73,346 research outputs found
Asymptotic integration and dispersion for hyperbolic equations
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 -trajectory by perturbation theory in a running coupling II: the continuous renormalization group
The renormalized trajectory of massless -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
-interaction. We use an exact functional differential equation for its
dependence on the running -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
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 boundary. We assume that at least one of the
material parameters is for some . Using regularity
theory for second order elliptic partial differential equations, we derive
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
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
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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