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On smallest triangles
[article]

2002
*
arXiv
*
pre-print

It is proved in addition that, if mu is the uniform probability measure

arXiv:math/0208044v1
fatcat:vnvxp4fhwrbufmruj7vh3oeteu
*on*the region S, then c(mu) <= 2/|S|, where |S| denotes the area of S. ... This result, and related conclusions, are proved using standard arguments of Poisson approximation, and may be extended to functionals more general than the area of a*triangle*. ... Some of this work was done at Eurandom, during the attendance by the authors at a meeting*on*discrete probability. David Stirzaker kindly indicated the relevance of the work of Morgan Crofton. ...##
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On smallest triangles

2003
*
Random structures & algorithms (Print)
*

It is proved in addition that, if µ is the uniform probability measure

doi:10.1002/rsa.10092
fatcat:q3bko4snlvarrepgdekztpc7ku
*on*the region S, then κ(µ) ≤ 2/|S|, where |S| denotes the area of S. ... This result, and related conclusions, are proved using standard arguments of Poisson approximation, and may be extended to functionals more general than the area of a*triangle*. ... Some of this work was done at Eurandom, during the attendance by the authors at a meeting*on*discrete probability. David Stirzaker kindly indicated the relevance of the work of Morgan Crofton. ...##
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Largest and Smallest Area Triangles on Imprecise Points
[article]

2018
*
arXiv
*
pre-print

*smallest*

*triangle*is NP-hard; but minimizing the

*smallest*

*triangle*can be done in O(n^2) time. ... Assume we are given a set of parallel line segments in the plane, and we wish to place a point

*on*each line segment such that the resulting point set maximizes or minimizes the area of the largest or

*smallest*... Consequently the

*smallest*largest-area

*triangle*of L equals to the

*smallest*largest-area

*triangle*of L . ...

##
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The smallest triangular cover for triangles of diameter one

2005
*
Journal of Applied Mathematics and Computing
*

In this paper, we find the

doi:10.1007/bf02936039
fatcat:rxo3krjfrnfornf23w5pfjqay4
*smallest*triangular cover of any prescribed shape for the family S of all*triangles*of diameter 1. ... A convex region covers a family of curves if it contains a congruent copy of each curve in the family, and a "worm problem" for that family is to find the convex region of*smallest*area. ... Now, we present our main results*on*the*smallest*triangular cover, denoted as T , of prescribed shape for*triangles*of diameter*one*. Theorem 1. ...##
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On a smallest topological triangle free (n_4) point-line configuration

2020
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The Art of Discrete and Applied Mathematics
*

An (n k ) configuration is a set of n points and n lines such that every point lies

doi:10.26493/2590-9770.1355.f3d
fatcat:efpioe5wszgvfpeso4muebko2q
*on*precisely k of these lines and every line contains precisely k of these points. We distinguish three concepts. ... A topological (n 4 ) configuration for n < 40 must contain a*triangle*, so our*triangle*free example is minimal. A c c e p t e d m a n u s c r i p t 2 Art Discrete Appl. Math. Definition 1.1. ... It is the*smallest*case for which we do not know a result like the previous theorem, and it is small enough to possibly behave exceptionally. ...##
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ON SMALLEST ORDER OF TRIANGLE-FREE GRAPHS WITH PRESCRIBED (3, k)-DEFECTIVE CHROMATIC NUMBER

2018
*
Far East Journal of Mathematical Sciences (FJMS)
*

Let f(m,k) be the

doi:10.17654/ms103040717
fatcat:xpznpo5afrge7kb72pq3h7jvli
*smallest*order of a*triangle*-free graph G such that X k (G) = m. In this paper we study the problem of determining f( m, 1). ... A graph is (m,k)-colourable if its vertices can be coloured with m colours such that the maximum degree of the subgraph induced*on*vertices receiving the same colour is at most k. ... Let f(m,k) be the*smallest*order of a*triangle*-free graph G such that x'k (G) = m. The determination of f(m,O) is still an open problem (see Toft [19] , Problem 29). ...##
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AN OPTIMAL PARALLEL ALGORITHM FOR FINDING THE SMALLEST ENCLOSING TRIANGLE ON A MESH-CONNECTED COMPUTER∗

1993
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Parallel Algorithms and Applications
*

In this paper, we consider the problem of nding the

doi:10.1080/10637199308915435
fatcat:jqgarbcvtjduvpjry6c6gehkue
*smallest**triangle*circumscribing a convex polygon with n edges. ... Since the nontrivial operation*on*MCC requires ( p n), the time complexity is optimal within constant time factor. ... , and nd the*smallest**one*among all the P-anchored*triangles*. ...##
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Fitting triangle into regular octagon

2013
*
Applied Mathematical Sciences
*

In this paper, we determine the

doi:10.12988/ams.2013.35263
fatcat:6t4hxvw65rc5pekmqzoiam3blu
*smallest*regular octagon which contains all*triangles*of perimeter two. Mathematics Subject Classification: 52C15 ... The Moser's worm problem [2] searches the region of*smallest*area which contains a congruent copy of every unit arc*on*the plane. A popular shape of arcs is a*triangle*. ... Results Theorem The*smallest*regular octagon cover for the family of all*triangles*of perimeter two is the regular octagon of diameter*one*(the area is 1 2 ). Proof. ...##
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Regular hexagon cover for isoperimetric triangles

2013
*
Applied Mathematical Sciences
*

In this paper, we search for the

doi:10.12988/ams.2013.13141
fatcat:z5j74kiq5bgu5jykmql3acq43q
*smallest*regular hexagon which covers the family of all*triangles*of perimeter two. ... The Moser's worm problem [2] searches the region of*smallest*area which contains a congruent copy of every unit arc*on*the plane. A popular shape of arcs is a*triangle*. ... In 2009, Zhang and Yuan [7] gave the*smallest*regularized parallelogram cover, whose length of the smaller diagonal is not less than*one*, for the family of all*triangles*of perimeter two. ...##
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Lepp terminal centroid method for quality triangulation

2010
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Computer-Aided Design
*

Longest edge based algorithms [18] [19] [20] 3, 4, 6] were designed to take advantage of the following mathematical properties

doi:10.1016/j.cad.2008.11.004
fatcat:usgcnqozc5hv7bvnxvgpbhvfi4
*on*the quality of*triangles*generated by iterative longest edge bisection ...*smallest*edges over the boundary. ... Furthermore when the equality*on*the bound*on*the largest angle holds for*one*of the*triangles*(say*triangle*ABC), then the other*triangle*is equilateral, the 4 vertices are cocircular and the distance ...##
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Triangles in regular heptagons

2013
*
Applied Mathematical Sciences
*

In this paper, we show that the regular heptagon of diameter

doi:10.12988/ams.2013.13143
fatcat:6g4ceblryfeopns3lsilk3j6ti
*one*is the*smallest*regular heptagon which contains all*triangles*of perimeter two. Mathematics Subject Classification: 52C15 ... In this paper, we show that the regular heptagon of diameter*one*is the*smallest*regular heptagon which contains all*triangles*of perimeter two. ... Results Theorem The*smallest*regular heptagon cover for the family of all*triangles*of perimeter two is the regular heptagon of diameter*one*(the area is approximately equal to 0.71974). Proof. ...##
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A Study on Size-Optimal Longest Edge Refinement Algorithms
[chapter]

2013
*
Proceedings of the 21st International Meshing Roundtable
*

Until now, these algorithms did not provide theoretical guarantees

doi:10.1007/978-3-642-33573-0_8
dblp:conf/imr/BedregalR12
fatcat:tmblm7i22za5hhmkcf6hhvtbxe
*on*the size of the triangulation obtained. ... Q*on*its*smallest*edge CA. ... Refining*triangles*by the*smallest*edge When a non-conforming*triangle*is produced after adding a point*on*its*smallest*edge (as described in Case 3 of Sec. 4.1), the triangulation is made valid after ...##
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90.80 Boxes for triangles of diameter one

2006
*
Mathematical Gazette
*

Since the equilateral

doi:10.1017/s0025557200180465
fatcat:cz4gqz4u7jez5ncvhuagon2z5u
*triangle*with side length 1 has altitude ^V3, it is easy to see that the breadth of a*triangle*of diameter*one*is at most \\J?>. ... A 'worm problem' for that family is to find the box of*smallest*area. Suppose X is a bounded convex set of points in the plane. ... Thus R covers every*triangle*of diameter 1, giving Lemma 1. Now we determine the*smallest*box for*triangles*of diameter*one*. ...##
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Regular pentagon cover for triangles of perimeter two

2013
*
Applied Mathematical Sciences
*

Wetzel determined the

doi:10.12988/ams.2013.13142
fatcat:cmtheuqgwncrvcarcinzafyf7y
*smallest*rectangle which covers the family of all*triangles*with perimeter two. ... In this paper, we search for the*smallest*right trapezoid which covers the family of all*triangles*of perimeter two. ... However, we can scale between*triangles*with perimeter*one*and*triangles*with perimeter two. ...##
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Longest-edge algorithms for size-optimal refinement of triangulations

2014
*
Computer-Aided Design
*

the

doi:10.1016/j.cad.2013.08.040
fatcat:ct4wzt2ymbeffnopqrxf6momx4
*smallest*angle). ... We prove that the iterative application of the algorithm gradually reduces the average extent of the propagation per target*triangle*, tending to affect only two*triangles*. ... In exchange, for non-conforming midpoints situated over the*smallest*edge of t * , some additional points in the interior and/or over the edges of t * are introduced, depending*on*the geometric*triangle*...
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