Abstract
Recently, Postnikov introduced Bert Kostant’s game to build the maximal positive root associated with the quadratic form of a simple graph. This result, and some other games based on Cartan matrices, give a new version of Gabriel’s theorem regarding algebras classification. In this paper, as a variation of Bert Kostant’s game, we introduce a wargame based on a missile defense system (MDS). In this case, missile trajectories are interpreted as suitable paths of a quiver (directed graph). The MDS protects a region of the Euclidean plane by firing missiles from a ground-based interceptor (GBI) located at the point (0,0). In this case, a missile success interception occurs if a suitable positive number associated with the launches of the enemy army can be written as a mixed sum of triangular and square numbers.
Highlights
In 2017, Green and Schroll introduced Brauer configuration algebras (BCAs) as a generalization of Brauer graph algebras, which are biserial algebras whose theory of representation is induced by a Brauer configuration containing combinatorial data suitable to establish their theory of representation
Interactions between research regarding Gabriel’s theorem on the classification of algebras, BCAs, and universal sums of square and triangular numbers are used to define a wargame based on the behavior of a missile defense system (MDS); a player in this game is declared a winner if some suitable positive integer can be expressed as a mixed sum of triangular and square numbers
We define a wargame as a variation of Bert Kostant’s game, whose outcomes can be described in terms of mixed sums of triangular and square numbers
Summary
In 2017, Green and Schroll introduced Brauer configuration algebras (BCAs) as a generalization of Brauer graph algebras, which are biserial algebras whose theory of representation is induced by a Brauer configuration containing combinatorial data suitable to establish their theory of representation. Interactions between research regarding Gabriel’s theorem on the classification of algebras, BCAs, and universal sums of square and triangular numbers are used to define a wargame based on the behavior of a missile defense system (MDS); a player in this game is declared a winner if some suitable positive integer can be expressed as a mixed sum of triangular and square numbers. The idea behind this game arises from a series of lectures [5] held by Postnikov on some combinatorial topics in 2018 at the Massachusetts. A graph neural network (GNN) is implemented to model the chain of command and communications in wargames
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