18,112 research outputs found
Negative Screenings in Liouville Theory
We demonstrate how negative powers of screenings arise as a nonperturbative
effect within the operator approach to Liouville theory. This explains the
origin of the corresponding poles in the exact Liouville three point function
proposed by Dorn/Otto and (DOZZ) and leads to a
consistent extension of the operator approach to arbitrary integer numbers of
screenings of both types. The general Liouville three point function in this
setting is computed without any analytic continuation procedure, and found to
support the DOZZ conjecture. We point out the importance of the concept of free
field expansions with adjustable monodromies - recently advocated by Petersen,
Rasmussen and Yu - in the present context, and show that it provides a unifying
interpretation for two types of previously constructed local observables.Comment: 41 pages, LaTe
Negative powers of Laguerre operators
We study negative powers of Laguerre differential operators in , .
For these operators we prove two-weight estimates, with ranges of
depending on . The case of the harmonic oscillator (Hermite operator) has
recently been treated by Bongioanni and Torrea by using a straightforward
approach of kernel estimates. Here these results are applied in certain
Laguerre settings. The procedure is fairly direct for Laguerre function
expansions of Hermite type, due to some monotonicity properties of the kernels
involved. The case of Laguerre function expansions of convolution type is less
straightforward. For half-integer type indices we transfer the desired
results from the Hermite setting and then apply an interpolation argument based
on a device we call the {\sl convexity principle} to cover the continuous range
of . Finally, we investigate negative powers of the
Dunkl harmonic oscillator in the context of a finite reflection group acting on
and isomorphic to . The two weight estimates we
obtain in this setting are essentially consequences of those for Laguerre
function expansions of convolution type.Comment: 30 page
Application of asymptotic expansions of maximum likelihood estimators errors to gravitational waves from binary mergers: the single interferometer case
In this paper we describe a new methodology to calculate analytically the
error for a maximum likelihood estimate (MLE) for physical parameters from
Gravitational wave signals. All the existing litterature focuses on the usage
of the Cramer Rao Lower bounds (CRLB) as a mean to approximate the errors for
large signal to noise ratios. We show here how the variance and the bias of a
MLE estimate can be expressed instead in inverse powers of the signal to noise
ratios where the first order in the variance expansion is the CRLB. As an
application we compute the second order of the variance and bias for MLE of
physical parameters from the inspiral phase of binary mergers and for noises of
gravitational wave interferometers . We also compare the improved error
estimate with existing numerical estimates. The value of the second order of
the variance expansions allows to get error predictions closer to what is
observed in numerical simulations. It also predicts correctly the necessary SNR
to approximate the error with the CRLB and provides new insight on the
relationship between waveform properties SNR and estimation errors. For example
the timing match filtering becomes optimal only if the SNR is larger than the
kurtosis of the gravitational wave spectrum
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