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COUNTING HERON TRIANGLES WITH CONSTRAINTS [chapter]

Pantelimon Stănică, Santanu Sarkar, Sourav Sen Gupta, Subhamoy Maitra, Nirupam Kar
Integers  
We also provide a count for the number of Heron triangles with a fixed height (there exists only one such when the height is prime).  ...  In this paper, we give some estimates for the number of Heron triangles with two of their sides fixed.  ...  We also count Heron triangles with a fixed height and provide an estimate of the number of Heron triangles with a fixed prime height.  ... 
doi:10.1515/9783110298161.24 fatcat:lcvsxika4vgytiszpc3ge4ob44

Classifying Diophantine parallelepipeds [article]

Randall L. Rathbun
2018 arXiv   pre-print
It is proposed to name these classes: acute (triclinic), obtuse (triclinic), 1-ortho (biclinic), 2-ortho (monoclinic), and rectangular, according to the count of rights angles which may exist.  ...  A computer search of 1,981,336,681 tetrahedrons with 6 rational face diagonals uncovers interesting examples of pipeds, including the perfect parallelepiped of Sawyer-Reiter (and 5 others), and the rectangular  ...  In fact triangles with this property have been given a special name: 'Heron triangles'.  ... 
arXiv:1812.01107v1 fatcat:jilw2jglfjawhaqiqmbadq6rk4

Causal dynamical triangulations in the spincube model of quantum gravity

Marko Vojinović
2016 Physical Review D  
By relating the edge-lengths to the integer areas of triangles, the simplicity constraint imposes very strong restrictions between them, ultimately leading to a requirement that all 4-simplices in the  ...  We study the implications of the simplicity constraint in the spincube model of quantum gravity.  ...  Here A H (l) is the Heron formula for the area of the trianglewith edge-lengths l, A H (l 1 , l 2 , l 3 ) = s(s − l 1 )(s − l 2 )(s − l 3 ) , (6) where s ≡ (l 1 + l 2 + l 3 )/2, while γ is a dimensionless  ... 
doi:10.1103/physrevd.94.024058 fatcat:rzk33ytzgzdxfdkdhm65gpplhq

How does the sun shine?

J.N. Bahcall, M. Fukugita, P.I. Krastev
1996 Physics Letters B  
Assuming that MSW neutrino oscillations occur and ignoring all solar physics except for the constraint that nuclear fusion produces the solar luminosity, we show that new solar neutrino experiments are  ...  solutions [16, 17] are indicated by dark triangles.  ...  These solutions are inconsistent with the standard solar model, but they are consistent with the luminosity constraint, i. e., the fusion energy release to the star associated with the neutrino production  ... 
doi:10.1016/0370-2693(96)00187-6 fatcat:ppoxyja2bnhh7laf5hcnt6km5q

The Smallest Shape Spaces. III. Triangles in 2- and 3-d [article]

Edward Anderson
2018 arXiv   pre-print
We finally make insightful comparison of triangles in 2-d with a) Part II's 4 points on the line. b) Triangles in 3-d, which are particularly significant as the smallest model exhibiting stratification  ...  Medians, and regular triangles bounding between tall and flat triangles, also play prominent roles.  ...  Reza Tavakol, Malcolm MacCallum, Enrique Alvarez and Jeremy Butterfield for support with my career.  ... 
arXiv:1711.10115v2 fatcat:37wal4e2evbfracarc3ij7rpae

How does the Sun shine?

John Bahcall
1989 Nature  
Assuming that MSW neutrino oscillations occur and ignoring all solar physics except for the constraint that nuclear fusion produces the solar luminosity, we show that new solar neutrino experiments are  ...  solutions [16, 17] are indicated by dark triangles.  ...  These solutions are inconsistent with the standard solar model, but they are consistent with the luminosity constraint, i. e., the fusion energy release to the star associated with the neutrino production  ... 
doi:10.1038/340265a0 fatcat:5btv6gmibra3dka42eiq3b7tuy

Bifurcations of relative equilibria in the 4- and 5-body problem

Kenneth R. Meyer, Dieter S. Schmidt
1988 Ergodic Theory and Dynamical Systems  
We show that families of isosceles triangle relative equilibria bifurcate from the equilateral triangle family as m passes through the degenerate value.  ...  The equilateral triangle family of relative equilibria of the 4-body problem consists of three particles of mass 1 at the vertices of an equilateral triangle and the fourth particle of arbitrary mass m  ...  This is the three-dimensional generalization of Heron of Alexandria's formula for the area of a triangle given its three sides.  ... 
doi:10.1017/s0143385700009433 fatcat:z5poxho6xzae5dza766lp5ne4a

'Meaning in a Bororo jaguar skin'

Gordon Brotherston
2001 Revista do Museu de Arqueologia e Etnologia  
In both stories, Adugo appears with other vertebrates in sequences that involve counting and numerical logic.  ...  The fives to the left are counted cumulatively, i.e. through all the numbers up to and including five, to give twice 15 (1+2+3+4+5), in what may be called 'sigma' count, in line with the Greek letter which  ... 
doi:10.11606/issn.2448-1750.revmae.2001.109421 fatcat:jr6kferivbdj3o5or5dyraqmwi

The Area of a Polygon with an Inscribed Circle [article]

Marshall W. Buck, Robert L. Siddon
2012 arXiv   pre-print
Heron's formula states that the area K of a triangle with sides a, b, and c is given by K=√(s(s-a) (s-b) (s-c)) where s is the semiperimeter (a+b+c)/2.  ...  In this paper we derive formulas giving the areas of any n-gon, with odd n, in terms of the ordered list of side lengths, if the n-gon is circumscribed about a circle (instead of being inscribed in a circle  ...  He starts with any right triangle, makes a mirror image copy of the triangle about the hypotenuse, joins the two triangles to form a quadrilateral with an inscribed circle.  ... 
arXiv:1203.3438v1 fatcat:2tysi2g4uzck3pbsitdkikyjhq

Streaming Temporal Graphs: Subgraph Matching

Eric L. Goodman, Dirk Grunwald
2019 2019 IEEE International Conference on Big Data (Big Data)  
Apache Flink has an advantage when triangles are rare, with max aggregate throughput for Flink at 32 nodes greater than the max achievable rate of SAM.  ...  We find that SAM is able to scale to 128 nodes or 2560 cores, while Apache Flink has max throughput with 32 nodes and degrades thereafter.  ...  In this example, we have defined a triangle. The second part of the query defines the temporal constraints.  ... 
doi:10.1109/bigdata47090.2019.9006429 dblp:conf/bigdataconf/GoodmanG19 fatcat:bij7zwsodbaqdhb26gla6ceppq

Self-duality of the 6j-symbol and Fisher zeros for the Tetrahedron [article]

Valentin Bonzom, Etera R. Livine
2019 arXiv   pre-print
formula for the asymptotics for the 6j-symbol at large spins in terms of the tetrahedron geometry, we obtain a geometric formula for the zeros of the (inhomogeneous) Ising partition function in terms of triangle  ...  3 − l 2 5 − l 2 6 ) + (l 2 2 − l 2 3 )(l 2 6 − l 2 5 ) 16 A 123 A 156 . (88) Here A ijk is the area of the triangle with edges (ijk) given in terms of the edge lengths by the Heron formula, as in (37  ...  This is exactly the equation (70) This constraint reflects that we are working with seven cycle variables L C instead of the six edge variables Y e , so the L C have to satisfy one constraint.  ... 
arXiv:1905.00348v1 fatcat:iyawo46u35hdbgkitmadi54mmm

Constructing 7-clusters [article]

Sascha Kurz, Landon Curt Noll, Randall Rathbun, Chuck Simmons
2013 arXiv   pre-print
On the way, we exhaustively determined all Heronian triangles with largest edge length up to 6· 10^6.  ...  We determine the smallest existent 7-cluster with respect to its diameter.  ...  Let count(x) denote the number of primitive Heronian triangles with diameter between (x − 1) · 10 000 + 1 and x · 10 000.  ... 
arXiv:1312.2318v1 fatcat:lhwyvzgtfne4rcxwcnc2ic5gxa

Translanguaging in Translation: Evidence from Japanese Mimetics

Eriko Sato
2017 International Journal of Linguistics & Communication  
To explore the possibility of translanguaging in translation, a corpus of seven English translations of a Japanese novel, GingaTetsudō no Yoru, was quantitatively and qualitatively studied with a focus  ...  See Abdulla (2001) and Dahlgren (2005) for the significance of keeping reiterative device in translation.22 Herons are medium to large sized birds with long legs and necks.  ...  Herons ...an ..falling downward like snowflakes. many heron, squawking "gyagya" gently landed, falling like snow.  ... 
doi:10.15640/ijlc.v5n1a2 fatcat:d2sagaei2naarl6ik5xs5w7mly

Identifying important habitats for waterbird conservation at a Greek Regional Nature Park

Vasilios Liordos, Vasileios J. Kontsiotis
2020 Avian Research  
help assess the importance of habitats and improve management strategies to benefit waterbirds, especially in areas with similar Mediterranean-type habitats and climate.  ...  Mud with rocky substrate was used by nine species and selected by eight species. Medium (2–4 m) and deep (4–6 m) open water habitats were used by seven species and selected by four species.  ...  The inlet holds nationally important wintering populations of four waterbird species (comparisons with country-wide mid-winter counts in Liordos 2011): the Common Shelduck (Tadorna tadorna; 55 individuals  ... 
doi:10.1186/s40657-020-00224-7 fatcat:fdxseros75dhpbyysrupghvad4

Mathematics teaching development as a human practice: identifying and drawing the threads

Barbara Jaworski
2012 ZDM: Mathematics Education  
They could count it, talk about it, work with it and manipulate it.  ...  They counted sides in the perimeter and counted again. It seemed to add up to 13.  ... 
doi:10.1007/s11858-012-0437-7 fatcat:yzhqhuxqevgdtju3dabqfnprsy
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