Superposition Formulas for Darboux Integrable Exterior Differential Systems

In this paper we present a far-reaching generalization of E. Vessiot's analysis of the Darboux integrable partial differential equations in one dependent and two independent variables. Our approach provides new insights into this classical method, uncovers the fundamental geometric invariants of Darboux integrable systems, and provides for systematic, algorithmic integration of such systems. This work is formulated within the general framework of Pfaffian exterior differential systems and, as such, has applications well beyond those currently found in the literature. In particular, our integration method is applicable to systems of hyperbolic PDE such as the Toda lattice equations, 2 dimensional wave maps and systems of overdetermined PDE.Comment: 80 page report. Updated version with some new sections, and major improvements to other

Invariants of Artinian Gorenstein Algebras and Isolated Hypersurface Singularities

We survey our recently proposed method for constructing biholomorphic invariants of quasihomogeneous isolated hypersurface singularities and, more generally, invariants of graded Artinian Gorenstein algebras. The method utilizes certain polynomials associated to such algebras, called nil-polynomials, and we compare them with two other classes of polynomials that have also been used to produce invariants.Comment: 13 page

Cosmology, cohomology, and compactification

Ashtekar and Samuel have shown that Bianchi cosmological models with compact spatial sections must be of Bianchi class A. Motivated by general results on the symmetry reduction of variational principles, we show how to extend the Ashtekar-Samuel results to the setting of weakly locally homogeneous spaces as defined, e.g., by Singer and Thurston. In particular, it is shown that any m-dimensional homogeneous space G/K admitting a G-invariant volume form will allow a compact discrete quotient only if the Lie algebra cohomology of G relative to K is non-vanishing at degree m.Comment: 6 pages, LaTe

Gravitational Waves: Just Plane Symmetry

We present some remarkable properties of the symmetry group for gravitational plane waves. Our main observation is that metrics with plane wave symmetry satisfy every system of generally covariant vacuum field equations except the Einstein equations. The proof uses the homothety admitted by metrics with plane wave symmetry and the scaling behavior of generally covariant field equations. We also discuss a mini-superspace description of spacetimes with plane wave symmetry.Comment: 10 pages, TeX, uses IOP style file

Covariants,joint invariants and the problem of equivalence in the invariant theory of Killing tensors defined in pseudo-Riemannian spaces of constant curvature

The invariant theory of Killing tensors (ITKT) is extended by introducing the new concepts of covariants and joint invariants of (product) vector spaces of Killing tensors defined in pseudo-Riemannian spaces of constant curvature. The covariants are employed to solve the problem of classification of the orthogonal coordinate webs generated by non-trivial Killing tensors of valence two defined in the Euclidean and Minkowski planes. Illustrative examples are provided.Comment: 60 pages. to appear in J. Math. Phy

Differential effects of long-term aerobic versus cognitively-engaging physical activity on children's visuospatial working memory related brain activation:A cluster RCT

Different types of physical activity are thought to differentially affect children's brain activation, via physiological mechanisms, or by activating similar brain areas during physical and cognitive tasks. Despite many behavioral studies relying on these mechanisms, they have been rarely studied. This study looks at both mechanisms simultaneously, by examining effects of two physical activity interventions (aerobic vs. cognitively-engaging) on children's brain activation. Functional Magnetic Resonance Imaging (fMRI) data of 62 children (48.4% boys, mean age 9.2 years) was analyzed. Children's visuospatial working memory related brain activity patterns were tested using a Spatial Span Task before and after the 14-week interventions consisting of four physical education lessons per week. The control group followed their regular program of two lessons per week. Analyses of activation patterns in SPM 12.0 revealed no activation changes between pretest and posttest (p > .05), and no differences between the three conditions in pretest-posttest changes in brain activation (p > .05). Large inter-individual differences were found, suggesting that not every child benefited from the interventions in the same way. To get more insight into the assumed mechanisms, further research is needed to understand whether, when, for whom, and how physical activity results in changed brain activation patterns

Semiclassical States in Quantum Cosmology: Bianchi I Coherent States

We study coherent states for Bianchi type I cosmological models, as examples of semiclassical states for time-reparametrization invariant systems. This simple model allows us to study explicitly the relationship between exact semiclassical states in the kinematical Hilbert space and corresponding ones in the physical Hilbert space, which we construct here using the group averaging technique. We find that it is possible to construct good semiclassical physical states by such a procedure in this model; we also discuss the sense in which the original kinematical states may be a good approximation to the physical ones, and the situations in which this is the case. In addition, these models can be deparametrized in a natural way, and we study the effect of time evolution on an "intrinsic" coherent state in the reduced phase space, in order to estimate the time for this state to spread significantly.Comment: 21 pages, 1 figure; Version to be published in CQG; The discussion has been slightly reorganized, two references added, and some typos correcte

Cardiovascular Fitness and Executive Functioning in Primary School-aged Children

Previous research in children has shown that higher cardiovascular fitness is related to better executive functioning. However, the available literature is hampered by methodological limitations. The present study investigates the relationship between cardiovascular fitness and executive functioning in a large sample of healthy children (N = 814). Cardiovascular fitness was assessed with estimated VO2Max from 20 m Shuttle Run Test performance. Executive functioning was assessed using a set of computerized neurocognitive tasks aimed at executive functions (working memory, motor inhibition, interference control) and lower-level neurocognitive functions (information processing and attention). Dependent measures derived from the neurocognitive tests were subjected to principal component analysis. Mixed model analyses tested the relation between cardiovascular fitness and neurocognitive functioning components. Results showed that children with higher cardiovascular fitness performed better on the neurocognitive function components Information Processing and Control, Visuospatial Working Memory and Attention Efficiency. The following measures contained in these components contributed to the observed relations: information processing measures, visuospatial working memory, and speed of alerting attention. No relationship was found between cardiovascular fitness and the other components: Verbal Working Memory, Attention Accuracy, and Interference Control. The present study suggests that there is a relationship between cardiovascular fitness and a specific set of executive functions and lower level neurocognitive functions. These findings highlight the importance of cardiovascular fitness for the overall health of school-aged children

Quantization of Midisuperspace Models

We give a comprehensive review of the quantization of midisuperspace models. Though the main focus of the paper is on quantum aspects, we also provide an introduction to several classical points related to the definition of these models. We cover some important issues, in particular, the use of the principle of symmetric criticality as a very useful tool to obtain the required Hamiltonian formulations. Two main types of reductions are discussed: those involving metrics with two Killing vector fields and spherically symmetric models. We also review the more general models obtained by coupling matter fields to these systems. Throughout the paper we give separate discussions for standard quantizations using geometrodynamical variables and those relying on loop quantum gravity inspired methods.Comment: To appear in Living Review in Relativit