Related papers: Hardinian Arrays
Here we prove some conjectures on the monotony of combinatorial sequences from the recent preprint of Zhi--Wei Sun.
The celebrated Thue-Morse sequence, or the Prouhet-Thue-Morse sequence (A010060 in the OEIS), has a number of interesting properties and is a rich source to many (counter)examples. We introduce two different square-free sequences on three…
In the book `What is Mathematics?' Richard Courant and Herbert Robbins presented a solution of a Whitney's problem of an inverted pendulum on a railway carriage moving on a straight line. Since the appearance of the book in 1941 the…
We introduce and study a family of random processes on trees we call hipster random walks, special instances of which we heuristically connect to the min-plus binary trees introduced by Robin Pemantle and studied by Auffinger and Cable…
This expository survey is based on my online talk at the ICCM 2020. It aims to sketch key steps of the recent proof of the uniform Mordell-Lang conjecture for curves embedded into Jacobians (a question of Mazur). The full version of this…
For the OEIS sequence A001711, contributed by N. J. A. Sloane long before the on-line era and identified there as the diagonal $T(n+4, 4)$ of a generalized-Stirling triangle, R. J. Mathar contributed in February 2020 the conjectured order-2…
In recent years, the study of holomorphic correspondences as dynamical systems that can display behaviors of both rational maps and Kleinian groups has gained a good amount of attention. This phenomenon is related to the Sullivan…
We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer \cite{meyer2013nonlinear}, rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional…
The autocorrelation of a sequence is a useful criterion, among all, of resistance to cryptographic attacks. The behavior of the autocorrelations of random Boolean functions (studied by Florian Caullery, Eric F\'erard and Fran\c{c}ois Rodier…
Partially commutative monoids provide a powerful tool to study graphs, viewingwalks as words whose letters, the edges of the graph, obey a specific commutation rule. A particularclass of traces emerges from this framework, the hikes, whose…
We introduce the notion of unavoidable (complete) sets of word patterns, which is a refinement for that of words, and study certain numerical characteristics for unavoidable sets of patterns. In some cases we employ the graph of pattern…
Graph theory is a branch of mathematics in which pair-wise relations between objects are studied. My PhD thesis, supervised by David R. Wood, introduces and investigates a new family of graphs, called link graphs, that generalises the…
In a surprising recent work, Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the…
We prove distributional convergence for a family of random processes on $\mathbb{Z}$, which we call asymmetric cooperative motions. The model generalizes the "totally asymmetric hipster random walk" introduced in [Addario-Berry, Cairns,…
We derive a new coupling of the running maximum of an Ornstein-Uhlenbeck process and the running maximum of an explicit i.i.d. sequence. We use this coupling to verify a conjecture of Darling and Erdos (1956).
Wu in 1999 conjectured that if $H$ is a subgraph of the complete graph $K_{2n+1}$ with $n$ edges, then there is a Hamiltonian cycle decomposition of $K_{2n+1}$ such that each edge of $H$ is in a separate Hamiltonian cycle. The conjecture…
We establish an inclusion relation between two uniform models of random $k$-graphs (for constant $k \ge 2$) on $n$ labeled vertices: $\mathbb G^{(k)}(n,m)$, the random $k$-graph with $m$ edges, and $\mathbb R^{(k)}(n,d)$, the random…
The classical Erd\H{o}s-Szekeres theorem dating back almost a hundred years states that any sequence of $(n-1)^2+1$ distinct real numbers contains a monotone subsequence of length $n$. This theorem has been generalised to higher dimensions…
We continue our development of the invariant theory of genus one curves with the aim of computing certain twists of the universal family of elliptic curves parametrised by the modular curve X(n) for n = 2,3,4,5. Our construction makes use…
The correlation measure of order $k$ is a measure of pseudorandomness that quantifies the similarity between a sequence and its shifts. It is known that the correlation of order 4 is large for the Rudin-Shapiro sequence despite having nice…