On the representation of kk-free integers by binary forms

Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$ and let $r$ denote the largest degree of an irreducible factor of $F$ over the rationals. Let $k$ be an integer with $k \geq 2$ and suppose that there is no prime $p$ such that $p^k$ divides $F(a,b)$ for all pairs of integers $(a,b)$. Let $R_{F,k}(Z)$ denote the number of $k$-free integers of absolute value at most $Z$ which are represented by $F$. We prove that there is a positive number $C_{F,k}$ such that $R_{F,k}(Z)$ is asymptotic to $C_{F,k} Z^{\frac{2}{d}}$ provided that $k$ exceeds $\frac{7r}{18}$ or $(k,r)$ is $(2,6)$ or $(3,8)$.Comment: 25 pages, revised. arXiv admin note: text overlap with arXiv:1605.0342

Probing Gluon Saturation through Dihadron Correlations at an Electron-Ion Collider

Two-particle azimuthal angle correlations have been proposed to be one of the most direct and sensitive probes to access the underlying gluon dynamics involved in hard scatterings. In anticipation of an Electron-Ion Collider (EIC), detailed studies of dihadron correlation measurements in electron-proton and electron-ion collisions at an EIC have been performed. The impact of such measurements on the understanding of the different gluon distribution functions, as a clean signature for gluon saturation and to constrain saturation models further, has been explored. It is shown that dihadron correlation measurements will be one of the key methods to probe gluon saturation phenomena at a future EIC.Comment: 13 pages, 13 eps figure

Estimating coherence measures from limited experimental data available

Quantifying coherence has received increasing attention, and considerable work has been directed towards finding coherence measures. While various coherence measures have been proposed in theory, an important issue following is how to estimate these coherence measures in experiments. This is a challenging task, since the state of a system is often unknown in practical applications and the accessible measurements in a real experiment are typically limited. In this Letter, we put forward an approach to estimate coherence measures of an unknown state from any limited experimental data available. Our approach is not only applicable to coherence measures but can be extended to other resource measures.Comment: 7 pages, 2 figure

Measure-Independent Freezing of Quantum Coherence

We find that all measures of coherence are frozen for an initial state in a strictly incoherent channel if and only if the relative entropy of coherence is frozen for the state. Our finding reveals the existence of measure-independent freezing of coherence, and provides an entropy-based dynamical condition in which the coherence of an open quantum system is totally unaffected by noise.Comment: 5 pages, no figures, accepted by Physical Review A as Rapid Communicatio

Driving light pulses with light in two-level media

A two-level medium, described by the Maxwell-Bloch (MB) system, is engraved by establishing a standing cavity wave with a linearly polarized electromagnetic field that drives the medium on both ends. A light pulse, polarized along the other direction, then scatters the medium and couples to the cavity standing wave by means of the population inversion density variations. We demonstrate that control of the applied amplitudes of the grating field allows to stop the light pulse and to make it move backward (eventually to drive it freely). A simplified limit model of the MB system with variable boundary driving is obtained as a discrete nonlinear Schroedinger equation with tunable external potential. It reproduces qualitatively the dynamics of the driven light pulse