Convex polytopes: Extremal constructions and 𝑓-vector shapes

The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials

Scientific Research Journal ◽

10.24191/srj.v16i2.9346 ◽

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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Bounds on the partition dimension of convex polytopes

Combinatorial Chemistry & High Throughput Screening ◽

10.2174/1386207323666201204144422 ◽

2020 ◽

Vol 23 ◽

Author(s):

Jia-Bao Liu ◽

Muhammad Faisal Nadeem ◽

Mohammad Azeem

Keyword(s):

Combinatorial Optimization ◽

Computer Science ◽

Facility Location ◽

Robot Navigation ◽

Convex Polytopes ◽

Pharmaceutical Chemistry ◽

Resolving Set ◽

Minimum Number ◽

Partition Dimension ◽

Np Problem

Aims and Objective: The idea of partition and resolving sets plays an important role in various areas of engineering, chemistry and computer science such as robot navigation, facility location, pharmaceutical chemistry, combinatorial optimization, networking, and mastermind game. Method: In a graph to obtain the exact location of a required vertex which is unique from all the vertices, several vertices are selected this is called resolving set and its generalization is called resolving partition, where selected vertices are in the form of subsets. Minimum number of partitions of the vertices into sets is called partition dimension. Results: It was proved that determining the partition dimension a graph is nondeterministic polynomial time (NP) problem. In this article, we find the partition dimension of convex polytopes and provide their bounds. Conclusion: The major contribution of this article is that, due to the complexity of computing the exact partition dimension we provides the bounds and show that all the graphs discussed in results have partition dimension either less or equals to 4, but it cannot been be greater than 4.

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A short note on extreme points of certain polytopes

Special Matrices ◽

10.1515/spma-2020-0005 ◽

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.

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