Folding, Tiling, and Multidimensional Coding

Folding a sequence $S$ into a multidimensional box is a method that is used to construct multidimensional codes. The well known operation of folding is generalized in a way that the sequence $S$ can be folded into various shapes. The new definition of folding is based on lattice tiling and a direction in the $D$-dimensional grid. There are potentially $\frac{3^D-1}{2}$ different folding operations. Necessary and sufficient conditions that a lattice combined with a direction define a folding are given. The immediate and most impressive application is some new lower bounds on the number of dots in two-dimensional synchronization patterns. This can be also generalized for multidimensional synchronization patterns. We show how folding can be used to construct multidimensional error-correcting codes and to generate multidimensional pseudo-random arrays

Self-avoiding and plane-filling properties for terdragons and other triangular folding curves

We consider $n$-folding triangular curves, or $n$-folding t-curves, obtained by folding $n$ times a strip of paper in $3$, each time possibly left then right or right then left, and unfolding it with $\pi /3$ angles. An example is the well known terdragon curve. They are self-avoiding like $n$-folding curves obtained by folding $n$ times a strip of paper in two, each time possibly left or right, and unfolding it with $\pi /2$ angles. We also consider complete folding t-curves, which are the curves without endpoint obtained as inductive limits of $n$-folding t-curves. We show that each of them can be extended into a unique covering of the plane by disjoint such curves, and this covering satisfies the local isomorphism property introduced to investigate aperiodic tiling systems. Two coverings are locally isomorphic if and only if they are associated to the same sequence of foldings. Each class of locally isomorphic coverings contains exactly $2^{\omega }$ (resp. $2^{\omega }$, $2$ or $5$, $0$) isomorphism classes of coverings by $1$ (resp. $2$, $3$, $\geq 4$) curves. These properties are partly similar to those of complete folding curves.Comment: 15 pages, 3 figure

Detecting Repetitions and Periodicities in Proteins by Tiling the Structural Space

The notion of energy landscapes provides conceptual tools for understanding the complexities of protein folding and function. Energy Landscape Theory indicates that it is much easier to find sequences that satisfy the "Principle of Minimal Frustration" when the folded structure is symmetric (Wolynes, P. G. Symmetry and the Energy Landscapes of Biomolecules. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 14249-14255). Similarly, repeats and structural mosaics may be fundamentally related to landscapes with multiple embedded funnels. Here we present analytical tools to detect and compare structural repetitions in protein molecules. By an exhaustive analysis of the distribution of structural repeats using a robust metric we define those portions of a protein molecule that best describe the overall structure as a tessellation of basic units. The patterns produced by such tessellations provide intuitive representations of the repeating regions and their association towards higher order arrangements. We find that some protein architectures can be described as nearly periodic, while in others clear separations between repetitions exist. Since the method is independent of amino acid sequence information we can identify structural units that can be encoded by a variety of distinct amino acid sequences

Tiling Billards on Triangle Tilings, and Interval Exchange Transformations

We consider the dynamics of light rays in triangle tilings where triangles are transparent and adjacent triangles have equal but opposite indices of refraction. We find that the behavior of a trajectory on a triangle tiling is described by an orientation-reversing three-interval exchange transformation on the circle, and that the behavior of all the trajectories on a given triangle tiling is described by a polygon exchange transformation. We show that, for a particular choice of triangle tiling, certain trajectories approach the Rauzy fractal, under rescaling.Comment: 31 pages, 19 figures, 2 appendices. Comments welcome

A generating algorithm for ribbon tableaux and spin polynomials

We describe a general algorithm for generating various families of ribbon tableaux and computing their spin polynomials. This algorithm is derived from a new matricial coding. An advantage of this new notation lies in the fact that it permits one to generate ribbon tableaux with skew shapes. This algorithm permits us to compute quickly big LLT polynomials in MuPAD-Combinat