6,825 research outputs found
The geometric mean is a Bernstein function
In the paper, the authors establish, by using Cauchy integral formula in the
theory of complex functions, an integral representation for the geometric mean
of positive numbers. From this integral representation, the geometric mean
is proved to be a Bernstein function and a new proof of the well known AG
inequality is provided.Comment: 10 page
Anomalous diffusion of optical vortices in random wavefields
We investigate the dynamic behavior of optical vortices, or phase
singularities, in random wavefields and demonstrate the direct experimental
observation of the anomalous diffusion of optical vortices. The observed
subdiffusion of optical vortices show excellent agreement with the fractional
Brownian motion, a Gaussian process. Paradoxically, the vortex displacements
are observed exhibiting a non-Gaussian heavy-tailed distribution. We also tune
the extent of subdiffusion and non-Gaussianity of optical vortex by varying the
viscoelasticity of light scattering media. This complex motion of optical
vortices is reminiscent of particles in viscoelastic environments suggesting a
vortex tracking based microrheology approach. The fractional Brownian yet
non-Gaussian subdiffusion of optical vortices may not only offer insights into
the dynamics of phase singularities, but also contribute to the understanding
certain general physics, including vortex diffusion in fluids and the
decoupling between Brownian and Gaussian
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