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Journal articles on the topic 'Simplicial complexes and polytopes'

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Author: Grafiati

Published: 25 May 2024

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1

Bruns, W., and J. Gubeladze. "Combinatorial Invariance of Stanley–Reisner Rings." gmj 3, no. 4 (August 1996): 315–18. http://dx.doi.org/10.1515/gmj.1996.315.

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Abstract In this short note we show that Stanley–Reisner rings of simplicial complexes, which have had a “dramatic application” in combinatorics [Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw Publications, 1992, p. 41], possess a rigidity property in the sense that they determine their underlying simplicial complexes.

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2

Erokhovets, N. Yu. "Buchstaber invariant theory of simplicial complexes and convex polytopes." Proceedings of the Steklov Institute of Mathematics 286, no. 1 (October 2014): 128–87. http://dx.doi.org/10.1134/s008154381406008x.

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3

Pournin, Lionel. "Lifting simplicial complexes to the boundary of convex polytopes." Discrete Mathematics 312, no. 19 (October 2012): 2849–62. http://dx.doi.org/10.1016/j.disc.2012.06.005.

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4

Joswig, Michael. "Projectivities in simplicial complexes and colorings of simple polytopes." Mathematische Zeitschrift 240, no. 2 (June 1, 2002): 243–59. http://dx.doi.org/10.1007/s002090100381.

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5

Deza, Michel, and Mathieu Dutour Sikirić. "Generalized cut and metric polytopes of graphs and simplicial complexes." Optimization Letters 14, no. 2 (November 13, 2018): 273–89. http://dx.doi.org/10.1007/s11590-018-1358-3.

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6

Ayzenberg, A. A. "Substitutions of polytopes and of simplicial complexes, and multigraded betti numbers." Transactions of the Moscow Mathematical Society 74 (April 9, 2014): 175–202. http://dx.doi.org/10.1090/s0077-1554-2014-00224-7.

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7

Santos, Francisco. "Recent progress on the combinatorial diameter of polytopes and simplicial complexes." TOP 21, no. 3 (October 2013): 426–60. http://dx.doi.org/10.1007/s11750-013-0295-7.

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8

Bahri, Anthony, Soumen Sarkar, and Jongbaek Song. "Infinite families of equivariantly formal toric orbifolds." Forum Mathematicum 31, no. 2 (March 1, 2019): 283–301. http://dx.doi.org/10.1515/forum-2018-0019.

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AbstractThe simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and more generally, polyhedral products. In this paper we extend the analysis to include toric orbifolds. Our main results yield infinite families of toric orbifolds, derived from a given one, whose integral cohomology is free of torsion and is concentrated in even degrees, a property which might be termed integrally equivariantly formal. In all cases, it is possible to give a description of the cohomology ring and to relate it to the cohomology of the original orbifold.

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9

Beben, Piotr, and Jelena Grbić. "LS-category of moment-angle manifolds and higher order Massey products." Forum Mathematicum 33, no. 5 (August 26, 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 show that the LS-category closely relates to vanishing of Massey products in H * ⁢ ( 𝒵 K ) {H^{*}(\mathcal{Z}_{K})} , and through this connection we describe first structural properties of Massey products in moment-angle manifolds. Some of the further applications include calculations of the LS-category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and k-neighborly complexes, which double as important examples of hyperbolic manifolds.

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10

De Loera, Jesús A. "Comments on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes." TOP 21, no. 3 (October 2013): 474–81. http://dx.doi.org/10.1007/s11750-013-0291-y.

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11

Eisenbrand, Friedrich. "Comments on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes." TOP 21, no. 3 (October 2013): 468–71. http://dx.doi.org/10.1007/s11750-013-0292-x.

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12

Terlaky, Tamás. "Comments on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes." TOP 21, no. 3 (October 2013): 461–67. http://dx.doi.org/10.1007/s11750-013-0293-9.

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13

Hiriart-Urruty, Jean-Baptiste. "Comments on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes." TOP 21, no. 3 (October 2013): 472–73. http://dx.doi.org/10.1007/s11750-013-0294-8.

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14

Santos, Francisco. "Rejoinder on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes." TOP 21, no. 3 (October 2013): 482–84. http://dx.doi.org/10.1007/s11750-013-0296-6.

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15

Zakharevich, Inna. "Simplicial polytope complexes and deloopings of $K$-theory." Homology, Homotopy and Applications 15, no. 2 (2013): 301–30. http://dx.doi.org/10.4310/hha.2013.v15.n2.a18.

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16

Miranda, Manuel, Gissell Estrada-Rodriguez, and Ernesto Estrada. "What Is in a Simplicial Complex? A Metaplex-Based Approach to Its Structure and Dynamics." Entropy 25, no. 12 (November 29, 2023): 1599. http://dx.doi.org/10.3390/e25121599.

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Geometric realization of simplicial complexes makes them a unique representation of complex systems. The existence of local continuous spaces at the simplices level with global discrete connectivity between simplices makes the analysis of dynamical systems on simplicial complexes a challenging problem. In this work, we provide some examples of complex systems in which this representation would be a more appropriate model of real-world phenomena. Here, we generalize the concept of metaplexes to embrace that of geometric simplicial complexes, which also includes the definition of dynamical systems on them. A metaplex is formed by regions of a continuous space of any dimension interconnected by sinks and sources that works controlled by discrete (graph) operators. The definition of simplicial metaplexes given here allows the description of the diffusion dynamics of this system in a way that solves the existing problems with previous models. We make a detailed analysis of the generalities and possible extensions of this model beyond simplicial complexes, e.g., from polytopal and cell complexes to manifold complexes, and apply it to a real-world simplicial complex representing the visual cortex of a macaque.

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17

Juhnke-Kubitzke, Martina, and Lorenzo Venturello. "Graded Betti Numbers of Balanced Simplicial Complexes." Acta Mathematica Vietnamica 46, no. 4 (October 1, 2021): 839–71. http://dx.doi.org/10.1007/s40306-021-00449-8.

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AbstractWe prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings S/I, where S is a polynomial ring and $I\subseteq S$ I ⊆ S is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.

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18

Kushnirenko, Anatoly. "Arnold’s Piecewise Linear Filtrations, Analogues of Stanley–Reisner Rings and Simplicial Newton Polyhedra." Mathematics 10, no. 23 (November 24, 2022): 4445. http://dx.doi.org/10.3390/math10234445.

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In 1974, the author proved that the codimension of the ideal (g1,g2,…,gd) generated in the group algebra K[Zd] over a field K of characteristic 0 by generic Laurent polynomials having the same Newton polytope Γ is equal to d!×Volume(Γ). Assuming that Newtons polytope is simplicial and super-convenient (that is, containing some neighborhood of the origin), the author strengthens the 1974 result by explicitly specifying the set Bsh of monomials of cardinality d!×Volume(Γ), whose equivalence classes form a basis of the quotient algebra K[Zd]/(g1,g2,…,gd). The set Bsh is constructed inductively from any shelling sh of the polytope Γ. Using the Bsh structure, we prove that the associated graded K -algebra grΓ(K[Zd]) constructed from the Arnold–Newton filtration of K -algebra K[Zd] has the Cohen–Macaulay property. This proof is a generalization of B. Kind and P. Kleinschmitt’s 1979 proof that Stanley–Reisner rings of simplicial complexes admitting shelling are Cohen–Macaulay. Finally, we prove that for generic Laurent polynomials (f1,f2,⋯,fd) with the same Newton polytope Γ, the set Bsh defines a monomial basis of the quotient algebra K[Zd]/(g1,g2,…,gd).

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19

Chen, Beifang. "Weight functions, double reciprocity laws, and volume formulas for lattice polyhedra." Proceedings of the National Academy of Sciences 95, no. 16 (August 4, 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 lattice polytope especially, the double reciprocity law with a special parameter reduces to the reciprocity law of Ehrhart. Several volume formulas for lattice polyhedra are obtained from the properties of the double reciprocity law. Moreover, the idea of weight and boundary weight leads to a new homology that is not homotopy invariant, but only homeomorphic invariant.

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20

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 (June 7, 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 vertices, there is a direct extension of the Buchstaber–Panov moment-angle complex. That extension denoted Z(K;(X,A)) is known as the polyhedral product functor, terminology due to Bill Browder, and agrees with the Buchstaber–Panov moment-angle complex in the special case (X,A) = (D2, S1) [1, 2]. A decomposition theorem was proven which splits the suspension of Z(K; (X, A)) into a bouquet of spaces determined by the full sub-complexes of K.This paper is a study of the cup-product structure for the cohomology ring of Z(K; (X, A)). The new result in the current paper is that the structure of the cohomology ring is given in terms of this geometric decomposition arising from the “stable” decomposition of Z(K; (X, A)) [1, 2]. The methods here give a determination of the cohomology ring structure for many new values of the polyhedral product functor as well as retrieve many known results.Explicit computations are made for families of suspension pairs and for the cases where Xi is the cone on Ai. These results complement and extend those of Davis–Januszkiewicz [6], Buchstaber–Panov [3, 4], Panov [13], Baskakov–Buchstaber–Panov, [3], Franz, [8, 9], as well as Hochster [12]. Furthermore, under the conditions stated below (essentially the strong form of the Künneth theorem), these theorems also apply to any cohomology theory.

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21

Glazyrin, A. A. "On simplicial partitions of polytopes." Mathematical Notes 85, no. 5-6 (June 2009): 799–806. http://dx.doi.org/10.1134/s0001434609050228.

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22

Pilaud, Vincent, Guillermo Pineda-Villavicencio, and Julien Ugon. "Edge connectivity of simplicial polytopes." European Journal of Combinatorics 113 (October 2023): 103752. http://dx.doi.org/10.1016/j.ejc.2023.103752.

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23

Volodin, V. D. "Combinatorics of flag simplicial 3-polytopes." Russian Mathematical Surveys 70, no. 1 (February 28, 2015): 168–70. http://dx.doi.org/10.1070/rm2015v070n01abeh004940.

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24

Linhart, Johann. "VOLUME AND CIRCUMRADIUS OF SIMPLICIAL POLYTOPES." Annals of the New York Academy of Sciences 440, no. 1 Discrete Geom (May 1985): 97–105. http://dx.doi.org/10.1111/j.1749-6632.1985.tb14543.x.

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25

Basu, Amitabh, Gérard Cornuéjols, and Matthias Köppe. "Unique Minimal Liftings for Simplicial Polytopes." Mathematics of Operations Research 37, no. 2 (May 2012): 346–55. http://dx.doi.org/10.1287/moor.1110.0536.

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26

Nevo, Eran, Guillermo Pineda-Villavicencio, Julien Ugon, and David Yost. "Almost Simplicial Polytopes: The Lower and Upper Bound Theorems." Canadian Journal of Mathematics 72, no. 2 (May 21, 2019): 537–56. http://dx.doi.org/10.4153/s0008414x18000123.

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AbstractWe study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex facet with, say, $d+s$ vertices, called almost simplicial polytopes. We provide tight lower and upper bound theorems for these polytopes as functions of $d,n$, and $s$, thus generalizing the classical Lower Bound Theorem by Barnette and the Upper Bound Theorem by McMullen, which treat the case where $s=0$. We characterize the minimizers and provide examples of maximizers for any $d$. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest.

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27

Hähnle, Nicolai, Steven Klee, and Vincent Pilaud. "Obstructions to weak decomposability for simplicial polytopes." Proceedings of the American Mathematical Society 142, no. 9 (June 10, 2014): 3249–57. http://dx.doi.org/10.1090/s0002-9939-2014-12101-0.

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28

Gonska, Bernd. "About f-Vectors of Inscribed Simplicial Polytopes." Discrete & Computational Geometry 55, no. 3 (February 29, 2016): 497–521. http://dx.doi.org/10.1007/s00454-016-9764-8.

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29

Finbow, W. "Simplicial neighbourly 5-polytopes with nine vertices." Boletín de la Sociedad Matemática Mexicana 21, no. 1 (April 11, 2014): 39–51. http://dx.doi.org/10.1007/s40590-014-0013-y.

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30

Brandts, J. "Gradient superconvergence on uniform simplicial partitions of polytopes." IMA Journal of Numerical Analysis 23, no. 3 (July 1, 2003): 489–505. http://dx.doi.org/10.1093/imanum/23.3.489.

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31

Juhnke-Kubitzke, Martina, and Satoshi Murai. "Balanced generalized lower bound inequality for simplicial polytopes." Selecta Mathematica 24, no. 2 (October 19, 2017): 1677–89. http://dx.doi.org/10.1007/s00029-017-0363-1.

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32

Nevo, Eran, and Eyal Novinsky. "A characterization of simplicial polytopes with g2=1." Journal of Combinatorial Theory, Series A 118, no. 2 (February 2011): 387–95. http://dx.doi.org/10.1016/j.jcta.2009.12.011.

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33

Zhilinskii, Boris. "On the number ofk-faces of primitive parallelohedra." Acta Crystallographica Section A Foundations and Advances 71, no. 2 (February 4, 2015): 212–15. http://dx.doi.org/10.1107/s205327331402806x.

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Dehn–Sommerville relations for simple (simplicial) polytopes are applied to primitive parallelohedra. New restrictions on numbers ofk-faces of non-principal primitive parallelohedra are explicitly formulated for five-, six- and seven-dimensional parallelohedra.

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34

Pąk, Karol. "Abstract Simplicial Complexes." Formalized Mathematics 18, no. 1 (January 1, 2010): 95–106. http://dx.doi.org/10.2478/v10037-010-0013-y.

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Abstract Simplicial Complexes In this article we define the notion of abstract simplicial complexes and operations on them. We introduce the following basic notions: simplex, face, vertex, degree, skeleton, subdivision and substructure, and prove some of their properties.

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35

Even-Zohar, Chaim, Michael Farber, and Lewis Mead. "Ample simplicial complexes." European Journal of Mathematics 8, no. 1 (January 17, 2022): 1–32. http://dx.doi.org/10.1007/s40879-021-00521-5.

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AbstractMotivated by potential applications in network theory, engineering and computer science, we study r-ample simplicial complexes. These complexes can be viewed as finite approximations to the Rado complex which has a remarkable property of indestructibility, in the sense that removing any finite number of its simplexes leaves a complex isomorphic to itself. We prove that an r-ample simplicial complex is simply connected and 2-connected for r large. The number n of vertexes of an r-ample simplicial complex satisfies $$\exp \bigl (\Omega \bigl (\frac{2^r}{\sqrt{r}}\bigr )\bigr )$$ exp ( Ω ( 2 r r ) ) . We use the probabilistic method to establish the existence of r-ample simplicial complexes with n vertexes for any $$n>r 2^r 2^{2^r}$$ n > r 2 r 2 2 r . Finally, we introduce the iterated Paley simplicial complexes, which are explicitly constructed r-ample simplicial complexes with nearly optimal number of vertexes.

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36

Charalambous, Hara. "Pointed simplicial complexes." Illinois Journal of Mathematics 41, no. 1 (March 1997): 1–9. http://dx.doi.org/10.1215/ijm/1255985839.

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37

Muhmood, Shahid, Imran Ahmed, and Adnan Liaquat. "Gallai simplicial complexes." Journal of Intelligent & Fuzzy Systems 36, no. 6 (June 11, 2019): 5645–51. http://dx.doi.org/10.3233/jifs-181478.

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38

Mnukhin, V. B., and J. Siemons. "Saturated simplicial complexes." Journal of Combinatorial Theory, Series A 109, no. 1 (January 2005): 149–79. http://dx.doi.org/10.1016/j.jcta.2004.08.003.

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39

Nagel, Uwe, and Tim Römer. "Glicci simplicial complexes." Journal of Pure and Applied Algebra 212, no. 10 (October 2008): 2250–58. http://dx.doi.org/10.1016/j.jpaa.2008.03.005.

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40

Matsuoka, Naoyuki, and Satoshi Murai. "Uniformly Cohen–Macaulay simplicial complexes and almost Gorenstein* simplicial complexes." Journal of Algebra 455 (June 2016): 14–31. http://dx.doi.org/10.1016/j.jalgebra.2016.02.005.

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41

Firsching, Moritz. "Realizability and inscribability for simplicial polytopes via nonlinear optimization." Mathematical Programming 166, no. 1-2 (February 10, 2017): 273–95. http://dx.doi.org/10.1007/s10107-017-1120-0.

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42

Stanley, Richard P. "THE NUMBER OF FACES OF SIMPLICIAL POLYTOPES AND SPHERES." Annals of the New York Academy of Sciences 440, no. 1 Discrete Geom (May 1985): 212–23. http://dx.doi.org/10.1111/j.1749-6632.1985.tb14556.x.

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43

Chin, F. Y. L., S. P. Y. Fung, and C. -A. Wang. "Approximation for Minimum Triangulations of Simplicial Convex 3-Polytopes." Discrete & Computational Geometry 26, no. 4 (January 2001): 499–511. http://dx.doi.org/10.1007/s00454-001-0045-8.

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44

Talman, A. J. J., and Y. Yamamoto. "A Simplicial Algorithm for Stationary Point Problems on Polytopes." Mathematics of Operations Research 14, no. 3 (August 1989): 383–99. http://dx.doi.org/10.1287/moor.14.3.383.

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45

Bjorner, Anders. "A Comparison Theorem for f-vectors of Simplicial Polytopes." Pure and Applied Mathematics Quarterly 3, no. 1 (2007): 347–56. http://dx.doi.org/10.4310/pamq.2007.v3.n1.a12.

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46

Klee, Steven, Eran Nevo, Isabella Novik, and Hailun Zheng. "A Lower Bound Theorem for Centrally Symmetric Simplicial Polytopes." Discrete & Computational Geometry 61, no. 3 (March 1, 2018): 541–61. http://dx.doi.org/10.1007/s00454-018-9978-z.

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47

Below, A., U. Brehm, J. A. De Loera, and and J. Richter-Gebert. "Minimal Simplicial Dissections and Triangulations of Convex 3-Polytopes." Discrete & Computational Geometry 24, no. 1 (January 2000): 35–48. http://dx.doi.org/10.1007/s004540010058.

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48

Adin, R. M. "On Face Numbers of Rational Simplicial Polytopes with Symmetry." Advances in Mathematics 115, no. 2 (October 1995): 269–85. http://dx.doi.org/10.1006/aima.1995.1057.

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49

Ziegler, Günter M. "Additive structures on f-vector sets of polytopes." Advances in Geometry 20, no. 2 (April 28, 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, the set of f-vectors of all 4-polytopes forms an “approximate affine semigroup”. The proof relies on the fact for d = 4 every d-polytope, or its dual, has a “small facet”. This fails for d > 4.We also describe a two further modified addition operations on f-vectors that can be geometrically realized by glueing corresponding polytopes. The second one of these may yield a semigroup structure on the f-vector set of all 4-polytopes.

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50

Böröczky, Károly J., and Rolf Schneider. "The Mean Width of Circumscribed Random Polytopes." Canadian Mathematical Bulletin 53, no. 4 (December 1, 2010): 614–28. http://dx.doi.org/10.4153/cmb-2010-067-5.

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AbstractFor a given convex body K in ℝd, a random polytope K(n) is defined (essentially) as the intersection of n independent closed halfspaces containing K and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of K(n) and K as n tends to infinity. For a simplicial polytope P, a precise asymptotic formula for the difference of the mean widths of P(n) and P is obtained.

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