Related papers: Couplings and Matchings: Combinatorial notes on St…
We introduce a sequent calculus with a simple restriction of Lambek's product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a…
Reasoning modulo equivalences is natural for everyone, including mathematicians. Unfortunately, in proof assistants based on type theory, equality is appallingly syntactic and, as a result, exploiting equivalences is cumbersome at best.…
We state and prove a new closure theorem closely related to the classical closure theorems of Poncelet and Steiner. Along the way, we establish a number of theorems concerning conic sections.
We present a case study in {\it experimental} yet {\it rigorous} mathematics by describing an algorithm, fully implemented in both Mathematica and Maple, that {\it automatically conjectures}, and then {\it automatically proves}, closed-form…
We discuss recent progress many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.
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…
In the stable marriage problem N men and N women have to be matched by pairs under the constraint that the resulting matching is stable. We study the statistical properties of stable matchings in the large N limit using both numerical and…
We extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. D\"obler and G. Peccati (2017). In particular, by adapting the exchangeable pairs couplings construction introduced by I.…
This note contains a new combinatorial proof of Cramer's rule based on the Gessel-Viennot-Lindstrom Lemma.
The Taylor expansion is a widely used and powerful tool in all branches of Mathematics, both pure and applied. In Probability and Mathematical Statistics, however, a stronger version of Taylor's classical theorem is often needed, but only…
The Collatz hypothesis is a theorem of the algorithmic theory of natural numbers. We prove the (algorithmic) formula that expresses the halting property of Collatz algorithm. The observation that Collatz's theorem cannot be proved in any…
We show that if a numerical method is posed as a sequence of operators acting on data and depending on a parameter, typically a measure of the size of discretization, then consistency, convergence and stability can be related by a…
In this paper we present new, short and elementary proofs of the famous projection and section theorems that are used in Stochastic Calculus.
We give a series of combinatorial results that can be obtained from any two collections (both indexed by $\Z\times \N$) of left and right pointing arrows that satisfy some natural relationship. When applied to certain self-interacting…
This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to be influential in computational geometry and in…
This paper presents an offering of some of the myriad connections between Combinatorics and Probability, directed in particular toward combinatorialists. The choice of material was dictated by the author's own interests, tastes and…
We present an adequacy theorem for a concurrent extension of probabilistic GCL. The underlying denotational semantics is based on the so-called mixed powerdomains, which combine non-determinism with probabilistic behaviour. The theorem…
We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for $s$-harmonic functions as a linear superposition of weighted classical harmonic…
We prove a new and unified GAGA theorem. This recovers all analytic and formal GAGA results in the literature, and is also valid in the non-noetherian setting. Our method can also be used to establish various Lefschetz theorems and…
Chang's Lemma is a widely employed result in additive combinatorics. It gives bounds on the dimension of the large spectrum of probability distributions on finite abelian groups. Recently, Bloom (2016) presented a powerful variant of…