Captures of stars by a massive black hole: Investigations in numerical stellar dynamics

Among the astrophysical systems targeted by LISA, stars on relativistic orbits around massive black holes (MBHs) are particularly promising sources. Unfortunately, the prediction for the number and characteristics of such sources suffers from many uncertainties. Stellar dynamical Monte Carlo simulations of the evolution of galactic nucleus models allow more realistic estimates of these quantities. The computations presented here strongly suggest that the closest such extreme mass-ratio binary to be detected by LISA could be a low-mass MS star (MSS) orbiting the MBH at the center of our Milky Way. Only compact stars contribute to the expected detections from other galaxies because MSSs are disrupted by tidal forces too early.Comment: 4 pages, 2 figures, to appear in the proceedings of "The Astrophysics of Gravitational Wave Sources", a workshop held at the University of Maryland, April 24-26, 200

Don’t Judge the S&P 500 by its Cover: When Expectations Meet Regression

This study aims to answer whether the health of the United States equities market is a representation of the well-being of the macro-economy. There are two primary goals in this study. The first is to model the determinants of fluctuating stock prices in the short-run. For this reason, data is collected in monthly units intended to represent short-term fluctuations in the S&P price over time. The second is to model the expected impact of recessionary pressures on the performance of equities

Algebraic relations between solutions of Painlev\'e equations

We calculate model theoretic ranks of Painlev\'e equations in this article, showing in particular, that any equation in any of the Painlev\'e families has Morley rank one, extending results of Nagloo and Pillay (2011). We show that the type of the generic solution of any equation in the second Painlev\'e family is geometrically trivial, extending a result of Nagloo (2015). We also establish the orthogonality of various pairs of equations in the Painlev\'e families, showing at least generically, that all instances of nonorthogonality between equations in the same Painlev\'e family come from classically studied B{\"a}cklund transformations. For instance, we show that if at least one of $\alpha, \beta$ is transcendental, then $P_{II} (\alpha)$ is nonorthogonal to $P_{II} ( \beta )$ if and only if $\alpha+ \beta \in \mathbb Z$ or $\alpha - \beta \in \mathbb Z$. Our results have concrete interpretations in terms of characterizing the algebraic relations between solutions of Painlev\'e equations. We give similar results for orthogonality relations between equations in different Painlev\'e families, and formulate some general questions which extend conjectures of Nagloo and Pillay (2011) on transcendence and algebraic independence of solutions to Painlev\'e equations. We also apply our analysis of ranks to establish some orthogonality results for pairs of Painlev\'e equations from different families. For instance, we answer several open questions of Nagloo (2016), and in the process answer a question of Boalch (2012).Comment: This manuscript replaces and greatly expands a portion of arXiv:1608.0475