It is shown that the (infinite) tiling problem byWang tiles is undecidable even if the given tile set is deterministic by all four corners, i.e. a tile is uniquely determined by the colors of any two adjacent edges. The reduction is done from the Turing machine halting problem and uses the aperiodic tile set of Kari and Papasoglu. © 2008 Elsevier B.V. All rights reserved.

Co-evolution of phytoplankton C:N:P stoichiometry and the deep ocean N:P ratio T. M. Lenton and C. A. Klausmeier School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, UK W. K. Kellogg Biological Station, Michigan State University, Hickory Corners, MI 49060, USA Received: 16 June 2006 – Accepted: 4 July 2006 – Published: 17 July 2006 Correspondence to: T. M. Lenton (t.len...

First, a classical example: the relationship between differentiable and continuous functions. It is clear that continuity is not enough for differentiability, a curve can have “corners” such as the absolute value function at 0. With a little bit of effort one can manufacture a continuous function that has countably many corners in the unit interval. E.g., one could have a kink at all positions ...

It was shown by Carbery, Christ, and Wright that any measurable set E in the unit square in R 2 not containing the corners of a rectangle with area greater than has measure bounded by O(q log 1). We remove the log under the additional assumption that the set does not contain the corners of any axis-parallel, possibly self-crossing hexagon with unsigned area bigger than. Our proof may be viewed ...

The isophote curvature times the gradient magnitude to some power has been studied in the literature as a measure of cornerness in images. Stability in terms of sampling noise has been proposed by selecting corners at high scale and locating them at fine scale. We will examine the problem of tracking extrema of these measures in the Linear Scale-Space using catastrophe theory. This enables us t...

Contour integration refers to the ability of the visual system to bind disjoint local elements into coherent global shapes. In cluttered images containing randomly oriented elements a contour becomes salient when its elements are coaligned with a smooth global trajectory, as described by the Gestalt law of good continuation. Abrupt changes of curvature strongly diminish contour salience. Here w...

It is known that given a directed graph E and a subset X of vertices, the sum Σ v∈X P v of vertex projections in the C *-algebra of E converges strictly in the multiplier algebra to a projection P X. Here we give a construction which, in certain cases, produces a directed graph F such that C * (F) is isomorphic to the corner P X C * (E)P X. Corners of this type arise naturally as the fixed poin...

This note is an illustration of the density-increment method used in the proof of the density Hales-Jewett theorem for k = 3. (Polymath project [2]) I will repeat the argument applying it to a problem which is easier than DHJ. In the last section I will describe the proof of the density Hales-Jewett theorem for k = 3. The results stated here are direct interpretations of the project’s results, ...

We develop a framework for dealing with smooth approximations to billiards with corners in the two-dimensional setting. Let a polygonal trajectory in a billiard start and end up at the same billiard’s corner point. We prove that smooth Hamiltonian flows which limit to this billiard have a nearby periodic orbit if and only if the polygon angles at the corner are ‘‘acceptable.’’ The criterion for...

The simplest modeling of planar quantum waveguides is the Dirichlet eigenproblem for the Laplace operator in unbounded open sets which are uniformly thin in one direction. Here we consider V-shaped guides. Their spectral properties depend essentially on a sole parameter, the opening of the V. The free energy band is a semi-infinite interval bounded from below. As soon as the V is not flat, ther...