A small aperiodic set of Wang tiles

A new aperiodic tile set containing only 14 Wang tiles is presented. The construction is based on Mealy machines that multiply Beatty sequences of real numbers by rational constants.     

An aperiodic set of 13 Wang tiles

A new aperiodic tile set containing only 13 tiles over 5 colors is presented. Its construction is based on a recent technique developed by Kari. The tilings simulate the behavior of sequential machines that multiply real numbers in balanced representations by real constants.     

Percolation in the hyperbolic plane

We study percolation in the hyperbolic plane and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and Poisson-Voronoi-Bernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, , there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, , there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, , there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of in the Poisson-Voronoi-Bernoulli percolation process on the intensity of the underlying Poisson process.

     

Coordinates in the hyperbolic plane

Abstract This note shows that in the hyperbolic plane three kinds of coordinates are possible. Keywords: Hyperbolic plane, Quasi-regular quadrangle, Pseudo-parallelogram, Hypercycle, Hypocycle Mathematics Subject Classification (2000): 20N05     

Each four-celled animal tiles the plane

An animal A is a set of unit squares in the plane, parallel to the axes, and with corners at integer lattice points. We show that any animal A with four cells tiles the plane, in the sense that infinitely many copies of A, translated by integer vectors and possibly rotated through 90°, 180°, or 270°, can be placed so as to fill plane exactly without overlap.     

The closure of the set of roots of strongly connected reliability polynomials is the entire complex plane

The strongly connected reliabilityscRel(D,p) of a digraph D is the probability that the spanning subgraph of D consisting of the operational arcs is strongly connected, given that the vertices always operate, but each arc is independently operational with probability p∈[0,1]. We show that the closure of the set of roots of strongly connected reliability polynomials is the whole complex plane.     

Projective-type axioms for the hyperbolic plane

It is shown that the Laws of Pappus and Desargues may replace the Axiom of Projectivities in Menger's development of hyperbolic geometry from axioms of alignment.     

A special set of eigenvectors for the hyperbolic Sutherland systems

We construct the integrals of motion for Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (su(n) spins) interacting with an external field of the Morse potential of an arbitrary strength τ ² . These systems are confined if certain constraints are imposed on τ, the pair coupling constant λ, and the number of particles. The ground state is described by a wave function of the Jastrow form.     

A classification and construction of entirely circular cubics in the hyperbolic plane

If each intersection point of a third order curve with the absolute conic of the hyperbolic plane is a tangential point, this curve will be called an entirely circular cubic. According to this definition a rough classification of such curves is given into four main types and nine sub-types. Some of them are constructed by a (1,2) or (1,1) mapping and the others are constructed by the generalized quadratic hyperbolic inversion. Thus we extend and complete Palman's paper [5] in a synthetic way. This revised version was published online in June 2006 with corrections to the Cover Date.     

A note on the distance set problem in the plane

We use a simple geometric-combinatorial argument to establish a quantitative relation between the generalized Hausdorff measure of a set and its distance set, extending a result originally due to Falconer.