Let Ω be an invariant nonrecurrent Fatou component associated with the automorphism F : C → C. Assume that all of the limit maps of {F|Ω} are constant. We prove the following theorem. If there is more than one such limit map then there are uncountably many. The images of these limit maps form a closed set in the boundary of Ω containing no isolated points. Additionally there cannot be more than...

Cognitive mapping is assumed to be through hippocampus-dependent place learning rather than striatum-dependent response learning. However, we proposed that either type of spatial learning, as long as it involves encoding metric relations between locations and reference points, could lead to a cognitive map. Furthermore, the fewer reference points to specify individual locations, the more accura...

Grafting a measured lamination on a hyperbolic surface defines a self-map of Teichmüller space, which is a homeomorphism by a result of Scannell and Wolf. In this paper we study the large-scale behavior of pruning, which is the inverse of grafting. Specifically, for each conformal structure X ∈ T (S), pruning X gives a map ML (S) → T (S). We show that this map extends to the Thurston compactifi...

This paper addresses the task of nuclei segmentation in high-resolution histopathological images. We propose an automatic end-to-end deep neural network algorithm for segmentation of individual nuclei. A nucleus-boundary model is introduced to predict nuclei and their boundaries simultaneously using a fully convolutional neural network. Given a color normalized image, the model directly outputs...

A simple model for two-dimensional photonic crystal devices consists of a finite number of possibly different circular cylinders centered on lattice points of a square or triangular lattice and surrounded by a homogeneous or layered background medium. The Dirichlet-to-Neumann (DtN) map method is a special method for analyzing the scattering of an incident wave by such a structure. It is more ef...

Let ∆n be the unit polydisc in Cn and let f be a holomorphic self map of ∆n. When n = 1, it is well known, by Schwarz’s lemma, that f has at most one fixed point in the unit disc. If no such point exists then f has a unique boundary point, call it x ∈ ∂∆, such that every horocycle E(x, R) of center x and radius R > 0 is sent into itself by f . This boundary point is called the Wolff point of f....

The phase flow method was originally introduced in [28] which can efficiently solve autonomous ordinary differential equations. In [13], the method was generalized to solve Hamiltonian system where the Hamiltonian function was discontinuous. However, both these methods require phase flow map constructed on an invariant manifold. This can increase computational cost when the invariant domain is ...

Abstract. Let B1 be the unit open disk in R2 and M be a closed Riemannian manifold. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in H1([0, T ]×B1,M) whose energy is non-increasing in time, given initial data u0 ∈ H(B1,M) and boundary data γ = u0|∂B1 . Previously, this uniqueness result was obtained by Rivière (when M is the round sphere and the en...

In Gold et al. (1996) a simple polygon digitizing technique was described where ‘fringe’ points were digitized around the interior of each map polygon, the Voronoi diagram constructedand the relevant polygon boundary segments extracted from the Voronoi cells. Gold (1997a) extended this to work with scanned map input, where the fringe points were generated automatically, and complete polygon bou...

It is shown that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses automorphisms of free groups with boundaries which play the role of mapping class groups of surfaces with several boundary components.