Kleinian Schottky groups, Patterson-Sullivan measures and Fourier decay

Let $\Gamma$ be a Zariski dense Kleinian Schottky subgroup of PSL2(C). Let $\Lambda(\Gamma)$ be its limit set, endowed with a Patterson-Sullivan measure $\mu$ supported on $\Lambda(\Gamma)$. We show that the Fourier transform $\widehat{\mu}(\xi)$ enjoys polynomial decay as $\vert \xi \vert$ goes to infinity. This is a PSL2(C) version of the result of Bourgain-Dyatlov [8], and uses the decay of exponential sums based on Bourgain-Gamburd sum-product estimate on C. These bounds on exponential sums require a delicate non-concentration hypothesis which is proved using some representation theory and regularity estimates for stationary measures of certain random walks on linear groups.Comment: 2 figure

A Hybrid Vortex Solution for Radial Equilibrium in Axial Compressors

A hybrid vortex solution using the radial equilibrium equation for three dimensional design in axial compressors is generated. One of the most common used vortex solutions is Free Vortex. However, it ignores the fact that axial velocity varies with radius. The Hybrid Vortex includes axial velocity distribution with radius, which gives a more effective design. A single stage is first designed using the Free Vortex design method. A low hub-to-tip ratio is set to ensure subsonic flow. The axial velocity profile is exported from the CFX solver of the inlet diffuser. Using the Hybrid Vortex solution to the radial equilibrium equation, a redesign is conducted by altering the circumferential velocity distribution to adhere to the imported axial velocity distribution and the newly derived method. A tip-strong pressure distribution is also used in new design to adjust loading on the blade. CFX simulations are generated after 1D design, meanline design, throughflow design and blade design. One of the key factors to evaluate compressor operation is off-design performance, which can be represented by the compressor map. Compressor maps are also generated and compared for each blade to show the advantage of the new design approach. It can be said that, by introducing real axial velocity profiles, complete with 3D effects, into the early stages of design and incorporating it with the new vortex solution, this new design approach delivers airfoils that are better aligned to the real boundary conditions with enhanced surge and stability margins, which is verified by CFD results

Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps

Let $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{SO}(d+1,1)^{\circ}$ with parabolic elements. We establish exponential mixing of the geodesic flow on the unit tangent bundle $\operatorname{T}^1(\Gamma\backslash \mathbb{H}^{d+1})$ with respect to the Bowen-Margulis-Sullivan measure, which is the unique probability measure on $\operatorname{T}^1(\Gamma\backslash \mathbb{H}^{d+1})$ with maximal entropy. As an application, we obtain a resonance free region for the resolvent of the Laplacian on $\Gamma\backslash \mathbb{H}^{d+1}$. Our approach is to construct a coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator

Blind spheres of paramagnetic dopants in solid state NMR

Solid-state NMR on paramagnetically doped crystal structures gives information about the spatial distribution of dopants in the host. Paramagnetic dopants may render NMR active nuclei virtually invisible by relaxation, paramagnetic broadening or shielding. In this contribution blind sphere radii r(0) have been reported, which could be extracted through fitting the NMR signal visibility function f (x) = exp(-ar(0)(3)x) to experimental data obtained on several model compound series: La(1-x)Ln(x)PO(4) (Ln = Nd, Sm, Gd, Dy, Ho, Er, Tm, Yb), Sr1-xEuxGa2S4 and (Zn1-xMnx)(3)(PO4)(2)center dot 4H(2)O. Radii were extracted for H-1, P-31 and Ga-71, and dopants like Nd3+, Gd3+, Dy3+, Ho3+, Er3+, Tm3+, Yb3+ and Mn2+. The observed radii determined differed in all cases and covered a range from 5.5 to 13.5 angstrom. While these radii were obtained from the amount of invisible NMR signal, we also show how to link the visibility function to lineshape parameters. We show under which conditions empirical correlations of linewidth and doping concentration can be used to extract blind sphere radii from second moment or linewidth parameter data. From the second moment analysis of La1-xSmxPO4 P-31 MAS NMR spectra for example, a blind sphere size of Sm3+ can be determined, even though the visibility function remains close to 100% over the entire doping range. Dependence of the blind sphere radius r(0) on the NMR isotope and on the paramagnetic dopant could be suggested and verified: for different nuclei, r(0) shows a 3 root gamma-dependence, gamma being the gyromagnetic ratio. The blind sphere radii r(0) for different paramagnetic dopants in a lanthanide series could be predicted from the pseudo-contact term