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Assimilation into numerical weather models of the refractivity, Abel-retrieved from radio occultations, as the local refractivity at ray tangent point may result in large errors in the presence of strong horizontal gradients (atmospheric fronts, strong convection). To reduce these errors, other authors suggested modeling the Abel-retrieved refractivity as a nonlocal linear function of the 3D re...

An Abel di erential equation y p x y q x y is said to have a center at a pair of complex numbers a b if y a y b for any its solution y x with the initial value y a small enough Let p q be polynomials and let P R p Q R q P and Q satisfy Polynomial Composition condition if there exist polynomials P Q and W such that P x P W x Q x Q W x and W a W b The main result of this paper is that for a xed p...

The Anti-Brownian ELectrophoretic trap (ABEL trap) is a new device that allows a user to trap and manipulate fluorescent objects as small as 20 nm freely diffusing in solution. We describe in detail how to build an ABEL trap. CHALLENGES IN NANOMANIPULATION One of the outstanding challenges of nanotechnology is to develop a means to trap and manipulate individual molecules in solution. To study ...

Legendre coefficients of an integrable function f(x) are proved to coincide with the Fourier a nonnegative index suitable Abel-type transform itself. The numerical computation N can thus be carried out efficiently in O(NlogN) operations by means single fast f(x). Symmetries associated exploited further reduce computational complexity. dual problem calculating sum expansions at prescribed set po...