Related papers: Multiplicative recurrence of M\"obius transformati…
We show that the M\"obius disjointess of zero entropy dynamical systems implies the existence of an increasing sequence of positive integers along which the Chowla conjecture on autocorrelations of the M\"obius function holds.
We observe that the Connes--Kreimer Hopf-algebraic approach to perturbative renormalisation works not just for Hopf algebras but more generally for filtered bialgebras $B$ with the property that $B_0$ is spanned by group-like elements (e.g.…
For the M\"obius spheres $S^{q,p}$, we give alternative elementary proofs of the recursive formulas for GJMS-operators and $Q$-curvatures due to the first author [Geom. Funct. Anal. 23, (2013), 1278-1370; arXiv:1108.0273]. These proofs make…
In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and…
Using the integral representations of the solutions of Schr\"odinger equation, which are the essential ingredients of the Gel'fand-Levitan and Marchenko integral equations of inverse scattering theory, we obtain a general theorem on the…
We study sets of recurrence, in both measurable and topological settings, for actions of $(\mathbb{N},\times)$ and $(\mathbb{Q}^{>0},\times)$. In particular, we show that autocorrelation sequences of positive functions arising from…
We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We derive some perturbation result of the Moore-Penorse inverse of a maximal accretive operator. As an application we give a factorization…
We consider the application of permutation orbifold constructions towards a new possible understanding of the genus zero property in Monstrous and Generalized Moonshine. We describe a theory of twisted Hecke operators in this setting and…
We present natural conjectural generalizations of the `positivity and integrality of mirror maps' phenomenon, encompassing the mirror maps appearing in the Batyrev--Borisov construction of mirror Calabi--Yau complete intersections in Fano…
We establish a Crapo complementation formula for the M\"obius function $\mu^X$ in a general decomposition space $X$ in terms of a convex subspace $K$ and its complement: $\mu^X \simeq \mu^{X\setminus K} + \mu^X*\zeta^K*\mu^X$. We work at…
The M\"obius inversion formula, introduced during the 19th century in number theory, was generalized to a wide class of monoids called locally finite such as the free partially commutative, plactic and hypoplactic monoids for instance. In…
Let $\lambda$ and $\mu$ denote the Liouville and M\"obius functions respectively. Hildebrand showed that all eight possible sign patterns for $(\lambda(n), \lambda(n+1), \lambda(n+2))$ occur infinitely often. By using the recent result of…
For a wide class of integer linear recurrence sequences $\left(u(n)\right)_{n=1}^\infty$, we give an upper bound on the number of $s$-tuples $\left(n_1, \ldots, n_s\right) \in \left(\mathbb Z\cap [M+1,M+ N]\right)^s$ such that the…
This note defines a family of Laurent polynomials (indexed in the rational projective line) which generalize the Markoff numbers and relate to the character variety of the one-cusped torus. We describe which monomials appear in each…
Unitary rotations of polychromatic images on finite two-dimensional pixellated screens provide invertibility, group composition, and thus conservation of information. Rotations have been applied on monochromatic image data sets, where we…
We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs,…
Gessel conjectured that the two-sided Eulerian polynomial, recording the common distribution of the descent number of a permutation and that of its inverse, has non-negative integer coefficients when expanded in terms of the gamma basis.…
We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete `multiplicative' zero-one law is established akin to the `simultaneous' zero-one laws of Cassels and Gallagher. As a…
We show that Sarnak's conjecture on M\"obius disjointness holds for all subshifts given by bijective substitutions and some other similar dynamical systems, e.g.\ those generated by Rudin-Shapiro type sequences.
In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL_2(F_{p^n}) occurs as the Galois group of some finite extension…