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Reducing the Influence of Tiny Normwise Relative Errors on Performance Profiles
No full textIt is a widespread but little-noticed phenomenon that the
normwise relative error
of vectors and of floating point numbers,
where is an approximation to ,
can be many orders of magnitude smaller than the unit roundoff.
We analyze this phenomenon and show that in the -norm
it happens precisely when
has components of widely varying magnitude
and every
component of of largest magnitude agrees with the corresponding
component of .
Performance profiles are a popular way to compare competing algorithms
according to particular measures of performance.
We show that performance profiles based on
normwise relative errors
can give a misleading impression
due to the influence of zero or tiny errors.
We propose a transformation
that reduces the influence of these extreme errors in a controlled manner,
while preserving the monotonicity of the underlying data and leaving
the performance profile unchanged at its left end-point.
Numerical examples with both artificial and genuine data illustrate
the benefits of the transformation
Endomorphisms of the Steenrod algebra and of its odd subalgebra
Get PDFWe characterise those algebra endomorphisms of the Steenrod algebra over the field of two elements, and those of its odd subalgebra, which send Steenrod squares to Steenrod squares or to 0. Two such maps appear in the literature, an epimorphism of the Steenrod algebra in the book of Steenrod and Epstein which halves superscripts on Steenrod squares and a monomorphism of the odd subalgebra in a paper of Monks. In the latter context a new map, an epimorphism, arises which has contrasting features to those of the endomorphism of Monks. Formulae for the endomorphisms are indicated both for the admissible and the Milnor bases
Fault Detection and Diagnosis for General Discrete-time Stochastic Systems Using Output Probability Density Estimation.
Get PDFA new approach of fault detection and diagnosis (FDD) for general stochastic systems in discrete-time is studied.
Our work on this problem is motivated by the fact that
most of the nonlinear control laws are implemented as digital controllers in reality. Different from the formulation of classical FDD problem, it is supposed that the measured information for the FDD is the probability density functions (PDFs) of the system output rather than its measured value. A radial basis function (RBF) neural network technique is proposed so that the output PDFs can be formulated in terms of the dynamic weighting of the RBFs neural network. Feasible criteria to detect and diagnose the system fault are provided by using linear matrix inequality (LMI) techniques. An illustrated example is included to demonstrate the ef�ciency of the proposed
algorithm, and satisfactory results are obtained
Minimum Entropy Approach For Robot Manipulator
Get PDFIn this paper, a new algorithm for an adaptive PI controller for nonlinear systems subject to stochastic non-
Gaussian disturbance is studied. The minimum entropy control
is applied to decrease the closed-loop tracking error on an ILC basis. The key issue here is to divide the control horizon into a number of equal time intervals called batches. Within each interval, there are a �xed number of sample points. The design procedure is divided into two main algorithms, within each batch and between any two adjacent batches. A D-type ILC law is employed to tune the PI controller coef�cients between two adjacent batches. However, within each batch, the PI coef�cients are �xed. A suf�cient condition is established to guarantee the stability of the closed-loop system. An analysis of the ILC
convergence is carried out. Two-link robot manipulator example is included to demonstrate the use of the control algorithm, and satisfactory results are obtained
Pulsatile Jets
Get PDFWe consider the evolution of high-Reynolds-number, planar, pulsatile jets in an
incompressible viscous fluid. The source of the jet flow comprises a mean-flow component with a superposed temporally periodic pulsation, and we address the spatiotemporal evolution of the resulting system. The analysis is presented for both a free symmetric jet and a wall jet. In both cases, pulsation of the source flow leads to a downstream short-wave linear instability, which triggers a breakdown of the boundary-layer structure in the nonlinear regime. We extend the work of Riley,Sanchez-Sans & Watson (J. Fluid Mech., vol. 638, 2009, p. 161) to show that the linear instability takes the form of a wave that propagates with the underlying jet flow, and may be viewed as a (spatially growing) weakly non-parallel analogue of the (temporally growing) short-wave modes identified by Cowley, Hocking & Tutty (Phys. Fluids, vol. 28, 1985, p. 441). The nonlinear evolution of the instability leads to wave steepening, and ultimately a singular breakdown of the jet is obtained at a critical downstream position. We speculate that the form of the breakdown is associated with the formation of a �pseudo-shock� in the jet, indicating a failure of the (long-length scale) boundary-layer scaling. The numerical results that we present disagree with the recent results of Riley et al. (2009) in the case of a free jet, together with other previously published works in this area
Modelling human balance using switched systems with linear feedback control
Get PDFWe are interested in understanding the mechanisms behind and the
character of the sway motion of healthy human subjects during quiet
standing with eyes closed. We assume that a human body can be modelled
as a single-link inverted pendulum, and the balance is achieved using liner
feedback control. Using these assumptions we derive a switched model
which we then investigate. Stable periodic motions (limit cycles) about an
upright position are found. The existence of these limit cycles is studied
as a function of system parameters. The exploration of the parameter
space leads to the detection of multistability and homoclinic bifurcation
On spherical classes in
Get PDFWe give an upper bound on the set of spherical classes in when . This is related
to the Curtis conjecture on spherical classes in . The results also provide some control over the
bordism classes on of immersions when is a Thom complex
The dimension of weakly mean porous measures: a probabilistic approach
Get PDFUsing probabilistic ideas, we prove that the packing dimension of a mean porous measure is strictly smaller than the dimension of the ambient space. Moreover, we give an explicit bound for the packing dimension, which is asymptotically sharp in the case of small porosity. This result was stated in [D. B. Beliaev and S. K. Smirnov, "On dimension of porous measures", Math. Ann. 323 (2002) 123-141], but the proof given there is not correct. We also give estimates on the dimension of weakly mean porous measures, which improve another result of Beliaev and Smirnov
The Expected Lifetime of an Extraction Project
No full textWhen a mining company begins extraction from a finite resource, it does so in the
presence of numerous uncertainties. One key uncertainty is the future price of the
commodity being extracted, since a large enough drop in price can make a resource
no longer cost-effective to extract, resulting in the mine being closed down. By specifying
a stochastic price process, and implementing a financial-type model which
leads to the use of partial differential equations, this paper creates the framework
for efficiently capturing the probability of a mine remaining open throughout its
planned extraction period, and derives the associated expected lifetime of extraction.
An approximation to the abandonment price is described, which enables a
closed-form solution to be derived for the probability of operational success and expected
lifetime. This approximation compares well with the full solution obtained
using a semi-Lagrangian numerical technique