Andre Permutations, Lexicographic Shellability and the cd-Index of a Convex Polytope

1993 ◽  
Vol 338 (1) ◽  
pp. 77 ◽  
Author(s):  
Mark Purtill
Keyword(s):  
Convex Polytope ◽  
Lexicographic Shellability

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

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 $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

scholarly journals 10 Points in Dimension 4 not Projectively Equivalent to the Vertices of a Convex Polytope

2001 ◽  
Vol 22 (5) ◽  
pp. 705-708 ◽  
Author(s):  
David Forge ◽  
Michel Las Vergnas ◽  
Peter Schuchert
Keyword(s):  
Convex Polytope

scholarly journals On the extreme points of a certain convex polytope

1970 ◽  
Vol 8 (4) ◽  
pp. 417-423 ◽  
Author(s):  
M. Katz
Keyword(s):  
Convex Polytope ◽  
Extreme Points

scholarly journals Calculus rules for combinations of ellipsoids and applications

1993 ◽  
Vol 47 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alberto Seeger
Keyword(s):  
Convex Hull ◽  
Convex Polytope ◽  
Finite Family ◽  
Minkowski Sum ◽  
Calculus Rules ◽  
Ellipsoidal Approximations

We derive formulas for the Minkowski sum, the convex hull, the intersection, and the inverse sum of a finite family of ellipsoids. We show how these formulas can be used to obtain inner and outer ellipsoidal approximations of a convex polytope.

scholarly journals Geometry of Multiprimary Display Colors II: Metameric Control Sets and Gamut Tiling Color Control Functions

2021 ◽  
Author(s):  
Carlos Rodriguez-Pardo ◽  
Gaurav Sharma
Keyword(s):  
Broad Class ◽  
Convex Polytope ◽  
Building Blocks ◽  
Complete Characterization ◽  
Mathematical Framework ◽  
Control Functions ◽  
Control Sets ◽  
Multiple Alternative ◽  
Color Control

<div>For multiprimary displays that have four or more primaries, a color may be reproduced using multiple alternative control vectors. We provide a complete characterization of the Metameric Control Set (MCS), i.e., the set of control vectors that reproduce a given color on the display. Specifically, we show that MCS is a convex polytope whose vertices are control vectors obtained from (parallelepiped) tilings of the gamut, i.e., the range of colors that the display can produce. The mathematical framework that we develop: (a) characterizes gamut tilings in terms of fundamental building blocks called facet spans, (b) establishes that the vertices of the MCS are fully characterized by the tilings of the gamut, and (c) introduces a methodology for the efficient enumeration of gamut tilings. The framework reveals the fundamental inter-relations between the geometry of the MCS and the geometry of the gamut developed in a companion Part I paper, and provides insight into alternative strategies for color control. Our characterization of tilings and the strategy for their enumeration also advance knowledge in geometry, providing new approaches and computational results for the enumeration of tilings for a broad class of zonotopes in R<sup>3</sup>.</div>

scholarly journals Integration on a convex polytope

1998 ◽  
Vol 126 (8) ◽  
pp. 2433-2441 ◽  
Author(s):  
Jean B. Lasserre
Keyword(s):  
Convex Polytope

Free Extensions of Chiral Polytopes

1995 ◽  
Vol 47 (3) ◽  
pp. 641-654 ◽  
Author(s):  
Egon Schulte ◽  
Asia Ivić Weiss
Keyword(s):  
Classical Theory ◽  
Convex Polytope ◽  
Regular Polytopes ◽  
Geometric Structures ◽  
Maximal Symmetry ◽  
Abstract Polytopes ◽  
Chiral Polytopes ◽  
Classical Notion ◽  
Extension Result ◽  
Amalgamated Product

AbstractAbstract polytopes are discrete geometric structures which generalize the classical notion of a convex polytope. Chiral polytopes are those abstract polytopes which have maximal symmetry by rotation, in contrast to the abstract regular polytopes which have maximal symmetry by reflection. Chirality is a fascinating phenomenon which does not occur in the classical theory. The paper proves the following general extension result for chiral polytopes. If 𝒦 is a chiral polytope with regular facets 𝓕 then among all chiral polytopes with facets 𝒦 there is a universal such polytope 𝓟, whose group is a certain amalgamated product of the groups of 𝒦 and 𝓕. Finite extensions are also discussed.

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