Nontangential limits and Fatou-type theorems on post-critically finite self-similar sets

In this paper we study the boundary limit properties of harmonic functions on $\mathbb R_+\times K$, the solutions $u(t,x)$ to the Poisson equation $$\frac{\partial^2 u}{\partial t^2} + \Delta u = 0,$$ where $K$ is a p.c.f. set and $\Delta$ its Laplacian given by a regular harmonic structure. In particular, we prove the existence of nontangential limits of the corresponding Poisson integrals, and the analogous results of the classical Fatou theorems for bounded and nontangentially bounded harmonic functions.Comment: 22 page

Intramedullary nailing versus sliding hip screw for A1 and A2 trochanteric hip fractures

AIMS: This study evaluated variation in the surgical treatment of stable (A1) and unstable (A2) trochanteric hip fractures among an international group of orthopaedic surgeons, and determined the influence of patient, fracture, and surgeon characteristics on choice of implant (intramedullary nailing (IMN) versus sliding hip screw (SHS)). METHODS: A total of 128 orthopaedic surgeons in the Science of Variation Group evaluated radiographs of 30 patients with Type A1 and A2 trochanteric hip fractures and indicated their preferred treatment: IMN or SHS. The management of Type A3 (reverse obliquity) trochanteric fractures was not evaluated. Agreement between surgeons was calculated using multirater kappa. Multivariate logistic regression models were used to assess whether patient, fracture, and surgeon characteristics were independently associated with choice of implant. RESULTS: The overall agreement between surgeons on implant choice was fair (kappa = 0.27 (95% confidence interval (CI) 0.25 to 0.28)). Factors associated with preference for IMN included USA compared to Europe or the UK (Europe odds ratio (OR) 0.56 (95% CI 0.47 to 0.67); UK OR 0.16 (95% CI 0.12 to 0.22); p < 0.001); exposure to IMN only during training compared to surgeons that were exposed to both (only IMN during training OR 2.6 (95% CI 2.0 to 3.4); p < 0.001); and A2 compared to A1 fractures (Type A2 OR 10 (95% CI 8.4 to 12); p < 0.001). CONCLUSION: In an international cohort of orthopaedic surgeons, there was a large variation in implant preference for patients with A1 and A2 trochanteric fractures. This is due to surgeon bias (country of practice and aspects of training). The observation that surgeons favoured the more expensive implant (IMN) in the absence of convincing evidence of its superiority suggests that surgeon de-biasing strategies may be a useful focus for optimizing patient outcomes and promoting value-based healthcare. Cite this article: Bone Joint J 2021;103-B(4):775-781

Comparisons of Supergranule Characteristics During the Solar Minima of Cycles 22/23 and 23/24

Supergranulation is a component of solar convection that manifests itself on the photosphere as a cellular network of around 35 Mm across, with a turnover lifetime of 1-2 days. It is strongly linked to the structure of the magnetic field. The horizontal, divergent flows within supergranule cells carry local field lines to the cell boundaries, while the rotational properties of supergranule upflows may contribute to the restoration of the poloidal field as part of the dynamo mechanism that controls the solar cycle. The solar minimum at the transition from cycle 23 to 24 was notable for its low level of activity and its extended length. It is of interest to study whether the convective phenomena that influences the solar magnetic field during this time differed in character to periods of previous minima. This study investigates three characteristics (velocity components, sizes and lifetimes) of solar supergranulation. Comparisons of these characteristics are made between the minima of cycles 22/23 and 23/24 using MDI Doppler data from 1996 and 2008, respectively. It is found that whereas the lifetimes are equal during both epochs (around 18 h), the sizes are larger in 1996 (35.9 +/- 0.3 Mm) than in 2008 (35.0 +/- 0.3 Mm), while the dominant horizontal velocity flows are weaker (139 +/- 1 m/s in 1996; 141 +/- 1 m/s in 2008). Although numerical differences are seen, they are not conclusive proof of the most recent minimum being inherently unusual.Comment: 22 pages, 5 figures. Solar Physics, in pres

Active region formation through the negative effective magnetic pressure instability

The negative effective magnetic pressure instability operates on scales encompassing many turbulent eddies and is here discussed in connection with the formation of active regions near the surface layers of the Sun. This instability is related to the negative contribution of turbulence to the mean magnetic pressure that causes the formation of large-scale magnetic structures. For an isothermal layer, direct numerical simulations and mean-field simulations of this phenomenon are shown to agree in many details in that their onset occurs at the same depth. This depth increases with increasing field strength, such that the maximum growth rate of this instability is independent of the field strength, provided the magnetic structures are fully contained within the domain. A linear stability analysis is shown to support this finding. The instability also leads to a redistribution of turbulent intensity and gas pressure that could provide direct observational signatures.Comment: 19 pages, 10 figures, submitted to Solar Physic

Concentration Inequalities and Confidence Bands for Needlet Density Estimators on Compact Homogeneous Manifolds

Let $X_1,...,X_n$ be a random sample from some unknown probability density $f$ defined on a compact homogeneous manifold $\mathbf M$ of dimension $d \ge 1$. Consider a 'needlet frame' $\{\phi_{j \eta}\}$ describing a localised projection onto the space of eigenfunctions of the Laplace operator on $\mathbf M$ with corresponding eigenvalues less than $2^{2j}$, as constructed in \cite{GP10}. We prove non-asymptotic concentration inequalities for the uniform deviations of the linear needlet density estimator $f_n(j)$ obtained from an empirical estimate of the needlet projection $\sum_\eta \phi_{j \eta} \int f \phi_{j \eta}$ of $f$. We apply these results to construct risk-adaptive estimators and nonasymptotic confidence bands for the unknown density $f$. The confidence bands are adaptive over classes of differentiable and H\"{older}-continuous functions on $\mathbf M$ that attain their H\"{o}lder exponents.Comment: Probability Theory and Related Fields, to appea

Objectively measured physical activity and sedentary behaviour and ankle brachial index: Cross-sectional and longitudinal associations in older men

AbstractBackgroundAssociations between bouts of physical activity (PA), sedentary behaviour (SB) and cardiovascular disease, and their mutual independence are not well defined. A low ankle brachial index (ABI ≤0.9) indicates peripheral arterial disease (PAD) and is predictive of cardiovascular events and functional impairment. We investigated the independence of PA and SB and the importance of bout duration in relation to ABI using objective measures.Methods945 men from the British Regional Heart Study, mean age 78.4 y, had concurrent measurements of ABI (Vicorder) and physical activity (Actigraph GT3X accelerometer); 427 men also had accelerometer measurements one year previously and contributed data to longitudinal analyses.Results and conclusionIn cross-sectional analyses, after adjusting for covariates each extra 10 min of moderate and vigorous PA per day was associated with an OR of 0.81 (95% CI 0.72, 0.91) for a low ABI, a stronger association than for light PA (OR 0.85, 95% CI 0.75, 0.98). Each extra 30 min of SB was associated with an OR of 1.19 (95% CI 1.07, 1.33) for a low ABI. Associations between moderate and vigorous PA and ABI persisted after adjustment for light PA or SB. Bout lengths for PA and SB were not associated with a low ABI. One year changes in PA or SB were not associated with low ABI.All physical activity and lower levels of SB, regardless of bout duration were inversely associated with ABI; more intense PA showed a stronger association. No associations between changes in PA and ABI were observed, but power may have been limited