28,837 research outputs found
Discrete time piecewise affine models of genetic regulatory networks
We introduce simple models of genetic regulatory networks and we proceed to
the mathematical analysis of their dynamics. The models are discrete time
dynamical systems generated by piecewise affine contracting mappings whose
variables represent gene expression levels. When compared to other models of
regulatory networks, these models have an additional parameter which is
identified as quantifying interaction delays. In spite of their simplicity,
their dynamics presents a rich variety of behaviours. This phenomenology is not
limited to piecewise affine model but extends to smooth nonlinear discrete time
models of regulatory networks. In a first step, our analysis concerns general
properties of networks on arbitrary graphs (characterisation of the attractor,
symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc).
In a second step, focus is made on simple circuits for which the attractor and
its changes with parameters are described. In the negative circuit of 2 genes,
a thorough study is presented which concern stable (quasi-)periodic
oscillations governed by rotations on the unit circle -- with a rotation number
depending continuously and monotonically on threshold parameters. These regular
oscillations exist in negative circuits with arbitrary number of genes where
they are most likely to be observed in genetic systems with non-negligible
delay effects.Comment: 34 page
Interacting Intersections
Intersecting p-branes can be viewed as higher-dimensional interpretations of
multi-charge extremal p-branes, where some of the individual p-branes undergo
diagonal dimensional oxidation, while the others oxidise vertically. Although
the naive vertical oxidation of a single p-brane gives a continuum of p-branes,
a more natural description arises if one considers a periodic array of p-branes
in the higher dimension, implying a dependence on the compactification
coordinates. This still reduces to the single lower-dimensional p-brane when
viewed at distances large compared with the period. Applying the same logic to
the multi-charge solutions, we are led to consider more general classes of
intersecting p-brane solutions, again depending on the compactification
coordinates, which turn out to be described by interacting functions rather
than independent harmonic functions. These new solutions also provide a more
satisfactory interpretation for the lower-dimensional multi-charge p-branes,
which otherwise appear to be nothing more than the improbable coincidence of
charge-centres of individual constituents with zero binding energy.Comment: 20 pages, Latex, references adde
Dynamical Anomalous Subvarieties: Structure and Bounded Height Theorems
According to Medvedev and Scanlon, a polynomial
of degree is called disintegrated if it is not linearly conjugate to
or (where is the Chebyshev polynomial of degree
). Let , let be
disintegrated polynomials of degrees at least 2, and let
be the corresponding coordinate-wise
self-map of . Let be an irreducible subvariety of
of dimension defined over . We define
the \emph{-anomalous} locus of which is related to the
\emph{-periodic} subvarieties of . We prove that
the -anomalous locus of is Zariski closed; this is a dynamical
analogue of a theorem of Bombieri, Masser, and Zannier \cite{BMZ07}. We also
prove that the points in the intersection of with the union of all
irreducible -periodic subvarieties of of
codimension have bounded height outside the -anomalous locus of
; this is a dynamical analogue of Habegger's theorem \cite{Habegger09} which
was previously conjectured in \cite{BMZ07}. The slightly more general self-maps
where each is a
disintegrated rational map are also treated at the end of the paper.Comment: Minor mistakes corrected, slight reorganizatio
A semi-numerical method for periodic orbits in a bisymmetrical potential
We use a semi-numerical method to find the position and period of periodic
orbits in a bisymmetrical potential, made up of a two dimensional harmonic
oscillator, with an additional term of a Plummer potential, in a number of
resonant cases. The results are compared with the outcomes obtained by the
numerical integration of the equations of motion and the agreement is good.
This indicates that the semi-numerical method gives general and reliable
results. Comparison with other methods of locating periodic orbits is also
made.Comment: Published in Mechanics Research Communication journal, 6 pages, 5
figures and 6 table
Multidimensional extension of the Morse--Hedlund theorem
A celebrated result of Morse and Hedlund, stated in 1938, asserts that a
sequence over a finite alphabet is ultimately periodic if and only if, for
some , the number of different factors of length appearing in is
less than . Attempts to extend this fundamental result, for example, to
higher dimensions, have been considered during the last fifteen years. Let
. A legitimate extension to a multidimensional setting of the notion of
periodicity is to consider sets of \ZZ^d definable by a first order formula
in the Presburger arithmetic . With this latter notion and using a
powerful criterion due to Muchnik, we exhibit a complete extension of the
Morse--Hedlund theorem to an arbitrary dimension $d$ and characterize sets of
$\ZZ^d$ definable in in terms of some functions counting recurrent
blocks, that is, blocks occurring infinitely often
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