Sharp measure contraction property for generalized H-type Carnot groups

We prove that H-type Carnot groups of rank $k$ and dimension $n$ satisfy the $\mathrm{MCP}(K,N)$ if and only if $K\leq 0$ and $N \geq k+3(n-k)$. The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in this paper, and for which we compute explicitly the optimal synthesis. This constitutes the largest class of Carnot groups for which the curvature exponent coincides with the geodesic dimension. We stress that generalized H-type Carnot groups have step 2, include all corank 1 groups and, in general, admit abnormal minimizing curves. As a corollary, we prove the absolute continuity of the Wasserstein geodesics for the quadratic cost on all generalized H-type Carnot groups.Comment: 18 pages. This article extends the results of arXiv:1510.05960. v2: revised and improved version. v3: final version, to appear in Commun. Contemp. Mat

CFD modelling of wind turbine airfoil aerodynamics

This paper reports the first findings of an ongoing research programme on wind turbine computational aerodynamics at the University of Glasgow. Several modeling aspects of wind turbine airfoil aerodynamics based on the solution of the Reynoldsaveraged Navier-Stokes (RANS) equations are addressed. One of these is the effect of an a priori method for structured grid adaptation aimed at improving the wake resolution. Presented results emphasize that the proposed adaptation strategy greatly improves the wake resolution in the far-field, whereas the wake is completely diffused by the non-adapted grid with the same number and distribution of grid nodes. A grid refinement analysis carried out with the adapted grid shows that the improvements of flow resolution thus achieved are of a smaller magnitude with respect to those accomplished by adapting the grid keeping constant the number of nodes. The proposed adaptation approach can be easily included in the structured generation process of both commercial and in-house structured mesh generators systems. The study also aims at quantifying the solution inaccuracy arising from not modeling the laminar-to-turbulent transition. It is found that the drag forces obtained by considering the flow as transitional or fully turbulent may differ by 50 %. The impact of various turbulence models on the predicted aerodynamic forces is also analyzed. All these issues are investigated using a special-purpose hyperbolic grid generator and a multi-block structured finitevolume RANS code. The numerical experiments consider the flow field past a wind turbine airfoil for which an exhaustive campaign of steady and unsteady experimental measurements was conducted. The predictive capabilities of the CFD solver are validated by comparing experimental data and numerical predictions for selected flow regimes. The incompressible analysis and design code XFOIL is also used to support the findings of the comparative analysis of numerical RANS-based results and experimental data

The role of fundamental solution in Potential and Regularity Theory for subelliptic PDE

In this survey we consider a general Hormander type operator, represented as a sum of squares of vector fields plus a drift and we outline the central role of the fundamental solution in developing Potential and Regularity Theory for solutions of related PDEs. After recalling the Gaussian behavior at infinity of the kernel, we show some mean value formulas on the level sets of the fundamental solution, which are the starting point to obtain a comprehensive parallel of the classical Potential Theory. Then we show that a precise knowledge of the fundamental solution leads to global regularity results, namely estimates at the boundary or on the whole space. Finally in the problem of regularity of non linear differential equations we need an ad hoc modification of the parametrix method, based on the properties of the fundamental solution of an approximating problem

Portrayal of psychiatric genetics in Australian print news media, 1996-2009

Objective: To investigate how Australian print news media portray psychiatric genetics. Design and setting: Content and framing analysis of a structured sample of print news items about psychiatric genetics published in Australian newspapers between 1996 and 2009. Main outcome measures: Identify dominant discourses about aetiology of mental illness, and perceived clinical outcomes and implications of psychiatric genetics research. Results: We analysed 406 eligible items about the genetics of psychiatric disorders. News coverage of psychiatric genetics has steadily increased since 1996. Items attributing the aetiology of psychiatric disorders to gene-environment interactions (51%) outnumbered items attributing only genetic (30%) or only environmental factors (20%). Of items that referred to heritability of mental illness, frames of genetic determinism (78%) occurred more frequently than probabilistic frames (22%). Of frames related to genetic prophesy, genetic optimism frames (78%) were used more frequently than frames of genetic pessimism (22%). Psychosocial and ethical implications of psychiatric genetics received comparatively relatively little coverage (23%). The analysis identified 22 predictions about psychiatric genetic discoveries and the availability of molecular-based interventions in psychiatry, most of which (20/ 22, 91%) failed to manifest by the predicted year. Conclusions: Excessive optimism about the power of genetic technology in psychiatric health care, perceived clinical benefits, and largely unfulfilled predictions about availability of these benefits could encourage unrealistic expectations about future molecular-based treatment options for mental health

Approximations of Sobolev norms in Carnot groups

This paper deals with a notion of Sobolev space $W^{1,p}$ introduced by J.Bourgain, H.Brezis and P.Mironescu by means of a seminorm involving local averages of finite differences. This seminorm was subsequently used by A.Ponce to obtain a Poincar\'e-type inequality. The main results that we present are a generalization of these two works to a non-Euclidean setting, namely that of Carnot groups. We show that the seminorm expressd in terms of the intrinsic distance is equivalent to the $L^p$ norm of the intrinsic gradient, and provide a Poincar\'e-type inequality on Carnot groups by means of a constructive approach which relies on one-dimensional estimates. Self-improving properties are also studied for some cases of interest

On the Hausdorff volume in sub-Riemannian geometry

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4 on every smooth curve) but in general not C^5. These results answer to a question addressed by Montgomery about the relation between two intrinsic volumes that can be defined in a sub-Riemannian manifold, namely the Popp and the Hausdorff volume. If the nilpotent approximation depends on the point (that may happen starting from dimension 5), then they are not proportional, in general.Comment: Accepted on Calculus and Variations and PD

Harnack inequality for fractional sub-Laplacians in Carnot groups

In this paper we prove an invariant Harnack inequality on Carnot-Carath\'eodory balls for fractional powers of sub-Laplacians in Carnot groups. The proof relies on an "abstract" formulation of a technique recently introduced by Caffarelli and Silvestre. In addition, we write explicitly the Poisson kernel for a class of degenerate subelliptic equations in product-type Carnot groups

Investigations on offshore wind turbine inflow modelling using numerical weather prediction coupled with local-scale computational fluid dynamics

The computational power available nowadays to industry and research paves the way to increasingly more accurate systems for the wind resource prediction. A promising approach is to support the mesoscale numerical weather prediction (NWP) with high fidelity computational fluid dynamics (CFD). This approach aims at increasing the spatial resolution of the wind prediction by not only accounting for the complex and multiphysics aspects of the atmosphere over a large geographical region, but also including the effects of the fine scale turbulence and the interaction of the wind flow with the sea surface. In this work, we test a set of model setups for both the mesoscale (NWP) and local scale (CFD) simulations employed in a multi-scale modelling framework. The method comprises a one-way coupling interface to define boundary conditions for the local scale simulation (based on the Reynolds Averaged Navier–Stokes equations) using the mesoscale wind given by the NWP system. The wind prediction in an offshore site is compared with LiDAR measurements, testing a set of mesoscale planetary boundary layer schemes, and different model choices for the local scale simulation, which include steady and unsteady approaches for simulation and boundary conditions, different turbulence closure constants, and the effect of the wave motion of the sea surface. The resulting wind is then used for the simulation of a large wind turbine, showing how a realistic wind profile and an ideal exponential law profile lead to different predictions of wind turbine rotor performance and loads

An embedded shock-fitting technique on unstructured dynamic grids

In this paper, a new shock-fitting technique based on unstructured dynamic grids is proposed to improve the performances of the unstructured “boundary” shock-fitting technique developed by Liu and co-workers in [1, 2]. The main feature of this new technique, which we call the “embedded” shock-fitting technique, is its capability to insert or remove shocks or parts thereof during the calculation. This capability is enabled by defining subsets of grid-points (mutually connected by lines) which behave as either “common”- or “shock”-points, shock-waves being made of an ordered collection of shock-points. Two different sets of flow variables, corresponding to the upstream and downstream sides of the shocks, are assigned to the shock-points, which may be switched to common- and back to shock-points, a feature that allows to vary the length of the existing shocks and/or make new shock-branches appear. This paper illustrates the algorithmic features of this new technique and presents the results obtained when simulating both steady and un-steady, two-dimensional flows