On symmetries of Chern-Simons and BF topological theories

We describe constructing solutions of the field equations of Chern-Simons and topological BF theories in terms of deformation theory of locally constant (flat) bundles. Maps of flat connections into one another (dressing transformations) are considered. A method of calculating (nonlocal) dressing symmetries in Chern-Simons and topological BF theories is formulated

Homogeneous variational problems: a minicourse

A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension $m$. In this minicourse we discuss these problems from a geometric point of view.Comment: This paper is a written-up version of the major part of a minicourse given at the sixth Bilateral Workshop on Differential Geometry and its Applications, held in Ostrava in May 201

Asymptotic normality of the Parzen-Rosenblatt density estimator for strongly mixing random fields

We prove the asymptotic normality of the kernel density estimator (introduced by Rosenblatt (1956) and Parzen (1962)) in the context of stationary strongly mixing random fields. Our approach is based on the Lindeberg's method rather than on Bernstein's small-block-large-block technique and coupling arguments widely used in previous works on nonparametric estimation for spatial processes. Our method allows us to consider only minimal conditions on the bandwidth parameter and provides a simple criterion on the (non-uniform) strong mixing coefficients which do not depend on the bandwith.Comment: 16 page

Symmetries of Helmholtz forms and globally variational dynamical forms

Invariance properties of classes in the variational sequence suggested to Krupka et al. the idea that there should exist a close correspondence between the notions of variationality of a differential form and invariance of its exterior derivative. It was shown by them that the invariance of a closed Helmholtz form of a dynamical form is equivalent with local variationality of the Lie derivative of the dynamical form, so that the latter is locally the Euler--Lagrange form of a Lagrangian. We show that the corresponding local system of Euler--Lagrange forms is variationally equivalent to a global Euler--Lagrange form.Comment: Presented at QTS7 - Quantum Theory and Symmetries VII, Prague 7-13/08/201

Adaptive density estimation for stationary processes

We propose an algorithm to estimate the common density $s$ of a stationary process $X_1,...,X_n$. We suppose that the process is either $\beta$ or $\tau$-mixing. We provide a model selection procedure based on a generalization of Mallows' $C_p$ and we prove oracle inequalities for the selected estimator under a few prior assumptions on the collection of models and on the mixing coefficients. We prove that our estimator is adaptive over a class of Besov spaces, namely, we prove that it achieves the same rates of convergence as in the i.i.d framework

Maternal-child co-participation in physical activity-related behaviours: prevalence and cross-sectional associations with mothers and children\u27s objectively assessed physical activity levels

BACKGROUND: Co-participation in physical activity may be a useful strategy for increasing physical activity in mothers and their young children, yet little empirical evidence exists on this topic for young families. This study aimed to identify the prevalence of mother-child co-participation in physical activity and examine the association between co-participatory behaviours and objectively-assessed physical activity in young children and their mothers. METHODS: One-hundred twenty-three 4-6 year-old children and their mothers were recruited from preschools in Belgium between November 2010 and January 2011. Mothers completed a questionnaire assessing the frequency of co-participation in five activities. Both mothers and children wore ActiGraph GT1M accelerometers concurrently for 7 days to assess the time spent in moderate-to vigorous-intensity physical activity (MVPA) and light- to vigorous-intensity physical activity (LMVPA). Descriptive statistics (means, frequencies) were used to determine the prevalence of co-participation. Separate multiple linear regression analyses examined the association between co-participation and mothers\u27 and children\u27s physical activity on weekdays and weekends. RESULTS: Most mothers reported infrequent co-participation in physical activities with their children. On weekdays, walking or cycling for short trips was positively associated with children\u27s MVPA while attending a park or similar more than once per week was negatively associated with children\u27s MVPA and LMVPA. Going to an indoor play centre together once or more per week was negatively associated with mother\u27s LMVPA. On weekends, walking or cycling with their child in their free time was positively associated with both children\u27s and mothers\u27 MVPA and childrens\u27 LMVPA. Going to an indoor play centre together 1-3 times/month was negatively associated with children\u27s weekend MVPA. CONCLUSIONS: Reported rates of co-participation in mothers and their preschool children were low. The association with maternal and child physical activity may be dependent on the co-participatory behaviour assessed and may differ between weekday and weekends. Promoting walking and cycling together during leisure time may be an effective strategy to increase both preschool children\u27s and mothers\u27 MVPA