Spectral analysis and zeta determinant on the deformed spheres

We consider a class of singular Riemannian manifolds, the deformed spheres $S^N_k$, defined as the classical spheres with a one parameter family $g[k]$ of singular Riemannian structures, that reduces for $k=1$ to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian $\Delta_{S^N_k}$, we study the associated zeta functions $\zeta(s,\Delta_{S^N_k})$. We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in $\zeta(s,\Delta_{S^N_k})$. An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular $\zeta(0,\Delta_{S^N_k})$ and $\zeta'(0,\Delta_{S^N_k})$. We give explicit formulas for the zeta regularized determinant in the low dimensional cases, $N=2,3$, thus generalizing a result of Dowker \cite{Dow1}, and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter $k$.Comment: 1 figur

Daily Physical Activities and Sports in Adult Survivors of Childhood Cancer and Healthy Controls: A Population-Based Questionnaire Survey

BACKGROUND: Healthy lifestyle including sufficient physical activity may mitigate or prevent adverse long-term effects of childhood cancer. We described daily physical activities and sports in childhood cancer survivors and controls, and assessed determinants of both activity patterns. METHODOLOGY/PRINCIPAL FINDINGS: The Swiss Childhood Cancer Survivor Study is a questionnaire survey including all children diagnosed with cancer 1976-2003 at age 0-15 years, registered in the Swiss Childhood Cancer Registry, who survived ≥5 years and reached adulthood (≥20 years). Controls came from the population-based Swiss Health Survey. We compared the two populations and determined risk factors for both outcomes in separate multivariable logistic regression models. The sample included 1058 survivors and 5593 controls (response rates 78% and 66%). Sufficient daily physical activities were reported by 52% (n = 521) of survivors and 37% (n = 2069) of controls (p<0.001). In contrast, 62% (n = 640) of survivors and 65% (n = 3635) of controls reported engaging in sports (p = 0.067). Risk factors for insufficient daily activities in both populations were: older age (OR for ≥35 years: 1.5, 95CI 1.2-2.0), female gender (OR 1.6, 95CI 1.3-1.9), French/Italian Speaking (OR 1.4, 95CI 1.1-1.7), and higher education (OR for university education: 2.0, 95CI 1.5-2.6). Risk factors for no sports were: being a survivor (OR 1.3, 95CI 1.1-1.6), older age (OR for ≥35 years: 1.4, 95CI 1.1-1.8), migration background (OR 1.5, 95CI 1.3-1.8), French/Italian speaking (OR 1.4, 95CI 1.2-1.7), lower education (OR for compulsory schooling only: 1.6, 95CI 1.2-2.2), being married (OR 1.7, 95CI 1.5-2.0), having children (OR 1.3, 95CI 1.4-1.9), obesity (OR 2.4, 95CI 1.7-3.3), and smoking (OR 1.7, 95CI 1.5-2.1). Type of diagnosis was only associated with sports. CONCLUSIONS/SIGNIFICANCE: Physical activity levels in survivors were lower than recommended, but comparable to controls and mainly determined by socio-demographic and cultural factors. Strategies to improve physical activity levels could be similar as for the general population