A geometric viewpoint on generalized hydrodynamics

Geometric Hydrodynamics via Madelung Transform

We introduce a geometric framework to study Newton’s equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important PDEs of hydrodynamical origin can be described in this framework in a natural way. In particular, the Madelung transform between the Schrödinger equation and Newton’s equations is a symplectomorphism ...

Dedekind sums: a combinatorial-geometric viewpoint

Spectral Clustering from a Geometric Viewpoint

Spectral methods have received attention as powerful theoretical and practical approaches to a number of machine learning problems. The methods are based on the solution of the eigenproblem of a similarity matrix formed from distance kernels. In this article we discuss three problems that are endemic in current implementations of spectral clustering: (1) the need to use another clustering metho...

Geometric Quantization from a Coherent State Viewpoint

A fully geometric procedure of quantization that utilizes a natural and necessary metric on phase space is reviewed and briefly related to the goals of the program of geometric quantization. Introduction and Background Purpose and achievements The goal of the present work is to present a conceptually simple, geometric prescription for quantization. Such a goal has been, and continues to be, the...

Mode coupling and generalized hydrodynamics

In this paper a general hydrodynamic mode coupling theory of equilibrium fluctuations in simple liquids is developed from molecular considerations. The approach developed hem avoids the shortcomings of previous mode coupling theories by adopting a complete hierarchy of equations for the slow modes of a hydrodynamic system and solving it formally through the N ordering approximation scheme devel...