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In previous works the authors have obtained an effective classification of planar locally minimal binary trees with convex boundaries. The main aim of the present paper is to find more subtle restrictions on the possible structure...
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In previous works the authors have obtained an effective classification of planar locally minimal binary trees with convex boundaries. The main aim of the present paper is to find more subtle restrictions on the possible structure of such trees in terms of the geometry of the given boundary set. Special attention is given to the case of quasiregular boundaries (that is, boundaries that are sufficiently close to regular ones in a certain sense). In particular, a series of quasiregular boundaries that cannot be spanned by a locally minimal binary tree is constructed.
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In this paper we study the structure of the set [G,#phi#]_#GAMMA# of immersed linear networks in R~N that are parallel to a given immersed linear network #GAMMA# : G -> R~N and whose boundary #phi# coincides with the boundary of #...
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In this paper we study the structure of the set [G,#phi#]_#GAMMA# of immersed linear networks in R~N that are parallel to a given immersed linear network #GAMMA# : G -> R~N and whose boundary #phi# coincides with the boundary of #GAMMA#. We prove that [G,#phi#]_#GAMMA# is a convex polyhedral subset in the configuration space of moving vertices of the graph G. We also calculate the dimension of this convex subset and estimate the number of its faces of maximal dimension. The results obtained are used to describe the space of all locally minimal (weighted minimal) networks in R~N with a fixed topology and a fixed boundary. In the case of planar networks in which the degrees of vertices are at most three (Steiner networks), this dimension is calculated in topological terms.
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The method of partial stabilization of locally minimal net. works, which was invented by Ivanov and Tuzhilin to construct examples of shortest trees with given topology, is developed. According to this method, boundary vertices of...
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The method of partial stabilization of locally minimal net. works, which was invented by Ivanov and Tuzhilin to construct examples of shortest trees with given topology, is developed. According to this method, boundary vertices of degree 2 are not added to all edges of the original locally minimal tree, but only to some of them. The problem of partial sta- bilization of locally minimal trees in a finite-dimensional Euclidean space is solved completely in the paper, that is,without any restrictions imposed on the number of edges remaining free of subdivision. A criterion for the realiz- ability of such stabilization is established. In addition, the general problem of searching for the shortest forest connecting a finite family of boundary compact sets in an arbitrary metric space is formalized; it is shown that such forests exist for any family of compact sets if and only if for any finite subset of the ambient space there exists a shortest tree connecting it. The theory developed here allows us to establish further generalizations of the stabilization theorem both for arbitrary metric spaces and for metric spaces with some special properties. Bibliography: 10 titles.
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We introduce the notion of an `immersed polygon', which naturally extends the notion of an ordinary planar polygon bounded by a closed (embedded) polygonal arc to the case when this arc may have self-intersections. We prove that e...
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We introduce the notion of an `immersed polygon', which naturally extends the notion of an ordinary planar polygon bounded by a closed (embedded) polygonal arc to the case when this arc may have self-intersections. We prove that every immersed polygon admits a diagonal triangulation and the closure of every embedded monotone polygonal arc bounds an immersed polygon. Given any non-degenerate planar linear tree, we construct an immersed polygon containing it.
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The first part of this work deals with the investigation and modeling of foliations generated by dynamic systems on their phase spaces and configuration spaces. In the second part, we speak in greater detail about curves and surfa...
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The first part of this work deals with the investigation and modeling of foliations generated by dynamic systems on their phase spaces and configuration spaces. In the second part, we speak in greater detail about curves and surfaces in three-dimensional space and their geometrical characteristics that can be useful for modeling polymer conformation.
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This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. Starting with a Markov model involving a structured probability density function, we de...
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This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. Starting with a Markov model involving a structured probability density function, we derive the non-local in time master equation and fractional equation for the probability of cell position. We derive the fractional Fokker-Planck equation for the density of cells and apply this equation to the anomalous chemotaxis problem. We show the structural instability of fractional subdiffusive equation with respect to the partial variations of anomalous exponent. We find the criteria under which the anomalous aggregation of cells takes place in the semi-infinite domain.
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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...
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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 ctriterion of extremality is presented, which shows that the extremality with respect to the Manhattan-length functional is a global topological property of networks.
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Abstract The behavior of extreme networks under deformations of their boundary sets is investigated. It is shown that analyticity of a deformation of the boundary set guarantees preservation of the network type for minimal spannin...
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Abstract The behavior of extreme networks under deformations of their boundary sets is investigated. It is shown that analyticity of a deformation of the boundary set guarantees preservation of the network type for minimal spanning trees, minimal fillings, and so-called stable shortest trees in the Euclidean space.
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The pair distribution function g(r) for a ferrofluid modeled by a bidisperse system of dipolar hard spheres is calculated. The influence of an external uniform magnetic field and polydispersity on g(r) and the related structure fa...
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The pair distribution function g(r) for a ferrofluid modeled by a bidisperse system of dipolar hard spheres is calculated. The influence of an external uniform magnetic field and polydispersity on g(r) and the related structure factor is studied. The calculation is performed by diagrammatic expansion methods within the thermodynamic perturbation theory in terms of the particle number density and the interparticle dipole- dipole interaction strength. Analytical expressions are provided for the pair distribution function to within the first order in number density and the second order in dipole-dipole interaction strength. The constructed theory is compared with the results of computer (Monte Carlo) simulations to determine the range of its validity. The scattering structure factor is determined using the Fourier transform of the pair correlation function g(r) - 1. The influence of the granulometric composition and magnetic field strength on the height and position of the first peak of the structure factor that is most amenable to an experimental study is analyzed. The data obtained can serve as a basis for interpreting the experimental small-angle neutron scattering results and determining the regularities in the behavior of the structure factor, its dependence on the fractional composition of a ferrofluid, interparticle correlations, and external magnetic field.
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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 tre...
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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).
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