Accuracy of the post-Newtonian approximation. II. Optimal asymptotic expansion of the energy flux for quasicircular, extreme mass-ratio inspirals into a Kerr black hole

We study the effect of black hole spin on the accuracy of the post-Newtonian approximation. We focus on the gravitational energy flux for the quasicircular, equatorial, extreme mass-ratio inspiral of a compact object into a Kerr black hole of mass M and spin J. For a given dimensionless spin a=J/M^2 (in geometrical units), the energy flux depends only on the orbital velocity v or (equivalently) on the Boyer-Lindquist orbital radius r. We investigate the formal region of validity of the Taylor post-Newtonian expansion of the energy flux (which is known up to order v^8 beyond the quadrupole formula), generalizing previous work by two of us. The "error function" used to determine the region of validity of the post-Newtonian expansion can have two qualitatively different kinds of behavior, and we deal with these two cases separately. We find that, at any fixed post-Newtonian order, the edge of the region of validity (as measured by v/v_{ISCO}, where v_{ISCO} is the orbital velocity at the innermost stable circular orbit) is only weakly dependent on a. Unlike in the nonspinning case, the lack of sufficiently high order terms does not allow us to determine if there is a convergent to divergent transition at order v^6. Independently of a, the inclusion of angular multipoles up to and including l=5 in the numerical flux is necessary to achieve the level of accuracy of the best-known (N=8) PN expansion of the energy flux.Comment: 9 pages, 8 figures. Minor changes to match published versio

Reconstructing DNA copy number by joint segmentation of multiple sequences

The variation in DNA copy number carries information on the modalities of genome evolution and misregulation of DNA replication in cancer cells; its study can be helpful to localize tumor suppressor genes, distinguish different populations of cancerous cell, as well identify genomic variations responsible for disease phenotypes. A number of different high throughput technologies can be used to identify copy number variable sites, and the literature documents multiple effective algorithms. We focus here on the specific problem of detecting regions where variation in copy number is relatively common in the sample at hand: this encompasses the cases of copy number polymorphisms, related samples, technical replicates, and cancerous sub-populations from the same individual. We present an algorithm based on regularization approaches with significant computational advantages and competitive accuracy. We illustrate its applicability with simulated and real data sets.Comment: 54 pages, 5 figure

Gravitational Waves From Perturbed Kerr Black Holes: Two Investigations

This thesis includes two main projects. The first project is a study of the effect of black hole spin on the accuracy of the post-Newtonian approximation. We focus on the gravitational energy produced by the quasicircular, equatorial, extreme mass-ratio inspiral of a compact object into a Kerr black hole of mass M and spin J. For a given dimensionless spin a ≡ J/M2 (in geometrical units G = c = 1 ), the energy flux depends only on the orbital velocity v or (equivalently) on the Boyer-Lindquist orbital radius r. We investigate the formal region of validity of the Taylor post-Newtonian expansion of the energy flux (which is known up to order v8 beyond the quadrupole formula) by comparing the expansion to numerical calculations of the flux. We find that, at any fixed post-Newtonian order, the edge of the region of validity (as measured by v/vISCO, where v ISCO is the orbital velocity at the innermost stable circular orbit) is only weakly dependent on a. Independently of a, the inclusion of angular multipoles up to and including ℓ = 5 in the numerical flux is necessary to achieve the level of accuracy of the best-known (v8) expansion of the energy flux. In the second project we study the excitation of Kerr black holes produced by infalling particles. Such a study requires an accurate knowledge of the Green\u27s function describing the response of the black hole to external perturbations. Relying on the formalism developed by Mano, Suzuki and Takasugi, we improve and extend previous calculations of the contribution to the Green\u27s function coming from quasinormal mode residues in the complex frequency plane (“excitation factors Bq”). Using these results we compute the “excitation coefficients” C q (the mode amplitudes) when the source of the perturbations is a particle falling into the black hole along the symmetry axis. We compare this calculation with numerical integrations of the perturbation equations, and we show quantitatively how the addition of higher overtones improves the agreement with the numerical waveforms

Positive solutions of a derivative dependent second-order problem subject to Stieltjes integral boundary conditions

In this paper, we investigate the derivative dependent second-order problem subject to Stieltjes integral boundary conditions \begin{equation*} \begin{cases}-u''(t)=f(t,u(t),u'(t)),\quad t\in[0,1],\\ au(0)-bu'(0)=\alpha[u],\ cu(1)+du'(1)=\beta[u],\end{cases} \end{equation*} where $f$: $[0,1]\times \mathbb{R}^{+}\times \mathbb{R}\rightarrow \mathbb{R}^{+}$ is continuous, $\alpha[u]$ and $\beta[u]$ are linear functionals involving Stieltjes integrals. Some inequality conditions on nonlinearity $f$ and the spectral radius condition of linear operator are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index. Not only is the general case considered but a large range of coefficients can be chosen to weaken the conditions in previous work for some special cases. The conditions allow that $f(t,x_{1},x_{2})$ has superlinear or sublinear growth in $x_{1}, x_{2}$. Two examples are provided to illustrate the theorems under multi-point and integral boundary conditions with sign-changing coefficients

Reconstructing DNA copy number by penalized estimation and imputation

Recent advances in genomics have underscored the surprising ubiquity of DNA copy number variation (CNV). Fortunately, modern genotyping platforms also detect CNVs with fairly high reliability. Hidden Markov models and algorithms have played a dominant role in the interpretation of CNV data. Here we explore CNV reconstruction via estimation with a fused-lasso penalty as suggested by Tibshirani and Wang [Biostatistics 9 (2008) 18--29]. We mount a fresh attack on this difficult optimization problem by the following: (a) changing the penalty terms slightly by substituting a smooth approximation to the absolute value function, (b) designing and implementing a new MM (majorization--minimization) algorithm, and (c) applying a fast version of Newton's method to jointly update all model parameters. Together these changes enable us to minimize the fused-lasso criterion in a highly effective way. We also reframe the reconstruction problem in terms of imputation via discrete optimization. This approach is easier and more accurate than parameter estimation because it relies on the fact that only a handful of possible copy number states exist at each SNP. The dynamic programming framework has the added bonus of exploiting information that the current fused-lasso approach ignores. The accuracy of our imputations is comparable to that of hidden Markov models at a substantially lower computational cost.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS357 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org