Related papers: Input/output coloring and Gr\"obner basis for diop…

A tree $T$, in an edge-colored graph $G$, is called {\em a rainbow tree} if no two edges of $T$ are assigned the same color. A {\em $k$-rainbow coloring}of $G$ is an edge coloring of $G$ having the property that for every set $S$ of $k$…

We introduce a functor ${\sf As}$ from the category of posets to the category of nonsymmetric binary and quadratic operads, establishing a new connection between these two categories. Each operad obtained by the construction ${\sf As}$…

We investigate algebras with one operation. We study when these algebras form a monoidal category and analyze Koszulness and cyclicity of the corresponding operads. We also introduce a new kind of symmetry for operads, the dihedrality,…

We introduce a vector differential operator $\mathbf{P}$ and a vector boundary operator $\mathbf{B}$ to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This…

In the first part, we give an explicit description of the cotangent complex of differential graded (dg) operads, modeled as an operadic infinitesimal bimodule. This leads to a uniform formula for the Quillen cohomology of their associated…

We construct growth bijections for bipolar oriented planar maps and for Schnyder woods. These give direct combinatorial proofs of several counting identities for these objects. Our method mainly uses two ingredients. First, a slit-slide-sew…

We introduce a Banach Lie group $G$ of unitary operators subject to a natural trace condition. We compute the homotopy groups of $G$, describe its cohomology and construct an $S^1$-central extension. We show that the central extension…

The set of common roots of a finite set $I$ (it is an ideal) of homogeneous polynomials is known as projective algebraic set $V$. In this article I show how to dualize such projective algebraic sets $V$ by elimination of variables from a…

We study differential operators associated with families of polynomials orthonormal with respect to certain measures. These operators, when applied to the Fourier transforms of such measures, produce basis functions for expansions of…

We provide general conditions under which the algebras for a coloured operad in a monoidal model category carry a Quillen model structure, and prove a Comparison Theorem to the effect that a weak equivalence between suitable such operads…

The equivalence between dg duality and Verdier duality has been established for cyclic operads earlier. We propose a generalization of this correspondence from cyclic operads and dg duality to twisted modular operads and Feynman transform.…

As observed by Joyal, the cohomology groups of the partition posets are naturally identified with the components of the operad encoding Lie algebras. This connection was explained in terms of operadic Koszul duality by Fresse, and later…

Based on a variation of the BIP operational semantics using the offer predicate introduced in our previous work, we extend the algebras used to model glue operators in BIP to encompass priorities. This extension uses the Algebra of Causal…

We present a special and attractive basis for the exceptional Lie algebra $G_2$, which turns $G_2$ into a $\mathbb{Z}_2^3$-graded Lie algebra. There are two basis elements for each degree of $\mathbb{Z}_2^3\setminus\{(0,0,0)\}$, thus…

The Lie algebra generated by $m\ $ $p$-dimensional Grassmannian Dirac operators and $m\ $ $p$-dimensional vector variables is identified as the orthogonal Lie algebra $\mathfrak{so}(2m+1)$. In this paper, we study the space $\mathcal{P}$ of…

We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space.…

We describe the proalgebraic groups represented by three Hopf algebras on planar binary trees previously introduced by the author and Christian Brouder in relation with the renormalization of quantum electrodynamics. Using two monoidal…

Consider a formally self-adjoint first order linear differential operator acting on pairs (2-columns) of complex-valued scalar fields over a 4-manifold without boundary. We examine the geometric content of such an operator and show that it…

We show that the edges of any graph $G$ containing two edge-disjoint spanning trees can be blue/red coloured so that the blue and red graphs are connected and the blue and red degrees at each vertex differ by at most four. This improves a…

The aim of this paper is to define cubic Dirac operators for colour Lie algebras. We give a necessary and sufficient condition to construct a colour Lie algebra from an $\epsilon$-orthogonal representation of an $\epsilon$-quadratic colour…