Poisson brackets in Hydrodynamics

This paper investigates different Poisson structures that have been proposed to give a Hamiltonian formulation to evolution equations issued from fluid mechanics. Our aim is to explore the main brackets which have been proposed and to discuss the difficulties which arise when one tries to give a rigorous meaning to these brackets. Our main interest is in the definition of a valid and usable bracket to study rotational fluid flows with a free boundary. We discuss some results which have emerged in the literature to solve some of the difficulties that arise. It appears to the author that the main problems are still open

F. A. Hayek as an ordo-liberal

Friedrich August von Hayek (1899-1992) is undoubtedly one of the most significant liberal thinkers of the past century. Born and raised in Vienna in the tradition of the Austrian School, he held academic positions i.a. in London, Chicago and Freiburg, thus uniting in his vita the four principal centers of neo-liberalism. His intellectual development is of special interest, since he shifts the focus of his research agenda several times, most notably from the field of business cycle research towards the broader field of social philosophy. According to this well-known break in his oeuvre, there is a classical division in secondary literature, splitting him into the two phases: Hayek I (the business cycle theorist) and Hayek II (the social philosopher). The present paper will try to show that this two-fold division is inadequate, or at least incomplete. Instead, a three-fold division seems more appropriate: here, Hayek I would again be the business cycle theorist, but Hayek II is seen as an ordo-liberal philosopher and Hayek III as the evolutionist philosopher. Regarding the time-span of the latter phases, the paper contends that the ordoliberal Hayek is to be seen in the 1930s and 1940s (the time of The Road to Serfdom and the founding of the Mont Pèlerin Society), whereas his evolutionist phase starts in the 1950s and continues to the end of his life. --

Least action principle for an integrable shallow water equation

For an integrable shallow water equation we describe a geometrical approach showing that any two nearby fluid configurations are successive states of a unique flow minimizing the kinetic energy.Comment: arXiv version is already officia

Integrability of invariant metrics on the diffeomorphism group of the circle

Each H^k Sobolev inner product defines a Hamiltonian vector field X_k on the regular dual of the Lie algebra of the diffeomorphism group of the circle. We show that only X_0 and X_1 are bi-Hamiltonian relatively to a modified Lie-Poisson structure

An Efficient Parallel Algorithm for Spectral Sparsification of Laplacian and SDDM Matrix Polynomials

For "large" class $\mathcal{C}$ of continuous probability density functions (p.d.f.), we demonstrate that for every $w\in\mathcal{C}$ there is mixture of discrete Binomial distributions (MDBD) with $T\geq N\sqrt{\phi_{w}/\delta}$ distinct Binomial distributions $B(\cdot,N)$ that $\delta$-approximates a discretized p.d.f. $\widehat{w}(i/N)\triangleq w(i/N)/[\sum_{\ell=0}^{N}w(\ell/N)]$ for all $i\in[3:N-3]$, where $\phi_{w}\geq\max_{x\in[0,1]}|w(x)|$. Also, we give two efficient parallel algorithms to find such MDBD. Moreover, we propose a sequential algorithm that on input MDBD with $N=2^k$ for $k\in\mathbb{N}_{+}$ that induces a discretized p.d.f. $\beta$, $B=D-M$ that is either Laplacian or SDDM matrix and parameter $\epsilon\in(0,1)$, outputs in $\widehat{O}(\epsilon^{-2}m + \epsilon^{-4}nT)$ time a spectral sparsifier $D-\widehat{M}_{N} \approx_{\epsilon} D-D\sum_{i=0}^{N}\beta_{i}(D^{-1} M)^i$ of a matrix-polynomial, where $\widehat{O}(\cdot)$ notation hides $\mathrm{poly}(\log n,\log N)$ factors. This improves the Cheng et al.'s [CCLPT15] algorithm whose run time is $\widehat{O}(\epsilon^{-2} m N^2 + NT)$. Furthermore, our algorithm is parallelizable and runs in work $\widehat{O}(\epsilon^{-2}m + \epsilon^{-4}nT)$ and depth $O(\log N\cdot\mathrm{poly}(\log n)+\log T)$. Our main algorithmic contribution is to propose the first efficient parallel algorithm that on input continuous p.d.f. $w\in\mathcal{C}$, matrix $B=D-M$ as above, outputs a spectral sparsifier of matrix-polynomial whose coefficients approximate component-wise the discretized p.d.f. $\widehat{w}$. Our results yield the first efficient and parallel algorithm that runs in nearly linear work and poly-logarithmic depth and analyzes the long term behaviour of Markov chains in non-trivial settings. In addition, we strengthen the Spielman and Peng's [PS14] parallel SDD solver