scholarly journals A variational solution of the A. D. Aleksandrov problem of existence of a convex polytope with prescribed Gauss curvature

2005 ◽  
Author(s):  
Vladimir Oliker
Keyword(s):  
Convex Polytope ◽  
Gauss Curvature ◽  
Variational Solution ◽  
Aleksandrov Problem

scholarly journals Immersed surfaces of prescribed Gauss curvature into Minkowski space

2000 ◽  
Vol 129 (7) ◽  
pp. 2093-2101
Author(s):  
Yuxin Ge
Keyword(s):  
Minkowski Space ◽  
Gauss Curvature ◽  
Immersed Surfaces

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

Convexified Gauss Curvature flow of Sets: A Stochastic Approximation

2004 ◽  
Vol 36 (2) ◽  
pp. 552-579 ◽  
Author(s):  
Hitoshi Ishii ◽  
Toshio Mikami
Keyword(s):  
Stochastic Approximation ◽  
Gauss Curvature ◽  
Curvature Flow ◽  
Gauss Curvature Flow

Horizontal Gauss curvature flow of graphs in Carnot groups

2011 ◽  
Vol 60 (4) ◽  
pp. 1267-1302 ◽  
Author(s):  
Erin Haller Martin
Keyword(s):  
Gauss Curvature ◽  
Curvature Flow ◽  
Carnot Groups ◽  
Gauss Curvature Flow

Regularity of almost extremal solutions of Monge–Ampère equations

1991 ◽  
Vol 117 (1-2) ◽  
pp. 21-29 ◽  
Author(s):  
John I. E. Urbas
Keyword(s):  
Large Class ◽  
Convex Domain ◽  
Gauss Curvature ◽  
Extremal Solutions ◽  
Classical Solutions ◽  
Uniformly Convex ◽  
Suitable Sense ◽  
Monge Ampere

SynopsisWe show that for a large class of Monge-Ampère equations, generalised solutions on a uniformly convex domain Ω⊂ℝn are classical solutions on any pre-assigned subdomain Ω′⋐Ω, provided the solution is almost extremal in a suitable sense. Alternatively, classical regularity holds on subdomains of Ω which are sufficiently distant from ∂Ω. We also show that classical regularity may fail to hold near ∂Ω in the nonextremal case. The main example of the class of equations considered is the equation of prescribed Gauss curvature.

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