On balanced planar graphs, following W. Thurston

Let $f:S^2\to S^2$ be an orientation-preserving branched covering map of degree $d\geq 2$, and let $\Sigma$ be an oriented Jordan curve passing through the critical values of $f$. Then $\Gamma:=f^{-1}(\Sigma)$ is an oriented graph on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced balanced planar graphs and showed that they combinatorially characterize all such $\Gamma$, where $f$ has $2d-2$ distinct critical values. We give a detailed account of this discussion, along with some examples and an appendix about Hurwitz numbers.Comment: 17 page

B\"ottcher coordinates

A well-known theorem of B\"ottcher asserts that an analytic germ f:(C,0)->(C,0) which has a superattracting fixed point at 0, more precisely of the form f(z) = az^k + o(z^k) for some a in C^*, is analytically conjugate to z->az^k by an analytic germ phi:(C,0)->(C,0) which is tangent to the identity at 0. In this article, we generalize this result to analytic maps of several complex variables

Purveyors of fine halos: Re-assessing globular cluster contributions to the Milky Way halo build-up with SDSS-IV

There is ample evidence in the Milky Way for globular cluster (GC) disruption. Hence one may expect that also part of the Galactic halo field stars may once have formed in GCs. We quantify the fraction of halo stars donated by GCs by searching for stars that bear the unique chemical fingerprints typical for a subset of GC stars often dubbed `second-generation stars'. These are stars showing light element abundance anomalies such as a pronounced CN-band strength accompanied by weak CH-bands. Based on this indicator, past studies have placed the fraction of halo stars with a GC origin between a few to up to 50%. Using low-resolution spectra from the most recent data release of the latest extension of the Sloan Digital Sky Survey (SDSS-IV), we were able to identify 118 metal-poor ($-1.8\le$[Fe/H]$\le -1.3$) CN-strong stars in a sample of 4470 halo giant stars out to 50 kpc. This results in an observed fraction of these stars of 2.6$\pm$0.2%. Using an updated formalism to account for the fraction of stars lost early on in the GCs' evolution we estimate the fraction of the halo that stems from disrupted clusters to be 11$\pm$1%. This number represents the case that stars lost from GCs were entirely from the first generation and is thus merely an upper limit. Our conclusions are sensitive to our assumptions of the mass lost early on from the first generation formed in the GCs, the ratio of first-to-second generation stars, and other GC parameters. We carefully test the influence of varying these parameters on the final result and find that, under realistic scenarios, the above fraction depends on the main assumptions at less than 10%. We further recover a flat trend in this fraction with Galactocentric radius, with a marginal indication of a rise beyond 30 kpc that could reflect the ex-situ origin of the outer halo. (abridged)Comment: 13 pages, 11 figures, accepted for publication in Astronomy & Astrophysic

Development of a Clinical Assessment Tool for Seizure Observation

The utility of long term video-EEG monitoring is well established and has diagnostic, prognostic, and therapeutic functions. Patients are admitted to the Epilepsy Monitoring Unit (EMU) to have medications lowered for seizure provocation. Electrographic and clinical information from the seizures are analyzed for the purpose of classifying and treating epilepsy. Clinical or ictal assessment is an interactive and demanding skill. Factors inherent in the seizures often limit the accuracy and detail of an ictal assessment. The literature suggests that an observational tool for use during ictal assessment may help to improve accuracy. To date, a standardized tool for use in seizure observation or ictal assessment has not been developed. The purpose of this article is to synthesize the current recommendations regarding the components of seizure observation and describe how they can be organized to formulate a standardized assessment tool. An observation tool that was developed with these recommendations in mind and is currently being used on a 10-bed EMU is described

Braid Group Actions on Rational Maps

Rational maps are maps from the Riemann sphere to itself that are defined by ratios of polynomials. A special type of rational map is the ones where the forward orbit of the critical points is finite. That is, under iteration, the critical points all eventually cycle in some periodic orbit. In the 1980s Thurston proved the surprising result that (except for a well-understood set of exceptions) when the post-critical set is finite the rational map is determined by the “combinatorics” of how the map behaves on the post-critical set. Recently, there has been interest in the question: what happens if we just fix the degree and impose the condition that only one critical orbit is finite. In that case, the family of rational maps defined by the combinatorics is a complex manifold naturally acted on by subgroups of the pure spherical braid group on n-strands where n depends on the order of the orbit and the degree, In this talk, we discuss the question: what is the global topology of this manifold

Axiomatic Interpretability for Multiclass Additive Models

Generalized additive models (GAMs) are favored in many regression and binary classification problems because they are able to fit complex, nonlinear functions while still remaining interpretable. In the first part of this paper, we generalize a state-of-the-art GAM learning algorithm based on boosted trees to the multiclass setting, and show that this multiclass algorithm outperforms existing GAM learning algorithms and sometimes matches the performance of full complexity models such as gradient boosted trees. In the second part, we turn our attention to the interpretability of GAMs in the multiclass setting. Surprisingly, the natural interpretability of GAMs breaks down when there are more than two classes. Naive interpretation of multiclass GAMs can lead to false conclusions. Inspired by binary GAMs, we identify two axioms that any additive model must satisfy in order to not be visually misleading. We then develop a technique called Additive Post-Processing for Interpretability (API), that provably transforms a pre-trained additive model to satisfy the interpretability axioms without sacrificing accuracy. The technique works not just on models trained with our learning algorithm, but on any multiclass additive model, including multiclass linear and logistic regression. We demonstrate the effectiveness of API on a 12-class infant mortality dataset.Comment: KDD 201

COMPUTING DYNAMICAL DEGREES OF RATIONAL MAPS ON MODULI SPACE

The dynamical degrees of a rational map f:X⇢X are fundamental invariants describing the rate of growth of the action of iterates of f on the cohomology of X. When f has non-empty indeterminacy set, these quantities can be very difficult to determine. We study rational maps f:XN⇢XN, where XN is isomorphic to the Deligne–Mumford compactification M¯¯¯¯0,N+3. We exploit the stratified structure of XN to provide new examples of rational maps, in arbitrary dimension, for which the action on cohomology behaves functorially under iteration. From this, all dynamical degrees can be readily computed (given enough book-keeping and computing time). In this paper, we explicitly compute all of the dynamical degrees for all such maps f:XN⇢XN, where dim(XN)≤3 and the first dynamical degrees for the mappings where dim(XN)≤5. These examples naturally arise in the setting of Thurston’s topological characterization of rational maps