scholarly journals The short toric polynomial

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Gábor Hetyei
Keyword(s):  
Lower Bound ◽  
Reflection Principle ◽  
Lattice Path ◽  
Simple Polytope ◽  
Independent Form ◽  
International Audience ◽  
Bound Theorem ◽  
Lattice Path Enumeration ◽  
Nous Utilisons ◽  
Eulerian Poset

International audience We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as Stanley's pair of toric polynomials, but allows different algebraic manipulations. Stanley's intertwined recurrence may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric h-vector in terms of the cd-index, may be stated in a rank-independent form, and it may be shown using weighted lattice path enumeration and the reflection principle. We use our techniques to derive a formula expressing the toric h-vector of a dual simplicial Eulerian poset in terms of its f-vector. This formula implies Gessel's formula for the toric h-vector of a cube, and may be used to prove that the nonnegativity of the toric h-vector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for simplicial polytopes. Nous introduisons le polynôme torique court associé à un ensemble ordonné Eulérien. Ce polynôme contient la même information que le couple de polynômes toriques de Stanley, mais il permet des manipulations algébriques différentes. La récurrence entrecroisée de Stanley peut être remplacée par une seule récurrence dans laquelle le degré des termes écartés est indépendant du rang. La variante torique courte de la formule de Bayer et Ehrenborg, qui exprime le vecteur torique d'un ensemble ordonné Eulérien en termes de son cd-index, est énoncée sous une forme qui ne dépend pas du rang et qui peut être démontrée en utilisant une énumération des chemins pondérés et le principe de réflexion. Nous utilisons nos techniques pour dériver une formule exprimant le vecteur h-torique d'un ensemble ordonné Eulérien dont le dual est simplicial, en termes de son f-vecteur. Cette formule implique la formule de Gessel pour le vecteur h-torique d'un cube, et elle peut être utilisée pour démontrer que la positivité du vecteur h-torique d'un polytope simple est une conséquence du Théorème de la Borne Inférieure Généralisé appliqué aux polytopes simpliciaux.

scholarly journals Almost simplicial polytopes: the lower and upper bound theorems

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Eran Nevo ◽  
Guillermo Pineda-Villavicencio ◽  
Julien Ugon ◽  
David Yost
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Upper Bound Theorem ◽  
Simplicial Polytopes ◽  
Full Version ◽  
The Face ◽  
International Audience ◽  
Bound Theorem

International audience this is an extended abstract of the full version. We 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 bounds for the face numbers of these polytopes as functions of d, n and s, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case s = 0. We characterize the minimizers and provide examples of maximizers, for any d.

scholarly journals Polytopes and $C^1$-convex bodies

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Karim Adiprasito ◽  
José Alejandro Samper
Keyword(s):  
Lower Bound ◽  
Convex Body ◽  
Hausdorff Distance ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
Simplicial Polytopes ◽  
The Face ◽  
International Audience ◽  
Bound Theorem ◽  
Corps Convexe

International audience The face numbers of simplicial polytopes that approximate $C^1$-convex bodies in the Hausdorff metric is studied. Several structural results about the skeleta of such polytopes are studied and used to derive a lower bound theorem for this class of polytopes. This partially resolves a conjecture made by Kalai in 1994: if a sequence $\{P_n\}_{n=0}^{\infty}$ of simplicial polytopes converges to a $C^1$-convex body in the Hausdorff distance, then the entries of the $g$-vector of $P_n$ converge to infinity. Nous étudions les nombres de faces de polytopes simpliciaux qui se rapprochent de $C^1$-corps convexes dans la métrique Hausdorff. Plusieurs résultats structurels sur le skeleta de ces polytopes sont recherchées et utilisées pour calculer un théorème limite inférieure de cette classe de polytopes. Cela résout partiellement une conjecture formulée par Kalai en 1994: si une suite $\{P_n\}_{n=0}^{\infty}$ de polytopes simpliciaux converge vers une $C^1$-corps convexe dans la distance Hausdorff, puis les entrées du $g$-vecteur de $P_n$ convergent vers l’infini.

scholarly journals A Continuous Analogue of Lattice Path Enumeration

10.37236/8788 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Quang-Nhat Le ◽  
Sinai Robins ◽  
Christophe Vignat ◽  
Tanay Wakhare
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lattice Paths ◽  
Lattice Path ◽  
Continuous Version ◽  
Continuous Analog ◽  
Continuous Analogue ◽  
Lattice Path Enumeration ◽  
Multinomial Coefficients ◽  
General Method

Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.  

scholarly journals On Greedy Trie Execution

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Zbigniew Gołębiewski ◽  
Filip Zagórski
Keyword(s):  
Lower Bound ◽  
Greedy Algorithm ◽  
International Audience ◽  
The Greedy Algorithm

International audience In the paper "How to select a looser'' Prodinger was analyzing an algorithm where $n$ participants are selecting a leader by flipping <underline>fair</underline> coins, where recursively, the 0-party (those who i.e. have tossed heads) continues until the leader is chosen. We give an answer to the question stated in the Prodinger's paper – what happens if not a 0-party is recursively looking for a leader but always a party with a smaller cardinality. We show the lower bound on the number of rounds of the greedy algorithm (for <underline>fair</underline> coin).

Accelerated Limit Loads Using Repeated Elastic Finite Element Analyses

2003 ◽  
Author(s):  
Prasad Mangalaramanan
Keyword(s):  
Finite Element ◽  
Lower Bound ◽  
Limit Load ◽  
Finite Element Analyses ◽  
Limit Loads ◽  
Bound Theorem

This paper demonstrates the limitations of repeated elastic finite element analyses (REFEA) based limit load determination that uses the classical lower bound theorem. The r-node method is prescribed as an alternative for obtaining better limit load estimates. Lower bound aspects pertaining to r-nodes are also discussed.

scholarly journals Limit Analysis of Geotechnical Problems By Applying Lower-Bound Theorem

1985 ◽  
Vol 25 (4) ◽  
pp. 37-48 ◽  
Author(s):  
Katsuhiko Arai ◽  
Keiichi Tagyo
Keyword(s):  
Lower Bound ◽  
Limit Analysis ◽  
Bound Theorem ◽  
Geotechnical Problems

A distance on the equivalence classes of spherical curves generated by deformations of type RI

2018 ◽  
Vol 27 (12) ◽  
pp. 1850066 ◽  
Author(s):  
Yukari Funakoshi ◽  
Megumi Hashizume ◽  
Noboru Ito ◽  
Tsuyoshi Kobayashi ◽  
Hiroko Murai
Keyword(s):  
Lower Bound ◽  
Technical Condition ◽  
Equivalence Classes ◽  
Spherical Curve ◽  
Bound Theorem ◽  
Local Deformations ◽  
Distance Formula

In this paper, we introduce a distance [Formula: see text] on the equivalence classes of spherical curves under deformations of type RI and ambient isotopies. We obtain an inequality that estimate its lower bound (Theorem 1). In Theorem 2, we show that if for a pair of spherical curves [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] and [Formula: see text] satisfy a certain technical condition, then [Formula: see text] is obtained from [Formula: see text] by a single weak RIII only. In Theorem 3, we show that if [Formula: see text] and [Formula: see text] satisfy other conditions, then [Formula: see text] is ambient isotopic to a spherical curve that is obtained from [Formula: see text] by a sequence of a particular local deformations, which realizes [Formula: see text].

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