We prove uniqueness and existence of the weak solutions of Euler equations with helical symmetry, with initial vorticity in L ∞ under "no vorticity stretching" geometric constraint. Our article follows the argument of the seminal work of Yudovich. We adjust the argument to resolve the difficulties which are specific to the helical symmetry.

We give a sharp counter example to local existence of low regularity solutions to Einstein's equations in wave coordinates. We show that there are initial data in H 2 satisfying the wave coordinate condition such that there is no solution in H 2 to Einstein's equations in wave coordinates for any positive time. This result is sharp since Klainerman-Rodnianski and Smith-Tataru proved existence for the same equations with slightly more regular initial data.

The inverse first passage time problem asks whether, for a Brownian motion B and a nonnegative random variable ζ, there exists a time-varying barrier b such that P{Bs > b(s), 0 ≤ s ≤ t} = P{ζ > t}. We study a "smoothed" version of this problem and ask whether therewhere λ is a killing rate parameter, and ψ : R → [0, 1] is a nonincreasing function. We prove that if ψ is suitably smooth, the function t → P{ζ > t} is twice continuously differentiable, and the condition 0 < − d log P{ζ>t} dt < λ holds for the hazard rate of ζ, then there exists a unique continuously differentiable function b solving the smoothed problem. We show how this result leads to flexible models of default for which it is possible to compute expected values of contingent claims.

We address in this paper the following two closely related problems: 1. How to represent functions with singularities (up to a prescribed accuracy) in a compact way? 2. How to reconstruct such functions from a small number of measurements? The stress is on a comparison of linear and nonlinear approaches. As a model case we use piecewise-constant functions on [0, 1], in particular, the Heaviside jump function H t = χ [0,t] . Considered as a curve in the Hilbert space L 2 ([0, 1]) it is completely characterized by the fact that any two its disjoint chords are orthogonal. We reinterpret this fact in a context of step-functions in one or two variables.Next we study the limitations on representability and reconstruction of piecewise-constant functions by linear and semi-linear methods. Our main tools in this problem are Kolmogorov's n-width and ǫ-entropy, as well as Temlyakov's (N, m)-width.On the positive side, we show that a very accurate non-linear reconstruction is possible. It goes through a solution of certain specific non-linear systems of algebraic equations. We discuss the form of these systems and methods of their solution, stressing their relation to Moment Theory and Complex Analysis.Finally, we informally discuss two problems in Computer Imaging which are parallel to the problems 1 and 2 above: compression of still images and video-sequences on one side, and image reconstruction from indirect measurement (for example, in Computer Tomography), on the other.----------------

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