Relevant bibliographies by topics / Rational tetrahedra

Academic literature on the topic 'Rational tetrahedra'

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Published: 4 June 2021
Last updated: 31 January 2023

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Journal articles on the topic "Rational tetrahedra"

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Chisholm, C., and J. A. MacDougall. "Rational and Heron tetrahedra." Journal of Number Theory 121, no. 1 (November 2006): 153–85. http://dx.doi.org/10.1016/j.jnt.2006.02.009.

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Mazėtis, Edmundas, and Grigorijus Melničenko. "Rational cuboids and Heron triangles II." Lietuvos matematikos rinkinys 60 (December 5, 2019): 34–38. http://dx.doi.org/10.15388/lmr.b.2019.15233.

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We study the connection of Heronian triangles with the problem of the existence of rational cuboids. It is proved that the existence of a rational cuboid is equivalent to the existence of a rectangular tetrahedron, which all sides are rational and the base is a Heronian triangle. Examples of rectangular tetrahedra are given, in which all sides are integer numbers, but the area of the base is irrational. The example of the rectangular tetrahedron is also given, which has lengths of one side irrational and the other integer, but the area of the base is integer.
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Buchholz, Ralph Heiner. "Perfect pyramids." Bulletin of the Australian Mathematical Society 45, no. 3 (June 1992): 353–68. http://dx.doi.org/10.1017/s0004972700030252.

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This paper discusses rational edged tetrahedra, in 3, 4 and n dimensions, with rational volume. The main results are (i) a proof of the existence of infinitely many tetrahedra with rational edge-lengths, face-areas and volume and (ii) a proof that there exist dimensions for which all regular hypertetrahedra with rational edge-lengths have rational hypervolume.
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Chisholm, C., and J. A. MacDougall. "Rational tetrahedra with edges in arithmetic progression." Journal of Number Theory 111, no. 1 (March 2005): 57–80. http://dx.doi.org/10.1016/j.jnt.2004.07.009.

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Chisholm, C., and J. A. MacDougall. "Rational tetrahedra with edges in geometric progression." Journal of Number Theory 128, no. 2 (February 2008): 251–62. http://dx.doi.org/10.1016/j.jnt.2007.07.002.

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Kolpakov, Alexander, and Sinai Robins. "Spherical tetrahedra with rational volume, and spherical Pythagorean triples." Mathematics of Computation 89, no. 324 (December 17, 2019): 2031–46. http://dx.doi.org/10.1090/mcom/3496.

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CHAIR, NOUREDDINE, and CHUAN-JIE ZHU. "TETRAHEDRA AND POLYNOMIAL EQUATIONS IN TOPOLOGICAL FIELD THEORY." International Journal of Modern Physics A 06, no. 20 (August 20, 1991): 3571–98. http://dx.doi.org/10.1142/s0217751x91001738.

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Some tetrahedra in SUk(2) Chern-Simons-Witten theory are computed. The results can be used to compute an arbitrary tetrahedron inductively by fusing with the fundamental representation. The results obtained are in agreement with those of quantum groups. By associating a (finite) topological field theory (FTFT) to every rational conformal field theory (RCFT), we show that the pentagon and hexagon equations in RCFT follow directly from some skein relations in FTFT. By generalizing the operation of surgery on links in FTFT, we also derive an explicit expression for the modular transformation matrix S(k) of the one-point conformal blocks on a torus in RCFT and the equations satisfied by S(k), in agreement with those required in RCFT. The implication of our results on the general program of classifying RCFT is also discussed.
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XU, GUOLIANG, CHANDRAJIT L. BAJAJ, and SUSAN EVANS. "C1 MODELING WITH HYBRID MULTIPLE-SIDED A-PATCHES." International Journal of Foundations of Computer Science 13, no. 02 (April 2002): 261–84. http://dx.doi.org/10.1142/s0129054102001084.

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We propose a new scheme for modeling a smooth interpolatory surface, from a surface discretization consisting of triangles, quadrilaterals and pentagons, by algebraic surface patches which are subsets of real zero contours of trivariate rational functions defined on a collection of tetrahedra and pyramids. The rational form of the modeling function provides enough degrees of freedom so that the number of the surface patches is significantly reduced, and the surface has quadratic recover property.
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Leininger, S., J. Fan, M. Schmitz, and P. J. Stang. "Archimedean solids: Transition metal mediated rational self-assembly of supramolecular-truncated tetrahedra." Proceedings of the National Academy of Sciences 97, no. 4 (February 4, 2000): 1380–84. http://dx.doi.org/10.1073/pnas.030264697.

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DURHUUS, BERGFINNUR, HANS PLESNER JAKOBSEN, and RYSZARD NEST. "TOPOLOGICAL QUANTUM FIELD THEORIES FROM GENERALIZED 6J-SYMBOLS." Reviews in Mathematical Physics 05, no. 01 (March 1993): 1–67. http://dx.doi.org/10.1142/s0129055x93000024.

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Given an associative algebra with a distinguished finite set of representations that is closed under a (deformed) tensor product, and satisfies some technical assumptions, we define generalized 6j-symbols, and show that they can be associated, in a natural way, with certain labeled tetrahedra. Given a 3-dimensional compact oriented manifold M with boundary ∂M = Σ we choose an arbitrary triangulation [Formula: see text] of M and exploit the above correspondence between 6j-symbols and labeled tetrahedra to construct a vectorspace UΣ and a vector Z(M) ∈ UΣ, independent of [Formula: see text], and fulfilling the axioms of a topological quantum field theory as formulated by Atiyah [11]. Examples covered by our approach are quantum groups corresponding to the classical simple Lie algebras as well as, expectedly, chiral algebras of 2-dimensional rational conformal field theories.
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Dissertations / Theses on the topic "Rational tetrahedra"

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Chisholm, Catherine Rachel. "Rational and Heron Tetrahedra." Thesis, 2004. http://hdl.handle.net/1959.13/24856.

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Rational tetrahedra are tetrahedra with rational edges. Heron tetrahedra are tetrahedra with integer edges, integer faces areas and integer volume --- the three-dimensional analogue of Heron triangles. Of course, if a rational tetrahedron has rational face areas and volume then it is easy to scale it up to get a Heron tetrahedron. So we also use `Heron tetrahedra' when we mean tetrahedra with rational edges, areas and volume. The work in this thesis is motivated by Buchholz's paper {\it Perfect Pyramids} [4]. Buchholz examined certain configurations of rational tetrahedra, looking first for tetrahedra with rational volume, and then for Heron tetrahedra. Buchholz left some of the cases he examined unsolved and Chapter 2 is largely devoted to the resolution of these. In Chapters 3 and 4 we expand upon some of Buchholz's results to find infinite families of Heron tetrahedra corresponding to rational points on certain elliptic curves. In Chapters 5 and 6 we blend the ideas of Buchholz in [4] and of Buchholz and MacDougall in [7], and consider rational tetrahedra with edges in arithmetic (AP) or geometric (GP) progression. It turns out that there are no Heron AP or GP tetrahedra, but AP tetrahedra can have rational volume. They can also have one rational face area, although only one AP tetrahedron has been found with a rational face area and rational volume. For GP tetrahedra there are still unsolved cases, but we show that if GP tetrahedra with rational volume exist, then there are only finitely many. The faces of a rational GP tetrahedron are never rational. Much of the work in these two chapters also appeared in the author's Honours thesis, but has been refined and extended here, and is included to give a more complete picture of the work on Heron tetrahedra which has been done to date. In the final chapter we use a different approach and concentrate on the face areas first, instead of the volume. To make it easier (hopefully) to find tetrahedra with all faces having rational area, we place restrictions on the types of faces and number of different faces the tetrahedra have.
Masters Thesis
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2

Chisholm, Catherine Rachel. "Rational and Heron Tetrahedra." 2004. http://hdl.handle.net/1959.13/24856.

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Rational tetrahedra are tetrahedra with rational edges. Heron tetrahedra are tetrahedra with integer edges, integer faces areas and integer volume --- the three-dimensional analogue of Heron triangles. Of course, if a rational tetrahedron has rational face areas and volume then it is easy to scale it up to get a Heron tetrahedron. So we also use `Heron tetrahedra' when we mean tetrahedra with rational edges, areas and volume. The work in this thesis is motivated by Buchholz's paper {\it Perfect Pyramids} [4]. Buchholz examined certain configurations of rational tetrahedra, looking first for tetrahedra with rational volume, and then for Heron tetrahedra. Buchholz left some of the cases he examined unsolved and Chapter 2 is largely devoted to the resolution of these. In Chapters 3 and 4 we expand upon some of Buchholz's results to find infinite families of Heron tetrahedra corresponding to rational points on certain elliptic curves. In Chapters 5 and 6 we blend the ideas of Buchholz in [4] and of Buchholz and MacDougall in [7], and consider rational tetrahedra with edges in arithmetic (AP) or geometric (GP) progression. It turns out that there are no Heron AP or GP tetrahedra, but AP tetrahedra can have rational volume. They can also have one rational face area, although only one AP tetrahedron has been found with a rational face area and rational volume. For GP tetrahedra there are still unsolved cases, but we show that if GP tetrahedra with rational volume exist, then there are only finitely many. The faces of a rational GP tetrahedron are never rational. Much of the work in these two chapters also appeared in the author's Honours thesis, but has been refined and extended here, and is included to give a more complete picture of the work on Heron tetrahedra which has been done to date. In the final chapter we use a different approach and concentrate on the face areas first, instead of the volume. To make it easier (hopefully) to find tetrahedra with all faces having rational area, we place restrictions on the types of faces and number of different faces the tetrahedra have.
Masters Thesis
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Conference papers on the topic "Rational tetrahedra"

1

Colmont, Marie, Olivier Mentré, and Marielle Huvé. "Rational Design of new oxo centred tetrahedra Bi-based compounds by Electron Microscopy." In 2008 MRS Fall Meetin. Materials Research Society, 2008. http://dx.doi.org/10.1557/proc-1148-pp15-04.

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Li, Yinsheng, Hiroto Itoh, Kunio Hasegawa, Kazuya Osakabe, and Hiroshi Okada. "Development of Stress Intensity Factors for Deep Surface Cracks in Plates and Cylinders." In ASME 2012 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/pvp2012-78109.

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A number of deep surface cracks have been detected in components of nuclear power plants in recent years. The depths of these cracks are even greater than the half of crack lengths. When a crack is detected during in-service inspections, methods provided in the ASME Boiler and Pressure Vessel Code Section XI or JSME Rules on Fitness-for-Service for Nuclear Power Plants can be used to assess the structural integrity of cracked components. The solution of the stress intensity factor is very important in the assessment of structural integrity. However, in the current codes, the solutions of the stress intensity factor are provided for semi-elliptical surface cracks with a limitation of a/l ≤ 0.5, where a is the crack depth, and l is the crack length. In this study, in order to assess the structural integrity in a more rational manner, the solutions of the stress intensity factor were calculated using finite element analysis with quadratic hexahedron elements for deep semi-elliptical surface cracks in plates, and for axial and circumferential semi-elliptical surface cracks in cylinders. The crack dimensions were focused on the range of a/l = 0.5 to 4.0. Solutions were provided at both the deepest and the surface points of the cracks. Furthermore, some of solutions were compared with the available existing studies and with solutions obtained using finite element analysis with quadratic tetrahedral elements and the virtual crack closure-integral method. As the conclusion, it is concluded that the solutions proposed in this paper are applicable in engineering applications.
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Journal articles
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