Related papers: General Board Geometry
Without further ado, we present the P_3-game. The P_3-game is decidable for elementary classes of graphs such as paths and cycles. From an algorithmic point of view, the connected P_3-game is fascinating. We show that the connected P_3-game…
In the case of simple graded manifolds utilized in supermechanics, supervector fields and exterior superforms are represented by global sections of smooth vector bundles.
Examining games from a fresh perspective we present the idea of game-inspired and game-based algorithms, dubbed "gamorithms".
A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…
We give a systematic construction of uniruled surfaces in positive characteristic. Using this construction, we find surfaces of general type with non-trivial vector fields, surfaces with arbitrarily non-reduced Picard schemes as well as…
The context of this paper is the use of formal methods for topology-based geometric modelling. Topology-based geometric modelling deals with objects of various dimensions and shapes. Usually, objects are defined by a graph-based topological…
We introduce a geometric construction of a gauge field theory of a complex adaptive system. It is based on a suitable simplicial formulation of a discrete geometry that manifests relevant properties valid in the classical differentiable…
The vertices of a $k$-token graph of a graph $G$ correspond to $k$ indistinguishable tokens placed on $k$ different vertices of $G$. Changing some conditions on both the nature of the tokens and the number of tokens allowed in each vertex…
By using the representational power of Chu spaces we define the notion of a generalized topological space (or GTS, for short), i.e., a mathematical structure that generalizes the notion of a topological space. We demonstrate that these…
Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of…
The problem of finding the general classification of geodetic graphs is still open. We believe that one of the obstacles to attain this goal is that geodetic graphs lack a structural description. In other words, their fundamental properties…
The generalized Markoff mod $p$ graph is defined via the equation $x^2+y^2+z^2=xyz+\kappa$ over the finite field $\mathbb{F}_p$ of prime order $p$. In this paper, we investigate the topological properties of the graph such as non-planarity,…
The notions of captured/lost vertices and dead edges in the Shannon game (Shannon switching game on nodes) are examined using graph theory. Simple methods are presented for identifying some dead edges and some captured sets of vertices,…
In this text, we outline a theory of schemes associated with a site, which generalizes a variety of geometries, such as manifolds, schemes, analytic spaces, simplicial complexes, and more. We present an abstract process of gluing model…
In this paper, we introduce a two-player impartial game on graphs, called a {\em feedback game}, which is a variant of the generalized geography. We study the feedback game on Eulerian graphs. In particular, we show that the…
In this article we discuss a connection between two famous constructions in mathematics: a Cayley graph of a group and a (rational) billiard surface. For each rational billiard surface, there is a natural way to draw a Cayley graph of a…
We define the Sign Game as a two-player game played on a simple undirected mathematical graph $G$. The players alternate turns, assigning vertices of $G$ either $1$ or $-1$, and edges take on the value of the product of their endvertices.…
Generalized cycles can be thought of as the extension of form-cycle duality between holomorphic forms and cycles, to meromorphic forms and generalized cycles. They appeared as an ubiquitous tool in the study of spectral curves and…
The notion of a generalized Lie bialgebroid (a generalization of the notion of a Lie bialgebroid) is introduced in such a way that a Jacobi manifold has associated a canonical generalized Lie bialgebroid. As a kind of converse, we prove…
We present the technical side of reasoning in Regular Boardgames (RBG) language -- a universal General Game Playing (GGP) formalism for the class of finite deterministic games with perfect information, encoding rules in the form of regular…