The Uncertainty Principle (up) as understood in this lecture is the following informal assertion: a non-zero “object” (a function, distribution, hyperfunction) and its Fourier image cannot be too small simultaneously. “The smallness” is understood in a very broad sense meaning fast decay (at infinity or at a point, bilateral or unilateral), perforated (or bounded, or semibounded) support etc. The UP becomes a theorem for many “smallnesses” and has a multitude of quite concrete quantitative forms. It plays a fundamental role as one of the major themes of classical Fourier analysis (and neighboring parts of analysis), but also in applications to physics and engineering. The lecture is a review of facts and techniques related to the UP; connections with local and non-local shift invariant operators are discussed at the end of the lecture (including some topical problems of potential theory). The lecture is intended for the general audience acquainted with basic facts of Fourier analysis on the line and circle, and rudiments of complex analysis.

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The uncertainty principle in harmonic analysis

A kinetic model of the solar wind based on Kappa velocity distribution functions for the electrons and protons es- capingoutofthecoronaispresented.Thehighvelocityparticles formingthetailofthesedistributionfunctionshaveanenhanced phase space density compared to a Maxwellian. The existence of such velocity distribution functions have been introduced in the pioneering work of Scudder (1992a,b) to explain the high temperature of the coronal plasma. The first results obtained with this new kinetic model of the solar wind are very encour- aging, indeed they fit better many major features observed in the solar wind than earlier models: e.g. the large bulk velocities observed in high speed streams emitted out of coronal regions where the plasma temperature is smaller, and the low speed so- lar wind originating in the hotter equatorial regions of the solar corona. This new kinetic model is also able to predict the high speed solar wind streams without unreasonably large coronal temperatures and without additional heating of the outer region ofthecorona,asitisneededinhydrodynamicmodelstoachieve the same solar wind speed.

http://aa.springer.de/papers/7324002/2300725.pdf

A kinetic model of the solar wind with Kappa distribution functions in the corona.

Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds

This paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed Using conformal mappings the free-boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable The new formulation leads to a regularity result and, by use of bifurcation theory, to the existence of waves of small amplitude even in the presence of stagnation points in the flow

Steady Periodic Water Waves with Constant Vorticity: Regularity and Local Bifurcation

The feedback capacity of additive stationary Gaussian noise channels is characterized as the solution to a variational problem in the noise power spectral density. When specialized to the first-order autoregressive moving-average noise spectrum, this variational characterization yields a closed-form expression for the feedback capacity. In particular, this result shows that the celebrated Schalkwijk-Kailath coding achieves the feedback capacity for the first-order autoregressive moving-average Gaussian channel, positively answering a long-standing open problem studied by Butman, Tiernan-Schalkwijk, Wolfowitz, Ozarow, Ordentlich, Yang-Kavc?ic?-Tatikonda, and others. More generally, it is shown that a k-dimensional generalization of the Schalkwijk-Kailath coding achieves the feedback capacity for any autoregressive moving-average noise spectrum of order k. Simply put, the optimal transmitter iteratively refines the receiver's knowledge of the intended message. This development reveals intriguing connections between estimation, control, and feedback communication.

Thomas Damon

Papers

Introduction to space