7,369 research outputs found
Adventures in Invariant Theory
We provide an introduction to enumerating and constructing invariants of
group representations via character methods. The problem is contextualised via
two case studies arising from our recent work: entanglement measures, for
characterising the structure of state spaces for composite quantum systems; and
Markov invariants, a robust alternative to parameter-estimation intensive
methods of statistical inference in molecular phylogenetics.Comment: 12 pp, includes supplementary discussion of example
Reconstructing Quantum Geometry from Quantum Information: Area Renormalisation, Coarse-Graining and Entanglement on Spin Networks
After a brief review of spin networks and their interpretation as wave
functions for the (space) geometry, we discuss the renormalisation of the area
operator in loop quantum gravity. In such a background independent framework,
we propose to probe the structure of a surface through the analysis of the
coarse-graining and renormalisation flow(s) of its area. We further introduce a
procedure to coarse-grain spin network states and we quantitatively study the
decrease in the number of degrees of freedom during this process. Finally, we
use these coarse-graining tools to define the correlation and entanglement
between parts of a spin network and discuss their potential interpretation as a
natural measure of distance in such a state of quantum geometry.Comment: 27 pages, 12 figures, RevTex
Extended Rate, more GFUN
We present a software package that guesses formulae for sequences of, for
example, rational numbers or rational functions, given the first few terms. We
implement an algorithm due to Bernhard Beckermann and George Labahn, together
with some enhancements to render our package efficient. Thus we extend and
complement Christian Krattenthaler's program Rate, the parts concerned with
guessing of Bruno Salvy and Paul Zimmermann's GFUN, the univariate case of
Manuel Kauers' Guess.m and Manuel Kauers' and Christoph Koutschan's
qGeneratingFunctions.m.Comment: 26 page
Euler integration over definable functions
We extend the theory of Euler integration from the class of constructible
functions to that of "tame" real-valued functions (definable with respect to an
o-minimal structure). The corresponding integral operator has some unusual
defects (it is not a linear operator); however, it has a compelling
Morse-theoretic interpretation. In addition, we show that it is an appropriate
setting in which to do numerical analysis of Euler integrals, with applications
to incomplete and uncertain data in sensor networks.Comment: 6 page
- …