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Cauchy's condensation test allows to determine the convergence of a monotone series by looking at a weighted subseries that only involves terms of the original series indexed by the powers of two. It is natural to ask whether the converse…
We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to…
We introduce two simplicial complexes, the noncrossing matching complex and the noncrossing bipartite complex. Both complexes are intimately related to the bubble lattice introduced in our earlier article "Bubble Lattices I: Structure"…
We construct and analyze several new families of permutons arising from random processes involving the Demazure product on the symmetric group. First, we consider Demazure products associated to random pipe dreams, generalizing the…
Cauchy's interlace theorem states that the characteristic polynomial of a symmetric matrix is interlaced by the characteristic polynomial of any principle submatrix. We prove this in two sentences using only the linearity of the…
We compute the rational Borel-Moore homology groups for affine determinantal varieties in the spaces of general, symmetric, and skew-symmetric matrices, solving a problem suggested by the work of Pragacz and Ratajski. The main ingredient is…
The paper considers the Cauchy problem for the system of partial differential equations of fractional order $D_t^{\mathcal{B}} {U}(t,x) + \mathbb{A}(D) {U} (t,x)=H(t,x) $. Here $U$ and $H$ are vector-functions, the $m\times m$ matrix of…
We prove that the permutations of $\{1,\dots, n\}$ having an increasing (resp., decreasing) subsequence of length $n-r$ index a subset of the set of all $r$th Kronecker powers of $n \times n$ permutation matrices which is a basis for the…
Motivated by questions in biological classification, we discuss some elementary combinatorial and computational properties of certain set systems that generalize hierarchies, namely, 'patchworks', 'weak patchworks', 'ample patchworks' and…
Covariance functions are the core of spatial statistics, stochastic processes, machine learning as well as many other theoretical and applied disciplines. The properties of the covariance function at small and large distances determine the…
In this paper, we study the staircase encoding of permutations, which maps a permutation to a staircase grid with cells filled with permutations. We consider many cases, where restricted to a permutation class, the staircase encoding…
The purpose of this article is to initiate a combinatorial study of the Bruhat-Chevalley ordering on certain sets of permutations obtained by omitting the parentheses from their standard cyclic notation. In particular, we show that these…
We recently introduced a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In particular, we introduced a subclass, which we called "self-dual". In this paper we…
We analyze some enumerative and asymptotic properties of Dyck paths under a line of slope 2/5.This answers to Knuth's problem \\#4 from his "Flajolet lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in June…
Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method). Especially, the usually hard average-case analysis is ammenable to this method. Here we…
In this paper the Mikhailov model is discretized by means of the Cauchy matrix approach. A pair of discrete Miura transformations are constructed. The discrete Mikhailov model is a coupled system, in which one equation comes from the…
Given an effective Cartier divisor D with simple normal crossing support on a smooth and proper scheme X over a perfect field of positive characteristic p, there is a natural notion of de Rham-Witt sheaves on X with zeros along D. We show…
Hypertrees and noncrossing trees are well-established objects in the combinatorics literature, but the hybrid notion of a noncrossing hypertree has received less attention. In this article I investigate the poset of noncrossing hypertrees…
A well-known and old result of Hazewinkel and Koszul states that the cohomology of a finite-dimensional Lie algebra is isomorphic, up to a suitable shift, to its twisted homology, a Lie-theoretical version of Poincare duality. This paper…
The deautonomisation of birational maps that have the singularity confinement property, i.e. the construction of nonautonomous versions of such maps that preserve the singularity properties of the original, has proven crucial in our…