Obtaining combinatorial structures associated with low-dimensional Leibniz algebras

In this paper, we analyze the relation between Leibniz algebras and combinatorial structures. More concretely, we study the properties to be satisfied by (pseudo)digraphs so that they are associated with low-dimensional Leibniz algebras. We present some results related to this association and show an algorithmic method to obtain them, which has been implemented with Maple

Higman-Neumann-Neumann extension and embedding theorems for Leibniz algebras

In this work we introduce the Higman-Neumann-Neumann (HNN)- extensions and the appropriate embedding theorems for dialgebras and Leibniz algebras. Due to the importance of the connection between the dialgebras and Leibniz algebras and the relationship between associative algebras and Lie algebras, we recall the theory of Groebner-Shirshov bases, and the Composition-Diamond Lemma in associative algebras and Lie algebras, as well as the theory of Groebner-Shirshov bases for dialgebras. As an application of the HNN-extensions of dialgebras and Leibniz algebras, we provide embedding theorems for dialgebras and Leibniz algebras, respectively: every dialgebra embeds inside its any HNN-extension and every Leibniz algebra embeds inside its any HNN-extension

Uniformly automatic classes of finite structures

We investigate the recently introduced concept of uniformly tree-automatic classes in the realm of parameterized complexity theory. Roughly speaking, a class of finite structures is uniformly tree-automatic if it can be presented by a set of finite trees and a tuple of automata. A tree t encodes a structure and an element of this structure is encoded by a labeling of t. The automata are used to present the relations of the structure. We use this formalism to obtain algorithmic meta-theorems for first-order logic and in some cases also monadic second-order logic on classes of finite Boolean algebras, finite groups, and graphs of bounded tree-depth. Our main concern is the efficiency of this approach with respect to the hidden parameter dependence (size of the formula). We develop a method to analyze the complexity of uniformly tree-automatic presentations, which allows us to give upper bounds for the runtime of the automata-based model checking algorithm on the presented class. It turns out that the parameter dependence is elementary for all the above mentioned classes. Additionally we show that one can lift the FPT results, which are obtained by our method, from a class C to the closure of C under direct products with only a singly exponential blow-up in the parameter dependence

Combinatorial functional and differential equations applied to differential posets

We give combinatorial proofs of the primary results developed by Stanley for deriving enumerative properties of differential posets. In order to do this we extend the theory of combinatorial differential equations developed by Leroux and Viennot.Facultad de Ciencias ExactasLaboratorio de Investigación y Formación en Informática Avanzad

PolyLogTools - Polylogs for the masses

We review recent developments in the study of multiple polylogarithms, including the Hopf algebra of the multiple polylogarithms and the symbol map, as well as the construction of single valued multiple polylogarithms and discuss an algorithm for finding fibration bases. We document how these algorithms are implemented in the Mathematica package PolyLogTools and show how it can be used to study the coproduct structure of polylogarithmic expressions and how to compute iterated parametric integrals over polylogarithmic expressions that show up in Feynman integal computations at low loop orders.Comment: Package URL: https://gitlab.com/pltteam/pl

The Combinatorics of Non-determinism

A deep connection exists between the interleaving semantics of concurrent processes and increasingly labelled combinatorial structures. In this paper we further explore this connection by studying the rich combinatorics of partially increasing structures underlying the operator of non-deterministic choice. Following the symbolic method of analytic combinatorics, we study the size of the computation trees induced by typical non-deterministic processes, providing a precise quantitative measure of the so-called "combinatorial explosion" phenomenon. Alternatively, we can see non-deterministic choice as encoding a family of tree-like partial orders. Measuring the (rather large) size of this family on average offers a key witness to the expressiveness of the choice operator. As a practical outcome of our quantitative study, we describe an efficient algorithm for generating computation paths uniformly at random

Non-Associative Algebraic Structures: Classification and Structure

These are detailed notes for a lecture on "Non-associative Algebraic Structures: Classification and Structure" which I presented as a part of my Agrega\c{c}\~ao em Matem\'atica e Applica\c{c}\~oes (University of Beira Interior, Covilh\~a, Portugal, 13-14/03/2023)

Combinatorial Conversion and Moment Bisimulation for Stochastic Rewriting Systems

We develop a novel method to analyze the dynamics of stochastic rewriting systems evolving over finitary adhesive, extensive categories. Our formalism is based on the so-called rule algebra framework and exhibits an intimate relationship between the combinatorics of the rewriting rules (as encoded in the rule algebra) and the dynamics which these rules generate on observables (as encoded in the stochastic mechanics formalism). We introduce the concept of combinatorial conversion, whereby under certain technical conditions the evolution equation for (the exponential generating function of) the statistical moments of observables can be expressed as the action of certain differential operators on formal power series. This permits us to formulate the novel concept of moment-bisimulation, whereby two dynamical systems are compared in terms of their evolution of sets of observables that are in bijection. In particular, we exhibit non-trivial examples of graphical rewriting systems that are moment-bisimilar to certain discrete rewriting systems (such as branching processes or the larger class of stochastic chemical reaction systems). Our results point towards applications of a vast number of existing well-established exact and approximate analysis techniques developed for chemical reaction systems to the far richer class of general stochastic rewriting systems