Smooth Approximation of Lipschitz functions on Riemannian manifolds

We show that for every Lipschitz function $f$ defined on a separable Riemannian manifold $M$ (possibly of infinite dimension), for every continuous $\epsilon:M\to (0,+\infty)$, and for every positive number $r>0$, there exists a $C^\infty$ smooth Lipschitz function $g:M\to\mathbb{R}$ such that $|f(p)-g(p)|\leq\epsilon(p)$ for every $p\in M$ and $\textrm{Lip}(g)\leq\textrm{Lip}(f)+r$. Consequently, every separable Riemannian manifold is uniformly bumpable. We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville-Godefroy-Zizler's smooth variational principle.Comment: 10 page

The impact of the COVID-19 lockdown on depression sufferers: a qualitative study from the province of Zaragoza, Spain

Background and purpose: The impact of COVID-19 and its control measures have exacerbated existing mental health conditions. Although the deleterious effects of mental health problems are well known, fewer studies have examined the links between the Social Determinants of Health (SDHs) and depression. This study provides insights into the relationship between SDHs and depression during the first strict lockdown in Spain, which lasted for a period of 7 weeks. Methods: Fifty-two structured interviews were conducted with people diagnosed with depression during June 2020 in the province of Zaragoza (Spain). Interviews were conducted by telephone due to lockdown constraints. Inductive thematic content analysis was used to explore, develop, and define emergent categories of analysis, which were mapped against the SDH framework. Results: Listening to people’s experiences of living with depression during lockdown provided insights into their concerns and coping strategies, which are greatly influenced by the conditions in which they live, their job and their age. Examples of these factors include access to and quality of physical spaces, including housing conditions and public spaces for socialising, social support, adverse working conditions which include caring responsibilities, and access to digital technologies and healthcare services. Conclusion: SDHs have played a fundamental role in shaping people’s health and well-being during the COVID-19 pandemic, and this study has shown that they have a considerable effect on depression outcomes. Governments should consider implementing social welfare programs to tackle both psychosocial problems and material need during crisis situations

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

Measuring health-related quality of life in men with osteoporosis or osteoporotic fracture

<p>Abstract</p> <p>Background</p> <p>Osteoporosis is a serious health problem that worsens the quality of life and the survival rate of individuals with this disease on account the osteoporotic fractures. Studies have long focused on women, and its presence in men has been underestimated. While many studies conducted in different countries mainly assess health-related quality of life and identify fracture risks factors in women, few data are available on a Spanish male population.</p> <p>Methods/Design</p> <p>Observational study.</p> <p>Study population</p> <p>Men ≥ 40 years of age with/without diagnosed osteoporosis and with/without osteoporotic fracture included by their family doctor.</p> <p>Measurements</p> <p>The relationship between customary clinical risk factors for osteoporotic fracture and health-related quality of life in a Spanish male population. A telephone questionnaire on health-related quality of life is made.</p> <p>Statistical analysis</p> <p>The association between qualitative variables will be assessed by the Chi-square test. The distribution of quantitative variables by Student's t-test. If the conditions for using this test are not met, the non-parametric Mann-Whitney's U test will be used.</p> <p>The validation of the results obtained by the FRAX™ tool will be performed by way of the Hosmer-Lemeshow test and by calculating the area under the Receiver Operating Characteristic (ROC) curve (AUC). All tests will be performed with a confidence intervals set at 95%.</p> <p>Discussion</p> <p>The applicability and usefulness of Health-related quality of life (HRQOL) studies are well documented in many countries. These studies allow implementing cost-effective measures in cases of a given disease and reducing the costly consequences derived therefrom. This study attempts to provide objective data on how quality of life is affected by the clinical aspects involved in osteoporosis in a Spanish male population and can be useful as well in cost utility analyses conducted by health authorities.</p> <p>The sample selected is not based on a high fracture risk group. Rather, it is composed of men in the general population, and accordingly comparisons should not lead to erroneous interpretations.</p> <p>A possible bias correction will be ensured by checking reported fractures against healthcare reports and X-rays, or by consulting health care centers as applicable.</p

Orion-KL Observations with the Extended Tuning Range of the New SEPIA660 APEX Facility Instrument

During Science Verification of the new SEPIA660 facility receiver at APEX, we carried out a shallow line survey of the archetypal Kleinmann- Low Nebula in the Orion star forming region (Orion-KL). These observations cover the tuning range towards the band edges, which has recently been extended beyond ALMA Band 9 specifications. At these frequencies, atmospheric transmission is very low but still sufficient to detect bright lines in Orion-KL. We present the collected spectra and compare with surveys from the literature, demonstrating the capabilities of the instrument

Incidence rate trends for colorectal cancer in Navarre (North of Spain) in the 1990-2005 period

Fundamento. En España, se ha observado un aumento de la incidencia de cáncer colorrectal (CCR) en ambos sexos en los últimos años, posiblemente debido a las mejoras diagnósticas, a la occidentalización de la dieta y al empeoramiento de los niveles de obesidad entre otros. En este trabajo se han estudiado las tendencias de la incidencia de CCR en las diferentes áreas de salud de Navarra (norte de España) durante el período 1990-2005. Métodos. Para cada sexo y área, se obtuvieron las tendencias de las tasas de incidencia y los correspondientes intervalos de confianza mediante modelos de P-splines. Resultados. Se observa una tendencia creciente de la incidencia de CCR en la mayoría de las áreas para ambos sexos, siendo menos pronunciada en las mujeres que en los hombres. En la zona centro de Pamplona (la capital) se observa una tendencia decreciente para los hombres durante el período estudiado. Conclusiones. Para cambiar las tendencias crecientes observadas en la mayoría de las áreas de la provincia, la prevención primaria es la mejor estrategia. Sin embargo, adquirir estilos de vida saludables tiene resultados a largo plazo por lo que un programa de detección temprana serviría como estrategia de prevención a más corto plazo.Background. In Spain, an increase in the incidence of colorectal cancer (CRC) has been observed in both sexes in recent years, probably due to an improved diagnostic, the westernization of dietary habits, and worse obesity levels, among others factors. In this work, CRC incidence rate trends in different health areas in Navarre (northern Spain) are studied during the 1990-2005 period. Methods. An estimated incidence trend curve for each health area and the corresponding confidence bands were obtained for each gender using P-spline models. Results. These results show an increasing trend of CRC in most of the areas in both sexes, being less pronounced in women than in men. In the central area of Pamplona (the capital city) a decreasing trend has been observed for men during the studied period. Conclusions. Primary prevention is the best strategy to change the increasing trend observed in most areas of the province of Navarre. However, a healthy lifestyle has long-term results, so it is important to have an early detection program that would serve as a short-term prevention strategy.Este trabajo ha sido financiado por el Ministerio de Ciencia e Innovación (Proyecto MTM2008- 03085 y MTM2011-22664) y por el CIBER de Epidemiología y Salud Pública (CIBERESP)

The SEDIGISM survey: First Data Release and overview of the Galactic structure

The SEDIGISM (Structure, Excitation and Dynamics of the Inner Galactic Interstellar Medium) survey used the APEX telescope to map 84 deg$^2$ of the Galactic plane between ℓ = −60° and +31° in several molecular transitions, including $^{13}$CO (2 – 1) and C$^{18}$O (2 – 1), thus probing the moderately dense (∼10$^3$ cm$^{-3}$) component of the interstellar medium. With an angular resolution of 30 arcsec and a typical 1σ sensitivity of 0.8–1.0 K at 0.25 km s$^{-1}$ velocity resolution, it gives access to a wide range of structures, from individual star-forming clumps to giant molecular clouds and complexes. The coverage includes a good fraction of the first and fourth Galactic quadrants, allowing us to constrain the large-scale distribution of cold molecular gas in the inner Galaxy. In this paper, we provide an updated overview of the full survey and the data reduction procedures used. We also assess the quality of these data and describe the data products that are being made publicly available as part of this First Data Release (DR1). We present integrated maps and position–velocity maps of the molecular gas and use these to investigate the correlation between the molecular gas and the large-scale structural features of the Milky Way such as the spiral arms, Galactic bar and Galactic Centre. We find that approximately 60 per cent of the molecular gas is associated with the spiral arms and these appear as strong intensity peaks in the derived Galactocentric distribution. We also find strong peaks in intensity at specific longitudes that correspond to the Galactic Centre and well-known star-forming complexes, revealing that the $^{13}$CO emission is concentrated in a small number of complexes rather than evenly distributed along spiral arms