73,346 research outputs found

    Asymptotic integration and dispersion for hyperbolic equations

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    The aim of this paper is to establish time decay properties and dispersive estimates for strictly hyperbolic equations with homogeneous symbols and with time-dependent coefficients whose derivatives are integrable. For this purpose, the method of asymptotic integration is developed for such equations and representation formulae for solutions are obtained. These formulae are analysed further to obtain time decay of Lp-Lq norms of propagators for the corresponding Cauchy problems. It turns out that the decay rates can be expressed in terms of certain geometric indices of the limiting equation and we carry out the thorough analysis of this relation. This provides a comprehensive view on asymptotic properties of solutions to time-perturbations of hyperbolic equations with constant coefficients. Moreover, we also obtain the time decay rate of the Lp-Lq estimates for equations of these kinds, so the time well-posedness of the corresponding nonlinear equations with additional semilinearity can be treated by standard Strichartz estimates.Comment: 30 page

    The renormalized ϕ44\phi^4_4-trajectory by perturbation theory in a running coupling II: the continuous renormalization group

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    The renormalized trajectory of massless ϕ4\phi^4-theory on four dimensional Euclidean space-time is investigated as a renormalization group invariant curve in the center manifold of the trivial fixed point, tangent to the ϕ4\phi^4-interaction. We use an exact functional differential equation for its dependence on the running ϕ4\phi^4-coupling. It is solved by means of perturbation theory. The expansion is proved to be finite to all orders. The proof includes a large momentum bound on amputated connected momentum space Green's functions.Comment: 26 pages LaTeX2

    Elliptic regularity theory applied to time harmonic anisotropic Maxwell's equations with less than Lipschitz complex coefficients

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    The focus of this paper is the study of the regularity properties of the time harmonic Maxwell's equations with anisotropic complex coefficients, in a bounded domain with C1,1C^{1,1} boundary. We assume that at least one of the material parameters is W1,3+δW^{1,3+\delta} for some δ>0\delta>0. Using regularity theory for second order elliptic partial differential equations, we derive W1,pW^{1,p} estimates and H\"older estimates for electric and magnetic fields up to the boundary. We also derive interior estimates in bi-anisotropic media.Comment: 19 page

    Synchronicity From Synchronized Chaos

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    The synchronization of loosely coupled chaotic oscillators, a phenomenon investigated intensively for the last two decades, may realize the philosophical notion of synchronicity. Effectively unpredictable chaotic systems, coupled through only a few variables, commonly exhibit a predictable relationship that can be highly intermittent. We argue that the phenomenon closely resembles the notion of meaningful synchronicity put forward by Jung and Pauli if one identifies "meaningfulness" with internal synchronization, since the latter seems necessary for synchronizability with an external system. Jungian synchronization of mind and matter is realized if mind is analogized to a computer model, synchronizing with a sporadically observed system as in meteorological data assimilation. Internal synchronization provides a recipe for combining different models of the same objective process, a configuration that may also describe the functioning of conscious brains. In contrast to Pauli's view, recent developments suggest a materialist picture of semi-autonomous mind, existing alongside the observed world, with both exhibiting a synchronistic order. Basic physical synchronicity is manifest in the non-local quantum connections implied by Bell's theorem. The quantum world resides on a generalized synchronization "manifold", a view that provides a bridge between nonlocal realist interpretations and local realist interpretations that constrain observer choice .Comment: 1) clarification regarding the connection with philosophical synchronicity in Section 2 and in the concluding section 2) reference to Maldacena-Susskind "ER=EPR" relation in discussion of role of wormholes in entanglement and nonlocality 3) length reduction and stylistic changes throughou

    The complexity of dynamics in small neural circuits

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    Mean-field theory is a powerful tool for studying large neural networks. However, when the system is composed of a few neurons, macroscopic differences between the mean-field approximation and the real behavior of the network can arise. Here we introduce a study of the dynamics of a small firing-rate network with excitatory and inhibitory populations, in terms of local and global bifurcations of the neural activity. Our approach is analytically tractable in many respects, and sheds new light on the finite-size effects of the system. In particular, we focus on the formation of multiple branching solutions of the neural equations through spontaneous symmetry-breaking, since this phenomenon increases considerably the complexity of the dynamical behavior of the network. For these reasons, branching points may reveal important mechanisms through which neurons interact and process information, which are not accounted for by the mean-field approximation.Comment: 34 pages, 11 figures. Supplementary materials added, colors of figures 8 and 9 fixed, results unchange
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