The support of the limit distribution of optimal Riesz energy points on sets of revolution in R3\mathbb{R}^{3}

Let A be a compact set in the right-half plane and $\Gamma(A)$ the set in $\mathbb{R}^{3}$ obtained by rotating A about the vertical axis. We investigate the support of the limit distribution of minimal energy point charges on $\Gamma(A)$ that interact according to the Riesz potential 1/r^{s}, 0<s<1, where r is the Euclidean distance between points. Potential theory yields that this limit distribution coincides with the equilibrium measure on $\Gamma(A)$ which is supported on the outer boundary of $\Gamma(A)$. We show that there are sets of revolution $\Gamma(A)$ such that the support of the equilibrium measure on $\Gamma(A)$ is {\bf not} the complete outer boundary, in contrast to the Coulomb case s=1. However, the support of the limit distribution on the set of revolution $\Gamma(R+A)$ as R goes to infinity, is the full outer boundary for certain sets A, in contrast to the logarithmic case (s=0)

The Riesz energy of the NN-th roots of unity: an asymptotic expansion for large NN

We derive the complete asymptotic expansion in terms of powers of $N$ for the Riesz $s$-energy of $N$ equally spaced points on the unit circle as $N\to \infty$. For $s\ge -2$, such points form optimal energy $N$-point configurations with respect to the Riesz potential $1/r^{s}$, $s\neq0$, where $r$ is the Euclidean distance between points. By analytic continuation we deduce the expansion for all complex values of $s$. The Riemann zeta function plays an essential role in this asymptotic expansion.Comment: Added reference to paper by Cohn and Kumar which shows optimality of roots of unity for -2<s<-

Synchronization of Sound Sources

Sound generation and -interaction is highly complex, nonlinear and self-organized. Already 150 years ago Lord Rayleigh raised the following problem: Two nearby organ pipes of different fundamental frequencies sound together almost inaudibly with identical pitch. This effect is now understood qualitatively by modern synchronization theory (M. Abel et al., J. Acoust. Soc. Am., 119(4), 2006). For a detailed, quantitative investigation, we substituted one pipe by an electric speaker. We observe that even minute driving signals force the pipe to synchronization, thus yielding three decades of synchronization -- the largest range ever measured to our knowledge. Furthermore, a mutual silencing of the pipe is found, which can be explained by self-organized oscillations, of use for novel methods of noise abatement. Finally, we develop a specific nonlinear reconstruction method which yields a perfect quantitative match of experiment and theory.Comment: 5 pages, 4 figure

Office-based Air-Fluid Exchange for Diabetic Post-Operative Vitreous Cavity Hemorrhage

Post-operative vitreous cavity hemorrhage (POVCH) is observed in 6-75% of eyes undergoing pars plana vitrectomy (PPV) for proliferative diabetic retinopathy (PDR). We describe our technique for office-based Air fluid exchange (AFX) in the treatment of POVCH. Sixteen eyes (15 patients) with PDR and POVCH undergoing office-based AFX between January 2006 and November 2016 were retrospectively identified. The pre- and post- procedure visual acuity (VA) and complications were compared between eyes with and without traction retinal detachment (TRD). Medicare charges for office-based AFX versus PPV were also analyzed. Mean age at the time of AFX was 55.31 (± 8.02) years. Nine eyes (56.25%) had TRD prior to PPV and 11 eyes (68.75%) were pseudophakic. The improvements in mean (± standard deviation [SD]) logMAR VA at the last postoperative visit (3 - 8 months) were 1.38 (± 0.99), 0.82 (± 0.91) and 2.09 (± 0.53) in all eyes, TRD eyes and non-TRD eyes, respectively. Complications included cataract progression, hypotony, and recurrence of TRD and ghost cell glaucoma. The total cost of outpatient AFX was $1,409.59 less than that of PPV. Office-based AFX is a cost-effective alternative treatment for non-clearing diabetic POVCH with an acceptable risk profile