On extremums of sums of powered distances to a finite set of points

The Steiner minimum tree and the minimum spanning tree are two important problems in combinatorial optimization. Let P denote a finite set of points, called terminals, in the Euclidean space. A Steiner minimum tree of P, denoted by SMT(P), is a network with minimum length to interconnect all terminals, and a minimum spanning tree of P, denoted by MST(P), is also a minimum network interconnecting all the points in P, however, subject to the constraint that all the line segments in it have to terminate at terminals. Therefore, SMT(P) may contain points not in P, but MST(P) cannot contain such kind of points. Let [Formula: see text] denote the n-dimensional Euclidean space. The Steiner ratio in [Formula: see text] is defined to be [Formula: see text], where Ls(P) and Lm(P), respectively, denote lengths of a Steiner minimum tree and a minimum spanning tree of P. The best previously known lower bound for [Formula: see text] in the literature is 0.615. In this paper, we show that [Formula: see text] for any n ≥ 2.

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Criteria for Embedded Eigenvalues for Discrete Schrödinger Operators

International Mathematics Research Notices ◽

10.1093/imrn/rnz262 ◽

2019 ◽

Cited By ~ 3

Author(s):

Wencai Liu

Keyword(s):

Upper Bound ◽

Schrödinger Operators ◽

Schrodinger Operators ◽

Embedded Eigenvalues ◽

Finite Set ◽

Discrete Schrödinger Operators ◽

Sign Type ◽

Free Operator ◽

Set Of Points ◽

Rational Type

Abstract In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n-1})+V(n)u(n). \end{equation*}$$We view $H$ as a perturbation of the free operator $H_0$, where $(H_0u)(n)= u({n+1})+u({n-1})$. For $H_0$ (no perturbation), $\sigma _{\textrm{ess}}(H_0)=\sigma _{\textrm{ac}}(H)=[-2,2]$ and $H_0$ does not have eigenvalues embedded into $(-2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $H_0+V$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(-2,2)$. We introduce the almost sign type potentials and develop the Prüfer transformation to address this problem, which leads to the following five results. 1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $\limsup _{n\to \infty } n|V(n)|=a<\infty .$ We obtain a lower/upper bound of $a$ such that $H_0+V$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$.4: Given any finite set of points $\{ E_j\}_{j=1}^N$ in $(-2,2)$ with $0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$, we construct the explicit potential $V(n)=\frac{O(1)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$.5: Given any countable set of points $\{ E_j\}$ in $(-2,2)$ with $0\notin \{ E_j\}+\{ E_j\}$, and any function $h(n)>0$ going to infinity arbitrarily slowly, we construct the explicit potential $|V(n)|\leq \frac{h(n)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}$.

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On the number of holomorphic mappings between Riemann surfaces of finite analytic type

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091509000637 ◽

2011 ◽

Vol 54 (3) ◽

pp. 711-730

Author(s):

Yoichi Imayoshi ◽

Manabu Ito ◽

Hiroshi Yamamoto

Keyword(s):

Riemann Surfaces ◽

Holomorphic Mappings ◽

Explicit Bound ◽

Finiteness Result ◽

Classical Statement ◽

Finite Set ◽

Compact Riemann Surfaces ◽

Set Of Points ◽

Topological Data

AbstractThe set of non-constant holomorphic mappings between two given compact Riemann surfaces of genus greater than 1 is always finite. This classical statement was made by de Franchis. Furthermore, bounds on the cardinality of the set depending only on the genera of the surfaces have been obtained by a number of mathematicians. The analysis is carried over in this paper to the case of Riemann surfaces of finite analytic type (i.e. compact Riemann surfaces minus a finite set of points) so that the finiteness result, together with a crude but explicit bound depending only on the topological data, may be extended for the number of holomorphic mappings between such surfaces.

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COMPUTATIONAL GEOMETRY COLUMN 27

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195996000083 ◽

1996 ◽

Vol 06 (01) ◽

pp. 123-125

Author(s):

JOSEPH O’ROURKE

Keyword(s):

Computational Geometry ◽

Perfect Matching ◽

Finite Set ◽

Set Of Points

The following new result is discussed: there is a perfect matching between the edges of any two triangulations of a finite set of points in the plane.

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An Algorithm of Vortex Patches Identification Based on Models of Point Vortices

UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES ◽

10.18522/1026-2237-2020-3-11-18 ◽

2020 ◽

pp. 11-18

Author(s):

V.N. Govorukhin

Keyword(s):

Mathematical Model ◽

Fluid Flow ◽

Flow Region ◽

Velocity Model ◽

Point Vortices ◽

Unknown Parameters ◽

Vortex System ◽

Vortex Patch ◽

Finite Set ◽

Set Of Points

An algorithm for identifying a plane vortex fluid flow from known velocity vectors at a finite set of points in the flow region is proposed. To describe the approximation of the vortex structure, a mathematical model of the system of point vortices is used. The parameters to be determined and characterizing the flow are the number of point vortices, their coordinates, and intensity. Unknown parameters are determined as a result of minimizing the functional that relates the known velocity vectors and the velocity model flows of the vortex system at the same points in the region. An implementation of the algorithm for the simplest case, one vortex patch, is presented. Computational experiments were carried out, which showed the effectiveness of the developed method.

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A Universal Separable Diversity

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2017-0008 ◽

2017 ◽

Vol 5 (1) ◽

pp. 138-151 ◽

Cited By ~ 1

Author(s):

David Bryant ◽

André Nies ◽

Paul Tupper

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Complete Metric Space ◽

Urysohn Space ◽

Fundamental Properties ◽

Finite Set ◽

Whole Space ◽

Set Of Points ◽

Separable Metric Space

AbstractThe Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.

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On the thickness of the complete bipartite graph

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100037385 ◽

1964 ◽

Vol 60 (1) ◽

pp. 01-05 ◽

Cited By ~ 33

Author(s):

Lowell W. Beineke ◽

Frank Harary ◽

John W. Moon

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Finite Set ◽

Planar Subgraphs ◽

Set Of Points

A graph consists of a finite set of points and a set of lines joining some pairs of these points. At most one line is permitted to join any two points and no point is joined to itself by a line. A graph G′ is a subgraph of the graph G if the points and lines of G′ are also points and lines of G. The union of several graphs having the same set of points is the graph formed by joining two points in this set if they are joined in at least one of the original graphs. A graph is planar if it can be drawn in the plane (or equivalently, on a sphere) so that no lines intersect. The thickness of a graph G is defined as the smallest integer t such that G is the union of t planar subgraphs.

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