Dmitry Berdinsky

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We prove that the dimension of trivariate tensor-product spline space of tri-degree (m,m,m) with maximal order of smoothness over a threedimensional 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(More)
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(More)
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(More)
In the present paper we extend the methods of the Weierstrass (or spinor) representation of surfaces in R [10, 11] and SU(2) = S [12] for surfaces in the three-dimensional Lie groups Nil , S̃L2, and Sol endowed with the so-called Thurston’s geometries [9]. The main feature of this approach is that the geometry of a surface is related to the spectral(More)
In this paper, we discuss the problem of instability in the dimension of a spline space over a T-mesh. For bivariate spline spaces S (5, 5, 3, 3) and S (4, 4, 2, 2), the instability in the dimension is shown over certain types of T-meshes. This result could be considered as an attempt to answer the question of how large the polynomial degree (m,m) should be(More)
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(More)
In this paper we study the spectral generalization of the Willmore functional for the surfaces in the three-dimensional nilpotent Lie group Nil with leftinvariant metric admitting four-dimensional isometry group, i.e. endowed by one of Thurston’s geometries. The Weierstrass representation for surfaces in Nil was introduced by us in [3] where following the(More)