Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function

Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function $N(T,\beta)$ defined to be the number of pairs $\gamma$ and $\gamma'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying $0<\gamma,\gamma'\leq T$ and $0 < \gamma'-\gamma \leq 2\pi \beta/\log T$ as $T\to \infty$. In this paper, assuming the Riemann hypothesis, we prove upper and lower bounds for $N(T,\beta)$, for all $\beta >0$, using Montgomery's formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval $[-\beta, \beta]$ in a way to minimize the $L^1\big(\mathbb{R}, \big\{1 - \big(\frac{\sin \pi x}{\pi x}\big)^2 \big\}\,dx\big)$-error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work by P. X. Gallagher in 1985, where the case $\beta \in \frac12 \mathbb{N}$ was considered using non-extremal majorants and minorants.Comment: to appear in J. Reine Angew. Mat

On a discrete version of Tanaka's theorem for maximal functions

In this paper we prove a discrete version of Tanaka's Theorem \cite{Ta} for the Hardy-Littlewood maximal operator in dimension $n=1$, both in the non-centered and centered cases. For the discrete non-centered maximal operator $\widetilde{M}$ we prove that, given a function $f: \mathbb{Z} \to \mathbb{R}$ of bounded variation, $$\textrm{Var}(\widetilde{M} f) \leq \textrm{Var}(f),$$ where $\textrm{Var}(f)$ represents the total variation of $f$. For the discrete centered maximal operator $M$ we prove that, given a function $f: \mathbb{Z} \to \mathbb{R}$ such that $f \in \ell^1(\mathbb{Z})$, $$\textrm{Var}(Mf) \leq C \|f\|_{\ell^1(\mathbb{Z})}.$$ This provides a positive solution to a question of Haj{\l}asz and Onninen \cite{HO} in the discrete one-dimensional case.Comment: V4 - Proof of Lemma 3 update

Toptet

Final states with four tops appear in various extensions of the Standard Model. Alas, top reconstruction faces combinatorial issues as they show up as large multiplicity events. In this paper, we present a new procedure to determine whether new physics is in fact due to a new source for tops. We establish the use of this procedure to separate the signal from background (primarily $t\bar{t}$+jets). Our analysis is model independent, in that it does not use any details of the four top production (such as possible missing energy), and does not require b-tagging.Comment: Modifications on the manuscrip

The gas metallicity gradient and the star formation activity of disc galaxies

We study oxygen abundance profiles of the gaseous disc components in simulated galaxies in a hierarchical universe. We analyse the disc metallicity gradients in relation to the stellar masses and star formation rates of the simulated galaxies. We find a trend for galaxies with low stellar masses to have steeper metallicity gradients than galaxies with high stellar masses at z ~0. We also detect that the gas-phase metallicity slopes and the specific star formation rate (sSFR) of our simulated disc galaxies are consistent with recently reported observations at z ~0. Simulated galaxies with high stellar masses reproduce the observed relationship at all analysed redshifts and have an increasing contribution of discs with positive metallicity slopes with increasing redshift. Simulated galaxies with low stellar masses a have larger fraction of negative metallicity gradients with increasing redshift. Simulated galaxies with positive or very negative metallicity slopes exhibit disturbed morphologies and/or have a close neighbour. We analyse the evolution of the slope of the oxygen profile and sSFR for a gas-rich galaxy-galaxy encounter, finding that this kind of events could generate either positive and negative gas-phase oxygen profiles depending on their state of evolution. Our results support claims that the determination of reliable metallicity gradients as a function of redshift is a key piece of information to understand galaxy formation and set constrains on the subgrid physics.Comment: 12 pages, 8 figures, accepted MNRA

Moist turbulent Rayleigh-Benard convection with Neumann and Dirichlet boundary conditions

Turbulent Rayleigh-Benard convection with phase changes in an extended layer between two parallel impermeable planes is studied by means of three-dimensional direct numerical simulations for Rayleigh numbers between 10^4 and 1.5\times 10^7 and for Prandtl number Pr=0.7. Two different sets of boundary conditions of temperature and total water content are compared: imposed constant amplitudes which translate into Dirichlet boundary conditions for the scalar field fluctuations about the quiescent diffusive equilibrium and constant imposed flux boundary conditions that result in Neumann boundary conditions. Moist turbulent convection is in the conditionally unstable regime throughout this study for which unsaturated air parcels are stably and saturated air parcels unstably stratified. A direct comparison of both sets of boundary conditions with the same parameters requires to start the turbulence simulations out of differently saturated equilibrium states. Similar to dry Rayleigh-Benard convection the differences in the turbulent velocity fluctuations, the cloud cover and the convective buoyancy flux decrease across the layer with increasing Rayleigh number. At the highest Rayleigh numbers the system is found in a two-layer regime, a dry cloudless and stably stratified layer with low turbulence level below a fully saturated and cloudy turbulent one which equals classical Rayleigh-Benard convection layer. Both are separated by a strong inversion that gets increasingly narrower for growing Rayleigh number.Comment: 19 pages, 13 Postscript figures, Figures 10,11,12,13, in reduced qualit