The Green's function and the Ahlfors map

The classical Green's function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green's function of a finitely connected domain in the plane in terms of a single Ahlfors mapping of the domain, which is a proper holomorphic mapping of the domain onto the unit disc that is the analogue of the Riemann map in the multiply connected setting.Comment: 14 page

Time's Arrow, A Halloween Concert, October 31, 1995

This is the concert program of the Time's Arrow, A Halloween Concert performance on Tuesday, October 31, 1995 at 8:00 p.m., at the Tsai Performance Center, 685 Commonwealth Avenue, Boston, Massachusetts. Works performed were An Idyll for the Misbegotten by George Crumb, La Vita Nuova by Nicholas Maw, Lucy and the Count by Jon Deak, and Mysteries of the Macabre by György Ligeti. Digitization for Boston University Concert Programs was supported by the Boston University Humanities Library Endowed Fund

Exact duality in semidefinite programming based on elementary reformulations

In semidefinite programming (SDP), unlike in linear programming, Farkas' lemma may fail to prove infeasibility. Here we obtain an exact, short certificate of infeasibility in SDP by an elementary approach: we reformulate any semidefinite system of the form Ai*X = bi (i=1,...,m) (P) X >= 0 using only elementary row operations, and rotations. When (P) is infeasible, the reformulated system is trivially infeasible. When (P) is feasible, the reformulated system has strong duality with its Lagrange dual for all objective functions. As a corollary, we obtain algorithms to generate the constraints of {\em all} infeasible SDPs and the constraints of {\em all} feasible SDPs with a fixed rank maximal solution.Comment: To appear, SIAM Journal on Optimizatio

The structure of the semigroup of proper holomorphic mappings of a planar domain to the unit disc

Given a bounded n-connected domain in the plane bounded by non-intersecting Jordan curves, and given one point on each boundary curve, L. Bieberbach proved that there exists a proper holomorphic mapping of the domain onto the unit disc that is an n-to-one branched covering with the properties that it extends continuously to the boundary and maps each boundary curve one-to-one onto the unit circle, and it maps each given point on the boundary to the point 1 in the unit circle. We modify a proof by H. Grunsky of Bieberbach's result to show that there is a rational function of 2n+2 complex variables that generates all of these maps. We also show how to generate all the proper holomorphic mappings to the unit disc via the rational function.Comment: 17 page

Exploring Different Dimensions of Attention for Uncertainty Detection

Neural networks with attention have proven effective for many natural language processing tasks. In this paper, we develop attention mechanisms for uncertainty detection. In particular, we generalize standardly used attention mechanisms by introducing external attention and sequence-preserving attention. These novel architectures differ from standard approaches in that they use external resources to compute attention weights and preserve sequence information. We compare them to other configurations along different dimensions of attention. Our novel architectures set the new state of the art on a Wikipedia benchmark dataset and perform similar to the state-of-the-art model on a biomedical benchmark which uses a large set of linguistic features.Comment: accepted at EACL 201

Spectral goodness of fit for network models

We introduce a new statistic, 'spectral goodness of fit' (SGOF) to measure how well a network model explains the structure of an observed network. SGOF provides an absolute measure of fit, analogous to the standard R-squared in linear regression. Additionally, as it takes advantage of the properties of the spectrum of the graph Laplacian, it is suitable for comparing network models of diverse functional forms, including both fitted statistical models and algorithmic generative models of networks. After introducing, defining, and providing guidance for interpreting SGOF, we illustrate the properties of the statistic with a number of examples and comparisons to existing techniques. We show that such a spectral approach to assessing model fit fills gaps left by earlier methods and can be widely applied