Minimum-Area ellipse containing a finite set of points. II

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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