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Total diameter and area of closed submanifolds
[article]

2015
*
arXiv
*
pre-print

The total diameter

arXiv:1312.1384v2
fatcat:amu7kettgfazrdisyeljpft6tm
*of*a closed*planar*curve C⊂ R^2 is the*integral**of*its antipodal*chord**lengths*. We show that this quantity is bounded below by twice the area*of*C. ... Furthermore, when C is*convex*or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the*convex*case only when C is a circle. ... We also thank Igor Belegradek*for*locating the reference [21] . Finally, thanks to the anonymous referee*for*suggesting improvements to an earlier draft*of*this work. ...##
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The chord length distribution function of a non-convex hexagon

2018
*
Communications in Applied and Industrial Mathematics
*

In this paper we obtain the

doi:10.1515/caim-2018-0002
fatcat:if7acxvzfra3xofjaoz3sbk5rm
*chord**length*distribution function*of*a non-*convex*equilateral hexagon and then derive the associated density function. ... Finally, we calculate the expected value*of*the*chord**length*. ... The*chord**length*distribution functions*for*a number*of**planar**convex**figures*are already known. ...##
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Page 1589 of Mathematical Reviews Vol. , Issue 86d
[page]

1986
*
Mathematical Reviews
*

Voss, Klaus (DDR-HUMB) 86d:52001

*Integrals**of**chord**length**powers**for**planar**convex**figures*. (German and Russian summaries) Elektron. Informationsverarb. Kybernet. 20 (1984), no. 7-9, 488-494. ... Let s(g) be the*length**of*the*chord*Fg, where g is a line and F a*convex*body, both in the Euclidean plane. ...##
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Circles Minimize most Knot Energies
[article]

2001
*
arXiv
*
pre-print

The proof is based on a theorem

arXiv:math/0105138v1
fatcat:wzaapjcbfvdqpd4vkpcyoxoofa
*of*G. Luko on average*chord**lengths**of*closed curves. ... Most*of*O'Hara's knot energies belong to this class. This proves two conjectures*of*O'Hara and*of*Freedman, He, and Wang. We also find energies not minimized by a round circle. ... We thank Kostya Oskolkov*for*pointing out that the complex form*of*Fourier series would simplify the first proof*of*Theorem 2.2. ...##
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Page 1492 of Mathematical Reviews Vol. , Issue 84d
[page]

1984
*
Mathematical Reviews
*

The

*chord**powers*S,, (i.e., perimeter*length*times nth moment*of*an invariant random*chord*) are evaluated*for*n=5, 6, 7 and 8*for*certain symmetric*planar**convex**figures*via the principal moments*of*inertia ... [Voss, Klaus]*Powers**of**chords**for**convex*sets. (German summary) Biometrical J. 2A (1982), no. 5, 513-516. ...##
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EARLY HISTORY OF GEOMETRIC PROBABILITY AND STEREOLOGY

2012
*
Image Analysis and Stereology
*

The paper provides an account

doi:10.5566/ias.v31.p1-16
fatcat:5pxthnj7xfbdpbaswuzmmjpnti
*of*the history*of*geometric probability and stereology from the time*of*Newton to the early 20th century. ... On the other hand, practical rules*of*the estimation*of*area or volume fraction and other characteristics, easily deducible from geometric probability theory, were proposed without the knowledge*of*this ... ACKNOWLEDGEMENTS The authors acknowledge the financial support*for*this work provided by GAAV, grant IAA801240901, and by the grant MSM 0021620839. ...##
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Characterization of balls by generalized Riesz energy
[article]

2017
*
arXiv
*
pre-print

is defined as the double

arXiv:1707.02405v1
fatcat:jlwrwtq63bhfreajr5ws3n7qzu
*integral**of*some*power**of*the distance between pairs*of*points. ... We show that balls, circles and 2-spheres can be identified by generalized Riesz energy among compact submanifolds*of*the Euclidean space that are either closed or with codimension 0, where the Riesz energy ... In fact,*for**convex*bodies, Mallows and Clark [MC] gave a pair*of*non-congruent*convex**planar*polygons with the same*chord**length*distribution as illustrated in*Figure*1 Waksman [W] pointed out that ...##
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Gauss equation and injectivity radii for subspaces in spaces of curvature bounded above
[article]

2005
*
arXiv
*
pre-print

A Gauss equation is proved

arXiv:math/0511570v1
fatcat:h4xf2usrarhcfdn3okethfm5bm
*for*subspaces*of*Alexandrov spaces*of*curvature bounded above by K. ... That is, a subspace*of*extrinsic curvature less than or equal to A, defined by a cubic inequality on the difference*of*arc and*chord*, has intrinsic curvature less than or equal to K+A^2. ... Acknowledgments Our interest in a Gauss Equation in Alexandrov spaces*of*curvature bounded above stems from discussions with David Berg and Igor Nikolaev in the mid-90's,*for*which we thank them. ...##
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The elastica problem under area constraint
[article]

2015
*
arXiv
*
pre-print

The proof is

arXiv:1411.6100v2
fatcat:2ga6qvbrw5aq3js62bmjg5wtfa
*of*a geometric nature and deforms parts*of*γ in a finite number*of*steps to construct some related*convex*sets with smaller energy. ... We show that the elastic energy E(γ)*of*a closed curve γ has a minimizer among all plane simple regular closed curves*of*given enclosed area A(γ), and that the minimum is attained*for*a circle. ... We thank A.Henrot*for*helpful discussions during a visit to Napoli in 2013. ...##
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Lower and upper bounds for chord power integrals of ellipsoids

2014
*
Applied Mathematical Sciences
*

First we discuss different representations

doi:10.12988/ams.2014.411913
fatcat:lbjf4ah2mvahzahswh46g55zd4
*of**chord**power**integrals*I p (K)*of*any order p ≥ 0*for**convex*bodies K ⊂ R d with inner points. ... A further inequality conjectured in Davy (1984) is proved*for*ellipsoids. Some remarks on*chord**power**integrals**of*superellipsoids and simplices round off the topic. ...*Chord**Power**Integrals*-Definition and Basics Let K be a*convex*body in R d with interior points and S d−1 = ∂B d the boundary*of*the Euclidean unit ball B d = {x ∈ R d : x ≤ 1}. ...##
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On the geometric dilation of closed curves, graphs, and point sets

2007
*
Computational geometry
*

The proof relies on halving pairs, pairs

doi:10.1016/j.comgeo.2005.07.004
fatcat:u3hewviktbf7nlsmna435jq4qu
*of*points dividing a given closed curve C in two parts*of*equal*length*, and their minimum and maximum distances h and H. ... The detour between two points u and v (on edges or vertices)*of*an embedded*planar*graph whose edges are curves is the ratio between the shortest path in in the graph between u and v and their Euclidean ... Acknowledgement We would like to thank John Sullivan and Salvador Segura-Gomis*for*helpful discussions and the anonymous referees*for*their valuable comments. ...##
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Evaluation of integral forces and pressure fields from planar velocimetry data for incompressible and compressible flows

2007
*
Experiments in Fluids
*

The approach to determine pressure fields and

doi:10.1007/s00348-007-0261-y
fatcat:w5tkgbi74ncnjdxxytydhly5sy
*integral*loads from*planar*velocimetry data is discussed, in relation to the implementation*for*incompressible and compressible flows around twodimensional ... The second topic considers the extension*of*the method to steady compressible flow, with the supersonic flow around a bi-*convex*airfoil as experimental test case. ... Tests were carried out on a full span bi-*convex*airfoil with a*chord**of*100 mm and a thickness*of*12 mm. ...##
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A Survey of Shape Feature Extraction Techniques
[chapter]

2008
*
Pattern Recognition Techniques, Technology and Applications
*

*Chord*distribution The basic idea

*of*

*chord*distribution is to calculate the

*lengths*

*of*all

*chords*in the shape (all pair-wise distances betwee boundary points) and to build a histogram

*of*their

*lengths*...

*Chord*

*length*function The

*chord*

*length*function is derived from shape boundary without using any reference point. ... This chapter is distributed under the terms

*of*the Creative Commons Attribution-NonCommercial-ShareAlike-3.0 License, which permits use, distribution and reproduction

*for*non-commercial purposes, provided ...

##
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Conceptual Design and Performance Optimization of a Tip Device for a Regional Turboprop Aircraft

2019
*
Aerospace (Basel)
*

The installation

doi:10.3390/aerospace6100107
fatcat:k5pm5oe6iva7zms3w5gjd5hgwq
*of*wing tip devices has not been a popular choice*for*regional turboprop aircraft, and the novelty*of*the current study is to investigate the feasibility*of*retrofitting the British Aerospace ... Since successful winglet retrofit programs*for*typical short to medium-range narrow-body aircraft report even more than 3%*of*block fuel improvements, undertaking the project*of*installing an optimal winglet ... Acknowledgments: The authors would like to thank Luuk van der Schaft, Odeh Dababneh, and Vishagen Ramasamy*for*contributing with ideas and advice. ...##
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Introducing symplectic billiards
[article]

2017
*
arXiv
*
pre-print

As opposed to usual/Birkhoff billiards, where

arXiv:1708.07395v1
fatcat:eu2q5t3znjdmpgakbq5qpzr3o4
*length*is the generating function,*for*symplectic billiards symplectic area is the generating function. ... We explore basic properties and exhibit several similarities, but also differences*of*symplectic billiards to Birkhoff billiards. ... It remains to use the Lusternik-Schnirelman lower bound*for*the number*of*critical points given by the category*of*the lens space L. ...
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