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

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

Significance Tests for Periodogram Peaks

We discuss methods currently in use for determining the significance of peaks in the periodograms of time series. We discuss some general methods for constructing significance tests, false alarm probability functions, and the role played in these by independent random variables and by empirical and theoretical cumulative distribution functions. We also discuss the concept of "independent frequencies" in periodogram analysis. We propose a practical method for estimating the significance of periodogram peaks, applicable to all time series irrespective of the spacing of the data. This method, based on Monte Carlo simulations, produces significance tests that are tailor-made for any given astronomical time series.Comment: 22 pages, 11 Encapsulated Postscript figures, AAS LaTeX v5.2 Submitted to Ap

The Principle of Symmetric Criticality in General Relativity

We consider a version of Palais' Principle of Symmetric Criticality (PSC) that is applicable to the Lie symmetry reduction of Lagrangian field theories. PSC asserts that, given a group action, for any group-invariant Lagrangian the equations obtained by restriction of Euler-Lagrange equations to group-invariant fields are equivalent to the Euler-Lagrange equations of a canonically defined, symmetry-reduced Lagrangian. We investigate the validity of PSC for local gravitational theories built from a metric. It is shown that there are two independent conditions which must be satisfied for PSC to be valid. One of these conditions, obtained previously in the context of transverse symmetry group actions, provides a generalization of the well-known unimodularity condition that arises in spatially homogeneous cosmological models. The other condition seems to be new. The conditions that determine the validity of PSC are equivalent to pointwise conditions on the group action alone. These results are illustrated with a variety of examples from general relativity. It is straightforward to generalize all of our results to any relativistic field theory.Comment: 46 pages, Plain TeX, references added in revised versio

Rufinamide in refractory childhood epileptic encephalopathies other than Lennox-Gastaut syndrome.

Background: To report on the first multicenter Italian experience with rufinamide as adjunctive drug in children, adolescents and young adults with refractory childhoodonset epileptic encephalopathies other than Lennox–Gastaut syndrome. Methods: Thirty-eight patients (19 males, 19 females), aged between 4 and 34 (mean 13.7 ± 8.3, median 12.5), all affected by different types of childhood-onset refractory epileptic encephalopathies other than Lennox–Gastaut syndrome, were treated with rufinamide as adjunctive drug for a mean period of 11.4 months (range 3–26 months). Results: Fifteen of 38 patients (39.5%) had a ‡50% seizure reduction in countable seizures. Complete seizure freedom was achieved in one of these patients (2.6%). Three patients (7.9%) had a 25–49% seizure reduction, whilst seizure frequency remained unchanged in 15 (39.5%) and increased in five patients (13.1%). Eleven patients (28.9%) reported adverse side effects. Vomiting was reported in five patients (13.1%); drowsiness, decreased appetite and irritability with migraine manifested in other four patients. They were transient and mild in all cases. Conclusion: Rufinamide may be an effective and well-tolerated adjunctive drug for the treatment of refractory childhood-onset epileptic encephalopathies other than Lennox–Gastaut syndrome. Rufinamide was most effective in patients with dropattacks and (bi)frontal spike–wave discharges. Introduction Rufinamide is a structurally triazole-derivative (1-[2,6- difluorophenyl)methyl]-1hydro-1,2,3-triazole-carboxamide) novel antiepileptic drug, structurally unrelated to the existing antiepileptic drugs, and approved by the Food and Drug Administration for the treatment of Lennox–Gastaut syndrome in patients aged 4 and over, and for the treatment of partial seizures in adults and adolescents. The proposed mechanism of action is the limitation of excessiv

Killing Vector Fields in Three Dimensions: A Method to Solve Massive Gravity Field Equations

Killing vector fields in three dimensions play important role in the construction of the related spacetime geometry. In this work we show that when a three dimensional geometry admits a Killing vector field then the Ricci tensor of the geometry is determined in terms of the Killing vector field and its scalars. In this way we can generate all products and covariant derivatives at any order of the ricci tensor. Using this property we give ways of solving the field equations of Topologically Massive Gravity (TMG) and New Massive Gravity (NMG) introduced recently. In particular when the scalars of the Killing vector field (timelike, spacelike and null cases) are constants then all three dimensional symmetric tensors of the geometry, the ricci and einstein tensors, their covariant derivatives at all orders, their products of all orders are completely determined by the Killing vector field and the metric. Hence the corresponding three dimensional metrics are strong candidates of solving all higher derivative gravitational field equations in three dimensions.Comment: 25 pages, some changes made and some references added, to be published in Classical and Quantum Gravit

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