Related papers: A variation on the theme of Nicomachus
In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. According to this conjecture, any set of $n$…
On any closed Riemannian 3-manifold which is not a torus bundle, every nonvanishing analytic solution of the stationary Euler equations has a periodic trajectory. This result is originally due to A. Rechtman (arXiv:0904.2719) and K.…
Two Beatty sequences are recorded by athletes running in opposite directions in a round stadium. This approach suggests a nice interpretation for well known partitioning criteria: such sequences (eventually) partition the integers…
We give a continued-fraction characterization of Stieltjes moment sequences for which there exists a representing measure with support in $[\xi, \infty)$. The proof is elementary.
Stolarsky's invariance principle quantifies the deviation of a subset of a metric space from the uniform distribution. Classically derived for spherical sets, it has been recently studied in a number of other situations, revealing a general…
For the multiple Fourier series of the periodization of some radial functions on $\mathbb{R}^d$, we investigate the behavior of the spherical partial sum. We show the Gibbs-Wilbraham phenomenon, the Pinsky phenomenon and the third…
We prove an upper bound for the least prime in an irrational Beatty sequence. This result may be compared with Linnik's theorem on the least prime in an arithmetic progression.
I review some standard theory of convex bodies in $\mathbb{R}^3$ and rephrase it in a formalism of Ted Newman to show the relation between the St\"utzfunktion of the former theory and the cut function introduced by Ted. This leads to a…
Let $\alpha>1$ be an irrational number of finite type $\tau$. In this paper, we introduce and study a zeta function $Z_\alpha^\sharp(r,q;s)$ that is closely related to the Lipschitz-Lerch zeta function and is naturally associated with the…
We discuss cases where Voevodsky's smash nilpotence conjecture is known, and give a few new ones. In particular we explain a theorem of the cube for $1$-cycles, which is due to Oussama Ouriachi.
We consider Steinhaus cake dividing game.
We prove some partial results on the periodicity of billiard systems on graphs. The results specialize to the case of $n$ billiards with equal mass on the unit interval or circle traveling at the same speed.
The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…
It is well known that Sturmian sequences are the aperiodic sequences that are balanced over a 2-letter alphabet. They are also characterized by their complexity: they have exactly $(n+1)$ factors of length $n$. One possible generalization…
We present a generalization of the classical Nicomachus' identity for the sum of the first $n$ cubes. Unlike previous generalizations, it has three rather than two terms, and involves not just one, but two distinct triangular numbers, and…
The compelling original idea of a time crystal has referred to a structure that repeats in time as well as in space, an idea that has attracted significant interest recently. While obstructions to realize such structures became apparent…
We prove the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinite-dimensional Banach space contains an infinite subset $A$ with the property that $\|x\pm y\| > 1$ for distinct elements…
Ben-Zvi--Sakellaridis--Venkatesh described a conjectural extension of the geometric Satake equivalence to spherical varieties, whose spectral decomposition is described by Hamiltonian varieties. The goal of this article is to study their…
Let $\alpha>1$ be irrational and of finite type, $\beta\in\mathbb{R}$. In this paper, it is proved that for $R\geqslant13$ and any fixed $c\in(1,c_R)$, there exist infinitely many primes in the intersection of Beatty sequence…
We consider a toy FRW universe with the exotic topology $S^1 \times S^3$. We show that for a specific choice of quantum field content, the semi-classical Friedman equations are consistent with temporal periodicity as required by the $S^1$…