Dmitry Berdinsky

Dimensions and bases of hierarchical tensor-product splines

We prove that the dimension of trivariate tensor-product spline space of tri-degree (m,m,m) with maximal order of smoothness over a three-dimensional domain coincides with the number of tensor-product B-spline basis functions acting effectively on the domain considered. A domain is required to belong to a certain class. This enables us to show that, for a certain assumption about the configuration of a hierarchical mesh, hierarchical B-splines span the spline space. This paper presents an extension to three-dimensional hierarchical meshes of results proposed recently by Giannelli and Juttler for two-dimensional hierarchical meshes.

On Automatic Transitive Graphs

We study infinite automatic transitive graphs. In particular we investigate automaticity of certain Cayley graphs. We provide examples of infinite automatic transitive graphs that are not Cayley graphs. We prove that Cayley graphs of Baumslag–Solitar groups and the restricted wreath products of automatic transitive graphs with ℤ are automatic.

Bases of T-meshes and the refinement of hierarchical B-splines

Abstract In this paper we consider spaces of bivariate splines of bi-degree ( m , n ) with maximal order of smoothness over domains associated to a two-dimensional grid. We define admissible classes of domains for which suitable combinatorial technique allows us to obtain the dimension of such spline spaces and the number of tensor-product B-splines acting effectively on these domains. Following the strategy introduced recently by Giannelli and Juttler, these results enable us to prove that under certain assumptions about the configuration of a hierarchical T-mesh the hierarchical B-splines form a basis of bivariate splines of bi-degree ( m , n ) with maximal order of smoothness over this hierarchical T-mesh. In addition, we derive a sufficient condition about the configuration of a hierarchical T-mesh that ensures a weighted partition of unity property for hierarchical B-splines with only positive weights.

Cayley Automatic Representations of Wreath Products

We construct the representations of Cayley graphs of wreath products using finite automata, pushdown automata and nested stack automata. These representations are in accordance with the notion of Cayley automatic groups introduced by Kharlampovich, Khoussainov and Miasnikov and its extensions introduced by Elder and Taback. We obtain the upper and lower bounds for a length of an element of a wreath product in terms of the representations constructed.

Iterative refinement of hierarchical T-meshes for bases of spline spaces with highest order smoothness

In this paper we propose a strategy for generating consistent hierarchical T-meshes which allow local refinement and offer a way to obtain spline basis functions with highest order smoothness incrementally. We describe the required ordering of line-segments during refinement and the construction of spline basis functions. We give our strategy for generating consistent hierarchical T-meshes over any shape of a two-dimensional domain.

On orthogonal curvilinear coordinate systems in constant curvature spaces

We describe a method for constructing an n-orthogonal coordinate system in constant curvature spaces. The construction proposed is actually a modification of the Krichever method for producing an orthogonal coordinate system in the n-dimensional Euclidean space. To demonstrate how this method works, we construct some examples of orthogonal coordinate systems on the two-dimensional sphere and the hyperbolic plane, in the case when the spectral curve is reducible and all irreducible components are isomorphic to a complex projective line.

Measuring Closeness Between Cayley Automatic Groups and Automatic Groups

In this paper we introduce a way to estimate a level of closeness of Cayley automatic groups to the class of automatic groups using a certain numerical characteristic. We characterize Cayley automatic groups which are not automatic in terms of this numerical characteristic and then study it for the lamplighter group, the Baumslag--Solitar groups and the Heisenberg group.

Cayley Automatic Groups and Numerical Characteristics of Turing Transducers

This paper is devoted to the problem of finding characterizations for Cayley automatic groups. The concept of Cayley automatic groups was recently introduced by Kharlampovich, Khoussainov and Miasnikov. We address this problem by introducing three numerical characteristics of Turing transducers: growth functions, Folner functions and average length growth functions. These three numerical characteristics are the analogs of growth functions, Folner functions and drifts of simple random walks for Cayley graphs of groups. We study these numerical characteristics for Turing transducers obtained from automatic presentations of labeled directed graphs.

On constant mean curvature surfaces in the Heisenberg group

This work is devoted to the theory of surfaces of constant mean curvature in the three-dimensional Heisenberg group. It is proved that each surface of such a kind locally corresponds to some solution of the system of a sine-Gordon type equation and a first order partial differential equation.