Approximate Capacities of Two-Dimensional Codes by Spatial Mixing

We apply several state-of-the-art techniques developed in recent advances of counting algorithms and statistical physics to study the spatial mixing property of the two-dimensional codes arising from local hard (independent set) constraints, including: hard-square, hard-hexagon, read/write isolated memory (RWIM), and non-attacking kings (NAK). For these constraints, the strong spatial mixing would imply the existence of polynomial-time approximation scheme (PTAS) for computing the capacity. It was previously known for the hard-square constraint the existence of strong spatial mixing and PTAS. We show the existence of strong spatial mixing for hard-hexagon and RWIM constraints by establishing the strong spatial mixing along self-avoiding walks, and consequently we give PTAS for computing the capacities of these codes. We also show that for the NAK constraint, the strong spatial mixing does not hold along self-avoiding walks

Annulus decomposition of handlebody-knots

By Thurston's hyperbolization theorem, handlebody-knots of genus $2$, like classical knots, can be classified into four classes based on essential surfaces of non-negative Euler characteristic in their exteriors. The class analogous to that of torus knots comprises irreducible, atoroidal, cylindrical handlebody-knots, which are characterized by the absence of essential disks and tori and the existence of an essential annulus in their exteriors. By Funayoshi-Koda, they are precisely those non-hyperbolic handlebody-knots with a finite symmetry group, and are the main subject of the paper. The paper purposes a classification scheme for such handlebody-knots building on Johannson's characteristic submanifold theory and the Koda-Ozawa classification of essential annuli in handlebody-knot exteriors. We introduce the notion of annulus diagrams for such handlebody-knots, and classify their possible shapes. In terms of annulus diagrams, a classification result for irreducible, atoroidal handlebody-knots whose exteriors contain a type $2$ annulus and a characterization of the simplest non-trivial handlebody-knot are obtained. Implications thereof for handlebody-knot symmetries are also discussed.Comment: 30 pages, 25 figure

Global well-posedness for radial extremal hypersurface equation in (1+3)\left(1+3 \right)-dimensional Minkowski space-time in critical Sobolev space

In this article, we prove the global well-posedness in the critical Sobolev space $H_{rad}^2\left(\mathbb{R}^2\right) \times H_{rad}^1 \left(\mathbb{R}^2\right)$ for the radial time-like extremal hypersurface equation in $\left(1+3\right)$- dimensional Minkowski space-time. This is achieved by deriving a new div-curl type lemma and combined it with energy and ``momentum" balance law to get some space-time estimates of the nonlinearity.Comment: 34 pages, this version reduces some printing mistake