Surface registration between cortical surfaces is crucial in medical imaging for performing systematic comparisons between brains. Landmark-matching registration that matches anatomical features, called the sulcal landmarks, is often required to obtain a meaningful 1-1 correspondence between brain surfaces. This is commonly done by parameterizing the surface onto a simple parameter domain, such as the unit sphere, in which the sulcal landmarks are consistently aligned. Landmark matching surface registration can then be obtained from the landmark aligned parameterizations. For genus-0 closed brain surfaces, the optimized spherical harmonic parameterization, which aligns landmarks to consistent locations on the sphere, has been widely used. This approach is limited by the loss of bijectivity under large deformations and the slow computation. In this paper, we propose FLASH, a fast algorithm to compute the optimized spherical harmonic parameterization with consistent landmark alignment. This is achieved by formulating the optimization problem to ℂ and thereby linearizing the problem. Errors introduced near the pole are corrected using quasiconformal theories. Also, by adjusting the Beltrami differential of the mapping, a diffeomorphic (1-1, onto) spherical parameterization can be effectively obtained. The proposed algorithm has been tested on 38 human brain surfaces. Experimental results demonstrate that the computation of the landmark aligned spherical harmonic parameterization is significantly accelerated using the proposed algorithm.

We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold $\check{X}$ using T-duality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold $\check{X}$, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series expansions in terms of the K\"ahler parameters of $X$ have integral coefficients. Applying the results in \cite{Chan10} and \cite{LLW10}, we compute the open Gromov-Witten invariants in terms of local BPS invariants and give evidences of our conjecture for several 3-dimensional examples including $K_{\proj^2}$ and $K_{\proj^1\times\proj^1}$.

In the description of isolated gravitating systems in General Relativity a spacelike timeslice has the structure of a complete Riemannian three-manifold with an asymptotically at end. Let (N; g) denote an end of such a complete Riemannian three-manifold, ie N is diffeomorphic to R3\B1 (0), and the metric on N asymptotically approaches the Euclidean metric near infinity: gij=

A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. The minimal two species periodic assembly is the one with the least energy per lattice cell area. There is a parameter $b$ in $[0,1]$ and the type of the lattice associated with the minimal assembly varies depending on $b$. There are several threshold defined by a number $B=0.1867...$ If $b \in [0, B)$, the minimal assembly is associated with a rectangular lattice whose ratio of the longer side and the shorter side is in $[\sqrt{3}, 1)$; if $b \in [B, 1-B]$, the minimal assembly is associated with a square lattice; if $b \in (1-B, 1]$, the minimal assembly is associated with a rhombic lattice with an acute angle in $[\frac{\pi}{3}, \frac{\pi}{2})$. Only when $b=1$, this rhombic lattice is the hexagonal lattice. None of the other values of $b$ yields the hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where the hexagonal lattice is ubiquitously observed.

The paper presents a study of Fuglede’s $p$ -module of systems of measures in condensers in polarizable Carnot groups. In particular, we calculate the $p$ -module of measures in spherical ring domains, find the extremal measures, and finally, extend a theorem by Rodin to these groups.