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 Almost Simplicial Polytopes: The Lower and Upper Bound Theorems

2019 ◽  
Vol 72 (2) ◽  
pp. 537-556
Author(s):  
Eran Nevo ◽  
Guillermo Pineda-Villavicencio ◽  
Julien Ugon ◽  
David Yost
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Independent Interest ◽  
Upper Bound Theorem ◽  
Moment Curve ◽  
Simplicial Polytopes ◽  
Cyclic Polytopes ◽  
Bound Theorem ◽  
The Moment

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.

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 The flag upper bound theorem for 3- and 5-manifolds

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Hailun Zheng
Keyword(s):  
Upper Bound ◽  
Upper Bound Theorem ◽  
3 Dimensional ◽  
First Case ◽  
The Face ◽  
International Audience ◽  
Bound Theorem ◽  
Unique Maximizer

International audience We prove that among all flag 3-manifolds on n vertices, the join of two circles with [n 2] and [n 2] vertices respectively is the unique maximizer of the face numbers. This solves the first case of a conjecture due to Lutz and Nevo. Further, we establish a sharp upper bound on the number of edges of flag 5-manifolds and characterize the cases of equality. We also show that the inequality part of the flag upper bound conjecture continues to hold for all flag 3-dimensional Eulerian complexes and characterize the cases of equality in this class.

scholarly journals A Computational Perspective on Network Coding

2010 ◽  
Vol 2010 ◽  
pp. 1-11
Author(s):  
Qin Guo ◽  
Mingxing Luo ◽  
Lixiang Li ◽  
Yixian Yang
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Lower Bound ◽  
Network Coding ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Multicast Networks ◽  
Computational Perspective ◽  
Source Rate

From the perspectives of graph theory and combinatorics theory we obtain some new upper bounds on the number of encoding nodes, which can characterize the coding complexity of the network coding, both in feasible acyclic and cyclic multicast networks. In contrast to previous work, during our analysis we first investigate the simple multicast network with source rateh=2, and thenh≥2. We find that for feasible acyclic multicast networks our upper bound is exactly the lower bound given by M. Langberg et al. in 2006. So the gap between their lower and upper bounds for feasible acyclic multicast networks does not exist. Based on the new upper bound, we improve the computational complexity given by M. Langberg et al. in 2009. Moreover, these results further support the feasibility of signatures for network coding.

Minimization of Bounded Cable Tensions in Cable-Based Parallel Manipulators

2007 ◽  
Author(s):  
Mahir Hassan ◽  
Amir Khajepour
Keyword(s):  
Lower Bound ◽  
Numerical Method ◽  
Upper Bound ◽  
Task Force ◽  
Parallel Manipulators ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Alternating Projection ◽  
Alternating Projection Method ◽  
Cable Forces

In this work, the application of the Dykstra’s alternating projection method to find the minimum-2-norm solution for actuator forces is discussed in the case when lower and upper bounds are imposed on the actuator forces. The lower bound is due to specified pretension desired in the cables and the upper bound is due to the maximum allowable forces in the cables. This algorithm presents a systematic numerical method to determine whether or not a solution exists to the cable forces within these bounds and, if it does exist, calculate the minimum-2-norm solution for the cable forces for a given task force. This method is applied to an example 2-DOF translational cable-driven manipulator and a geometrical demonstration is presented.

Asymptotic bounds for the fluid queue fed by sub-exponential On/Off sources

2000 ◽  
Vol 32 (01) ◽  
pp. 244-255 ◽  
Author(s):  
V. Dumas ◽  
A. Simonian
Keyword(s):  
Lower Bound ◽  
Finite Number ◽  
Upper Bound ◽  
Content Distribution ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Fluid Queue ◽  
Asymptotic Bounds ◽  
Multiplicative Factor ◽  
Buffer Content

We consider a fluid queue fed by a superposition of a finite number of On/Off sources, the distribution of the On period being subexponential for some of them and exponential for the others. We provide general lower and upper bounds for the tail of the stationary buffer content distribution in terms of the so-called minimal subsets of sources. We then show that this tail decays at exponential or subexponential speed according as a certain parameter is smaller or larger than the ouput rate. If we replace the subexponential tails by regularly varying tails, the upper bound and the lower bound are sharp in that they differ only by a multiplicative factor.

scholarly journals Iterative estimation of total π-electron energy

2005 ◽  
Vol 70 (10) ◽  
pp. 1193-1197 ◽  
Author(s):  
Lemi Türker ◽  
Ivan Gutman
Keyword(s):  
Lower Bound ◽  
Electron Energy ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Benzenoid Hydrocarbons ◽  
Total Electron ◽  
Iterative Estimation ◽  
Total Electron Energy ◽  
Almost All

In this work, the lower and upper bounds for total ?-electron energy (E) was studied. A method is presented, by means of which, starting with a lower bound EL and an upper bound EU for E, a sequence of auxiliary quantities E0 E1, E2,? is computed, such that E0 = EL, E0 < E1 < E2 < ?, and E = EU. Therefore, an integer k exists, such that Ek E < Ek+1. If the estimates EL and EU are of the McClelland type, then k is called the McClelland number. For almost all benzenoid hydrocarbons, k = 3.

Concerning Analysis of Central Burst in Metal Forming

1978 ◽  
Vol 100 (3) ◽  
pp. 386-387 ◽  
Author(s):  
E. H. Lee ◽  
R. M. McMeeking
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Metal Forming ◽  
Driving Force ◽  
Extrusion Process ◽  
Work Piece ◽  
Upper Bound Theorem ◽  
Forming Defect ◽  
Correct Analysis ◽  
Bound Theorem

A criterion for the generation of internal cracks normal to and astride the central axis in a drawing or extrusion process has been based on the conditions requiring less driving force with such cracks than without. Application of the upper bound theorem of limit analysis to approximate these forces yields a criterion for central burst. However, use of the lower bound theorem indicates that the cracked work-piece will always deform with less force, so that such a criterion would always predict the occurrence of this forming defect. Energy required to initiate and extend the crack must be included for a correct analysis.

BOUNDS ON THE MINIMAL SUMSET SIZE FUNCTION IN GROUPS

2007 ◽  
Vol 03 (04) ◽  
pp. 503-511 ◽  
Author(s):  
SHALOM ELIAHOU ◽  
MICHEL KERVAIRE
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Solvable Groups ◽  
Upper Bounds ◽  
Minimal Size ◽  
Lower And Upper Bounds ◽  
Numerical Function ◽  
Size Function

In this paper, we give lower and upper bounds for the minimal size μG(r,s) of the sumset (or product set) of two finite subsets of given cardinalities r,s in a group G. Our upper bound holds for solvable groups, our lower bound for arbitrary groups. The results are expressed in terms of variants of the numerical function κG(r,s), a generalization of the Hopf–Stiefel function that, as shown in [6], exactly models μG(r,s) for G abelian.

scholarly journals New bounds and algorithms for on-line scheduling: two identical processors, known sum and upper bound on the tasks

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Enrico Angelelli ◽  
Maria Grazia Speranza ◽  
Zsolt Tuza
Keyword(s):  
Upper Bound ◽  
Completion Time ◽  
Optimal Algorithm ◽  
Multiprocessor Scheduling ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
International Audience ◽  
On Line ◽  
Maximum Completion Time ◽  
New Algorithms

International audience In this paper we study a semi on-line version of the classical multiprocessor scheduling problem on two identical processors. We assume that the sum of the tasks and an upper bound gamma on the size of each task are known. Each task has to be assigned upon arrival and the assignment cannot be changed later. The objective is the minimization of the maximum completion time on the processors. In this paper we propose new algorithms and improve known lower and upper bounds on the competitive ratio. Algorithms and bounds depend on the value of gamma. An optimal algorithm is obtained for gamma in the interval [ 1/n,2(n+1)/n(2n+1) ] and gamma = (2n-1)/2n(n-1), where n is any integer value larger or equal 2.

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