149 research outputs found
Laplacians with point interactions -- expected and unexpected spectral properties
We study the one-dimensional Laplace operator with point interactions on the
real line identified with two copies of the half-line . All
possible boundary conditions that define generators of -semigroups on
are characterized.
Here, the Cayley transform of the boundary conditions plays an important role
and using an explicit representation of the Green's functions, it allows us to
study invariance properties of semigroups
A Poincar\'e-Birkhoff theorem for tight Reeb flows on
We consider Reeb flows on the tight -sphere admitting a pair of closed
orbits forming a Hopf link. If the rotation numbers associated to the
transverse linearized dynamics at these orbits fail to satisfy a certain
resonance condition then there exist infinitely many periodic trajectories
distinguished by their linking numbers with the components of the link. This
result admits a natural comparison to the Poincar\'e-Birkhoff theorem on
area-preserving annulus homeomorphisms. An analogous theorem holds on
and applies to geodesic flows of Finsler metrics on .Comment: 67 pages. To appear in Inventiones Mathematica
Acceleration of generalized hypergeometric functions through precise remainder asymptotics
We express the asymptotics of the remainders of the partial sums {s_n} of the
generalized hypergeometric function q+1_F_q through an inverse power series z^n
n^l \sum_k c_k/n^k, where the exponent l and the asymptotic coefficients {c_k}
may be recursively computed to any desired order from the hypergeometric
parameters and argument. From this we derive a new series acceleration
technique that can be applied to any such function, even with complex
parameters and at the branch point z=1. For moderate parameters (up to
approximately ten) a C implementation at fixed precision is very effective at
computing these functions; for larger parameters an implementation in higher
than machine precision would be needed. Even for larger parameters, however,
our C implementation is able to correctly determine whether or not it has
converged; and when it converges, its estimate of its error is accurate.Comment: 36 pages, 6 figures, LaTeX2e. Fixed sign error in Eq. (2.28), added
several references, added comparison to other methods, and added discussion
of recursion stabilit
The use of normal forms for analysing nonlinear mechanical vibrations.
A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations
Surfaces away from horizons are not thermodynamic
Since the 1970s, it has been known that black-hole (and other) horizons are truly thermodynamic. More generally, surfaces which are not horizons have also been conjectured to behave thermodynamically. Initially, for surfaces microscopically expanded from a horizon to so-called stretched horizons, and more recently, for more general ordinary surfaces in the emergent gravity program. To test these conjectures we ask whether such surfaces satisfy an analogue to the first law of thermodynamics (as do horizons). For static asymptotically flat spacetimes we find that such a first law holds on horizons. We prove that this law remains an excellent approximation for stretched horizons, but counter-intuitively this result illustrates the insufficiency of the laws of black-hole mechanics alone from implying truly thermodynamic behavior. For surfaces away from horizons in the emergent gravity program the first law fails (except for spherically symmetric scenarios), thus undermining the key thermodynamic assumption of this program
Epistemic and Ontic Quantum Realities
Quantum theory has provoked intense discussions about its interpretation since its pioneer days. One of the few scientists who have been continuously engaged in this development from both physical and philosophical perspectives is Carl Friedrich von Weizsaecker. The questions he posed were and are inspiring for many, including the authors of this contribution. Weizsaecker developed Bohr's view of quantum theory as a theory of knowledge. We show that such an epistemic perspective can be consistently complemented by Einstein's ontically oriented position
Modelling the underlying principles of human aesthetic preference in evolutionary art
Our understanding of creativity is limited, yet there is substantial research trying to mimic human creativity in artificial systems and in particular to produce systems that automatically evolve art appreciated by humans. We propose here to study human visual preference through observation of nearly 500 user sessions with a simple evolutionary art system. The progress of a set of aesthetic measures throughout each interactive user session is monitored and subsequently mimicked by automatic evolution in an attempt to produce an image to the liking of the human user
Revisiting detrended fluctuation analysis
Half a century ago Hurst introduced Rescaled Range (R/S) Analysis to study fluctuations in time series. Thousands of works have investigated or applied the original methodology and similar techniques, with Detrended Fluctuation Analysis becoming preferred due to its purported ability to mitigate nonstationaries. We show Detrended Fluctuation Analysis introduces artifacts for nonlinear trends, in contrast to common expectation, and demonstrate that the empirically observed curvature induced is a serious finite-size effect which will always be present. Explicit detrending followed by measurement of the diffusional spread of a signals' associated random walk is preferable, a surprising conclusion given that Detrended Fluctuation Analysis was crafted specifically to replace this approach. The implications are simple yet sweeping: there is no compelling reason to apply Detrended Fluctuation Analysis as it 1) introduces uncontrolled bias; 2) is computationally more expensive than the unbiased estimator; and 3) cannot provide generic or useful protection against nonstationaries
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