Sign changes in the prime number theorem

The Ramanujan Journal - Let V(T) denote the number of sign changes in $$\psi (x) - x$$ for $$x\in [1, T]$$ . We show that $$\liminf _{T\rightarrow \infty } V(T)/\log T\ge \gamma _{1}/\pi +...     

Nonequilibrium dynamics of a diffusion-limited reaction driven by a cluster-memory mechanism

The diffusion-limited reaction A+AA+B is studied in general dimension. The asymptotic decay of the system is found to depend nontrivially upon the initial concentration of A particles for certain ranges of the diffusion constant, backward reaction rate, and total concentration of particles. This nonequilibrium behavior is due to the formation of clusters centered about the initial A particles. A perturbative analysis ind=1 shows that the transition to the nonequilibrium dynamics is sharp and is quite similar to another previously studied reaction A+AA. Ford>1, a scaling argument is presented which describes the dependence of the asymptotic decay on the initial concentration of A particles and the equilibrium concentration for large backward reaction rates. Monte Carlo data are shown which confirm the analytic work ind=1, 2, and 3.     

Sulforhodamine Adsorbed Langmuir-Blodgett Layers on Silver Island Films: Effect of Probe Distance on the Metal-Enhanced Fluorescence

Metal-Enhanced Fluorescence (MEF) has become an important method in biomedical sensing. In this paper, we present the distance-dependent MEF of sulforhodamine B (SRB) monolayer on silver island films (SIFs). SRB is electrostatically incorporated into the Langmuir-Blodgett (LB) layers of octadecylamine (ODA) deposited on glass and SIFs substrates. The distances between SRB and SIFs or glass surfaces are controlled by depositing a varied number of inert stearic acid (SA) spacer layers. SRB is incorporated into positively charged LB layers of ODA by immersing the ODA deposited substrates into aqueous solution of SRB. Dye incorporated ODA layers with 10 nm separation distance from the SIFs surface show maximum metal-enhanced fluorescence intensity; ~7-fold increase in intensity as compared to that from the glass surface. The corresponding enhancement factor is reduced with increasing or decreasing the probe distance from the SIFs surface. Additionally, SRB on SIF surfaces show reduced lifetimes. We observed the shortest lifetime from the SRB with 5 nm distance from the SIF surfaces and the lifetime increased consistently with increasing the distances between the fluorophore and the SIFs surface. These observed spectral changes, increase in fluorescence intensity and decreased fluorescence lifetimes, are in accordance with the expected effects due to near-field interactions between the silver nanoparticles and fluorophores. We have also analyzed the complex fluorescence heterogeneous decays on metallic nanostructured surfaces using continuous distributions of decay times. The decay-time distributions appear to be sensitive to the distance between the metal and fluorophore and represent the underlying heterogeneity of the samples. The present systematic study provides significant information on the effect of fluorophore distance on the metal-enhanced fluorescence phenomenon.