45,091 research outputs found
Polynomial Norms
In this paper, we study polynomial norms, i.e. norms that are the
root of a degree- homogeneous polynomial . We first show
that a necessary and sufficient condition for to be a norm is for
to be strictly convex, or equivalently, convex and positive definite. Though
not all norms come from roots of polynomials, we prove that any
norm can be approximated arbitrarily well by a polynomial norm. We then
investigate the computational problem of testing whether a form gives a
polynomial norm. We show that this problem is strongly NP-hard already when the
degree of the form is 4, but can always be answered by testing feasibility of a
semidefinite program (of possibly large size). We further study the problem of
optimizing over the set of polynomial norms using semidefinite programming. To
do this, we introduce the notion of r-sos-convexity and extend a result of
Reznick on sum of squares representation of positive definite forms to positive
definite biforms. We conclude with some applications of polynomial norms to
statistics and dynamical systems
Issues in the design of switched linear systems : a benchmark study
In this paper we present a tutorial overview of some of the issues that arise in the design of switched linear control systems. Particular emphasis is given to issues relating to stability and control system realisation. A benchmark regulation problem is then presented. This problem is most naturally solved by means of a switched control design. The challenge to the community is to design a control system that meets the required performance specifications and permits the application of rigorous analysis techniques. A simple design solution is presented and the limitations of currently available analysis techniques are illustrated with reference to this example
Output Reachable Set Estimation and Verification for Multi-Layer Neural Networks
In this paper, the output reachable estimation and safety verification
problems for multi-layer perceptron neural networks are addressed. First, a
conception called maximum sensitivity in introduced and, for a class of
multi-layer perceptrons whose activation functions are monotonic functions, the
maximum sensitivity can be computed via solving convex optimization problems.
Then, using a simulation-based method, the output reachable set estimation
problem for neural networks is formulated into a chain of optimization
problems. Finally, an automated safety verification is developed based on the
output reachable set estimation result. An application to the safety
verification for a robotic arm model with two joints is presented to show the
effectiveness of proposed approaches.Comment: 8 pages, 9 figures, to appear in TNNL
Disease spread over randomly switched large-scale networks
In this paper we study disease spread over a randomly switched network, which
is modeled by a stochastic switched differential equation based on the so
called -intertwined model for disease spread over static networks. Assuming
that all the edges of the network are independently switched, we present
sufficient conditions for the convergence of infection probability to zero.
Though the stability theory for switched linear systems can naively derive a
necessary and sufficient condition for the convergence, the condition cannot be
used for large-scale networks because, for a network with agents, it
requires computing the maximum real eigenvalue of a matrix of size exponential
in . On the other hand, our conditions that are based also on the spectral
theory of random matrices can be checked by computing the maximum real
eigenvalue of a matrix of size exactly
String Formation Beyond Leading Colour
We present a new model for the hadronisation of multi-parton systems, in
which colour correlations beyond leading are allowed to influence the
formation of confining potentials (strings). The multiplet structure of
is combined with a minimisation of the string potential energy, to decide
between which partons strings should form, allowing also for "baryonic"
configurations (e.g., two colours can combine coherently to form an
anticolour). In collisions, modifications to the leading-colour picture
are small, suppressed by both colour and kinematics factors. But in
collisions, multi-parton interactions increase the number of possible
subleading connections, counteracting their naive suppression.
Moreover, those that reduce the overall string lengths are kinematically
favoured. The model, which we have implemented in the PYTHIA 8 generator, is
capable of reaching agreement not only with the important
\left(n_\mathrm{charged}) distribution but also with measured
rates (and ratios) of kaons and hyperons, in both and collisions.
Nonetheless, the shape of their spectra remains challenging to
explain.Comment: 48 pages, 22 figure
Functionals of Brownian bridges arising in the current mismatch in D/A-converters
Digital-to-analog converters (DAC) transform signals from the abstract
digital domain to the real analog world. In many applications, DAC's play a
crucial role.
Due to variability in the production, various errors arise that influence the
performance of the DAC. We focus on the current errors, which describe the
fluctuations in the currents of the various unit current elements in the DAC. A
key performance measure of the DAC is the Integrated Non-linearity (INL), which
we study in this paper.
There are several DAC architectures. The most widely used architectures are
the thermometer, the binary and the segmented architectures. We study the two
extreme architectures, namely, the thermometer and the binary architectures. We
assume that the current errors are i.i.d. normally distributed, and reformulate
the INL as a functional of a Brownian bridge. We then proceed by investigating
these functionals. For the thermometer case, the functional is the maximal
absolute value of the Brownian bridge, which has been investigated in the
literature. For the binary case, we investigate properties of the functional,
such as its mean, variance and density.Comment: 22 pages, 4 figures. Version 2 with Section 3.6 added, and Section 4
revised. To appear in "Probability in the Engineering and Informational
Sciences
Feedback cooling of atomic motion in cavity QED
We consider the problem of controlling the motion of an atom trapped in an
optical cavity using continuous feedback. In order to realize such a scheme
experimentally, one must be able to perform state estimation of the atomic
motion in real time. While in theory this estimate may be provided by a
stochastic master equation describing the full dynamics of the observed system,
integrating this equation in real time is impractical. Here we derive an
approximate estimation equation for this purpose, and use it as a drive in a
feedback algorithm designed to cool the motion of the atom. We examine the
effectiveness of such a procedure using full simulations of the cavity QED
system, including the quantized motion of the atom in one dimension.Comment: 22 pages, 17 figure
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