Abstract:
In this paper we apply to gravitational waves from non-spinning binary systems a recently intro- duced frequentist methodology to calculate analytically the error for a maximum likelihood estimate (MLE) of physical parameters. While existing literature focuses on using the Cramer Rao Lower bound (CRLB) and Monte Carlo simulations, we use a power expansion of the bias and covariance in inverse powers of the signal to noise ratio. The use of higher order derivatives of the likelihood function in the expansions makes the prediction also sensitive to the secondary lobes of the MLE probability distribution. We discuss conditions for validity of the CRLB and predict new features in regions of the parameter space currently not explored. For example, we see how the bias can become the most important contributor to the parameters' errors for high mass systems (200M and above).

Abstract:
This paper describes the most accurate analytical frequentist assessment to date of the uncertainties in the estimation of physical parameters from gravitational waves generated by non spinning binary systems and Earth-based networks of laser interferometers. The paper quantifies how the accuracy in estimating the intrinsic parameters mostly depends on the network signal to noise ratio (SNR), but the resolution in the direction of arrival also strongly depends on the network geometry. We compare results for 6 different existing and possible global networks and two different choices of the parameter space. We show how the fraction of the sky where the one sigma angular resolution is below 2 square degrees increases about 3 times when transitioning from the Hanford (USA), Livingston (USA) and Cascina (Italy) network to possible 5 sites ones (while keeping the network SNR fixed). The technique adopted here is an asymptotic expansion of the uncertainties in inverse powers of the signal to noise ratio where the first order is the inverse Fisher information matrix. We show that a common approach to use simplified parameter spaces and only the Fisher information matrix can largely underestimate the uncertainties (by a factor ~7 for the one sigma sky uncertainty in square degrees at a network SNR of ~15).

Abstract:
A frequentist asymptotic expansion method for error estimation is employed for a network of gravitational wave detectors to assess the capability of gravitational wave observations, with Adv. LIGO and Adv. Virgo, to distinguish between the post-Einsteinian (ppE) description of coalescing binary systems and that of GR. When such errors are smaller than the parameter value, there is possibility to detect these violations from GR. A parameter space with inclusion of dominant dephasing ppE parameters is used for a study of first- and second-order (co)variance expansions, focusing on the inspiral stage of a nonspinning binary system of zero eccentricity detectible through Adv. LIGO and Adv. Virgo. Our procedure is more reliable than frequentist studies based only on Fisher information estimates and complements Bayesian studies. Second-order asymptotics indicate the possibility of constraining deviations from GR in low-SNR ($\rho \sim 15-17$) regimes. The errors on $\beta$ also increase errors of other parameters such as the chirp mass $\mathcal{M}$ and symmetric mass ratio $\eta$. Application is done to existing alternative theories of gravity, which include modified dispersion relation of the waveform, non-spinning models of quadratic modified gravity, and dipole gravitational radiation (i.e., Brans-Dicke type) modifications.

Abstract:
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.

Abstract:
We describe a general method to observationally exclude a theoretical model for gravitational wave (GW) emission from a transient astrophysical source (event) by using a null detection from a network of GW detectors. In the case of multiple astrophysical events with no GW detection, statements about individual events can be combined to increase the exclusion confidence. We frame and demonstrate the method using a population of hypothetical core collapse supernovae.

Abstract:
In this paper we reconsider, in a purely topological framework, the concept of bend-twist map previously studied in the analytic setting by Tongren Ding in (2007). We obtain some results about the existence and multiplicity of fixed points which are related to the classical Poincaré-Birkhoff twist theorem for area-preserving maps of the annulus; however, in our approach, like in Ding (2007), we do not require measure-preserving conditions. This makes our theorems in principle applicable to nonconservative planar systems. Some of our results are also stable for small perturbations. Possible applications of the fixed point theorems for topological bend-twist maps are outlined in the last section. 1. Introduction, Basic Setting and Preliminary Results The investigation of twist maps defined on annular domains can be considered as a relevant topic in the study of dynamical systems in two-dimensional manifolds. Twist maps naturally appear in a broad number of situations, and thus they have been widely considered both from the theoretical point of view and for their significance in various applications which range from celestial mechanics to fluid dynamics. One of the most classical examples of a fixed point theorem concerning twist maps on an annulus is the celebrated Poincaré-Birkhoff “twist theorem,” also known as the “Poincaré last geometric theorem.” It asserts the existence of at least two fixed points for an area-preserving homeomorphism of a closed planar annulus ( ) onto itself which leaves the inner boundary and the outer boundary invariant and rotates and in the opposite sense (this is the so-called twist condition at the boundary). The Poincaré-Birkhoff fixed point theorem was stated (and proved in some special cases) by Poincaré [1] in 1912, the year of his death. In 1913 [2], Birkhoff, with an ingenious application of the index of a vector field along a curve, gave a proof of the existence of a fixed point (see also [3]). A complete description of Birkhoff’s approach, with also the explanation how to obtain a second fixed point, can be found in the expository article by Brown and Neumann [4]. The history of the “twist” theorem and its generalizations and developments is quite interesting but impossible to summarize in few lines. After about hundred years of studies on this topic, some controversial “proofs” of its extensions have been settled only recently. We refer the interested reader to [5] where the part of the story concerning the efforts of avoiding the condition of boundary invariance is described. In this connection, we also recommend

Abstract:
We prove the existence of infinitely many periodic solutions and complicated dynamics, due to the presence of a topological horseshoe, for the classical Volterra predator-prey model with a periodic harvesting. The proof relies on some recent results about chaotic planar maps combined with the study of geometric features which are typical of linked twist maps.

Abstract:
We present some recent results on the existence of periodic solutions and chaotic like dynamics for second order scalar nonlinear ODEs. The equations under consideration belong to a simple class of perturbed planar Hamiltonian systems with slowly varying periodic coefficients, a typical example being given by the pendulum equation with moving support. Although there is already a broad literature on this subject, our approach, based on the concept of stretching along the paths, appears new in this context. In particular, our method is global in nature and stable with respect to small perturbations of the coefficients. Thus it applies even when some small friction terms are inserted into the equations. The main tool on which all our results are based is a topological lemma (that we call path crossing lemma) which was already implicitly used by Poincaré (1883-1884) [51], as well as by Butler (1976) [8] and Conley (1975) [12] and subsequently “rediscovered” and applied in many different contexts. For this reason, the first part of this paper is devoted to a detailed exposition of the Crossing Lemma and its connections with other topological results.

Abstract:
We show that for each $lambda > 0$, the problem $-Delta_p u = lambda f(u)$ in $Omega$, $u = 0$ on $partial Omega$ has a sequence of positive solutions $(u_n)_n$ with $max_{Omega} u_n$ decreasing to zero. We assume that $displaystyle{liminf_{so0^+}frac{F(s)}{s^p} = 0}$ and that $displaystyle{limsup_{so 0^+}frac{F(s)}{s^p} = +infty}$, where $F'=f$. We stress that no condition on the sign of $f$ is imposed.

Abstract:
We propose, in the general setting of topological spaces, a definition of two-dimensional oriented cell and consider maps which possess a property of stretching along the paths with respect to oriented cells. For these maps, we prove some theorems on the existence of fixed points, periodic points, and sequences of iterates which are chaotic in a suitable manner. Our results, motivated by the study of the Poincaré map associated to some nonlinear Hill's equations, extend and improve some recent work. The proofs are elementary in the sense that only well known properties of planar sets and maps and a two-dimensional equivalent version of the Brouwer fixed point theorem are used.