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Journal articles on the topic 'Z-polytopes'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

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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2

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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3

Beben, Piotr, and Jelena Grbić. "LS-category of moment-angle manifolds and higher order Massey products." Forum Mathematicum 33, no. 5 (2021): 1179–205. http://dx.doi.org/10.1515/forum-2021-0015.

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Abstract Using the combinatorics of the underlying simplicial complex K, we give various upper and lower bounds for the Lusternik–Schnirelmann (LS) category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} . We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS-category. In particular, we characterize the LS-category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} over triangulated d-manifolds K for d ≤ 2 {d\leq 2} , as well as higher-dimensional spheres built up via connected sum, join, and vertex doubling operations. We sh

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4

Hartley, Michael I. "Covers ℘ for Abstract Regular Polytopes $\mathcal {Q}$ such that $\mathcal{Q}=\mathcal{P}/\mathbf{Z}_{p}^{k}$." Discrete & Computational Geometry 44, no. 4 (2009): 844–59. http://dx.doi.org/10.1007/s00454-009-9234-7.

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5

Kabluchko, Zakhar, and Dmitry Zaporozhets. "Absorption probabilities for Gaussian polytopes and regular spherical simplices." Advances in Applied Probability 52, no. 2 (2020): 588–616. http://dx.doi.org/10.1017/apr.2020.7.

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AbstractThe Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in $\mathbb{R}^d$ . We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point $x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\mathcal P_{n,d}$ contains the point $\sigma X$ , where $\sigma\geq 0$ is constant and X is a standard normal vector independent of $\mathcal P_{n,d}$ . As a by-product, we also compute the expected number of k-faces and the expected volume of $\mathcal P_{n,d}$ , thus rec

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6

Cunningham, Gabe. "Mixing chiral polytopes." Journal of Algebraic Combinatorics 36, no. 2 (2011): 263–77. http://dx.doi.org/10.1007/s10801-011-0335-z.

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7

Stenson, Catherine. "Families of Tight Inequalities for Polytopes." Discrete & Computational Geometry 34, no. 3 (2005): 507–21. http://dx.doi.org/10.1007/s00454-005-1193-z.

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8

Bisztriczky, Tibor, K�roly B�r�czky, Jr., and David S. Gunderson. "Cyclic polytopes, hyperplanes, and Gray codes." Journal of Geometry 78, no. 1-2 (2003): 25–49. http://dx.doi.org/10.1007/s00022-003-1705-z.

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9

Barvinok, Alexander, Seung Jin Lee, and Isabella Novik. "Centrally symmetric polytopes with many faces." Israel Journal of Mathematics 195, no. 1 (2012): 457–72. http://dx.doi.org/10.1007/s11856-012-0107-z.

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10

Fredrickson, Karl. "Extremal Transitions from Nested Reflexive Polytopes." Communications in Mathematical Physics 335, no. 3 (2014): 1381–95. http://dx.doi.org/10.1007/s00220-014-2201-z.

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11

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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12

Billera, Louis J., Paul Filliman, and Bernd Sturmfels. "Constructions and complexity of secondary polytopes." Advances in Mathematics 83, no. 2 (1990): 155–79. http://dx.doi.org/10.1016/0001-8708(90)90077-z.

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13

De Loera, Jesús A., David C. Haws, and Matthias Köppe. "Ehrhart Polynomials of Matroid Polytopes and Polymatroids." Discrete & Computational Geometry 42, no. 4 (2008): 670–702. http://dx.doi.org/10.1007/s00454-008-9080-z.

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14

McMullen, Peter. "Regular Polytopes of Nearly Full Rank: Addendum." Discrete & Computational Geometry 49, no. 3 (2013): 703–5. http://dx.doi.org/10.1007/s00454-013-9487-z.

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15

Mészáros, Karola, and Alejandro H. Morales. "Volumes and Ehrhart polynomials of flow polytopes." Mathematische Zeitschrift 293, no. 3-4 (2019): 1369–401. http://dx.doi.org/10.1007/s00209-019-02283-z.

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16

Linke, Eva, and Eugenia Saorín Gómez. "Decomposition of polytopes using inner parallel bodies." Monatshefte für Mathematik 176, no. 4 (2014): 575–88. http://dx.doi.org/10.1007/s00605-014-0629-z.

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17

Mixer, Mark, and Egon Schulte. "Symmetric Graphs from Polytopes of High Rank." Graphs and Combinatorics 28, no. 6 (2011): 843–57. http://dx.doi.org/10.1007/s00373-011-1089-z.

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18

Cunningham, Gabe, María Del Río-Francos, Isabel Hubard, and Micael Toledo. "Symmetry Type Graphs of Polytopes and Maniplexes." Annals of Combinatorics 19, no. 2 (2015): 243–68. http://dx.doi.org/10.1007/s00026-015-0263-z.

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19

Gouveia, João, Pablo A. Parrilo, and Rekha R. Thomas. "Approximate cone factorizations and lifts of polytopes." Mathematical Programming 151, no. 2 (2014): 613–37. http://dx.doi.org/10.1007/s10107-014-0848-z.

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20

Ogata, Shoetsu. "Very ample but not normal lattice polytopes." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 54, no. 1 (2011): 291–302. http://dx.doi.org/10.1007/s13366-011-0077-z.

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21

Karpenkov, Oleg. "Classification of lattice-regular lattice convex polytopes." Functional Analysis and Other Mathematics 1, no. 1 (2007): 17–35. http://dx.doi.org/10.1007/s11853-007-0002-z.

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22

McMullen, Peter, and Egon Schulte. "Hermitian forms and locally toroidal regular polytopes." Advances in Mathematics 82, no. 1 (1990): 88–125. http://dx.doi.org/10.1016/0001-8708(90)90084-z.

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23

Weiss, Asia Ivić. "Some infinite families of finite incidence-polytopes." Journal of Combinatorial Theory, Series A 55, no. 1 (1990): 60–73. http://dx.doi.org/10.1016/0097-3165(90)90047-z.

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24

Borda, Bence. "Lattice Points in Algebraic Cross-polytopes and Simplices." Discrete & Computational Geometry 60, no. 1 (2017): 145–69. http://dx.doi.org/10.1007/s00454-017-9946-z.

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25

Legendre, Eveline. "Toric Kähler–Einstein Metrics and Convex Compact Polytopes." Journal of Geometric Analysis 26, no. 1 (2015): 399–427. http://dx.doi.org/10.1007/s12220-015-9556-z.

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26

Letchford, Adam N., and Michael M. Sørensen. "Binary positive semidefinite matrices and associated integer polytopes." Mathematical Programming 131, no. 1-2 (2010): 253–71. http://dx.doi.org/10.1007/s10107-010-0352-z.

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27

Nagano, Kiyohito, and Kazuyuki Aihara. "Equivalence of convex minimization problems over base polytopes." Japan Journal of Industrial and Applied Mathematics 29, no. 3 (2012): 519–34. http://dx.doi.org/10.1007/s13160-012-0083-z.

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28

Klee, Steven, Eran Nevo, Isabella Novik, and Hailun Zheng. "A Lower Bound Theorem for Centrally Symmetric Simplicial Polytopes." Discrete & Computational Geometry 61, no. 3 (2018): 541–61. http://dx.doi.org/10.1007/s00454-018-9978-z.

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29

Gao, Yibo, Benjamin Krakoff, and Lisa Yang. "The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes." Discrete & Computational Geometry 62, no. 1 (2019): 209–38. http://dx.doi.org/10.1007/s00454-019-00076-z.

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30

Muthiah, Dinakar, and Peter Tingley. "Affine PBW bases and MV polytopes in rank 2." Selecta Mathematica 20, no. 1 (2013): 237–60. http://dx.doi.org/10.1007/s00029-012-0117-z.

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31

Kohn, Kathlén, and Kristian Ranestad. "Projective Geometry of Wachspress Coordinates." Foundations of Computational Mathematics 20, no. 5 (2019): 1135–73. http://dx.doi.org/10.1007/s10208-019-09441-z.

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Abstract We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress (A rational finite element basis, Academic Press, New York, 1975). The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren (Adv Comput Math 6:97–108, 1996). This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint

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32

Vershik, A. M. "Classification of Finite Metric Spaces and Combinatorics of Convex Polytopes." Arnold Mathematical Journal 1, no. 1 (2015): 75–81. http://dx.doi.org/10.1007/s40598-014-0005-z.

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33

Ehrenborg, Richard, and Harold Fox. "Inequalities for cd -Indices of Joins and Products of Polytopes." Combinatorica 23, no. 3 (2003): 427–52. http://dx.doi.org/10.1007/s00493-003-0026-z.

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34

Starkov, Konstantin E. "On the Ultimate Dynamics of the Four-Dimensional Rössler System." International Journal of Bifurcation and Chaos 24, no. 11 (2014): 1450149. http://dx.doi.org/10.1142/s0218127414501491.

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In this paper, we construct the polytope which contains all compact ω-limit sets of the four-dimensional Rössler system which is a generalization of the hyperchaotic Rössler system for the case of positive parameters. Further, we find a few three-dimensional planes containing all compact ω-limit sets for bounded positive half-trajectories located in some subdomains in the half-space z > 0. Besides, we analyze one case in which all compact ω-limit sets in the half-space z > 0 are contained in one three-dimensional plane. Our approach is based on a combination of the LaSalle theorem and th

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35

Kaveh, Kiumars. "Vector Fields and the Cohomology Ring of Toric Varieties." Canadian Mathematical Bulletin 48, no. 3 (2005): 414–27. http://dx.doi.org/10.4153/cmb-2005-039-1.

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AbstractLetXbe a smooth complex projective variety with a holomorphic vector field with isolated zero setZ. From the results of Carrell and Lieberman there exists a filtrationF0⊂F1⊂ · · · ofA(Z), the ring of ℂ-valued functions onZ, such thatas graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra ofX.

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36

Øbro, Mikkel. "Classification of terminal simplicial reflexive d-polytopes with 3d − 1 vertices." manuscripta mathematica 125, no. 1 (2007): 69–79. http://dx.doi.org/10.1007/s00229-007-0133-z.

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37

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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38

Llanas, Bernardo. "Efficient Computation of the Hausdorff Distance Between Polytopes by Exterior Random Covering." Computational Optimization and Applications 30, no. 2 (2005): 161–94. http://dx.doi.org/10.1007/s10589-005-4560-z.

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39

Bisztriczky, T., F. Fodor, and D. Oliveros. "Separation in totally-sewn 4-polytopes with the decreasing universal edge property." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 53, no. 1 (2011): 123–38. http://dx.doi.org/10.1007/s13366-011-0069-z.

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40

BARVINOK, ALEXANDER. "What Does a Random Contingency Table Look Like?" Combinatorics, Probability and Computing 19, no. 4 (2010): 517–39. http://dx.doi.org/10.1017/s0963548310000039.

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Let R = (r1, . . ., rm) and C = (c1, . . ., cn) be positive integer vectors such that r1 + ⋯ + rm = c1 + ⋯ + cn. We consider the set Σ(R, C) of non-negative m × n integer matrices (contingency tables) with row sums R and column sums C as a finite probability space with the uniform measure. We prove that a random table D ∈ Σ(R, C) is close with high probability to a particular matrix (‘typical table’) Z defined as follows. We let g(x) = (x + 1)ln(x + 1) − x ln x for x ≥ 0 and let g(X) = ∑i,jg(xij) for a non-negative matrix X = (xij). Then g(X) is strictly concave and attains its maximum on the

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41

Jorge, Helena A. "Combinatorics of Polytopes with a Group of Linear Symmetries of Prime Power Order." Discrete and Computational Geometry 30, no. 4 (2003): 529–42. http://dx.doi.org/10.1007/s00454-003-2867-z.

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42

Barvinok, Alexander, Seung Jin Lee, and Isabella Novik. "Explicit Constructions of Centrally Symmetric $$k$$ -Neighborly Polytopes and Large Strictly Antipodal Sets." Discrete & Computational Geometry 49, no. 3 (2013): 429–43. http://dx.doi.org/10.1007/s00454-013-9495-z.

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43

Hörrmann, Julia, Joscha Prochno та Christoph Thäle. "On the Isotropic Constant of Random Polytopes with Vertices on an $$\ell _p$$ ℓ p -Sphere". Journal of Geometric Analysis 28, № 1 (2017): 405–26. http://dx.doi.org/10.1007/s12220-017-9826-z.

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44

Liu, Gaku. "Flip-Connectivity of Triangulations of the Product of a Tetrahedron and Simplex." Discrete & Computational Geometry 63, no. 1 (2019): 1–30. http://dx.doi.org/10.1007/s00454-019-00157-z.

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AbstractA flip is a minimal move between two triangulations of a polytope. The set of triangulations of a polytope was shown by Santos to not always be connected by flips, and it is an interesting problem to find large classes of polytopes for which it is. One such class which has received considerable attention is the product of two simplices. Santos proved that the set of triangulations of a product of two simplices is connected by flips when one of the simplices is a triangle. However, the author showed that it is not connected when one of the simplices is four-dimensional and the other has

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45

Kimura, Masatoshi, and Tetsuya Takine. "Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities." Advances in Applied Probability 52, no. 4 (2020): 1249–83. http://dx.doi.org/10.1017/apr.2020.40.

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AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$

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46

Abu-Qasmieh, Isam, and Ali Mohammad Alqudah. "Triad system for object's 3D localization using low-resolution 2D ultrasonic sensor array." International Review of Applied Sciences and Engineering 11, no. 2 (2020): 115–22. http://dx.doi.org/10.1556/1848.2020.20010.

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AbstractIn the recently published researches in the object localization field, 3D object localization takes the largest part of this research due to its importance in our daily life. 3D object localization has many applications such as collision avoidance, robotic guiding and vision and object surfaces topography modeling. This research study represents a novel localization algorithm and system design using a low-resolution 2D ultrasonic sensor array for 3D real-time object localization. A novel localization algorithm is developed and applied to the acquired data using the three sensors having

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47

BAHRI, A., M. BENDERSKY, F. R. COHEN, and S. GITLER. "Cup-products for the polyhedral product functor." Mathematical Proceedings of the Cambridge Philosophical Society 153, no. 3 (2012): 457–69. http://dx.doi.org/10.1017/s0305004112000230.

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AbstractDavis–Januszkiewicz introduced manifolds which are now known as moment-angle manifolds over a polytope [6]. Buchstaber–Panov introduced and extensively studied moment-angle complexes defined for any abstract simplicial complex K [4]. They completely described the rational cohomology ring structure in terms of the Tor-algebra of the Stanley-Reisner algebra [4].Subsequent developments were given in work of Denham–Suciu [7] and Franz [9] which were followed by [1, 2]. Namely, given a family of based CW-pairs X, A) = {(Xi, Ai)}mi=1 together with an abstract simplicial complex K with m vert

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48

Tsukerman, Emmanuel, and Lauren Williams. "Bruhat interval polytopes." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 27th..., Proceedings (2015). http://dx.doi.org/10.46298/dmtcs.2507.

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International audience Let $u$ and $v$ be permutations on $n$ letters, with $u$ &le; $v$ in Bruhat order. A <i>Bruhat interval polytope</i> $Q_{u,v}$ is the convex hull of all permutation vectors $z=(z(1),z(2),...,z(n))$ with $u$ &le; $z$ &le; $v$. Note that when $u=e$ and $v=w_0$ are the shortest and longest elements of the symmetric group, $Q_{e,w_0}$ is the classical permutohedron. Bruhat interval polytopes were studied recently in the 2013 paper “The full Kostant-Toda hierarchy on the positive flag variety” by Kodama and the second author, in the context of the Toda

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49

Hegedüs, Gábor, Akihiro Higashitani, and Alexander Kasprzyk. "Ehrhart Polynomial Roots of Reflexive Polytopes." Electronic Journal of Combinatorics 26, no. 1 (2019). http://dx.doi.org/10.37236/7780.

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Recent work has focused on the roots $z\in\mathbb{C}$ of the Ehrhart polynomial of a lattice polytope $P$. The case when $\Re{z}=-1/2$ is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when $\mathrm{dim}(P)\leq 7$. We also consider the "half-strip condition", where all roots $z$ satisfy $-\mathrm{dim}(P)/2\leq\Re{z}\leq \mathrm{dim}(P)/2-1$, and show that this holds for any reflexive polytope with $\mathrm{dim}(P)\leq 5$. We give an example of a $10$-dimensional reflexive polytope which violates the half-strip condition, t

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50

Mészáros, Karola, Connor Simpson, and Zoe Wellner. "Flow Polytopes of Partitions." Electronic Journal of Combinatorics 26, no. 1 (2019). http://dx.doi.org/10.37236/8114.

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Recent progress on flow polytopes indicates many interesting families with product formulas for their volume. These product formulas are all proved using analytic techniques. Our work breaks from this pattern. We define a family of closely related flow polytopes $F_{(\lambda, {\bf a})}$ for each partition shape $\lambda$ and netflow vector ${\bf a}\in Z^n_{> 0}$. In each such family, we prove that there is a polytope (the limiting one in a sense) which is a product of scaled simplices, explaining their product volumes. We also show that the combinatorial type of all polytopes in a fixed fam

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