Related papers: Garland Recurrences
Motivated by a recent work of Tr\"umper we consider the general Collatz word (up-down pattern) and the sequences following this pattern. The recurrences for the first and last sequence entries are given, obtained from repeated application…
The main contribution of this paper is a six-step semi-automatic algorithm that obtains a recursion satisfied by a family of determinants by systematically and iteratively applying Laplace expansion to the underlying matrix family. The…
Graphical models capture relations between entities in a wide range of applications including social networks, biology, and natural language processing, among others. Graph neural networks (GNN) are neural models that operate over graphs,…
Let $s$ be a finite sequence over a field of length $n$. It is well-known that if $s$ satisfies a linear recurrence of order $d$ with non-zero constant term, then the reverse of $s$ also satisfies a recurrence of order $d$ (with…
Prior to the introduction of Graph Neural Networks (GNNs), modeling and analyzing irregular data, particularly graphs, was thought to be the Achilles' heel of deep learning. The core concept of GNNs is to find a representation by…
Sorting algorithms have attracted a great deal of attention and study, as they have numerous applications to Mathematics, Computer Science and related fields. In this thesis, we first deal with the mathematical analysis of the Quicksort…
Recently, techniques for applying convolutional neural networks to graph-structured data have emerged. Graph convolutional neural networks (GCNNs) have been used to address node and graph classification and matrix completion. Although the…
A partial order $\prec$ on $[n]$ is naturally labelled (NL) if $x\prec y$ implies $x<y$. We establish a bijection between {3, 2+2}-free NL posets and 12-34-avoiding permutations, determine functional equations satisfied by their generating…
In this note we count linear arrangements that avoid certain patterns and show their connection to the derangement numbers. We discuss the sequence Dn, which counts linear arrangements that avoid patterns 12, 23, ..., (n-1)n, n1, and show…
Billey et al. [arXiv:1507.04976] have recently discovered a surprisingly simple formula for the number $a_n(\sigma)$ of leaf-labelled rooted non-embedded binary trees (also known as phylogenetic trees) with $n\geq 1$ leaves, fixed (for the…
In this article, we give a particular recreational application of the sequence A000533 and A261544 in "The On-line Encyclopedia of Integer Sequences" (OEIS). The recreational application provides a direct extension to "The Repetitious…
The appropriate selection of recurrence thresholds is a key problem in applications of recurrence quantification analysis and related methods across disciplines. Here, we discuss the distribution of pairwise distances between state vectors…
Let $Q$ be the matrix $\displaystyle \begin{pmatrix} a & b \\ 1 & 0 \end{pmatrix}$ in $GL_2(\mathbb{F}_q)$ where $\mathbb{F}_q$ is a finite field, and let $G$ be the finite cyclic group generated by $Q$. We consider the action of $G$ on the…
This paper considers generalised network, intended as networks where (a) the edges connecting the nodes are nonlinear, and (b) stochastic processes are continuously indexed over both vertices and edges. Such topological structures are…
We consider a certain linear recursive relation with integer parameters and study some of its algebraic and geometric properties, with the purpose of estimating the number of chains of valences in the Farey series.
Recent advances in Generative Adversarial Networks (GANs) have resulted in its widespread applications to multiple domains. A recent model, IRGAN, applies this framework to Information Retrieval (IR) and has gained significant attention…
Graph neural networks (GNNs) are typically applied to static graphs that are assumed to be known upfront. This static input structure is often informed purely by insight of the machine learning practitioner, and might not be optimal for the…
We study the properties of the third order sequence $(w_n)=\left(w_n(a,b,c; r, s,t)\right)$ defined by the recurrence relation $w_n = rw_{n - 1} + sw_{n - 2} + tw_{n - 3}\, (n \ge 3)$ with $w_0 = a,\,w_1 = b,\,w_2=c$, where $a$, $b$, $c$,…
Models of iterated computation, such as (completely) iterative monads, often depend on a notion of guardedness, which guarantees unique solvability of recursive equations and requires roughly that recursive calls happen only under certain…
The tangent number $T_{2n+1}$ is equal to the number of increasing labelled complete binary trees with $2n+1$ vertices. This combinatorial interpretation immediately proves that $T_{2n+1}$ is divisible by $2^n$. However, a stronger…