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 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.

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.

Linear bounds on characteristic polynomials of matroids

2019 ◽  
Vol 168 (3) ◽  
pp. 505-518
Author(s):  
SUIJIE WANG ◽  
YEONG–NAN YEH ◽  
FENGWEI ZHOU
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Upper Bound ◽  
Hyperplane Arrangement ◽  
Characteristic Polynomials ◽  
Combinatorial Interpretation ◽  
Upper Bound Theorem ◽  
Arrangement Of Hyperplanes ◽  
Bound Theorem ◽  
Image Position

AbstractLet χ(t) = a0tn – a1tn−1 + ⋯ + (−1)rartn−r be the chromatic polynomial of a graph, the characteristic polynomial of a matroid, or the characteristic polynomial of an arrangement of hyperplanes. For any integer k = 0, 1, …, r and real number x ⩾ k − r − 1, we obtain a linear bound of the coefficient sequence, that is \begin{align*} {r+x\choose k}\leqslant \sum_{i=0}^{k}a_{i}{x\choose k-i}\leqslant {m+x\choose k}, \end{align*} where m is the size of the graph, matroid, or hyperplane arrangement. It extends Whitney’s sign-alternating theorem, Meredith’s upper bound theorem, and Dowling and Wilson’s lower bound theorem on the coefficient sequence. In the end, we also propose a problem on the combinatorial interpretation of the above inequality.

Selection of the Optimal Distribution for the Upper Bound Theorem in Indentation Processes

2014 ◽  
Vol 797 ◽  
pp. 117-122 ◽  
Author(s):  
Carolina Bermudo ◽  
F. Martín ◽  
Lorenzo Sevilla
Keyword(s):  
Upper Bound ◽  
Geometric Distribution ◽  
Optimal Choice ◽  
Optimal Distribution ◽  
Upper Bound Theorem ◽  
Modular Model ◽  
Bound Theorem ◽  
Rigid Zones ◽  
Dimensionless Relation ◽  
Selection Of

It has been established, in previous studies, the best adaptation and solution for the implementation of the modular model, being the current choice based on the minimization of the p/2k dimensionless relation obtained for each one of the model, analyzed under the same boundary conditions and efforts. Among the different cases covered, this paper shows the study for the optimal choice of the geometric distribution of zones. The Upper Bound Theorem (UBT) by its Triangular Rigid Zones (TRZ) consideration, under modular distribution, is applied to indentation processes. To extend the application of the model, cases of different thicknesses are considered

Upper Bound Solutions to Plane-Strain Extrusion Problems

1970 ◽  
Vol 92 (1) ◽  
pp. 158-164 ◽  
Author(s):  
P. C. T. Chen
Keyword(s):  
Plane Strain ◽  
Upper Bound ◽  
Material Properties ◽  
Incompressible Material ◽  
Velocity Fields ◽  
Deformation Pattern ◽  
Upper Bound Theorem ◽  
Admissible Velocity ◽  
Die Geometry ◽  
Bound Theorem

A method for selecting admissible velocity fields is presented for incompressible material. As illustrations, extrusion processes through three basic types of curved dies have been treated: cosine, elliptic, and hyperbolic. Upper-bound theorem is used in obtaining mean extrusion pressures and also in choosing the most suitable deformation pattern for extrusion through square dies. Effects of die geometry, friction, and material properties are discussed.

Hardening effect analysis in indentation processes by modular configuration of the upper bound theorem

2017 ◽  
Vol 10 (1) ◽  
pp. 59
Author(s):  
Carolina Bermudo Gamboa ◽  
Francisco De Sales Martín Fernández ◽  
Lorenzo Sevilla Hurtado
Keyword(s):  
Upper Bound ◽  
Upper Bound Theorem ◽  
Effect Analysis ◽  
Hardening Effect ◽  
Bound Theorem

scholarly journals Influence of Anisotropy and Nonhomogeneity on Stability Analysis of Fissured Slopes Subjected to Seismic Action

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Zhidan Liu ◽  
Jingwu Zhang ◽  
Weiping Chen ◽  
Di Wu
Keyword(s):  
Upper Bound ◽  
Mathematical Optimization ◽  
Optimization Procedure ◽  
Seismic Action ◽  
Anisotropy Coefficient ◽  
Upper Bound Theorem ◽  
Bound Theorem ◽  
Software Codes ◽  
The Stability ◽  
The Impact

Based on the upper bound theorem of limit analysis, this paper presents a procedure for assessment of the influence of the soil anisotropy and nonhomogeneity on the stability of fissured slopes subjected to seismic action. By means of a mathematical optimization procedure written in Matlab software codes, the stability factors NS and λcφ are derived with respect to the best upper bound solutions. A series of stability charts are obtained in this paper, and then the critical locations of cracks are determined for cracks of known depth. The results demonstrate a significant influence of the soil anisotropy and nonhomogeneity on the stability of the fissured slopes and the location distribution of the cracks. In addition, the procedures for getting the factor of safety are put forward. It is shown that a decrease in the nonhomogeneity coefficient n0 and an increase in the anisotropy coefficient k could lead to the fissured slopes becoming unsafe. Finally, this article also illustrates the variation in the safety factor of fissured slopes under the impact of three factors (Kh, H1/H, and λ).

scholarly journals The Gravitational Compression of an Elastic Sphere

1955 ◽  
Vol 8 (1) ◽  
pp. 167
Author(s):  
A Keane
Keyword(s):  
Lower Bound ◽  
Elastic Material ◽  
Upper Bound ◽  
Moment Of Inertia ◽  
Elastic Sphere ◽  
The Moment ◽  
Gravitational Compression ◽  
Gravitational Equilibrium

On considering a sphere in hydrostatic gravitational equilibrium, composed of a homogeneous elastic material for which the variation of incompressibility x with pressure p is given by dx/dp=n, a constant, we find that there is an upper bound to the radius R of the sphere provided n=2, and that for all values of n there is a lower bound to the value of I/MR2, where I is the moment of inertia about a diameter and M is the mass of the sphere.

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