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Bounds for partial derivatives: necessity of UMD and sharp constants
We prove the necessity of the UMD condition, with a quantitative estimate of
the UMD constant, for any inequality in a family of bounds between
different partial derivatives of . In particular, we show that the estimate
characterizes the UMD property,
and the best constant is equal to one half of the UMD constant. This
precise value of seems to be new even for scalar-valued functions.Comment: v2: corrected typo in the reference
The fractional Bessel equation in H\"older spaces
Motivated by the Poisson equation for the fractional Laplacian on the whole
space with radial right hand side, we study global H\"older and Schauder
estimates for a fractional Bessel equation. Our methods stand on the so-called
semigroup language. Indeed, by using the solution to the Bessel heat equation
we derive pointwise formulas for the fractional operators. Appropriate H\"older
spaces, which can be seen as Campanato-type spaces, are characterized through
Bessel harmonic extensions and fractional Carleson measures. From here the
regularity estimates for the fractional Bessel equations follow. In particular,
we obtain regularity estimates for radial solutions to the fractional
Laplacian.Comment: 36 pages. To appear in Journal of Approximation Theor
High Resolution Ionization of Ultracold Neutral Plasmas
Collective effects, such as waves and instabilities, are integral to our
understanding of most plasma phenomena. We have been able to study these in
ultracold neutral plasmas by shaping the initial density distribution through
spatial modulation of the ionizing laser intensity. We describe a relay imaging
system for the photoionization beam that allows us to create higher resolution
features and its application to extend the observation of ion acoustic waves to
shorter wavelengths. We also describe the formation of sculpted density
profiles to create fast expansion of plasma into vacuum and streaming plasmas
Emergence of Kinetic Behavior in Streaming Ultracold Neutral Plasmas
We create streaming ultracold neutral plasmas by tailoring the photoionizing
laser beam that creates the plasma. By varying the electron temperature, we
control the relative velocity of the streaming populations, and, in conjunction
with variation of the plasma density, this controls the ion collisionality of
the colliding streams. Laser-induced fluorescence is used to map the spatially
resolved density and velocity distribution function for the ions. We identify
the lack of local thermal equilibrium and distinct populations of
interpenetrating, counter-streaming ions as signatures of kinetic behavior.
Experimental data is compared with results from a one-dimensional, two-fluid
numerical simulation.Comment: 8 pages, 6 figure
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