27 research outputs found
On Nonzero Kronecker Coefficients and their Consequences for Spectra
A triple of spectra (r^A, r^B, r^{AB}) is said to be admissible if there is a
density operator rho^{AB} with (Spec rho^A, Spec rho^B, Spec rho^{AB})=(r^A,
r^B, r^{AB}). How can we characterise such triples? It turns out that the
admissible spectral triples correspond to Young diagrams (mu, nu, lambda) with
nonzero Kronecker coefficient [M. Christandl and G. Mitchison, to appear in
Comm. Math. Phys., quant-ph/0409016; A. Klyachko, quant-ph/0409113]. This means
that the irreducible representation V_lambda is contained in the tensor product
of V_mu and V_nu. Here, we show that such triples form a finitely generated
semigroup, thereby resolving a conjecture of Klyachko. As a consequence we are
able to obtain stronger results than in [M. Ch. and G. M. op. cit.] and give a
complete information-theoretic proof of the correspondence between triples of
spectra and representations. Finally, we show that spectral triples form a
convex polytope.Comment: 13 page
Quantitative predictions on auxin-induced polar distribution of PIN proteins during vein formation in leaves
The dynamic patterning of the plant hormone auxin and its efflux facilitator
the PIN protein are the key regulator for the spatial and temporal organization
of plant development. In particular auxin induces the polar localization of its
own efflux facilitator. Due to this positive feedback auxin flow is directed
and patterns of auxin and PIN arise. During the earliest stage of vein
initiation in leaves auxin accumulates in a single cell in a rim of epidermal
cells from which it flows into the ground meristem tissue of the leaf blade.
There the localized auxin supply yields the successive polarization of PIN
distribution along a strand of cells. We model the auxin and PIN dynamics
within cells with a minimal canalization model. Solving the model analytically
we uncover an excitable polarization front that triggers a polar distribution
of PIN proteins in cells. As polarization fronts may extend to opposing
directions from their initiation site we suggest a possible resolution to the
puzzling occurrence of bipolar cells, such we offer an explanation for the
development of closed, looped veins. Employing non-linear analysis we identify
the role of the contributing microscopic processes during polarization.
Furthermore, we deduce quantitative predictions on polarization fronts
establishing a route to determine the up to now largely unknown kinetic rates
of auxin and PIN dynamics.Comment: 9 pages, 4 figures, supplemental information included, accepted for
publication in Eur. Phys. J.
Statistical mechanics of lossy data compression using a non-monotonic perceptron
The performance of a lossy data compression scheme for uniformly biased
Boolean messages is investigated via methods of statistical mechanics. Inspired
by a formal similarity to the storage capacity problem in the research of
neural networks, we utilize a perceptron of which the transfer function is
appropriately designed in order to compress and decode the messages. Employing
the replica method, we analytically show that our scheme can achieve the
optimal performance known in the framework of lossy compression in most cases
when the code length becomes infinity. The validity of the obtained results is
numerically confirmed.Comment: 9 pages, 5 figures, Physical Review
Modeling oscillatory Microtubule--Polymerization
Polymerization of microtubules is ubiquitous in biological cells and under
certain conditions it becomes oscillatory in time. Here simple reaction models
are analyzed that capture such oscillations as well as the length distribution
of microtubules. We assume reaction conditions that are stationary over many
oscillation periods, and it is a Hopf bifurcation that leads to a persistent
oscillatory microtubule polymerization in these models. Analytical expressions
are derived for the threshold of the bifurcation and the oscillation frequency
in terms of reaction rates as well as typical trends of their parameter
dependence are presented. Both, a catastrophe rate that depends on the density
of {\it guanosine triphosphate} (GTP) liganded tubulin dimers and a delay
reaction, such as the depolymerization of shrinking microtubules or the decay
of oligomers, support oscillations. For a tubulin dimer concentration below the
threshold oscillatory microtubule polymerization occurs transiently on the
route to a stationary state, as shown by numerical solutions of the model
equations. Close to threshold a so--called amplitude equation is derived and it
is shown that the bifurcation to microtubule oscillations is supercritical.Comment: 21 pages and 12 figure
Objective Functions for Topography: A Comparison of Optimal Maps
Topographic mappings are important in several contexts, including data visualization, connectionist representation, and cortical structure. Many different ways of quantifying the degree of topography of a mapping have been proposed. In order to investigate the consequences of the varying assumptions that these diierent approaches embody, we have optimized the mapping with respect to a number of different measures for a very simple problem- the mapping from a square to a line. The principal results are that (1) different objective functions can produce very different maps, (2) only a small number of these functions produce mappings which match common intuitions as to what a topographic mapping "should" actually look like for this problem, (3) the objective functions can be put into certain broad categories based on the overall form of the maps, and (4) certain categories of objective functions may be more appropriate for particular types of problem than other categories