Related papers: Counting compatible indexing systems for $C_{p^n}$
We introduce the three-Catalan triangle, highlighting the three-Catalan numbers along with their recurrence relation and combinatorial interpretation, which allows us to establish their log-convexity. Additionally, we prove that the rows of…
It has been shown by Nistor that given any extension of associative algebras over C, the connecting morphism in periodic cyclic homology is compatible, under the Chern-Connes character, with the index morphism in lower algebraic K-theory.…
We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural…
In this paper we shall relate computational complexity to the principle of natural selection. We shall do this by giving a philosophical account of complexity versus universality. It seems sustainable to equate universal systems to complex…
We study the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with ${\Bbb{F}}_p$-coefficients). This conjecture is essential for understanding the structure of the isotropic motivic…
Let $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$ be two $n$-tuples of positive integers, let $X$ be a set of positive integers, and let $g$ be a positive integer. In this work we show an algorithmic process in order to compute all the sets…
We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…
The cyclic sieving phenomenon of Reiner, Stanton, and White characterizes the stabilizers of cyclic group actions on finite sets using q-analogue polynomials. Eu and Fu demonstrated a cyclic sieving phenomenon on generalized cluster…
We introduce a new clustering method for the classification of functional data sets by their probabilistic law, that is, a procedure that aims to assign data sets to the same cluster if and only if the data were generated with the same…
We consider families of chain-cochain infinite complexes $\mathcal C$ of spaces with elements depending on a number of parameters, and endowed with a converging associative multiple product. The existence of left/right local/non-local…
In this paper, we consider the family of pattern-replacement equivalence relations referred to as the "indices and values adjacent" case. Each such equivalence is determined by a partition $P$ of a subset of $S_c$ for some $c$. In 2010,…
We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the…
We demonstrate that the checkable/synthesisable split in bidirectional typechecking coincides with existing dualities in polarised System L, also known as polarised $\mu\tilde{\mu}$-calculus. Specifically, positive terms and negative…
The family of cycle completable graphs has several cryptomorphic descriptions, the equivalence of which has heretofore been proven by a laborious implication-cycle that detours through a motivating matrix completion problem. We give a…
Three families of posets depending on a nonnegative integer parameter $m$ are introduced. The underlying sets of these posets are enumerated by the $m$-Fuss Catalan numbers. Among these, one is a generalization of Stanley lattices and…
By a very simple argument, we prove that if $l,m,n$ are nonnegative integers then $$\sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m} =\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}. On the basis of this…
The \emph{index set} of a computable structure $\mathcal{A}$ is the set of indices for computable copies of $\mathcal{A}$. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary…
We present new proofs of eight integral representations of the Catalan numbers. Then, we create analogous integral representations of the Motzkin numbers and obtain new results. Most integral representations of counting sequences found in…
A general explicit coupling for mutual synchronization of two arbitrary identical continuous systems is proposed. The synchronization is proved analytically. The coupling is given for all 19 systems from Sprott's collection. For one of the…
In unsupervised machine learning, agreement between partitions is commonly assessed with so-called external validity indices. Researchers tend to use and report indices that quantify agreement between two partitions for all clusters…