Schwarzian Derivative Criteria for Valence of Analytic and Harmonic Mappings

For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass--Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally certain classes of harmonic mappings are shown to have finite Schwarzian norm

Normalized Ricci flow on Riemann surfaces and determinants of Laplacian

In this note we give a simple proof of the fact that the determinant of Laplace operator in smooth metric over compact Riemann surfaces of arbitrary genus $g$ monotonously grows under the normalized Ricci flow. Together with results of Hamilton that under the action of the normalized Ricci flow the smooth metric tends asymptotically to metric of constant curvature for $g\geq 1$, this leads to a simple proof of Osgood-Phillips-Sarnak theorem stating that that within the class of smooth metrics with fixed conformal class and fixed volume the determinant of Laplace operator is maximal on metric of constant curvatute.Comment: a reference to paper math.DG/9904048 by W.Mueller and K.Wendland where the main theorem of this paper was proved a few years earlier is adde

How Do Homebuyers Value Different Types of Green Space?

It is important to understand tradeoffs in preferences for natural and constructed green space in semi-arid urban areas because these lands compete for scarce water resources. We perform a hedonic study using high resolution, remotely-sensed vegetation indices and house sales records. We find that homebuyers in the study area prefer greener lots, greener neighborhoods, and greener nearby riparian corridors, and they pay premiums for proximity to green space amenities. The findings have fundamental implications for the efficient allocation of limited water supplies between different types of green space and for native vegetation conservation in semi-arid metropolitan areas.hedonic model, locally weighted regression, spatial, open space, golf course, park, riparian, Consumer/Household Economics, Land Economics/Use,

Generation of parabolic similaritons in tapered silicon photonic wires: comparison of pulse dynamics at telecom and mid-IR wavelengths

We study the generation of parabolic self-similar optical pulses in tapered Si photonic nanowires (Si-PhNWs) both at telecom (\lambda=1.55 \mu m) and mid-IR (\lambda=2.2 \mu m) wavelengths. Our computational study is based on a rigorous theoretical model, which fully describes the influence of linear and nonlinear optical effects on pulse propagation in Si-PhNWs with arbitrarily varying width. Numerical simulations demonstrate that, in the normal dispersion regime, optical pulses evolve naturally into parabolic pulses upon propagating in millimeter-long tapered Si-PhNWs, with the efficiency of this pulse reshaping process being strongly dependent on the spectral and pulse parameter regime in which the device operates, as well as the particular shape of the Si-PhNW.Comment: 4 pages, 5 figure

Exploring a rheonomic system

A simple and illustrative rheonomic system is explored in the Lagrangian formalism. The difference between Jacobi's integral and energy is highlighted. A sharp contrast with remarks found in the literature is pointed out. The non-conservative system possess a Lagrangian not explicitly dependent on time and consequently there is a Jacobi's integral. The Lagrange undetermined multiplier method is used as a complement to obtain a few interesting conclusion

Achieving sub-diffraction imaging through bound surface states in negative-refracting photonic crystals at the near-infrared

We report the observation of imaging beyond the diffraction limit due to bound surface states in negative refraction photonic crystals. We achieve an effective negative index figure-of-merit [-Re(n)/Im(n)] of at least 380, ~125x improvement over recent efforts in the near-infrared, with a 0.4 THz bandwidth. Supported by numerical and theoretical analyses, the observed near-field resolution is 0.47 lambda, clearly smaller than the diffraction limit of 0.61 lambda. Importantly, we show this sub-diffraction imaging is due to the resonant excitation of surface slab modes, allowing refocusing of non-propagating evanescent waves

Wavelength conversion and parametric amplification of optical pulses via quasi-phase-matched FWM in long-period Bragg silicon waveguides

We present a theoretical analysis supported by comprehensive numerical simulations of quasi phase-matched four-wave mixing (FWM) of ultrashort optical pulses that propagate in weakly width-modulated silicon photonic nanowire gratings. Our study reveals that, by properly designing the optical waveguide such that the interacting pulses co-propagate with the same group-velocity, a conversion efficiency enhancement of more than 15 dB, as compared to a uniform waveguide, can readily be achieved. We also analyze the dependence of the conversion efficiency and FWM gain on the pulse width, time delay, walk-off parameter, and grating modulation depth.Comment: 4 pages, 5 figure

Asymptotics of relative heat traces and determinants on open surfaces of finite area

The goal of this paper is to prove that on surfaces with asymptotically cusp ends the relative determinant of pairs of Laplace operators is well defined. We consider a surface with cusps (M,g) and a metric h on the surface that is a conformal transformation of the initial metric g. We prove the existence of the relative determinant of the pair $(\Delta_{h},\Delta_{g})$ under suitable conditions on the conformal factor. The core of the paper is the proof of the existence of an asymptotic expansion of the relative heat trace for small times. We find the decay of the conformal factor at infinity for which this asymptotic expansion exists and the relative determinant is defined. Following the paper by B. Osgood, R. Phillips and P. Sarnak about extremal of determinants on compact surfaces, we prove Polyakov's formula for the relative determinant and discuss the extremal problem inside a conformal class. We discuss necessary conditions for the existence of a maximizer.Comment: This is the final version of the article before it gets published. 51 page