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Journal articles on the topic 'Lattice polytope'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

HEGEDÜS, GÁBOR, and ALEXANDER M. KASPRZYK. "THE BOUNDARY VOLUME OF A LATTICE POLYTOPE." Bulletin of the Australian Mathematical Society 85, no. 1 (2011): 84–104. http://dx.doi.org/10.1017/s0004972711002577.

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AbstractFor a d-dimensional convex lattice polytope P, a formula for the boundary volume vol(∂P) is derived in terms of the number of boundary lattice points on the first ⌊d/2⌋ dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulas for the f-vector of a smooth polytope in dimensions three, four, and five. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.

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2

HILLE, LUTZ, та HARALD SKARKE. "REFLEXIVE POLYTOPES IN DIMENSION 2 AND CERTAIN RELATIONS IN SL2(ℤ)". Journal of Algebra and Its Applications 01, № 02 (2002): 159–73. http://dx.doi.org/10.1142/s0219498802000124.

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It is well known that there are 16 two-dimensional reflexive polytopes up to lattice isomorphism. One can check directly from the list that the number of lattice points on the boundary of such a polytope plus the number of lattice points on the boundary of the dual polytope is always 12. It turns out that two-dimensional reflexive polytopes correspond to certain relations of two generators A and B of SL 2(ℤ) of length 12. We generalize this correspondence to reflexive configurations with winding number w and relations of length 12w.

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3

Ryshkov, S. S., and R. M. Erdahl. "The Empty Sphere Part II." Canadian Journal of Mathematics 40, no. 5 (1988): 1058–73. http://dx.doi.org/10.4153/cjm-1988-043-5.

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Blow up a sphere in one of the interstices of a lattice until it is held rigidly. There will be no lattice points in the interior and sufficiently many on the boundary so that their convex hull is a solid figure. Such a sphere was called an empty sphere by B. N. Delone in 1924 when he introduced his method for lattice coverings [3, 4]. The circumscribed polytope is called an L-polytope. Our interest in such matters stems from the following result [6, Theorems 2.1 and 2.3]: With a list of the L-polytopes for lattices of dimension ≦n one can give a geometrical description of the possible sets of

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4

Koca, Mehmet, Nazife Ozdes Koca, Abeer Al-Siyabi, and Ramazan Koc. "Explicit construction of the Voronoi and Delaunay cells ofW(An) andW(Dn) lattices and their facets." Acta Crystallographica Section A Foundations and Advances 74, no. 5 (2018): 499–511. http://dx.doi.org/10.1107/s2053273318007842.

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Voronoi and Delaunay (Delone) cells of the root and weight lattices of the Coxeter–Weyl groupsW(An) andW(Dn) are constructed. The face-centred cubic (f.c.c.) and body-centred cubic (b.c.c.) lattices are obtained in this context. Basic definitions are introduced such as parallelotope, fundamental simplex, contact polytope, root polytope, Voronoi cell, Delone cell,n-simplex,n-octahedron (cross polytope),n-cube andn-hemicube and their volumes are calculated. The Voronoi cell of the root lattice is constructed as the dual of the root polytope which turns out to be the union of Delone cells. It is

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5

Lalvani, Haresh. "Higher Dimensional Periodic Table Of Regular And Semi-Regular Polytopes." International Journal of Space Structures 11, no. 1-2 (1996): 155–71. http://dx.doi.org/10.1177/026635119601-222.

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This paper presents a higher-dimensional periodic table of regular and semi-regular n-dimensional polytopes. For regular n-dimensional polytopes, designated by their Schlafli symbol {p,q,r,…u,v,w}, the table is an (n-1)-dimensional hypercubic lattice in which each polytope occupies a different vertex of the lattice. The values of p,q,r,…u,v,w also establish the corresponding n-dimensional Cartesian co-ordinates (p,q,r,…u,v,w) of their respective positions in the hypercubic lattice. The table is exhaustive and includes all known regular polytopes in Euclidean, spherical and hyperbolic spaces, i

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Dutour Sikirić, Mathieu. "The Seven Dimensional Perfect Delaunay Polytopes and Delaunay Simplices." Canadian Journal of Mathematics 69, no. 5 (2017): 1143–68. http://dx.doi.org/10.4153/cjm-2016-013-7.

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AbstractFor a lattice L of ℝn, a sphere S(c, r) of center c and radius r is called empty if for any v ∈ L we have. Then the set S(c, r) ∩ L is the vertex set of a Delaunay polytope P = conv(S(c, r) ∩ L). A Delaunay polytope is called perfect if any aõne transformation ø such that ø(P) is a Delaunay polytope is necessarily an isometry of the space composed with an homothety.Perfect Delaunay polytopes are remarkable structures that exist only if n = 1 or n ≥ 6, and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Er

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7

Cho, Mi Ju, Jin Hong Kim, and Hwa Lee. "Criteria for Hattori–Masuda multi-polytopes via Duistermaat–Heckman functions and winding numbers." Advances in Geometry 18, no. 3 (2018): 355–72. http://dx.doi.org/10.1515/advgeom-2017-0023.

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AbstractA multi-fan (respectively multi-polytope), introduced first by Hattori and Masuda, is a purely combinatorial object generalizing an ordinary fan (respectively polytope) in algebraic geometry. It is well known that an ordinary fan or polytope is associated with a toric variety. On the other hand, we can geometrically realize multi-fans in terms of torus manifolds. However, it is unfortunate that two different torus manifolds may correspond to the same multi-fan. The goal of this paper is to give some criteria for a multi-polytope to be an ordinary polytope in terms of the Duistermaat–He

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8

Bayer, Margaret, and Bernd Sturmfels. "Lawrence Polytopes." Canadian Journal of Mathematics 42, no. 1 (1990): 62–79. http://dx.doi.org/10.4153/cjm-1990-004-4.

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In 1980 Jim Lawrence suggested a construction Λ which assigns to a given rank r oriented matroid M on n points a rank n + r oriented matroid Λ(M) on 2n points such that the face lattice of Λ(M) is polytopal if and only if M is realizable. The Λ-construction generalized a technique used by Perles to construct a nonrational polytope [10]. It was used by Lawrence to prove that the class of polytopal lattices is strictly contained in the class of face lattices of oriented matroids (unpublished) and by Billera and Munson to show that the latter class is not closed under polarity. See [4] for a disc

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9

Lagarias, Jeffrey C., and Günter M. Ziegler. "Bounds for Lattice Polytopes Containing a Fixed Number of Interior Points in a Sublattice." Canadian Journal of Mathematics 43, no. 5 (1991): 1022–35. http://dx.doi.org/10.4153/cjm-1991-058-4.

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AbstractA lattice polytope is a polytope in whose vertices are all in . The volume of a lattice polytope P containing exactly k ≥ 1 points in d in its interior is bounded above by . Any lattice polytope in of volume V can after an integral unimodular transformation be contained in a lattice cube having side length at most n˙n ! V. Thus the number of equivalence classes under integer unimodular transformations of lattice poly topes of bounded volume is finite. If S is any simplex of maximum volume inside a closed bounded convex body K in having nonempty interior, then K⊆ ( n + 2)S — (n+ l)s whe

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10

Ardila, Federico, Matthias Beck, and Jodi McWhirter. "The arithmetic of Coxeter permutahedra." Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales 44, no. 173 (2020): 1152–66. http://dx.doi.org/10.18257/raccefyn.1189.

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Ehrhart theory mesures a polytope P discretely by counting the lattice points inside its dilates P, 2P, 3P, ..... We compute the Ehrhart theory of four families of polytopes of great importance in several areas of mathematics: the standard Coxeter permutahedra for the classical Coxeter groups An, Bn, Cn, Dn. A central tool, of independent interest, is a description of the Ehrhart theory of a rational translate of an integer projection of a cube.

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11

Maehara, H. "Embedding a polytope in a lattice." Discrete & Computational Geometry 13, no. 3-4 (1995): 585–92. http://dx.doi.org/10.1007/bf02574065.

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12

Bagheri, Amir, and Rahim Rahmati-Asghar. "On Non-standard Hilbert Functions." Algebra Colloquium 25, no. 01 (2018): 71–80. http://dx.doi.org/10.1142/s1005386718000056.

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Let [Formula: see text] be a non-standard polynomial ring over a field k and let M be a finitely generated graded S-module. In this paper, we investigate the behaviour of Hilbert function of M and its relations with lattice point counting. More precisely, by using combinatorial tools, we prove that there exists a polytope such that the image of Hilbert function in some degree is equal to the number of lattice points of this polytope.

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13

Talis, Alexander, Ayal Everstov та Valentin Kraposhin. "Spiral tetrahedral packing in the β-Mn crystal as symmetry realization of the 8D E 8 lattice". Acta Crystallographica Section A Foundations and Advances 77, № 1 (2021): 7–18. http://dx.doi.org/10.1107/s2053273320012978.

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Experimental values of atomic positions in the β-Mn crystal permit one to distinguish among them a fragment of the helix containing 15 interpenetrating distorted icosahedra, 90 vertices and 225 tetrahedra. This fragment corresponds to the closed helix of 15 icosahedra in the 4D {3, 3, 5} polytope. The primitive cubic lattice of these icosahedral helices envelopes not only all atoms of β-Mn, but also all tetrahedra belonging to the tiling of the β-Mn structure. The 2D projection of all atomic positions in the β-Mn unit cells shows that they are situated (by neglecting small differences) on thre

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14

Dutour Sikirić, Mathieu, Achill Schürmann, and Frank Vallentin. "The Contact Polytope of the Leech Lattice." Discrete & Computational Geometry 44, no. 4 (2010): 904–11. http://dx.doi.org/10.1007/s00454-010-9266-z.

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15

Diaz, Ricardo, and Sinai Robins. "The Ehrhart Polynomial of a Lattice Polytope." Annals of Mathematics 145, no. 3 (1997): 503. http://dx.doi.org/10.2307/2951842.

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16

Haase, Christian, and Ilarion V. Melnikov. "The Reflexive Dimension of a Lattice Polytope." Annals of Combinatorics 10, no. 2 (2006): 211–17. http://dx.doi.org/10.1007/s00026-006-0283-9.

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17

Andrews, Lawrence C., and Herbert J. Bernstein. "The geometry of Niggli reduction:BGAOL–embedding Niggli reduction and analysis of boundaries." Journal of Applied Crystallography 47, no. 1 (2014): 346–59. http://dx.doi.org/10.1107/s1600576713031002.

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Niggli reduction can be viewed as a series of operations in a six-dimensional space derived from the metric tensor. An implicit embedding of the space of Niggli-reduced cells in a higher-dimensional space to facilitate calculation of distances between cells is described. This distance metric is used to create a program,BGAOL, for Bravais lattice determination. Results fromBGAOLare compared with results from other metric based Bravais lattice determination algorithms. This embedding depends on understanding the boundary polytopes of the Niggli-reduced coneNin the six-dimensional spaceG6. This a

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18

Diaz, Ricardo, and Sinai Robins. "Erratum: The Ehrhart Polynomial of a Lattice Polytope." Annals of Mathematics 146, no. 1 (1997): 237. http://dx.doi.org/10.2307/2951835.

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19

Chadder, Nathan, and Antoine Deza. "Computational determination of the largest lattice polytope diameter." Discrete Applied Mathematics 281 (July 2020): 106–10. http://dx.doi.org/10.1016/j.dam.2019.10.026.

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20

Chadder, Nathan, and Antoine Deza. "Computational determination of the largest lattice polytope diameter." Electronic Notes in Discrete Mathematics 62 (November 2017): 105–10. http://dx.doi.org/10.1016/j.endm.2017.10.019.

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21

Koca, Nazife O., Amal J. H. Al Qanobi, and Mehmet Koca. "4D Pyritohedral Symmetry." Sultan Qaboos University Journal for Science [SQUJS] 21, no. 2 (2016): 150. http://dx.doi.org/10.24200/squjs.vol21iss2pp150-161.

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We describe an extension of the pyritohedral symmetry in 3D to 4-dimensional Euclidean space and construct the group elements of the 4D pyritohedral group of order 576 in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W (F4) and W (H4), implying that it is a group relevant to the crystallographic as well as quasicrystallographic structures in 4-dimensions. We derive the vertices of the 24 pseudoicosahedra, 24 tetrahedra and the 96 triangular pyramids forming the facets of the pseudo snub 24-cell. It turns out that the relevant lattice is the

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22

Wang, Ren-hong, and Zhi-qiang Xu. "Discrete truncated powers and lattice points in rational polytope." Journal of Computational and Applied Mathematics 159, no. 1 (2003): 149–59. http://dx.doi.org/10.1016/s0377-0427(03)00561-2.

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23

Kannan, Ravi. "Lattice translates of a polytope and the Frobenius problem." Combinatorica 12, no. 2 (1992): 161–77. http://dx.doi.org/10.1007/bf01204720.

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24

Barvinok, A. I. "Computing the Ehrhart polynomial of a convex lattice polytope." Discrete & Computational Geometry 12, no. 1 (1994): 35–48. http://dx.doi.org/10.1007/bf02574364.

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25

Borovik, Alexandre V., Israel M. Gelfand, Andrew Vince, and Neil White. "The lattice of flats and its underlying flag matroid polytope." Annals of Combinatorics 1, no. 1 (1997): 17–26. http://dx.doi.org/10.1007/bf02558461.

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26

Koca, Mehmet, Nazife Ozdes Koca, and Abeer Al-Siyabi. "SU(5) grand unified theory, its polytopes and 5-fold symmetric aperiodic tiling." International Journal of Geometric Methods in Modern Physics 15, no. 04 (2018): 1850056. http://dx.doi.org/10.1142/s0219887818500561.

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We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell [Formula: see text] and the rectified 5-cell [Formula: see text] derived from the [Formula: see text] Coxeter–Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope [Formula: see text] whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the [Formula: see text] charge conservation. The Dynkin diagram symmetry of the [Formula: see text] diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root l

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27

Liu, Jia-Bao, Mobeen Munir, Qurat-ul-Ain Munir, and Abdul Rauf Nizami. "Some Metrical Properties of Lattice Graphs of Finite Groups." Mathematics 7, no. 5 (2019): 398. http://dx.doi.org/10.3390/math7050398.

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This paper is concerned with the combinatorial facts of the lattice graphs of Z p 1 × p 2 × ⋯ × p m , Z p 1 m 1 × p 2 m 2 , and Z p 1 m 1 × p 2 m 2 × p 3 1 . We show that the lattice graph of Z p 1 × p 2 × ⋯ × p m is realizable as a convex polytope. We also show that the diameter of the lattice graph of Z p 1 m 1 × p 2 m 2 × ⋯ × p r m r is ∑ i = 1 r m i and its girth is 4.

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KHOLODENKO, A. L. "NEW MODELS FOR VENEZIANO AMPLITUDES: COMBINATORIAL, SYMPLECTIC AND SUPERSYMMETRIC ASPECTS." International Journal of Geometric Methods in Modern Physics 02, no. 04 (2005): 563–84. http://dx.doi.org/10.1142/s0219887805000703.

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The bosonic string theory evolved as an attempt to find a physical/quantum mechanical model capable of reproducing Euler's beta function (Veneziano amplitude) and its multidimensional analogue. The multidimensional analogue of beta function was studied mathematically for some time from different angles by mathematicians such as Selberg, Weil and Deligne among many others. The results of their studies apparently were not taken into account in physics literature on string theory. In several recent publications, attempts were made to restore the missing links. As discussed in these publications,

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Lasserre, Jean B., and Eduardo S. Zeron. "An Alternative Algorithm for Counting Lattice Points in a Convex Polytope." Mathematics of Operations Research 30, no. 3 (2005): 597–614. http://dx.doi.org/10.1287/moor.1050.0145.

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30

Kato, Yuichi, and Masanori Yamanaka. "First Brillouin Polytope and Band Structure of Diamond Lattice in Four Dimensions." Journal of the Physical Society of Japan 86, no. 3 (2017): 033601. http://dx.doi.org/10.7566/jpsj.86.033601.

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31

Kaibel, Volker, and Marc E. Pfetsch. "Computing the face lattice of a polytope from its vertex-facet incidences." Computational Geometry 23, no. 3 (2002): 281–90. http://dx.doi.org/10.1016/s0925-7721(02)00103-7.

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32

Chen, Beifang. "Weight functions, double reciprocity laws, and volume formulas for lattice polyhedra." Proceedings of the National Academy of Sciences 95, no. 16 (1998): 9093–98. http://dx.doi.org/10.1073/pnas.95.16.9093.

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We extend the concept of manifold with boundary to weight and boundary weight functions. With the new concept, we obtained the double reciprocity laws for simplicial complexes, cubical complexes, and lattice polyhedra with weight functions. For a polyhedral manifold with boundary, if the weight function has the constant value 1, then the boundary weight function has the constant value 1 on the boundary and 0 elsewhere. In particular, for a lattice polyhedral manifold with boundary, our double reciprocity law with a special parameter reduces to the functional equation of Macdonald; for a lattic

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33

Salman Al-Najjar, Dr Shatha A. "An Ehrhart polynomial for a dual polytope and the number of lattice points." IOSR Journal of Mathematics 3, no. 2 (2012): 32–36. http://dx.doi.org/10.9790/5728-0323236.

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Zhao, Xuanyi, Jinggai Li, Shiqi He, and Chungang Zhu. "Geometric conditions for injectivity of 3D Bézier volumes." AIMS Mathematics 6, no. 11 (2021): 11974–88. http://dx.doi.org/10.3934/math.2021694.

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<abstract><p>The one-to-one property of injectivity is a crucial concept in computer-aided design, geometry, and graphics. The injectivity of curves (or surfaces or volumes) means that there is no self-intersection in the curves (or surfaces or volumes) and their images or deformation models. Bézier volumes are a special class of Bézier polytope in which the lattice polytope equals $ \Box_{m, n, l}, (m, n, l\in Z) $. Piecewise 3D Bézier volumes have a wide range of applications in deformation models, such as for face mesh deformation. The injectivity of 3D Bézier volumes means that

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35

Lindner, Marko, and Steffen Roch. "On the integer points in a lattice polytope: n-fold Minkowski sum and boundary." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 52, no. 2 (2011): 395–404. http://dx.doi.org/10.1007/s13366-011-0040-z.

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Grishukhin, Vyacheslav P. "The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice $ E_7$." Sbornik: Mathematics 203, no. 11 (2012): 1571–88. http://dx.doi.org/10.1070/sm2012v203n11abeh004276.

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Foissy, Loïc. "Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials." Advances in Pure and Applied Mathematics 10, no. 1 (2019): 27–63. http://dx.doi.org/10.1515/apam-2016-0051.

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Abstract To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra morphism with values in \mathbb{Q}[X] . We deduce from the interacting bialgebras an algebraic proof of the duality principle, a generalization and a new proof of a result on B-series due to Whright and Zhao, using a monoid of characters on quasi-posets, and a generalization of Faulhaber’s formula. We also give non-commutative versions of these results

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38

Talis, Alexander, Ayal Everstov, and Valentin Kraposhin. "Crystal structures of alpha and beta modifications of Mn as packing of tetrahedral helices extracted from a four-dimensional {3, 3, 5} polytope." Acta Crystallographica Section B Structural Science, Crystal Engineering and Materials 76, no. 5 (2020): 948–54. http://dx.doi.org/10.1107/s2052520620011154.

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The crystal structures of both α- and β-Mn modifications have been presented as packing of tetrahedral helices extracted from four-dimensional {3, 3, 5} polytope construction. Presentation of the β-Mn structure as a primitive cubic arrangement formed by double tetrahedral helices around a central tetrahedral Coxeter–Boerdijk helix (tetrahelix) enables the inclusion in the structure description not only all atoms but also all tetrahedra; these tetrahedra are not accounted for in the preceding models for the β-Mn structure. The tetrahelix periodicity arising by minimal deformations of tetrahedra

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Binyamini, Gal. "Bezout-type theorems for differential fields." Compositio Mathematica 153, no. 4 (2017): 867–88. http://dx.doi.org/10.1112/s0010437x17007035.

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We prove analogs of the Bezout and the Bernstein–Kushnirenko–Khovanskii theorems for systems of algebraic differential conditions over differentially closed fields. Namely, given a system of algebraic conditions on the first $l$ derivatives of an $n$-tuple of functions, which admits finitely many solutions, we show that the number of solutions is bounded by an appropriate constant (depending singly-exponentially on $n$ and $l$) times the volume of the Newton polytope of the set of conditions. This improves a doubly-exponential estimate due to Hrushovski and Pillay. We illustrate the applicatio

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VIÑA, ANDRÉS. "LIFTING HAMILTONIAN LOOPS TO ISOTOPIES IN FIBRATIONS." International Journal of Geometric Methods in Modern Physics 10, no. 10 (2013): 1350057. http://dx.doi.org/10.1142/s0219887813500576.

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Let G be a Lie group, H a closed subgroup and M the homogeneous space G/H. Each representation Ψ of H determines a G-equivariant principal bundle [Formula: see text] on M endowed with a G-invariant connection. We consider subgroups [Formula: see text] of the diffeomorphism group Diff (M), such that, each vector field [Formula: see text] admits a lift to a preserving connection vector field on [Formula: see text]. We prove that [Formula: see text]. This relation is applicable to subgroups [Formula: see text] of the Hamiltonian groups of the flag varieties of a semisimple group G. Let MΔ be the

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Ziegler, Günter M. "Additive structures on f-vector sets of polytopes." Advances in Geometry 20, no. 2 (2020): 217–31. http://dx.doi.org/10.1515/advgeom-2018-0025.

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AbstractWe show that the f-vector sets of d-polytopes have non-trivial additive structure: They span affine lattices and are embedded in monoids that we describe explicitly. Moreover, for many large subclasses, such as the simple polytopes, or the simplicial polytopes, there are monoid structures on the set of f-vectors by themselves: “addition of f-vectors minus the f-vector of the d-simplex” always yields a new f-vector. For general 4-polytopes, we show that the modified addition operation does not always produce an f-vector, but that the result is always close to an f-vector. In this sense,

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42

Kautny, Paul, Thomas Schwartz, Berthold Stöger, and Johannes Fröhlich. "An unusual case of OD-allotwinning: 9,9′-(2,5-dibromo-1,4-phenylene)bis[9H-carbazole]." Acta Crystallographica Section B Structural Science, Crystal Engineering and Materials 73, no. 1 (2017): 65–73. http://dx.doi.org/10.1107/s2052520616018291.

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9,9′-(2,5-Dibromo-1,4-phenylene)bis[9H-carbazole] (1) crystallizes as a category I order–disorder (OD) structure composed of non-polar layers of one kind with B2/m(1)1 layer symmetry. The crystals are made up of the two polytypes with a maximum degree of order (MDO). The monoclinic MDO1 polytype (B21/d) possesses an orthorhombic B-centered lattice and appears in two orientations, which are related by reflection at (100). The orthorhombic MDO2 polytype (F2dd) has a doubled b-axis and appears in two orientations, which are related by inversion. The crystal structures of both polytypes were deter

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Huang, Wei, Hui Jun Guo, Xi Liu, et al. "A Competitive Lattice Model Monte Carlo Method for Simulation of Competitive Growth of Different Polytypes in SiC Single Crystal." Materials Science Forum 858 (May 2016): 45–48. http://dx.doi.org/10.4028/www.scientific.net/msf.858.45.

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A competitive lattice model was developed for the Kinetic Monte Carlo (KMC) simulation of the competition of 4H and 6H polytypes in SiC crystal growth based on the on-lattice model. In the competitive lattice model, site positions are fixed at the perfect crystal lattice positions without any adjustment of the site positions. The effect of surface steps was investigated, and behavior similar to step-controlled homoepitaxy was observed in KMC simulation of PVT grown SiC. Maintaining the step growth mode is an important factor to maintain a stable single polytype during SiC growth.

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Clingher, Adrian, and Jae-Hyouk Lee. "Lorentzian Lattices and E-Polytopes." Symmetry 10, no. 10 (2018): 443. http://dx.doi.org/10.3390/sym10100443.

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We consider certain E n -type root lattices embedded within the standard Lorentzian lattice Z n + 1 ( 3 ≤ n ≤ 8 ) and study their discrete geometry from the point of view of del Pezzo surface geometry. The lattice Z n + 1 decomposes as a disjoint union of affine hyperplanes which satisfy a certain periodicity. We introduce the notions of line vectors, rational conic vectors, and rational cubics vectors and their relations to E-polytopes. We also discuss the relation between these special vectors and the combinatorics of the Gosset polytopes of type ( n − 4 ) 21 .

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Arakcheeva, Alla, Philip Pattison, Annette Bauer-Brandl, Henrik Birkedal, and Gervais Chapuis. "Cimetidine, C10H16N6S, form C: crystal structure and modelling of polytypes using the superspace approach." Journal of Applied Crystallography 46, no. 1 (2013): 99–107. http://dx.doi.org/10.1107/s0021889812048133.

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The H2 antagonist cimetidine forms many polymorphs, several of which have resisted structural analysis thus far. Using single-crystal X-ray measurements obtained from synchrotron radiation, the crystal structure of cimetidine form C has been solved. This layered structure crystallizes in space groupC2/cwith an unusually large lattice parameter,a= 82.904 Å. The thickness of each layerLis equal toa′ =a/6 = 13.82 Å, anda= 6a′ originates from a sixfoldLLLL′L′L′ sequence withLandL′ differing by 0.5b. This packing is reminiscent of polytypic stacking in metals. A (3 + 1)-dimensional superspace model

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Pezoldt, Jörg, and Volker Cimalla. "Imprinting the Polytype Structure of Silicon Carbide by Rapid Thermal Processing." Crystals 10, no. 6 (2020): 523. http://dx.doi.org/10.3390/cryst10060523.

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Silicon carbide is a material with a multistable crystallographic structure, i.e., a polytypic material. Different polytypes exhibit different band gaps and electronic properties with nearly identical basal plane lattice constants, making them interesting for heterostructures without concentration gradients. The controlled formation of this heterostructure is still a challenge. The ability to adjust a defined temperature–time profile using rapid thermal processing was used to imprint the polytype transitions by controlling the nucleation and structural evolution during the temperature ramp-up

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47

Zhang, Philip B. "Interlacing polynomials and the veronese construction for rational formal power series." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 1 (2019): 1–16. http://dx.doi.org/10.1017/prm.2018.76.

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AbstractFixing a positive integer r and $0 \les k \les r-1$, define $f^{\langle r,k \rangle }$ for every formal power series f as $ f(x) = f^{\langle r,0 \rangle }(x^r)+xf^{\langle r,1 \rangle }(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1 \rangle }(x^r).$ Jochemko recently showed that the polynomial $U^{n}_{r,k}\, h(x) := ( (1+x+\cdots +x^{r-1})^{n} h(x) )^{\langle r,k \rangle }$ has only non-positive zeros for any $r \ges \deg h(x) -k$ and any positive integer n. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial $h(x)$ of a lattice polytope of dimensi

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Prabhu, Nagabhushana. "Sections of simplices." International Journal of Mathematics and Mathematical Sciences 22, no. 2 (1999): 401–10. http://dx.doi.org/10.1155/s0161171299224015.

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We show that for⌊d/2⌋≤k≤d, the relative interior of everyk-face of ad-simplexΔdcan be intersected by a2(d−k)-dimensional affine flat. Bezdek, Bisztriczky, and Connelly's results [2] show that the conditionk≥⌊d/2⌋above cannot be dropped and hence raise the question of determining, for all0≤k,j<d, an upper bound on the functionc(j,k;d), defined as the smallest number ofj-flats,j<d, needed to intersect the relative interiors of all thek-faces ofΔd. Using probabilistic arguments, we show thatC( j,k;d)≤(d+1k+1)(w+1k+1)log(d+1k+1), where w=min(max(⌊j2⌋+k,j),d). (*)Finally, we consider the func

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Pikhurko, Oleg. "Lattice points in lattice polytopes." Mathematika 48, no. 1-2 (2001): 15–24. http://dx.doi.org/10.1112/s0025579300014339.

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Betke, U., and P. McMullen. "Lattice points in lattice polytopes." Monatshefte f�r Mathematik 99, no. 4 (1985): 253–65. http://dx.doi.org/10.1007/bf01312545.

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