10,326 research outputs found
On the Schoenberg Transformations in Data Analysis: Theory and Illustrations
The class of Schoenberg transformations, embedding Euclidean distances into
higher dimensional Euclidean spaces, is presented, and derived from theorems on
positive definite and conditionally negative definite matrices. Original
results on the arc lengths, angles and curvature of the transformations are
proposed, and visualized on artificial data sets by classical multidimensional
scaling. A simple distance-based discriminant algorithm illustrates the theory,
intimately connected to the Gaussian kernels of Machine Learning
Structure-Preserving Discretization of Incompressible Fluids
The geometric nature of Euler fluids has been clearly identified and
extensively studied over the years, culminating with Lagrangian and Hamiltonian
descriptions of fluid dynamics where the configuration space is defined as the
volume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewed
as a consequence of Noether's theorem associated with the particle relabeling
symmetry of fluid mechanics. However computational approaches to fluid
mechanics have been largely derived from a numerical-analytic point of view,
and are rarely designed with structure preservation in mind, and often suffer
from spurious numerical artifacts such as energy and circulation drift. In
contrast, this paper geometrically derives discrete equations of motion for
fluid dynamics from first principles in a purely Eulerian form. Our approach
approximates the group of volume-preserving diffeomorphisms using a finite
dimensional Lie group, and associated discrete Euler equations are derived from
a variational principle with non-holonomic constraints. The resulting discrete
equations of motion yield a structure-preserving time integrator with good
long-term energy behavior and for which an exact discrete Kelvin's circulation
theorem holds
Cluster varieties from Legendrian knots
Many interesting spaces --- including all positroid strata and wild character
varieties --- are moduli of constructible sheaves on a surface with
microsupport in a Legendrian link. We show that the existence of cluster
structures on these spaces may be deduced in a uniform, systematic fashion by
constructing and taking the sheaf quantizations of a set of exact Lagrangian
fillings in correspondence with isotopy representatives whose front projections
have crossings with alternating orientations. It follows in turn that results
in cluster algebra may be used to construct and distinguish exact Lagrangian
fillings of Legendrian links in the standard contact three space.Comment: 47 page
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