We begin by considering the one-dimensional analogue of Kac's question: a vibrating string of length L. Idealize this string by the interval [0,L] and represent the possible configuration of the vibrating string as a function f(x,t) defined for x (the position variable) in [0,L] and any non-negative number t (the time variable). Since the endpoints are fixed, f must satisfy the boundary conditi...

By fixing all parameters in a phylogenetic likelihood model except for one branch length, one obtains a one-dimensional likelihood function. In this work, we introduce a mathematical framework to characterize the shapes of such one-dimensional phylogenetic likelihood functions. This framework is based on analyses of algebraic structures on the space of all frequency patterns with respect to a p...

Traditionally, network analysis is based on local properties of vertices, like their degree or clustering, and their statistical behavior across the network in question. This paper develops an approach which is different in two respects. We investigate edge-based properties, and we define global characteristics of networks directly. The latter will provide our affirmative answer to the question...

Studies have shown that observers’ judgments of surface orientation and curvature are affected by the presence of surface texture pattern. However, the question of designing a texture pattern that does not hide the surface information nor conveys a misrepresentation of the surface remains unsolved. The answer to this question has important potential impact across a wide range of visualization a...

Image segmentation with one shape prior is an important problem in computer vision. Most algorithms not only share a similar energy definition, but also follow a similar optimization strategy. Therefore, they all suffer from the same drawbacks in practice such as slow convergence and difficult-to-tune parameters. In this paper, by reformulating the energy cost function, we establish an importan...

We show that the Bers embedding of the Teichmüller space of a once-punctured torus converges to the cardioid in the sense of Carathéodory up to rotation when the base torus goes to the boundary of its moduli space.

We discuss the line shape for an internal state transition for bosonic atoms confined as a one-dimensional gas, using a method by Haldane. Typical line shape is an edge singularity due to the absence of Bose-Einstein condensation in such systems.

Reconstructing past population size from present day genetic data is a major goal of population genetics. Recent empirical studies infer population size history using coalescent-based models applied to a small number of individuals. Here we provide tight bounds on the amount of exact coalescence time data needed to recover the population size history of a single, panmictic population at a certa...

Objects are three-dimensional entities, but visual observations are largely 2D. Inferring 3D properties from individual 2D views is thus a generically useful skill that is critical to object perception. We ask the question: can we learn useful image representations by explicitly training a system to infer 3D shape from 2D views? The few prior attempts at single view 3D reconstruction all target...

We match shapes, even under severe deformations, via a smooth re-parametrization of their integral invariant signatures. These robust signatures and correspondences are the foundation of a shape energy functional for variational image segmentation. Integral invariant shape templates do not require registration and allow for significant deformations of the contour, such as the articulation of th...