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Heights of characters in blocks
Get PDFWe give a brief survey of the role of the height of an irreducible character in a block, and describe some recent joint work with Alexander Moreto concerning the minimal non-zero height of an irreducible character in a block. In particular we present a new consequence of Dade's conjecture
Free centre-by-nilpotent-by-abelian groups
No full textWe prove that the free centre-by-(nilpotent-of-class-)-by-abelian
groups are torsion-free for . This is in
startling contrast to the cases when is a prime and when
, where these relatively free groups do contain non-trivial elements of
finite order
Oscillations in the NF-κB Signaling Pathway
Get PDFNF-κB oscillations were suggested by Hoffmann et al from electro-mobility shift assays (EMSA) in population studies of IκBα-/- embryonic fibroblasts and simulated in a computational model. NF-κB oscillations were also observed by Nelson et al at the single cell level. The Hoffmann model gave a fairly good pre- diction of Nelson et al oscillatory experimental data using fluorescent proteins. A common comment on the source of oscillations is the existence of negative feedback loops. Just from the point of mathematics, we can set up a simple system containing a negative feedback loop that possesses oscillating behaviour resembling the ones observed in the experiments like Fonslet et al did. However, different models with similar structures can have dramatically different dynam- ical behaviour. In order to understand biological mechanisms, it is necessary to work on those real models that are based on experimental data even though models may be very large. In this paper, we are able to analyze the dynamical properties of Hoffmann’s large computational model (containing 24 variables and 64 parameters) by using computational and analytical methods and give an explanation of the source of oscillations. We find that the computational model can be treated as a fast-slow system where the level of total IκB Kinase (IKK) is treated as a slow variable. If we consider the limit in which the level of total IKK does not change at all, then we can take the level of total IKK as a parameter. Since the total NF-κB is conserved in the model, we can also view the total NF-κB as a parameter. If the actual variation of IKK is sufficiently slow, then orbits in the true system trace attractors in the reduced model. We find that for some range of the level of NF-κB, the reduced system experiences Hopf bifurcation twice while varying the level of total IKK. The damped oscillations observed in the computational system come from the existence of stable limit cycles and stable spirals in the reduced system family
Fiedler Companion Linearizations and the Recovery of Minimal Indices
Get PDFA standard way of dealing with a matrix polynomial
is to convert it into an equivalent matrix pencil --
a process known as linearization.
For any regular matrix polynomial,
a new family of linearizations generalizing
the classical first and second Frobenius companion forms
has recently been introduced by Antoniou and Vologiannidis,
extending some linearizations previously defined
by Fiedler for scalar polynomials.
We prove that these pencils are linearizations
even when
is a singular square matrix polynomial,
and show explicitly how to recover the left and right minimal indices and minimal bases of the polynomial
from the minimal indices and bases of these linearizations.
In addition, we provide a simple way to recover the eigenvectors of a regular polynomial
from those of any of these linearizations,
without any computational cost.
The existence of an eigenvector recovery procedure
is essential for a linearization
to be relevant for applications
Steep capillary-gravity waves in oscillatory shear-driven flows
Get PDFWe study steep capillary-gravity waves that form at the interface between two stably
stratified layers of immiscible liquids in a horizontally oscillating vessel. The oscillatory
nature of the external forcing prevents the waves from overturning, and thus enables
the development of steep waves at large forcing. They arise through a supercritical
pitchfork bifurcation, characterized by the square root dependence of the height of
the wave on the excess vibrational Froude number (W, square root of the ratio of
vibrational to gravitational forces). At a critical value Wc, a transition to a linear
variation in W is observed. It is accompanied by sharp qualitative changes in the
harmonic content of the wave shape, so that trochoidal waves characterize the weakly
nonlinear regime, but ‘finger’-like waves form for W Wc. In this strongly nonlinear
regime, the wavelength is a function of the product of amplitude and frequency of
forcing, whereas for W <Wc, the wavelength exhibits an explicit dependence on the
frequency of forcing that is due to the effect of viscosity. Most significantly, the radius
of curvature of the wave crests decreases monotonically with W to reach the capillary
length for W =Wc, i.e. the lengthscale for which surface tension forces balance gravitational
forces. For W <Wc, gravitational restoring forces dominate, but for W Wc,
the wave development is increasingly defined by localized surface tension effects
Nilpotent blocks of quasisimple groups for the prime two
No full textWe investigate the nilpotent blocks of positive defect of the quasisimple groups for the prime 2. We show that every nilpotent block of a quasisimple group has abelian defect groups, and give explicit characterisations in many cases. A conjecture of Puig concerning the recognition of nilpotent blocks is also shown to hold for these groups
A new approach for MOR of second order Dynamical Systems
Get PDFWe consider a new idea for model reduction of second order dynamical systems.
It is based on a new theorem which shows under which conditions one can recover the second order form of a dynamical system. This theorem adds some constraints on the projection matrices that will be used to construct the reduced
model
Cylindrical Levy processes in Banach spaces
Get PDFCylindrical probability measures are finitely additive measures on Banach spaces that have sigma-additive projections to Euclidean spaces of all dimensions. They are naturally associated to notions of weak (cylindrical) random variable and hence weak (cylindrical)
stochastic processes. In this paper we focus on cylindrical
Levy processes. These have (weak) Levy-Ito
decompositions and an associated Levy-Khintchine formula. If
the process is weakly square integrable, its covariance operator can be used to construct a reproducing kernel Hilbert space in
which the process has a decomposition as an infinite series built from a sequence of uncorrelated bona fide one-dimensional Levy processes. This series is used to define cylindrical stochastic integrals from which cylindrical Ornstein-Uhlenbeck processes may
be constructed as unique solutions of the associated Cauchy
problem. We demonstrate that such processes are cylindrical Markov processes and study their (cylindrical) invariant measures
Intrinsic correlation in planar Poisson line processes
Get PDFThe polygons arising from a planar Poisson line process have an
exponential distribution of their side lengths and are known to be
more regular as their area, perimeter or number of sides increase.
Local regions with higher line density have smaller polygon side
lengths and conversely. Numerical analysis of computer generated
Poisson line processes shows that when pairs of adjacent polygon
sides (x,y) are sorted such that x < y they are correlated
with correlation coefficient ~ 0.616 as compared to 1/sqrt{5} ~
0.447 for independent sorted exponential (x,y) pairs. This
correlation is consistent with the observed regularity of polygons
in realizations of planar Poisson line processes
Variable Selection for Joint Mean and Covariance Models via Penalized Likelihood
Get PDFIn this paper, we propose a penalized maximum likelihood method for variable
selection in joint mean and covariance models for longitudinal data. Under certain
regularity conditions, we establish the consistency and asymptotic normality of the
penalized maximum likelihood estimators of parameters in the models. We further
show that the proposed estimation method can correctly identify the true models, as
if the true models would be known in advance. We also carry out real data analysis
and simulation studies to assess the small sample performance of the new procedure,
showing that the proposed variable selection method works satisfactoril