A note on the Fredholm properties of Toeplitz operators on weighted Bergman spaces with matrix-valued symbols

We characterize the essential spectra of Toeplitz operators Ta on weighted Bergman spaces with matrix-valued symbols; in particular we deal with two classes of symbols, the Douglas algebra C+H∞ and the Zhu class Q := L∞ ∩VMO∂ . In addition, for symbols in C+H∞ , we derive a formula for the index of Ta in terms of its symbol a in the scalar-valued case, while in the matrix-valued case we indicate that the standard reduction to the scalar-valued case fails to work analogously to the Hardy space case. Mathematics subject classification (2010): 47B35,

Hankel and Toeplitz transforms on H 1: continuity, compactness and Fredholm properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H 1 and give a new proof of the old result stating that the Hankel operator H a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H a to be compact on H 1. The Fredholm properties of Toeplitz operators on H 1 are studied for symbols in a Banach algebra similar to C + H ∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H 1 and H 2.

Pathogenic LRRK2 mutations do not alter gene expression in cell model systems or human brain tissue

Point mutations in LRRK2 cause autosomal dominant Parkinson's disease. Despite extensive efforts to determine the mechanism of cell death in patients with LRRK2 mutations, the aetiology of LRRK2 PD is not well understood. To examine possible alterations in gene expression linked to the presence of LRRK2 mutations, we carried out a case versus control analysis of global gene expression in three systems: fibroblasts isolated from LRRK2 mutation carriers and healthy, non-mutation carrying controls; brain tissue from G2019S mutation carriers and controls; and HEK293 inducible LRRK2 wild type and mutant cell lines. No significant alteration in gene expression was found in these systems following correction for multiple testing. These data suggest that any alterations in basal gene expression in fibroblasts or cell lines containing mutations in LRRK2 are likely to be quantitatively small. This work suggests that LRRK2 is unlikely to play a direct role in modulation of gene expression, although it remains possible that this protein can influence mRNA expression under pathogenic cicumstances.

Decay to equilibrium for jump processes with heavy tails

The case is made for a more careful analysis of the large time asymptotic of infinite particle systems in the thermodynamic limit beyond zero density. The insufficiency of current analysis even in the model case of free particles is demonstrated. Recent advances based on more sophisticated analytical tools like functions of mean variation and Hardy spaces are sketched.

The myths of modern primitivism

RE/Search Publications’ Modern Primitives (Vale and Juno 1989) changed countless lives, bringing what had been a localized and niche set of body modification practices, aesthetics and philosophies out of San Francisco to a global audience, dominating scholarly and popular discourse around body modification subculture for more than a decade afterwards. The voice of Fakir Musafar dominates the book. This article argues that modern primitives as Musafar defines them never really existed (and never could have existed) in the terms he suggests, and goes on to address an important sub-strand within Modern Primitives almost entirely ignored by critics and commentators, who have read the book as generally representative of the body modification culture as a whole. With specific reference to contributors such as infamous tattoo artist Don Ed Hardy who do not frame their practice in ‘primitive’ terms, the article concludes with a study of an alternative account presented by Vale and Juno’s book: body modification as artistic practice.

A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators

We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.

The Brownian traveller on manifolds

We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the at case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the at case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.

Dominant β-catenin mutations cause intellectual disability with recognizable syndromic features

The recent identification of multiple dominant mutations in the gene encoding β-catenin in both humans and mice has enabled exploration of the molecular and cellular basis of β-catenin function in cognitive impairment. In humans, β-catenin mutations that cause a spectrum of neurodevelopmental disorders have been identified. We identified de novo β-catenin mutations in patients with intellectual disability, carefully characterized their phenotypes, and were able to define a recognizable intellectual disability syndrome. In parallel, characterization of a chemically mutagenized mouse line that displays features similar to those of human patients with β-catenin mutations enabled us to investigate the consequences of β-catenin dysfunction through development and into adulthood. The mouse mutant, designated batface (Bfc), carries a Thr653Lys substitution in the C-terminal armadillo repeat of β-catenin and displayed a reduced affinity for membrane-associated cadherins. In association with this decreased cadherin interaction, we found that the mutation results in decreased intrahemispheric connections, with deficits in dendritic branching, long-term potentiation, and cognitive function. Our study provides in vivo evidence that dominant mutations in β-catenin underlie losses in its adhesion-related functions, which leads to severe consequences, including intellectual disability, childhood hypotonia, progressive spasticity of lower limbs, and abnormal craniofacial features in adults

Pathogenic Parkinson’s disease mutations across the functional domains of LRRK2 alter the autophagic/lysosomal response to starvation

LRRK2 is one of the most important genetic contributors to Parkinson’s disease (PD). Point mutations in this gene cause an autosomal dominant form of PD, but to date no cellular phenotype has been consis- tently linked with mutations in each of the functional domains (ROC, COR and Kinase) of the protein product of this gene. In this study, primary fibroblasts from individuals carrying pathogenic mutations in the three central domains of LRRK2 were assessed for alterations in the autophagy/lysosomal pathway using a combination of biochemical and cellular approaches. Mutations in all three domains resulted in alterations in markers for autophagy/lysosomal function compared to wild type cells. These data high- light the autophagy and lysosomal pathways as read outs for pathogenic LRRK2 function and as a marker for disease, and provide insight into the mechanisms linking LRRK2 function and mutations.

Association of vitamin D status with arterial blood pressure and hypertension risk: a mendelian randomisation study

BACKGROUND: Low plasma 25-hydroxyvitamin D (25[OH]D) concentration is associated with high arterial blood pressure and hypertension risk, but whether this association is causal is unknown. We used a mendelian randomisation approach to test whether 25(OH)D concentration is causally associated with blood pressure and hypertension risk. METHODS: In this mendelian randomisation study, we generated an allele score (25[OH]D synthesis score) based on variants of genes that affect 25(OH)D synthesis or substrate availability (CYP2R1 and DHCR7), which we used as a proxy for 25(OH)D concentration. We meta-analysed data for up to 108 173 individuals from 35 studies in the D-CarDia collaboration to investigate associations between the allele score and blood pressure measurements. We complemented these analyses with previously published summary statistics from the International Consortium on Blood Pressure (ICBP), the Cohorts for Heart and Aging Research in Genomic Epidemiology (CHARGE) consortium, and the Global Blood Pressure Genetics (Global BPGen) consortium. FINDINGS: In phenotypic analyses (up to n=49 363), increased 25(OH)D concentration was associated with decreased systolic blood pressure (β per 10% increase, -0·12 mm Hg, 95% CI -0·20 to -0·04; p=0·003) and reduced odds of hypertension (odds ratio [OR] 0·98, 95% CI 0·97-0·99; p=0·0003), but not with decreased diastolic blood pressure (β per 10% increase, -0·02 mm Hg, -0·08 to 0·03; p=0·37). In meta-analyses in which we combined data from D-CarDia and the ICBP (n=146 581, after exclusion of overlapping studies), each 25(OH)D-increasing allele of the synthesis score was associated with a change of -0·10 mm Hg in systolic blood pressure (-0·21 to -0·0001; p=0·0498) and a change of -0·08 mm Hg in diastolic blood pressure (-0·15 to -0·02; p=0·01). When D-CarDia and consortia data for hypertension were meta-analysed together (n=142 255), the synthesis score was associated with a reduced odds of hypertension (OR per allele, 0·98, 0·96-0·99; p=0·001). In instrumental variable analysis, each 10% increase in genetically instrumented 25(OH)D concentration was associated with a change of -0·29 mm Hg in diastolic blood pressure (-0·52 to -0·07; p=0·01), a change of -0·37 mm Hg in systolic blood pressure (-0·73 to 0·003; p=0·052), and an 8·1% decreased odds of hypertension (OR 0·92, 0·87-0·97; p=0·002). INTERPRETATION: Increased plasma concentrations of 25(OH)D might reduce the risk of hypertension. This finding warrants further investigation in an independent, similarly powered study.