and applications

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46241/1/208_2005_Article_BF01446636.pd

EU sports law: a uniform algorithm for regulatory rules

© 2017, T.M.C. Asser Instituut. In applying the EU economic provisions to the regulatory rules in sport, four different categories of “sporting exception” can be discerned in the jurisprudence of the Court. They include sporting rules that do not produce any economic effect, ‘purely sporting’ rules, inherent rules, and objectively justified rules. Based on the existing parameters of the EU sports law and policy, this article advances arguments in support of discarding the nuances in the Court’s analytical approach to sporting exception. Ordinary EU law, coupled by the concept of specificity of sport that is now included in Article 165(1) TFEU, already contains the all-encompassing, uniform analytical structure apt to accommodate all categories of regulatory rules in sports. In addition, the proposed uniform framework can be often be utilised to justify the challenged sporting rules in both internal market law and competition law, thus avoiding duplication of analysis. This is enabled by the high degree of convergence in their application to the rules of private regulatory bodies

Grids of stellar models with rotation VII : models from 0.8 to 300 M⊙ at supersolar metallicity (Z = 0.020)

We present a grid of stellar models at supersolar metallicity (Z = 0.020) extending the previous grids of Geneva models at solar and sub-solar metallicities. A metallicity of Z = 0.020 was chosen to match that of the inner Galactic disc. A modest increase of 43 per cent (= 0.02/0.014) in metallicity compared to solar models means that the models evolve similarly to solar models but with slightly larger mass-loss. Mass-loss limits the final total masses of the supersolar models to 35 M⊙ even for stars with initial masses much larger than 100 M⊙. Mass-loss is strong enough in stars above 20 M⊙ for rotating stars (25 M⊙ for non-rotating stars) to remove the entire hydrogen-rich envelope. Our models thus predict SNII below 20 M⊙ for rotating stars (25 M⊙ for non-rotating stars) and SNIb (possibly SNIc) above that. We computed both isochrones and synthetic clusters to compare our supersolar models to the Westerlund 1 (Wd1) massive young cluster. A synthetic cluster combining rotating and non-rotating models with an age spread between log10(age/yr) = 6.7 and 7.0 is able to reproduce qualitatively the observed populations of WR, RSG, and YSG stars in Wd1, in particular their simultaneous presence at log10(L/L⊙) = 5–5.5. The quantitative agreement is imperfect and we discuss the likely causes: synthetic cluster parameters, binary interactions, mass-loss and their related uncertainties. In particular, mass-loss in the cool part of the HRD plays a key role