scholarly journals Tropical Bisectors and Voronoi Diagrams

2021 ◽  
Author(s):  
Francisco Criado ◽  
Michael Joswig ◽  
Francisco Santos
Keyword(s):  
Euclidean Space ◽  
Open Subset ◽  
Voronoi Diagrams ◽  
Polyhedral Norms ◽  
Degenerate Cases

AbstractIn this paper we initiate the study of tropical Voronoi diagrams. We start out with investigating bisectors of finitely many points with respect to arbitrary polyhedral norms. For this more general scenario we show that bisectors of three points are homeomorphic to a non-empty open subset of Euclidean space, provided that certain degenerate cases are excluded. Specializing our results to tropical bisectors then yields structural results and algorithms for tropical Voronoi diagrams.

An asymptotic analysis for Hamilton–Jacobi equations with large Hamiltonian drift terms

2017 ◽  
Vol 0 (0) ◽  
Author(s):  
Taiga Kumagai
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Asymptotic Analysis ◽  
Open Subset ◽  
Dimensional Euclidean Space ◽  
Asymptotic Behavior Of Solutions ◽  
Two Dimensional ◽  
Jacobi Equations ◽  
Drift Terms

AbstractWe investigate the asymptotic behavior of solutions of Hamilton–Jacobi equations with large Hamiltonian drift terms in an open subset of the two-dimensional Euclidean space. The drift is given by

Continuity of solutions of Schrödinger equations

1991 ◽  
Vol 110 (3) ◽  
pp. 581-597
Author(s):  
Mitsuru Nakai
Keyword(s):  
Euclidean Space ◽  
Total Variation ◽  
Open Subset ◽  
Radon Measure ◽  
Dimensional Euclidean Space ◽  
Schrödinger Equations ◽  
Kato Class ◽  
Continuity Of Solutions ◽  
Image Position ◽  
Newtonian Kernel

We denote by N(x, y) the Newtonian kernel on the d-dimensional Euclidean space (where d ≥ 2) so that N(x, y) = log|x–y|-1 for d = 2 and N(x, y) = |x−y|2−d for d ≥ 3. A signed Radon measure μ on an open subset Ω in d is said to be of Kato class iffor every y in Ω. where |μ| is the total variation measure of μ.

Estimates for the constant in two nonlinear Korn inequalities

2020 ◽  
pp. 108128652095770
Author(s):  
Maria Malin ◽  
Cristinel Mardare
Keyword(s):  
Euclidean Space ◽  
Open Subset ◽  
Rigid Motion ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Square Roots ◽  
Korn Inequality ◽  
New Inequalities

A nonlinear Korn inequality estimates the distance between two immersions from an open subset of [Formula: see text] into the Euclidean space [Formula: see text], [Formula: see text], in terms of the distance between specific tensor fields that determine the two immersions up to a rigid motion in [Formula: see text]. We establish new inequalities of this type in two cases: when k = n, in which case the tensor fields are the square roots of the metric tensor fields induced by the two immersions, and when k = 3 and n = 2, in which case the tensor fields are defined in terms of the fundamental forms induced by the immersions. These inequalities have the property that their constants depend only on the open subset over which the immersions are defined and on three scalar parameters defining the regularity of the immersions, instead of constants depending on one of the immersions, considered as fixed, as up to now.

scholarly journals Applications of Nachbin's Theorem concerning Dense Subalgebras of Differentiable Functions

2018 ◽  
Vol 19 (3) ◽  
pp. 465
Author(s):  
Marcia Sayuri Kashimoto
Keyword(s):  
Euclidean Space ◽  
Open Subset ◽  
Differentiable Functions ◽  
Real Functions

In this paper, we give some applications of  Nachbin's Theorem  to approximation and interpolation in the the space of all k times continuously differentiable real functions on  any open subset of the  Euclidean space.

The Poisson Integral of a Singular Measure

1983 ◽  
Vol 35 (4) ◽  
pp. 735-749 ◽  
Author(s):  
Patrick Ahern
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Lebesgue Measure ◽  
Open Subset ◽  
Borel Measure ◽  
The Other ◽  
Poisson Integral ◽  
Singular Measure ◽  
Almost Everywhere ◽  
Image Position

Let σ be a finite positive singular Borel measure defined on Euclidean space RN. For w ∈ RN and y > 0, its Poisson integral is defined by the formulawhere CN is chosen so thatSince σ is singular, almost everywhere with respect to Lebesgue measure on RN. On the other hand, almost everywhere dσ. It follows that for all sufficiently small y,is a non-empty open subset of RN. If σ has compact support then |Ey| → 0 as y → 0, where |Ey| denotes the Lebesgue measure of Ey. In this paper we give a lower bound on the rate at which |Ey| may go to zero. The lower bound depends on the smoothness of the measure; the smoother the measure, the more slowly |Ey| may approach 0.

scholarly journals Voronoi Diagrams of Moving Points

1998 ◽  
Vol 08 (03) ◽  
pp. 365-379 ◽  
Author(s):  
Gerhard Albers ◽  
Leonidas J. Guibas ◽  
Joseph S. B. Mitchell ◽  
Thomas Roos
Keyword(s):  
Euclidean Space ◽  
Voronoi Diagram ◽  
Upper Bound ◽  
Voronoi Diagrams ◽  
Maximum Length ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Moving Points ◽  
Over Time

Consider a set of n points in d-dimensional Euclidean space, d ≥ 2, each of which is continuously moving along a given individual trajectory. As the points move, their Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Voronoi diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, at a cost of O( log n) per event, while showing that the number of topological events has an upper bound of O(ndλs(n)), where λs(n) is the (nearly linear) maximum length of a (n,s)-Davenport-Schinzel sequence, and s is a constant depending on the motions of the point sites. In addition, we show that if only k points are moving (while leaving the other n - k points fixed), there is an upper bound of O(knd-1λs(n)+(n-k)dλ s(k)) on the number of topological events.

Infinitesimal rigidity of hyperquadrics in semi-Euclidean space

2016 ◽  
Vol 13 (02) ◽  
pp. 1630001
Author(s):  
An Sook Shin ◽  
Hobum Kim ◽  
Hyelim Han
Keyword(s):  
Euclidean Space ◽  
Open Subset ◽  
Infinitesimal Rigidity

In this paper, we show that hyperquadrics are infinitesimally rigid in a semi-Euclidean space. We also show that hypersurfaces of hyperquadrics cut by hyperplanes not passing through the origin are infinitesimally rigid in the hyperquadrics, whereas those cut by hyperplanes through the origin are not infinitesimally rigid in hyperquadrics. Furthermore, we prove that any hypersurface in a semi-Euclidean space containing some open subset of a hyperplane is not infinitesimally rigid.

Integral means of subharmonic functions

1971 ◽  
Vol 69 (1) ◽  
pp. 151-152 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Euclidean Space ◽  
Open Subset ◽  
Integral Means ◽  
Subharmonic Functions ◽  
Mean Values ◽  
The Mean ◽  
Image Position

If u is subharmonic in some open subset of Euclidean space, RN (N ≥ 1), which contains the ball |x| ≤ ρ and if S(r) and B(r) denote the mean values of u taken over the sphere |x| = r and the ball |x| ≤ r respectively (0 ≤ r ≤ ρ) then, ((2), p. 196),

Continuity of homotopies

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Open Subset ◽  
Small Neighborhood ◽  
Unit Vector ◽  
Zariski Topology ◽  
Simple Point ◽  
Smooth Variety ◽  
Additional Material ◽  
Generic Type ◽  
Zariski Open Subset

This chapter includes some additional material on homotopies. In particular, for a smooth variety V, there exists an “inflation” homotopy, taking a simple point to the generic type of a small neighborhood of that point. This homotopy has an image that is properly a subset of unit vector V, and cannot be understood directly in terms of definable subsets of V. The image of this homotopy retraction has the merit of being contained in unit vector U for any dense Zariski open subset U of V. The chapter also proves the continuity of functions and homotopies using continuity criteria and constructs inflation homotopies before proving GAGA type results for connectedness. Additional results regarding the Zariski topology are given.

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