Alexey Avgustinovich Tuzhilin

Branching solutions to one-dimensional variational problems

Preliminary results networks extremality criteria linear networks in RN extremals of length type functionals - the case of parametric networks extremals of functionals generated by norms.

The Gromov–Hausdorff metric on the space of compact metric spaces is strictly intrinsic

It is proved that the Gromov-Hausdorff metric on the space of compact metric spaces considered up to an isometry is strictly intrinsic, i.e., the corresponding metric space is geodesic. In other words, each two points of this space (each two compact metric spaces) can be connected by a geodesic. For finite metric spaces a geodesic is constructed explicitly.

Steiner Ratio for Manifolds

The Steiner ratio characterizes the greatest possible deviation of the length of a minimal spanning tree from the length of the minimal Steiner tree. In this paper, estimates of the Steiner ratio on Riemannian manifolds are obtained. As a corollary, the Steiner ratio for flat tori, flat Klein bottles, and projective plane of constant positive curvature are computed.

Stabilization of Locally Minimal Trees

It is proved that any locally minimal tree on Euclidean space can be “stabilized” (i.e., rendered shortest) by adding boundary vertices without changing the initial tree as a set in space. This result is useful for constructing examples of shortest trees.

One-dimensional minimal fillings with negative edge weights

A.O. Ivanov and A.A. Tuzhilin proposed a particular case of Gromov’s minimal fillings problem generalized to the case of stratified manifolds using weighted graphs with a nonnegative weight function as minimal fillings of finite metric spaces. In this paper we introduce generalized minimal fillings, i.e., the minimal fillings where the weight function is not necessarily nonnegative. We prove that for any finite metric space its minimal filling has the minimum weight in the class of its generalized fillings.

The structure of minimal Steiner trees in the neighborhoods of the lunes of their edges

We give a complete description of small neighborhoods of the closures of lunes of the edges of Steiner minimal trees (Theorem 1.1); to this end, we prove a generalization of a stabilization theorem for embedded locally minimal trees [1]; the case of two such disjoint trees is considered (Theorem 2.2).

Minimal binary trees with a regular boundary: The case of skeletons with five endpoints

Locally minimal binary trees that span the vertices of regular polygons are studied. Their description is given in the dual language, that of diagonal triangulations of polygons. Diagonal triangulations of a special form, called skeletons, are considered. It is shown that planar binary trees dual to skeletons with five endpoints do not occur among locally minimal binary trees that span the vertices of regular polygons.

Planar Manhattan Local Minimal and Critical Networks

One-dimensional branching extremals of Lagrangian-type functionals are considered. Such extremals appear as solutions to the classical Steiner problem on a shortest network, i.e., a connected system of paths that has the smallest total length among all the networks spanning a given finite set of terminal points in the plane. In the present paper, the Manhattan-length functional is investigated, with Lagrangian equal to the sum of the absolute values of projections of the velocity vector onto the coordinate axes. Such functionals are useful in problems arising in electronics, robotics, chip design, etc. In this case, in contrast to the case of the Steiner problem, local minimality does not imply extremality (however, each extreme network is locally minimal). A criterion of extremality is presented, which shows that the extremality with respect to the Manhattan-length functional is a global topological property of networks. Bibliography: 95 titles.

Minimal Networks: A Review

Minimal Networks Theory is a branch of mathematics that goes back to 17th century and unites ideas and methods of metric, differential, and combinatorial geometry and optimization theory. It is still studied intensively, due to many important applications such as transportation problem, chip design, evolution theory, molecular biology, etc. In this review we point out several significant directions of the Theory. We also state some open problems which solution seems to be crucial for the further development of the Theory. Minimal Networks can be considered as one-dimensional minimal surfaces. The simplest example of such a network is a shortest curve or, more generally, a geodesic. The first ones are global minima of the length functional considered on the curves connecting fixed boundary points. The second ones are the curves such that each sufficiently small part of them is a shortest curve. A natural generalization of the problem appears, if the boundary consists of three and more points, and additional branching points are permitted. Steiner minimal trees are analogues of the shortest curves, and locally minimal networks are generalizations of geodesics. We also include some results concerning so-called minimal fillings and minimal networks in the spaces of compacts.

Critical Analysis of Amino Acids and Polypeptides Geometry

We present several geometrical and statistical methods analyzing information contained in Protein Data Bank for consistency and general theory fitting. In addition, we include a list of examples demonstrating the necessity of careful analysis and sampling of the PDB items. In particular, we show some examples visualizing the distribution of disturbance level of the famous Pauling plane low. Besides, we present a method of automatical helices finding based on discrete analogues of curvature and torsion of smooth curve.