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Evolutionary games on graphs play an important role in the study of evolution of cooperation in applied biology. Using rigorous mathematical concepts from a dynamical systems and graph theoretical point of view, we formalize the notions of…
Evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from Biology to Economics. Within this context, the approach of choice for many researchers is the so-called replicator…
Gradient vector fields are fundamental objects from both theoretical and practical perspectives, since various phenomena can be modeled within this framework. The ``moduli space'' of such vector fields provides the foundation for describing…
We construct a fully faithful functor from the category of graphs to the category of fields. Using this functor, we resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure S,…
We introduce the theory $\mathrm{PF}^{+,\times}$ of pseudofinite fields with generic additive and multiplicative character added as continuous logic predicates. Using the Weil bounds on character sums over finite fields as well as the…
The first-order theory of finite and infinite trees has been studied since the eighties, especially by the logic programming community. Following Djelloul, Dao and Fr\"uhwirth, we consider an extension of this theory with an additional…
We explicitly present expansions of the complex field which are models of the theories of green points in the multiplicative group case and in the case of an elliptic curve without complex multiplication defined over $\mathbb{R}$. In fact,…
There are several approaches for using computers in deriving mathematical proofs. For their illustration, we provide an in-depth study of using computer support for proving one complex combinatorial conjecture -- correctness of a strategy…
Combinatorial Game Theory is a branch of mathematics and theoretical computer science that studies sequential 2-player games with perfect information. Normal play is the convention where a player who cannot move loses. Here, we generalize…
Matrix field theory is a combinatorially non-local field theory which has recently been found to be a non-trivial but solvable QFT example. To generalize such non-perturbative structures to other models, a more combinatorial understanding…
This is a detailed survey -- with rigorous and self-contained proofs -- of some of the basics of elementary combinatorics and algebra, including the properties of finite sums, binomial coefficients, permutations and determinants. It is…
Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…
A new mathematical model for evolutionary games on graphs is proposed to extend the classical replicator equation to finite populations of players organized on a network with generic topology. Classical results from game theory,…
In this survey, we review some of the recent connections between the representation theory of (untwisted) quantum affine algebras and the representation theory of current algebras. We mainly focus on the finite-dimensional representations…
We establish a "diagonal" ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg's correspondence principle, prove that any "large" set in $K$ contains…
The strong-coupling character expansion of lattice models is reanalyzed in the perspective of its complete algorithmization. The geometric problem of identifying, counting, and grouping together all possible contributions is disentangled…
The first part of this note is a short introduction on continued fraction expansions for certain algebraic power series. In the last part, as an illustration, we present a family of algebraic continued fractions of degree 4, including a toy…
This work is a part of an ongoing effort to understand the relationships between properties used in theory combination. We here focus on including two properties that are related to shiny theories: the finite model property and stable…
Green's theorem states that the Hall algebra of the category of representations of a quiver over a finite field is a twisted bialgebra. Considering instead categories of orthogonal or symplectic quiver representations leads to a class of…
Some time ago, conformal data with affine fusion rules were found. Our purpose here is to realize some of these conformal data, using systems of free bosons and parafermions. The so constructed theories have an extended $W$ algebras which…