Related papers: The algebra of balanced dessins
Algebraic scheme for constructing deformations of structure constants for associative algebras generated by a deformation driving algebras (DDAs) is discussed. An ideal of left divisors of zero plays a central role in this construction.…
A non-aligned drawing of a graph is a drawing where no two vertices are in the same row or column. Auber et al. showed that not all planar graphs have non-aligned drawings that are straight-line, planar, and in the minimal-possible $n\times…
An abstract mathematical framework is presented in this paper as a unification of several deformed or generalized algebra proposed recently in the context of generalized statistical theories intended to treat certain complex thermodynamic…
In this paper we study the geometry of graph spaces endowed with a special class of graph edit distances. The focus is on geometrical results useful for statistical pattern recognition. The main result is the Graph Representation Theorem.…
This research addresses a new tool for data analysis known as Topological Data Analysis TDA It underlies an area of Mathematics known as Combinatorial Algebra or more recently Algebraic Topology which through making strong use of…
The Erd\H{o}s discrepancy problem, now a theorem by T. Tao, asks whether every sequence with values plus or minus one has unbounded discrepancy along all homogeneous arithmetic progressions. We establish weighted variants of this problem,…
We define nodal finite dimensional algebras and describe their structure over an algebraically closed field. For a special class of such algebras (type A) we find a criterion of tameness.
The goal of this paper is to experiment new math concepts and theories, especially if they run counter to the classical ones. To prove that contradiction is not a catastrophe, and to learn to handle it in an (un)usual way. To transform the…
The notion of "balance" is fundamental for sociologists who study social networks. In formal mathematical terms, it concerns the distribution of triad configurations in actual networks compared to random networks of the same edge density.…
We develop a linear-algebraic framework for dimensional analysis in systems with constraints, particularly when variables are numerous or related by implicit relations so that direct elimination is impractical. By expressing both…
The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the…
This paper focuses on defining a measure, appropriate for obtaining optimally sparse solutions to underdetermined systems of linear equations.* The general idea is the extension of metrics in n-dimensional spaces via the Cartesian product…
The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this…
The paper considers pseudo-differential boundary value control systems. The underlying operators form an algebra D with the help of which we are able to formulate typical boundary value control problems. The symbolic calculus gives tools to…
This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text…
We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that were used to construct canonical bases in…
Let (W,S) be a Coxeter system of affine type D, and let TL(W) the corresponding generalized Temperley-Lieb algebra. In this extended abstract we define an infinite dimensional associative algebra made of decorated diagrams which is…
We exhibit an adjunction between a category of abstract algebras of partial functions that we call difference-restriction algebras and a category of Hausdorff \'etale spaces. Difference-restriction algebras are those algebras isomorphic to…
Statistical graph models aim at modeling graphs as random realization among a set of possible graphs. One issue is to evaluate whether or not a graph is likely to have been generated by one particular model. In this paper we introduce the…
Directed mixed graphs permit directed and bidirected edges between any two vertices. They were first considered in the path analysis developed by Sewall Wright and play an essential role in statistical modeling. We introduce a matrix…