Related papers: Fa\`a di Bruno for operads and internal algebras
In this paper we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs,…
We introduce dicodensity monads: a generalisation of pointwise codensity monads generated by functors to monads generated by mixed-variant bifunctors. Our construction is based on the notion of strong dinaturality (also known as Barr…
Summation formulas are obtained for products of associated Lagurre polynomials by means of the Green's function K for the Hamiltonian H = -{d^2\over dx^2} + x^2 + Ax^{-2}, A > 0. K is constructed by an application of a Mercer type theorem…
This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional…
As a fundamental and ubiquitous combinatorial notion, species has attracted sustained interest, generalizing from set-theoretical combinatorial to algebraic combinatorial and beyond. The Rota-Baxter algebra is one of the algebraic…
The theme of this article is the algebraic combinatorics of leaf-labeled rooted binary trees and forests of such trees. The structure of a Hopf operad is defined on the vector spaces spanned by forests of leaf-labeled, rooted, binary trees.…
The natural Hopf algebra $\mathbf{N} \cdot \mathcal{O}$ of an operad $\mathcal{O}$ is a Hopf algebra whose bases are indexed by some words on $\mathcal{O}$. We construct polynomial realizations of $\mathbf{N} \cdot \mathcal{O}$ by using…
The discrete Green's functions are the pseudoinverse (or the inverse) of the Laplacian (or its variations) of a graph. In this paper, we will give combinatorial interpretations of Green's functions in terms of enumerating trees and forests…
The method of generating functional is generalized to the case of strongly correlated systems, and applied to the Hubbard model. For the electronic Green's function constructed for Hubbard operators, an equation using variational…
We introduce the $\mathcal{T}$-construction, an endofunctor on the category of generalized operads as a general mechanism by which various notions of plethystic substitution arise from more ordinary notions of substitution. In the special…
We prove sharp results about recurrent behaviour of orbits of forward compositions of inner functions, inspired by fundamental results about iterates of inner functions, and give examples to illustrate behaviours that cannot occur in the…
The aim of this paper is to further explore an idea from J.-L. Loday briefly exposed in [5]. We impose a natural and simple symmetry on a unit action over the most general quadratic relation which can be written. This leads us to two…
We explore monoids generated by operators on certain infinite partial orders. Our starting point is the work of Fomin and Greene on monoids satisfying the relations $(\u{r}+\u{r+1})\u{r+1}\u{r}=\u{r+1}\u{r}(\u{r}+\u{r+1})$ and…
We study a discrete model of the Laplacian in $\mathbb{R}^2$ that preserves the geometric structure of the original continual object. This means that, speaking of a discrete model, we do not mean just the direct replacement of differential…
We give a Hopf-algebraic formulation of the $R^*$-operation, which is a canonical way to render UV and IR divergent Euclidean Feynman diagrams finite. Our analysis uncovers a close connection to Brown's Hopf algebra of motic graphs. Using…
If h is a ring-valued function on a simplicial complex G we can define two matrices L and g, where the matrix entries are the h energy of homoclinic intersections. We know that the sum over all h values on G is equal to the sum of the Green…
These notes provide an informal introduction to a type of Mackey functor that arises naturally in algebraic topology in connection with Morava $K$-theory of classifying spaces of finite groups. The main aim is to identify key algebraic…
Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We…
We extend the Cisinski-Moerdijk-Weiss theory of $\infty$-operads to the equivariant setting to obtain a notion of $G$-$\infty$-operads that encode "equivariant operads with norm maps" up to homotopy. At the root of this work is the…
We argue that operads provide a general framework for dealing with polynomials and combinatory completeness of combinatory algebras, including the classical $\mathbf{SK}$-algebras, linear $\mathbf{BCI}$-algebras, planar…