For any positive integer k ě 1, we denote by dk the k–fold divisor function: for n a positive integer, dkpnq is the number of solutions of the equation n “ n1 . . . nk, where the ni are positive integers. The purpose of this paper is to investigate the exponent of distribution of the ternary divisor function d3 in arithmetic progressions. More generally, we will say that a real number Θ ą 0 is ...

We investigate extremal problems for quasirandom hypergraphs. We say that a 3-uniform hypergraph H “ pV,Eq is pd, η, q-quasirandom if for any subset X Ď V and every set of pairs P Ď V ˆV the number of pairs px, py, zqq P XˆP with tx, y, zu being a hyperedge of H is in the interval d |X| |P | ̆ η |V |. We show that for any ε ą 0 there exists η ą 0 such that every sufficiently large p1{2` ε, η, q...

We construct Peano curves γ : r0,8q Ñ R2 whose “footprints” γpr0, tsq, t ą 0, have C8 boundaries and are tangent to a common continuous line field on the punctured plane R2 r tγp0qu. Moreover, these boundaries can be taken C8-close to any prescribed smooth family of nested smooth Jordan curves contracting to a point.

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Extremal problems for 3-uniform hypergraphs are known to be very difficult and despite considerable effort the progress has been slow. We suggest a more systematic study of extremal problems in the context of quasirandom hypergraphs. We say that a 3-uniform hypergraph H “ pV,Eq is weakly pd, ηq-quasirandom if for any subset U Ď V the number of hyperedges of H contained in U is in the interval d...

The observed extragalactic background light (EBL) is affected by light attenuation due to absorption of light by galactic and intergalactic dust in the Universe. Even galactic opacity of 10-20 percent and minute universe intergalactic opacity of 0.01 mag h Gpc ́1 at the local Universe have a significant impact on the EBL because obscuration of galaxies and density of intergalactic dust increase ...

Abstract. Self-similarity of random trees is related to the operation of pruning. Pruning R cuts the leaves and their parental edges and removes the resulting chains of degree-two nodes from a finite tree. A Horton-Strahler order of a vertex v and its parental edge is defined as the minimal number of prunings necessary to eliminate the subtree rooted at v. A branch is a group of neighboring ver...

Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the wea...