Spontaneous symmetry breaking and the formation of columnar structures in the primary visual cortex II --- Local organization of orientation modules

Self-organization of orientation-wheels observed in the visual cortex is discussed from the view point of topology. We argue in a generalized model of Kohonen's feature mappings that the existence of the orientation-wheels is a consequence of Riemann-Hurwitz formula from topology. In the same line, we estimate partition function of the model, and show that regardless of the total number N of the orientation-modules per hypercolumn the modules are self-organized, without fine-tuning of parameters, into definite number of orientation-wheels per hypercolumn if N is large.Comment: 36 pages Latex2.09 and eps figures. Needs epsf.sty, amssym.def, and Type1 TeX-fonts of BlueSky Res. for correct typo in graphics file

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

On Dynamics of Cubic Siegel Polynomials

Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomials which have a fixed Siegel disk of rotation number theta, with theta being a given irrational number of Brjuno type. Our main goal is to prove that when theta is of bounded type, the boundary of the Siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial. We also prove that the locus of all cubics with both critical points on the boundary of their Siegel disk is a Jordan curve, which is in some sense parametrized by the angle between the two critical points. A main tool in the bounded type case is a related space of degree 5 Blaschke products which serve as models for our cubics. Along the way, we prove several results about the connectedness locus of these cubic polynomials.Comment: 58 pages. 20 PostScript figure

Conformally invariant scaling limits in planar critical percolation

This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLE(k)), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path converges in the scaling limit to chordal SLE(6). No prior knowledge is assumed beyond some general complex analysis and probability theory.Comment: 55 pages, 10 figure

Conformality in the self-organization network

AbstractThe conformality of the self-organizing network is studied in this work. We use multi-dimensional deformation analyses to interpret the self-organizing mapping. It can be shown that this mapping is quasi-conformal with a convergent deformation bound. Based on analyses, a deformation measure and a non-conformality measure are derived to indicate the evolution status of the network. These measures can serve as new criteria to evolve the network. We test these measures with simulations on surface mapping problems

Basic properties of SLE

SLE is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed $\kappa$. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions. The present paper attempts a first systematic study of SLE. It is proved that for all $\kappa\ne 8$ the SLE trace is a path; for $\kappa\in[0,4]$ it is a simple path; for $\kappa\in(4,8)$ it is a self-intersecting path; and for $\kappa>8$ it is space-filling. It is also shown that the Hausdorff dimension of the SLE trace is a.s. at most $1+\kappa/8$ and that the expected number of disks of size \eps needed to cover it inside a bounded set is at least \eps^{-(1+\kappa/8)+o(1)} for $\kappa\in[0,8)$ along some sequence \eps\to 0. Similarly, for $\kappa\ge 4$, the Hausdorff dimension of the outer boundary of the SLE hull is a.s. at most $1+2/\kappa$, and the expected number of disks of radius \eps needed to cover it is at least \eps^{-(1+2/\kappa)+o(1)} for a sequence \eps\to 0.Comment: Made several correction