Related papers: Geometric representation of interval exchange maps…
Polytope exchange transformations (PETs) are higher dimensional generalizations of interval exchange transformations (IETs) which have been well-studied for more than 40 years. A general method of constructing PETs based on multigraphs was…
Answering connectivity queries in real algebraic sets is a fundamental problem in effective real algebraic geometry that finds many applications in e.g. robotics where motion planning issues are topical. This computational problem is…
The joint degree matrix of a graph gives the number of edges between vertices of degree i and degree j for every pair (i,j). One can perform restricted swap operations to transform a graph into another with the same joint degree matrix. We…
The purpose of this paper is to provide a new account of multiplicity for finite morphisms between smooth projective varieties. Traditionally, this has been defined using commutative algebra in terms of the length of integral ring…
We study the dynamics of renormalisation of an interval exchange transformation which features exact scaling (the cubic Arnoux-Yoccoz model). Using a symbolic space that describes both dynamics and scaling, we characterize the periodic…
Transitive consistency is an intrinsic property for collections of linear invertible transformations between Euclidean coordinate frames. In practice, when the transformations are estimated from data, this property is lacking. This work…
Irrational numbers of bounded type have several equivalent characterizations. They have bounded partial quotients in terms of arithmetic characterization and in the dynamics of the circle rotation, the rescaled recurrence time to $r$-ball…
The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated…
Finite fields form an important chapter in abstract algebra, and mathematics in general. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a…
Let A, B, S be categories, let F:A-->S and G:B-->S be functors. We assume that for "many" objects a in A, there exists an object b in B such that F(a) is isomorphic to G(b). We establish a general framework under which it is possible to…
We present two new proofs of the exchange theorem for the Laplace transformation of vector-valued distributions. We then derive an explicit solution to the Dirichlet problem of the polyharmonic operator in a half-space. Finally, we obtain…
We characterize cutting sequences of infinite geodesics on square-tiled surfaces by considering interval exchanges on specially chosen intervals on the surface. These interval exchanges can be thought of as skew products over a rotation,…
The vector transform operators are investigated; these operators are used at the solution of boundary value problems in piecewise homogeneous spherically symmetric areas. In particular, examples of transformation operators for vector…
We prove that any over-twist pattern is conjugate to an interval exchange transformation with bounded number of segments of isometry, restricted on one of its cycles. The bound is independent of the period and over-rotation number of the…
In this work, we relate the geometry of chaotic attractors of typical analytic unimodal maps to the behavior of the critical orbit. Our main result is an explicit formula relating the combinatorics of the critical orbit with the exponents…
The modeling and control of networks over finite lattices are studied via the algebraic state space approach. Using the semi-tensor product of matrices, we obtain the algebraic state space representation of the dynamics of (control)…
In spatial networks vertices are arranged in some space and edges may cross. When arranging vertices in a 1-dimensional lattice edges may cross when drawn above the vertex sequence as it happens in linguistic and biological networks. Here…
We study the symmetries and geodesics of an infinite translation surface which arises as a limit of translation surfaces built from regular polygons, studied by Veech. We find the affine symmetry group of this infinite translation surface,…
We discuss a link between graph theory and geometry that arises when considering graph dynamical systems with odd interactions. The equilibrium set in such systems is not a collection of isolated points, but rather a union of manifolds,…
Vertex splitting is a graph modification operation in which a vertex is replaced by multiple vertices such that the union of their neighborhoods equals the neighborhood of the original vertex. We introduce and study vertex splitting as a…