21,543 research outputs found
Geometry of random interactions
It is argued that spectral features of quantal systems with random
interactions can be given a geometric interpretation. This conjecture is
investigated in the context of two simple models: a system of randomly
interacting d bosons and one of randomly interacting fermions in a j=7/2 shell.
In both examples the probability for a given state to become the ground state
is shown to be related to a geometric property of a polygon or polyhedron which
is entirely determined by particle number, shell size, and symmetry character
of the states. Extensions to more general situations are discussed
Data depth and floating body
Little known relations of the renown concept of the halfspace depth for
multivariate data with notions from convex and affine geometry are discussed.
Halfspace depth may be regarded as a measure of symmetry for random vectors. As
such, the depth stands as a generalization of a measure of symmetry for convex
sets, well studied in geometry. Under a mild assumption, the upper level sets
of the halfspace depth coincide with the convex floating bodies used in the
definition of the affine surface area for convex bodies in Euclidean spaces.
These connections enable us to partially resolve some persistent open problems
regarding theoretical properties of the depth
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