4,310 research outputs found
Determination of the prime bound of a graph
Given a graph , a subset of is a module of if for each
, is adjacent to all the elements of or to none
of them. For instance, , and () are
modules of called trivial. Given a graph , (respectively
) denotes the largest integer such that there is a module
of which is a clique (respectively a stable) set in with . A
graph is prime if and if all its modules are trivial. The
prime bound of is the smallest integer such that there is a prime
graph with , and . We establish the following. For every graph such that
and
is not an integer,
. Then, we prove that
for every graph such that where , or . Moreover if and only if or its complement
admits isolated vertices. Lastly, we show that for every non
prime graph such that and .Comment: arXiv admin note: text overlap with arXiv:1110.293
Feedback Control of Nonlinear Dissipative Systems by Finite Determining Parameters - A Reaction-diffusion Paradigm
We introduce here a simple finite-dimensional feedback control scheme for
stabilizing solutions of infinite-dimensional dissipative evolution equations,
such as reaction-diffusion systems, the Navier-Stokes equations and the
Kuramoto-Sivashinsky equation. The designed feedback control scheme takes
advantage of the fact that such systems possess finite number of determining
parameters (degrees of freedom), namely, finite number of determining Fourier
modes, determining nodes, and determining interpolants and projections. In
particular, the feedback control scheme uses finitely many of such observables
and controllers. This observation is of a particular interest since it implies
that our approach has far more reaching applications, in particular, in data
assimilation. Moreover, we emphasize that our scheme treats all kinds of the
determining projections, as well as, the various dissipative equations with one
unified approach. However, for the sake of simplicity we demonstrate our
approach in this paper to a one-dimensional reaction-diffusion equation
paradigm
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