108 research outputs found
Computational Methods for the Construction of a Class of Noetherian Operators
This paper presents some algorithmic techniques to compute explicitly the
noetherian operators associated to a class of ideals and modules over a
polynomial ring. The procedures we include in this work can be easily encoded
in computer algebra packages such as CoCoA and Singular
Augmented Sparse Reconstruction of Protein Signaling Networks
The problem of reconstructing and identifying intracellular protein signaling
and biochemical networks is of critical importance in biology today. We sought
to develop a mathematical approach to this problem using, as a test case, one
of the most well-studied and clinically important signaling networks in biology
today, the epidermal growth factor receptor (EGFR) driven signaling cascade.
More specifically, we suggest a method, augmented sparse reconstruction, for
the identification of links among nodes of ordinary differential equation (ODE)
networks from a small set of trajectories with different initial conditions.
Our method builds a system of representation by using a collection of integrals
of all given trajectories and by attenuating block of terms in the
representation itself. The system of representation is then augmented with
random vectors, and minimization of the 1-norm is used to find sparse
representations for the dynamical interactions of each node. Augmentation by
random vectors is crucial, since sparsity alone is not able to handle the large
error-in-variables in the representation. Augmented sparse reconstruction
allows to consider potentially very large spaces of models and it is able to
detect with high accuracy the few relevant links among nodes, even when
moderate noise is added to the measured trajectories. After showing the
performance of our method on a model of the EGFR protein network, we sketch
briefly the potential future therapeutic applications of this approach.Comment: 24 pages, 6 figure
Syzygies of modules and applications to propagation of regularity phenomena
Propagation of regularity is considered for solutions of rectangular systems of infinite order partial differential equations (resp. convolution equations) in spaces of hyperfunctions (resp. C∞ functions and distributions). Known results of this kind are recovered as particular cases, when finite order partial differential equations are considered
A Phragm\'en - Lindel\"of principle for slice regular functions
The celebrated 100-year old Phragmen-Lindelof principle is a far reaching
extension of the maximum modulus theorem for holomorphic functions of one
complex variable. In some recent papers there has been a resurgence of interest
in principles of this type for functions of a hypercomplex variable and for
solutions of suitable partial differential equations. In the present article we
obtain a Phragmen-Lindelof principle for slice regular functions, a class of
quaternion-valued functions of a quaternionic variable which has been recently
introduced.Comment: 10 page
Gauss sums, superoscillations and the Talbot carpet
We consider the evolution, for a time-dependent Schr\"odinger equation, of
the so called Dirac comb. We show how this evolution allows us to recover
explicitly (indeed optically) the values of the quadratic generalized Gauss
sums. Moreover we use the phenomenon of superoscillatory sequences to prove
that such Gauss sums can be asymptotically recovered from the values of the
spectrum of any sufficiently regular function compactly supported on . The
fundamental tool we use is the so called Galilean transform that was introduced
and studied in the context on non-linear time dependent Schr\"odinger
equations. Furthermore, we utilize this tool to understand in detail the
evolution of an exponential in the case of a Schr\"odinger
equation with time-independent periodic potential
Delay-Coordinates Embeddings as a Data Mining Tool for Denoising Speech Signals
In this paper we utilize techniques from the theory of non-linear dynamical
systems to define a notion of embedding threshold estimators. More specifically
we use delay-coordinates embeddings of sets of coefficients of the measured
signal (in some chosen frame) as a data mining tool to separate structures that
are likely to be generated by signals belonging to some predetermined data set.
We describe a particular variation of the embedding threshold estimator
implemented in a windowed Fourier frame, and we apply it to speech signals
heavily corrupted with the addition of several types of white noise. Our
experimental work seems to suggest that, after training on the data sets of
interest,these estimators perform well for a variety of white noise processes
and noise intensity levels. The method is compared, for the case of Gaussian
white noise, to a block thresholding estimator
Quantum harmonic oscillator with superoscillating initial datum
In this paper we study the evolution of superoscillating initial data for the
quantum driven harmonic oscillator. Our main result shows that
superoscillations are amplified by the harmonic potential and that the analytic
solution develops a singularity in finite time. We also show that for a large
class of solutions of the Schr\"odinger equation, superoscillating behavior at
any given time implies superoscillating behavior at any other time.Comment: 12 page
Apparent Correction to the Speed of Light in a Gravitational Potential
The effects of physical interactions are usually incorporated into the
quantum theory by including the corresponding terms in the Hamiltonian. Here we
consider the effects of including the gravitational potential energy of massive
particles in the Hamiltonian of quantum electrodynamics. This results in a
predicted correction to the speed of light that is proportional to the fine
structure constant. The correction to the speed of light obtained in this way
depends on the gravitational potential and not the gravitational field, which
is not gauge invariant and presumably nonphysical. Nevertheless, the predicted
results are in reasonable agreement with experimental observations from
Supernova 1987a.Comment: 25 pages, 6 figure
- …