High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities

We define a discrete Menger-type curvature of d+2 points in a real separable Hilbert space H by an appropriate scaling of the squared volume of the corresponding (d+1)-simplex. We then form a continuous curvature of an Ahlfors d-regular measure on H by integrating the discrete curvature according to the product measure. The aim of this work, continued in a subsequent paper, is to estimate multiscale least squares approximations of such measures by the Menger-type curvature. More formally, we show that the continuous d-dimensional Menger-type curvature is comparable to the ``Jones-type flatness''. The latter quantity adds up scaled errors of approximations of a measure by d-planes at different scales and locations, and is commonly used to characterize uniform rectifiability. We thus obtain a characterization of uniform rectifiability by using the Menger-type curvature. In the current paper (part I) we control the continuous Menger-type curvature of an Ahlfors d-regular measure by its Jones-type flatness.Comment: 47 pages, 13 figures. Minor revisions and the inclusion of figure

General Relativity, the Cosmological Constant and Modular Forms

Strong field (exact) solutions of the gravitational field equations of General Relativity in the presence of a Cosmological Constant are investigated. In particular, a full exact solution is derived within the inhomogeneous Szekeres-Szafron family of space-time line element with a nonzero Cosmological Constant. The resulting solution connects, in an intrinsic way, General Relativity with the theory of modular forms and elliptic curves. The homogeneous FLRW limit of the above space-time elements is recovered and we solve exactly the resulting Friedmann Robertson field equation with the appropriate matter density for generic values of the Cosmological Constant %Lambda and curvature constant K. A formal expression for the Hubble constant is derived. The cosmological implications of the resulting non-linear solutions are systematically investigated. Two particularly interesting solutions i) the case of a flat universe K=0, Lambda not= 0 and ii) a case with all three cosmological parameters non-zero, are described by elliptic curves with the property of complex multiplication and absolute modular invariant j=0 and 1728, respectively. The possibility that all non-linear solutions of General Relativity are expressed in terms of theta functions associated with Riemann-surfaces is discussed.Comment: LaTeX file, 34 pages plus 9 EPS figures, Accepted for Publication in Classical and Quantum Gravit

Successful captive rearing of an Egyptian vulture at Kalba Bird of Prey Centre, UAE

Egyptian vultures (Neophron percnopterus) are endangered across their range, and ex-situ conservation efforts focus on establishing captive breeding programmes, with the ultimate goal of releasing captive-bred individuals into secure habitat. To date, this species has not been successfully bred in captivity in the Arabian peninsula. Moreover, efforts to reduce the risk of human imprinting by captive-reared birds of other species have typically included the use of hand puppets and excluding visual contact with humans. This report documents the first successful captive rearing of an Egyptian vulture in the United Arab Emirates, and describes the successful return of the chick to its parents without the use of hand puppets. By temporarily returning the chick to its parents during daylight from 11 days of age, it was possible to maintain normal parenting behaviours in the adult birds, this leading to the successful dual-imprinting of the chick. The chick is now included in a flight demonstration as part of a conservation education programme, and will be included in a regional captive breeding programme upon maturity