Curvature based triangulation of metric measure spaces

We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role of the lower bound on Ricci curvature is replaced by the curvature-dimension condition ${\rm CD}(K,N)$. We show also that for weighted Riemannian manifolds the triangulation can be improved to become a thick one and that, in consequence, such manifolds admit weight-sensitive quasimeromorphic mappings. An application of this last result to information manifolds is considered. Further more, we extend to weak ${\rm CD}(K,N)$ spaces the results of Kanai regarding the discretization of manifolds, and show that the volume growth of such a space is the same as that of any of its discretizations.Comment: 24 pages, submitted for publicatio

Electrolytic refining of gold

At the request of the editor of ELECTROCHEMICAL INDUSTRY, I herewith give some notes on the electrolytic method of gold refining, to supplement the article of Dr. Tuttle (Vol. I, page 157, January, 1903)

Directed transport and Floquet analysis for a periodically kicked wavepacket at a quantum resonance

The dynamics of a kicked quantum mechanical wavepacket at a quantum resonance is studied in the framework of Floquet analysis. It is seen how a directed current can be created out of a homogeneous initial state at certain resonances in an asymmetric potential. The almost periodic parameter dependence of the current is found to be connected with level crossings in the Floquet spectrum.Comment: 8 pages, 4 figures, submitted to Phys. Rev.

Inviscid symmetry breaking with non-increasing energy

In a recent article, C. Bardos et. al. constructed weak solutions of the three-dimensional incompressible Euler equations which emerge from two-dimensional initial data yet become fully three-dimensional at positive times. They asked whether such symmetry-breaking solutions could also be constructed under the additional condition that they should have non-increasing energy. In this note, we give a positive answer to this question and show that such a construction is possible for a large class of initial data. We use convex integration techniques as developed by De Lellis-Sz\'ekelyhidi.Comment: To appear in C. R. Math. Acad. Sci. Pari

Root finding with threshold circuits

We show that for any constant d, complex roots of degree d univariate rational (or Gaussian rational) polynomials---given by a list of coefficients in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a uniform family of constant-depth polynomial-size threshold circuits). The basic idea is to compute the inverse function of the polynomial by a power series. We also discuss an application to the theory VTC^0 of bounded arithmetic.Comment: 19 pages, 1 figur