A useful semiclassical method to calculate eigenfunctions of the Schrödinger equation is the mapping to a well-known ordinary differential equation, as for example Airy’s equation. In this paper we generalize the mapping procedure to the nonlinear Schrödinger equation or Gross-Pitaevskii equation describing the macroscopic wave function of a Bose-Einstein condensate. The nonlinear Schrödinger e...

A theory of random walks on the mapping class group and its non-elementary subgroups is developed. We prove convergence of sample paths in the Thurston compact-iication and show that the space of projective measured foliations with the corresponding harmonic measure can be identiied with the Poisson boundary of random walks. The methods are based on an analysis of the asymptotic geometry of Tei...

Conformal mapping has been applied mostly to harmonic functions, i.e. solutions of Laplace’s equation. In this paper, it is noted that some other equations are also conformally invariant and thus equally well suited for conformal mapping in two dimensions. In physics, these include steady states of various nonlinear diffusion equations, the advection-diffusion equations for potential flows, and...

In this paper, we propose a new method for parameterizing a contractible domain (called the computational domain) which is defined by its boundary. Using a sequence of harmonic maps, we first build a mapping from the computational domain to the parameter domain, i.e., the unit square or unit cube. Then we parameterize the original domain by spline approximation of the inverse mapping. Numerical...

We have recently investigated a family of algorithms which use the underlying latent space model developed for the Generative Topographic mapping(GTM) but which train the parameters in a different manner. Our first model was the Topographic Product of Experts (ToPoE) which is fast but not so data-driven as our second model, the Harmonic Topographic Mapping (HaToM). However the HaToM is much slo...

We call a piecewise linear mapping from a planar triangulation to the plane a convex combination mapping if the image of every interior vertex is a convex combination of the images of its neighbouring vertices. Such mappings satisfy a discrete maximum principle and we show that they are oneto-one if they map the boundary of the triangulation homeomorphically to a convex polygon. This result can...

We study harmonic maps from a 3-manifold with boundary to $$\mathbb {S}^1$$ and prove special case of Gromov dihedral rigidity three-dimensional cubes whose angles are $$\pi / 2$$ . Furthermore, we give some applications mapping torus hyperbolic 3-manifolds.

In the paper I study the gradient eld of a harmonic function f in R3 in a neighbourhood of a critical point 0. I show that the ow of ∇f , as a mapping between level sets of f , is a strati ed mapping that gives, in our case, an answer to the problem of stratifying the space of orbits of the eld ∇f posed by R. Thom. I also show that the trajectories of ∇f having 0 as a limit point satisfy the ni...

We numerically demonstrate so-far undescribed features in ionization and high harmonic generation from bound states with nonvanishing electronic angular momentum. The states' modified response to a strong laser pulse can be exploited for novel measurement and pulse production schemes. It is shown that angularly asymmetric tunneling from the states can be mapped onto variations of high harmonic ...

ferroelectric Rama K. Vasudevan, Shujun Zhang, Jilai Ding, M. Baris Okatan, Stephen Jesse, Sergei V. Kalinin, and Nazanin Bassiri-Gharb Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA and Institute for Functional Imaging of Materials, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Department of Materials Science and Enginee...