2,289 research outputs found
Pfaffians and Shuffling Relations for the Spin Module
We present explicit formulas for a set of generators of the ideal of
relations among the pfaffians of the principal minors of the antisymmetric
matrices of fixed dimension. These formulas have an interpretation in terms of
the standard monomial theory for the spin module of orthogonal groups.Comment: 10 page
The ring of sections of a complete symmetric variety
We study the ring of sections A(X) of a complete symmetric variety X, that is
of the wonderful completion of G/H where G is an adjoint semi-simple group and
H is the fixed subgroup for an involutorial automorphism of G. We find
generators for Pic(X), we generalize the PRV conjecture to complete symmetric
varieties and construct a standard monomial theory for A(X) that is compatible
with G orbit closures in X. This gives a degeneration result and the rational
singularityness for A(X).Comment: 15 pages, Late
Pl\:ucker relations and spherical varieties: application to model varieties
A general framework for the reduction of the equations defining classes of
spherical varieties to (maybe infinite dimensional) grassmannians is proposed.
This is applied to model varieties of type A, B and C; in particular a standard
monomial theory for these varieties is presented.Comment: 15 pages, accepted for publication in Transformation Group
Front Propagation in Chaotic and Noisy Reaction-Diffusion Systems: a Discrete-Time Map Approach
We study the front propagation in Reaction-Diffusion systems whose reaction
dynamics exhibits an unstable fixed point and chaotic or noisy behaviour. We
have examined the influence of chaos and noise on the front propagation speed
and on the wandering of the front around its average position. Assuming that
the reaction term acts periodically in an impulsive way, the dynamical
evolution of the system can be written as the convolution between a spatial
propagator and a discrete-time map acting locally. This approach allows us to
perform accurate numerical analysis. They reveal that in the pulled regime the
front speed is basically determined by the shape of the map around the unstable
fixed point, while its chaotic or noisy features play a marginal role. In
contrast, in the pushed regime the presence of chaos or noise is more relevant.
In particular the front speed decreases when the degree of chaoticity is
increased, but it is not straightforward to derive a direct connection between
the chaotic properties (e.g. the Lyapunov exponent) and the behaviour of the
front. As for the fluctuations of the front position, we observe for the noisy
maps that the associated mean square displacement grows in time as in
the pushed case and as in the pulled one, in agreement with recent
findings obtained for continuous models with multiplicative noise. Moreover we
show that the same quantity saturates when a chaotic deterministic dynamics is
considered for both pushed and pulled regimes.Comment: 11 pages, 11 figure
Equations defining symmetric varieties and affine Grassmannians
Let be a simple involution of an algebraic semisimple group and
let be the subgroup of of points fixed by . If the restricted
root system is of type , or and is simply connected or if the
restricted root system is of type and is adjoint, then we describe a
standard monomial theory and the equations for the coordinate ring
using the standard monomial theory and the Pl\"ucker relations of an
appropriate (maybe infinite dimensional) Grassmann variety.Comment: 48 page
AI & Civil Liability
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Bimodality in gene expression without feedback: From Gaussian white noise to log-normal coloured noise
Extrinsic noise-induced transitions to bimodal dynamics have been largely
investigated in a variety of chemical, physical, and biological systems. In the
standard approach in physical and chemical systems, the key properties that
make these systems mathematically tractable are that the noise appears linearly
in the dynamical equations, and it is assumed Gaussian and white. In biology,
the Gaussian approximation has been successful in specific systems, but the
relevant noise being usually non-Gaussian, non-white, and nonlinear poses
serious limitations to its general applicability. Here we revisit the
fundamental features of linear Gaussian noise, pinpoint its limitations, and
review recent new approaches based on nonlinear bounded noises, which highlight
novel mechanisms to account for transitions to bimodal behaviour. We do this by
considering a simple but fundamental gene expression model, the repressed gene,
which is characterized by linear and nonlinear dependencies on external
parameters. We then review a general methodology introduced recently, so-called
nonlinear noise filtering, which allows the investigation of linear, nonlinear,
Gaussian and non-Gaussian noises. We also present a derivation of it, which
highlights its dynamical origin. Testing the methodology on the repressed gene
confirms that the emergence of noise-induced transitions appears to be strongly
dependent on the type of noise adopted, and on the degree of nonlinearity
present in the system.Comment: Review paper, 17 pages, 8 figure
Time-Reversal Symmetry in Open Classical and Quantum Systems
Deriving an arrow of time from time-reversal symmetric microscopic dynamics
is a fundamental open problem in physics. Here we focus on several derivations
of dissipative dynamics and the thermodynamic arrow of time to study precisely
how time-reversal symmetry is broken in open classical and quantum systems.
These derivations all involve the Markov approximation applied to a system
interacting with an infinite heat bath. We find that the Markov approximation
does not imply a violation of time-reversal symmetry. Our results show instead
that the time-reversal symmetry is maintained in standard dissipative equations
of motion, such as the Langevin equation and the Fokker-Planck equation in open
classical dynamics, and the Brownian motion, the Lindblad and the Pauli master
equations in open quantum dynamics. In all cases, the resulting equations of
motion describe thermalisation that occurs into the future as well as into the
past. As a consequence, we argue that the resulting dynamics are better
described by a definition of Markovianity that is symmetric with respect to the
future and the past.Comment: 14 pages, 3 figure
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