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

Bounds in Query Learning

We introduce new combinatorial quantities for concept classes, and prove lower and upper bounds for learning complexity in several models of query learning in terms of various combinatorial quantities. Our approach is flexible and powerful enough to enough to give new and very short proofs of the efficient learnability of several prominent examples (e.g. regular languages and regular $\omega$-languages), in some cases also producing new bounds on the number of queries. In the setting of equivalence plus membership queries, we give an algorithm which learns a class in polynomially many queries whenever any such algorithm exists. We also study equivalence query learning in a randomized model, producing new bounds on the expected number of queries required to learn an arbitrary concept. Many of the techniques and notions of dimension draw inspiration from or are related to notions from model theory, and these connections are explained. We also use techniques from query learning to mildly improve a result of Laskowski regarding compression schemes

Measuring Millennials: Teenage Idleness in the Digital Age

This research aims to model the relationship between factors contributing to situational privilege and teenage idleness. We will study the impact of race, income, household type, unemployment, and education on teenage idleness across 348 Metropolitan Statistical Areas within the United States. It is important to identify influential factors on teen idleness in order for government and community leaders to implement successful policies to get teenagers off the streets and into the workforce. Factors that were found to have a significant impact on teen idleness included the MSAs makeup of household types, race, median income, unemployment, and attainment of a bachelor’s degree or higher

A new Monte Carlo code for star cluster simulations: II. Central black hole and stellar collisions

We have recently written a new code to simulate the long term evolution of spherical clusters of stars. It is based on the pioneering Monte Carlo scheme proposed by Henon in the 70's. Our code has been devised in the specific goal to treat dense galactic nuclei. After having described how we treat relaxation in a first paper, we go on and include further physical ingredients that are mostly pertinent to galactic nuclei, namely the presence of a central (growing) black hole (BH) and collisions between MS stars. Stars that venture too close to the BH are destroyed by the tidal field. This process is a channel to feed the BH and a way to produce accretion flares. Collisions between stars have often been proposed as another mechanism to drive stellar matter into the central BH. To get the best handle on the role of this process in galactic nuclei, we include it with unpreceded realism through the use of a set of more than 10000 collision simulations carried out with a SPH (Smoothed Particle Hydrodynamics) code. Stellar evolution has also been introduced in a simple way, similar to what has been done in previous dynamical simulations of galactic nuclei. To ensure that this physics is correctly simulated, we realized a variety of tests whose results are reported here. This unique code, featuring most important physical processes, allows million particle simulations, spanning a Hubble time, in a few CPU days on standard personal computers and provides a wealth of data only rivalized by N-body simulations.Comment: 32 pages, 19 figures. Slightly shortened and clarified following referee's suggestions. Accepted for publication in A&A. Version with high quality figures available at http://obswww.unige.ch/~freitag/papers/article_MC2.ps.g