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Endomorphisms algebras can replace the concept of principal fiber bundle. Gauge theories are reformulated within this algebraic framework and further generalized to unify ordinary connections and Higgs fields. A 'noncommutative Maxwell'…

We consider the problem of estimating the intersection multiplicity between an algebraic variety and a Pfaffian foliation, at every point of the variety. We show that this multiplicity can be majorized at every point $p$ by the local…

We introduce priors and algorithms to perform Bayesian inference in Gaussian models defined by acyclic directed mixed graphs. Such a class of graphs, composed of directed and bi-directed edges, is a representation of conditional…

Let $\mathbb{Z}_{p^s}$ be the residue class ring of integers modulo $p^s$, where $p$ is a prime number and $s$ is a positive integer. Using matrix representation and the inner rank of a matrix, we study the intersection, join, dimension…

Universal cycles, such as De Bruijn cycles, are cyclic sequences of symbols that represent every combinatorial object from some family exactly once as a consecutive subsequence. Graph universal cycles are a graph analogue of universal…

An operation on species corresponding to the inner plethysm of their associated cycle index series is constructed. This operation, the inner plethysm of species, is generalized to n-sorted species. Polynomial maps on species are studied and…

Using the celebrated Witten-Kontsevich theorem, we prove a recursive formula of the $n$-point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us…

We show that recent determinant evaluations involving Catalan numbers and generalisations thereof have most convenient explanations by combining the Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a simple…

In this work we consider a suitable generalization of the Feynman path integral on a specific class of Riemannian manifolds consisting of compact Lie groups with bi-invariant Riemannian metrics. The main tools we use are the Cartan…

We prove the famous Faber intersection number conjecture and other more general results by using a recursion formula of $n$-point functions for intersection numbers on moduli spaces of curves. We also present some vanishing properties of…

The natural habitat of most Bayesian methods is data represented by exchangeable sequences of observations, for which de Finetti's theorem provides the theoretical foundation. Dirichlet process clustering, Gaussian process regression, and…

We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by F\"urer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of…

The particular structure of Galileon interactions allows for higher-derivative terms while retaining second order field equations for scalar fields and Abelian $p$-forms. In this work we introduce an index-free formulation of these…

This is a simple mathematical introduction into Feynman diagram technique, which is a standard physical tool to write perturbative expansions of path integrals near a critical point of the action. I start from a rigorous treatment of a…

We give a new proof of the trace formula for regular graphs. Our approach is inspired by path integral approach in quantum mechanics, and calculations are mostly combinatorial.

This works deals with the formal mathematical structure of so called Grassmann Numbers applied to Theoretical Physics, which is a basic concept on Berezin integration. To achieve this purpose we make use of some constructions from relative…

This work is devoted to the study of the relationships between graph theory and the qualitative analysis of ordinary differential equations, with a special focus on two-dimensional systems. In particular, we reinterpret classical results…

We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and…

Paragrassmann algebras with one and many paragrassmann variables are considered from the algebraic point of view without using the Green ansatz. Operators of differentiation with respect to paragrassmann variables and a covariant…

We present an exact formula for the ordinary generating series of the simple paths between any two vertices of a graph. Our formula involves the adjacency matrix of the connected induced subgraphs and remains valid on weighted and directed…