Comment on “A Gibbs Sampler for a Class of Random Convex Polytopes” by P.E. Jacob, R. Gong, P.T. Edlefsen and A.P. Dempster

2021 ◽  
Vol 116 (535) ◽  
pp. 1204-1205
Author(s):  
Fabrizio Ruggeri
Keyword(s):  
Gibbs Sampler ◽  
Convex Polytopes ◽  
Random Convex

scholarly journals A Gibbs Sampler for a Class of Random Convex Polytopes

2021 ◽  
pp. 1-12
Author(s):  
Pierre E. Jacob ◽  
Ruobin Gong ◽  
Paul T. Edlefsen ◽  
Arthur P. Dempster
Keyword(s):  
Gibbs Sampler ◽  
Convex Polytopes ◽  
Random Convex

Rejoinder—A Gibbs Sampler for a Class of Random Convex Polytopes

2021 ◽  
Vol 116 (535) ◽  
pp. 1211-1214
Author(s):  
Pierre E. Jacob ◽  
Ruobin Gong ◽  
Paul T. Edlefsen ◽  
Arthur P. Dempster
Keyword(s):  
Gibbs Sampler ◽  
Convex Polytopes ◽  
Random Convex

scholarly journals Comments on “A Gibbs Sampler for a Class of Random Convex Polytopes”

2021 ◽  
Vol 116 (535) ◽  
pp. 1206-1210
Author(s):  
Kentaro Hoffman ◽  
Jan Hannig ◽  
Kai Zhang
Keyword(s):  
Gibbs Sampler ◽  
Convex Polytopes ◽  
Random Convex

scholarly journals A Class of Random Convex Polytopes

1972 ◽  
Vol 43 (1) ◽  
pp. 260-272 ◽  
Author(s):  
A. P. Dempster
Keyword(s):  
Convex Polytopes ◽  
Random Convex

The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials

2019 ◽  
Vol 16 (2) ◽  
pp. 1
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’Aini Aris ◽  
Azlina Jumadi
Keyword(s):  
Convex Polytopes ◽  
Polynomial Systems ◽  
Matrix Formulation ◽  
Bivariate Polynomials ◽  
Homogeneous Coordinates ◽  
The Matrix ◽  
Projective Vector ◽  
Coordinate Rings ◽  
Resultant Matrix ◽  
The Given

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

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