The Morse-Sard theorem revisited

Let $n, m, k$ be positive integers with $k=n-m+1$. We establish an abstract Morse-Sard-type theorem which allows us to deduce, on the one hand, a previous result of De Pascale's for Sobolev $W^{k,p}_{\textrm{loc}}(\mathbb{R}^n, \mathbb{R}^m)$ functions with $p>n$ and, on the other hand, also the following new result: if $f\in C^{k-1}(\mathbb{R}^n, \mathbb{R}^m)$ satisfies $$\limsup_{h\to 0}\frac{|D^{k-1}f(x+h)-D^{k-1}f(x)|}{|h|}<\infty$$ for every $x\in\mathbb{R}^n$ (that is, $D^{k-1}f$ is a Stepanov function), then the set of critical values of $f$ is Lebesgue-null in $\mathbb{R}^m$. In the case that $m=1$ we also show that this limiting condition holding for every $x\in\mathbb{R}^n\setminus\mathcal{N}$, where $\mathcal{N}$ is a set of zero $(n-2+\alpha)$-dimensional Hausdorff measure for some $0<\alpha<1$, is sufficient to guarantee the same conclusion.Comment: We corrected some misprints and made some changes in the introductio

Can we make a Finsler metric complete by a trivial projective change?

A trivial projective change of a Finsler metric $F$ is the Finsler metric $F + df$. I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change. The problem actually came from lorentz geometry and mathematical relativity: it was observed that it is possible to understand the light-line geodesics of a (normalized, standard) stationary 4-dimensional space-time as geodesics of a certain Finsler Randers metric on a 3-dimensional manifold. The trivial projective change of the Finsler metric corresponds to the choice of another 3-dimensional slice, and the existence of a trivial projective change that is forward and backward complete is equivalent to the global hyperbolicity of the space-time.Comment: 11 pages, one figure, submitted to the proceedings of VI International Meeting on Lorentzian Geometry (Granada

G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion

The present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven by G-Brownian motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven by GBrownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of introducing G-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this paper is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian motion of independent interest

Fracture experience among participants from the FROCAT study: what thresholding is appropriate using the FRAX tool?

ObjectiveTo perform an external validation of FRAX algorithm thresholds for reporting level of risk of fracture in Spanish women (low &lt;5%; intermediate ?5% and &lt;7.5%; high ?7.5%) taken from a prospective cohort “FRIDEX”.MethodsA retrospective study of 1090 women aged ?40 and ?90 years old obtained from the general population (FROCAT cohort). FRAX was calculated with data registered in 2002. All fractures were validated in 2012. Sensitivity analysis was performed.ResultsWhen analyzing the cohort (884) excluding current or past anti osteoporotic medication (AOM), using our nominated thresholds, among the 621 (70.2%) women at low risk of fracture, 5.2% [CI95%: 3.4–7.6] sustained a fragility fracture; among the 99 at intermediate risk, 12.1% [6.4–20.2]; and among the 164 defined as high risk, 15.9% [10.6–24.2]. Sensitivity analysis against model risk stratification FRIDEX of FRAX Spain shows no significant difference. By including 206 women with AOM, the sensitivity analysis shows no difference in the group of intermediate and high risk and minimal differences in the low risk group.ConclusionsOur findings support and validate the use of FRIDEX thresholds of FRAX when discussing the risk of fracture and the initiation of therapy with patients

Epidemiology of fractures in Armenia: development of a country-specific FRAX model and comparison to its surrogate

Summary: Fracture probabilities derived from the surrogate FRAX model for Armenia were compared to those from the model based on regional estimates of the incidence of hip fracture. Disparities between the surrogate and authentic FRAX models indicate the importance of developing country-specific FRAX models. Despite large differences between models, differences in the rank order of fracture probabilities were minimal. Objective: Armenia has relied on a surrogate FRAX model based on the fracture epidemiology of Romania. This paper describes the epidemiology of fragility fractures in Armenia used to create an Armenia-specific FRAX model with an aim of comparing this new model with the surrogate model. Methods: We carried out a population-based study in two regions of Armenia (Ararat and Vayots Dzor representing approximately 11% of the country’s population). We aimed to identify all low-energy fractures: retrospectively from hospital registers in 2011–2012 and prospectively in 2013 with the inclusion of primary care sources. Results: The differences in incidence between the surveys with and without data from primary care suggested that 44% of patients sustaining a hip fracture did not receive specialized medical care. A similar proportion of forearm and humeral fractures did not come to hospital attention (48 and 49%, respectively). Only 57.7% of patients sustaining a hip fracture were hospitalized. In 2013, hip fracture incidence at the age of 50 years or more was 201/100,000 for women and 136/100,000 for men, and age- and sex-specific rates were incorporated into the new “authentic” FRAX model for Armenia. Compared to the surrogate model, the authentic model gave lower 10-year fracture probabilities in men and women aged less than 70 years but substantially higher above this age. Notwithstanding, there were very close correlations in fracture probabilities between the surrogate and authentic models ( >  0.99) so that the revisions had little impact on the rank order of risk. Conclusion: A substantial proportion of major osteoporotic fractures in Armenia do not come to hospital attention. The disparities between surrogate and authentic FRAX models indicate the importance of developing country-specific FRAX models. Despite large differences between models, differences in the rank order of fracture probabilities were minimal