4,425 research outputs found

    Approximation properties of certain operator-induced norms on Hilbert spaces

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    We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the L2 norm, and study their approximation properties over Hilbert subspaces of L2 . The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in reproducing kernel Hilbert spaces (RKHS). Our results have implications to the analysis of M-estimators in models based on finite-dimensional linear approximation of functions, and also to some related packing problems

    Curvature blow up in Bianchi VIII and IX vacuum spacetimes

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    The maximal globally hyperbolic development of non-Taub-NUT Bianchi IX vacuum initial data and of non-NUT Bianchi VIII vacuum initial data is C2 inextendible. Furthermore, a curvature invariant is unbounded in the incomplete directions of inextendible causal geodesics.Comment: 20 pages, no figures. Submitted to Classical and Quantum Gravit

    A geometric approach to scalar field theories on the supersphere

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    Following a strictly geometric approach we construct globally supersymmetric scalar field theories on the supersphere, defined as the quotient space S2∣2=UOSp(1∣2)/U(1)S^{2|2} = UOSp(1|2)/\mathcal{U}(1). We analyze the superspace geometry of the supersphere, in particular deriving the invariant vielbein and spin connection from a generalization of the left-invariant Maurer-Cartan form for Lie groups. Using this information we proceed to construct a superscalar field action on S2∣2S^{2|2}, which can be decomposed in terms of the component fields, yielding a supersymmetric action on the ordinary two-sphere. We are able to derive Lagrange equations and Noether's theorem for the superscalar field itself.Comment: 38 pages, 1 figur

    Measurements of the short-term stability of quartz crystal resonators: A window on future developments in crystal oscillators

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    Recent measurements of the inherent short-term stability of quartz crystal resonators are presented. These measurements show that quartz resonators are much more stable for times less than 1s than the best available commercial quartz oscillators. A simple model appears to explain the noise mechanism in crystal controlled oscillators and points the way to design changes which should permit more than 2 orders of magnitude improvement in their short-term stability. Calculations show that a reference signal at 1 THz, derived from frequency multiplying a 5 MHz source with the above measured crystal stability, should have an instantaneous or fast linewidth of order 1 Hz. These calculations explicitly include the noise contribution of our present multiplier chains and are briefly outlined

    Neuroscientists’ everyday experiences of ethics: The interplay of regulatory, professional, personal and tangible ethical spheres

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    Copyright @ 2013 The Authors. This article has been published using OnlineOpen. Re-use of this article is permitted in accordance with the Terms and Conditions set out at http://wileyonline library.com/onlineopen#OnlineOpen_Terms.The ethical issues neuroscience raises are subject to increasing attention, exemplified in the emergence of the discipline neuroethics. While the moral implications of neurotechnological developments are often discussed, less is known about how ethics intersects with everyday work in neuroscience and how scientists themselves perceive the ethics of their research. Drawing on observation and interviews with members of one UK group conducting neuroscience research at both the laboratory bench and in the clinic, this article examines what ethics meant to these researchers and delineates four specific types of ethics that shaped their day-to-day work: regulatory, professional, personal and tangible. While the first three categories are similar to those identified elsewhere in sociological work on scientific and clinical ethics, the notion of ‘tangible ethics’ emerged by attending to everyday practice, in which these scientists’ discursive distinctions between right and wrong were sometimes challenged. The findings shed light on how ethical positions produce and are, in turn, produced by scientific practice. Informing sociological understandings of neuroscience, they also throw the category of neuroscience and its ethical specificity into question, given that members of this group did not experience their work as raising issues that were distinctly neuro-ethical.Wellcome Trus

    Cosmic No Hair for Collapsing Universes

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    It is shown that all contracting, spatially homogeneous, orthogonal Bianchi cosmologies that are sourced by an ultra-stiff fluid with an arbitrary and, in general, varying equation of state asymptote to the spatially flat and isotropic universe in the neighbourhood of the big crunch singularity. This result is employed to investigate the asymptotic dynamics of a collapsing Bianchi type IX universe sourced by a scalar field rolling down a steep, negative exponential potential. A toroidally compactified version of M*-theory that leads to such a potential is discussed and it is shown that the isotropic attractor solution for a collapsing Bianchi type IX universe is supersymmetric when interpreted in an eleven-dimensional context.Comment: Extended discussion to include Kantowski-Sachs universe. In press, Classical and Quantum Gravit
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