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Taylor's Theorem for Matrix Functions with Applications to Condition Number Estimation
Get PDFWe derive an explicit formula for the remainder term of a
Taylor polynomial of a matrix function.
This formula generalizes a known result for the remainder of the
Taylor polynomial for an analytic function of a complex scalar.
We investigate some consequences of this result,
which culminate in new upper bounds for the level-1 and level-2
condition numbers of a matrix function in terms of the
pseudospectrum of the matrix.
Numerical experiments show that,
although the bounds can be pessimistic,
they can be computed much faster than the standard methods.
This makes the upper bounds ideal for a quick estimation of the
condition number whilst a more accurate (and expensive) method
can be used if further accuracy is required.
They are also easily applicable to more complicated matrix functions
for which no specialized condition number estimators are
currently available
On the sign characteristics of Hermitian matrix polynomials
Get PDFThe sign characteristics of Hermitian matrix polynomials are discussed, and in particular an appropriate definition of the sign characteristics associated with the eigenvalue infinity. The concept of sign characteristic arises in different forms in many scientific fields, and is essential for the stability analysis in Hamiltonian systems or the perturbation behavior of eigenvalues under structured
perturbations. We extend classical results by Gohberg, Lancaster, and Rodman to the case of infinite eigenvalues. We derive a systematic approach, studying how
sign characteristics behave after an analytic change of variables, including the important special case of Mobius transformations, and we prove a signature
constraint theorem. We also show that the sign characteristic at infinity stays invariant in a neighborhood under perturbations for even degree Hermitian matrix polynomials, while it may change for odd degree matrix polynomials. We argue that the non-uniformity can be resolved by introducing an extra zero leading matrix coefficient
Ranking the Importance of Nuclear Reactions for Activation and Transmutation Events
Get PDFPathways-reduced analysis is one of the techniques used by the Fispact-II nuclear activation and transmutation software to study the sensitivity of the computed inventories to uncertainties in reaction cross-sections. Although deciding which pathways are most important is very helpful in for example determining which nuclear data would benefit from further refinement, pathways-reduced analysis need not necessarily define the most critical reaction, since one reaction may contribute to several different pathways. This work examines three different techniques for ranking reactions in their order of importance in determining the final inventory, viz. a pathways based metric (PBM), the direct method and one based on the Pearson correlation coefficient. Reasons why the PBM is to be preferred are presented
Bifurcation from stable fixed point to two-dimensional attractor in the border collision normal form
Get PDFThe border collision normal form is a family of continuous two-dimensional piecewise smooth maps describing dynamics close to a critical parameter at which a fixed point intersects the switching surface. It is well known that if the fixed point is stable on one side of the bifurcation point then after the bifurcation the system may have stable periodic orbits and/or chaotic attractors with a quasi-one dimensional structure (robust chaos). We show that it is also possible to have a robust transition from a stable fixed point to an attractor with topological dimension two, i.e. the highest dimension possible in the phase spac
Aluminium foam data reconstruction using CGLS and TV Regularisation - 100 and 200 projection data.
Get PDFThe Henry Mosely X-ray Imaging Facility (HMXIF) is a suite of x-ray imaging systems located at the University of Manchester in the School of Materials Science. Most commercial x-ray CT machines provide an in-built variant of the filtered back-projection (FBP) algorithm as standard. These algorithms are fast and accurate but have specific requirements in terms of ray sampling and can lead to streak artifacts in the background of the reconstructed image, creating difficulties during image segmentation. Algebraic reconstruction algorithms are largely over-looked due to lower accuracy, slow speed and large memory requirements (historically). However, they are more flexible in terms of data sampling.
This study investigates the results of software implementations of two algebraic iterative algorithms, Conjugate Gradient Least Squares (CGLS) and CGLS with Total-Variation Regularisation (TV-Reg), applied to data collected using the Nikon Xtek Custom Bay: A large 3D imaging system. Comparisons are made with the in-built FBP reconstruction and CGLS/TV-Reg reconstructions, where the latter are applied to datasets with a significantly reduced number of projections. Subsequently, data-acquisition time is reduced and increased machine functionality in the way of time-lapse imaging becomes a possibility
Detecting and Reducing Redundancy in Alarm Networks
Get PDFAlarm systems are vital for the safe operation of almost all large-scale industrial and technical installations, such as chemical plants or power stations. The optimization of alarm systems has great potential to improve the safety of these installations, and also to increase their profitability through the reduction of automated shut-downs and suboptimal operation modes.
In this work we present a new approach to alarm system optimization through the identification of redundant alarms. Our approach is based on a ranking of alarms by their connectivity in the alarm network. We also propose an overall redundancy measure for the alarm system which can be used to monitor performance improvements after redundant alarms have been removed. We present an example demonstrating that our ranking technique provides operational staff with useful information, allowing them to enhance the effectiveness of their existing alarm systems
The Twin Continua of Inductive Methods
Get PDFAfter dominating the subject of Inductive Logic for over 50 years Carnap's Continuum of Inductive Methods has
in the past decade had its monopoly challenged by a second continuum of inductive methods, the NP-Continuum, which is also based on arguably rational principles. Does this mean there are (at least) two distinct notions of rational or logical probability? We describe the bases and key properties of both continua
A Freshwater Starvation Mechanism for Dansgaard-Oeschger Cycles
Get PDFIce core records indicate that the northern hemisphere underwent a series of cyclic climate changes during the
last glacial period known as Dansgaard-Oeschger cycles. The most distinctive feature of these is a rapid warming
event, often attributed to a sudden change in the strength of the Atlantic meridional overturning circulation
(AMOC). We suggest that such a change may have occurred as part of a natural oscillation, which resulted
from salinity changes driven by the temperature-controlled runoff from ice sheets. Contrary to many previous
studies, this mechanism does not require large freshwater pulses to the North Atlantic. Instead, steady changes
in ice-sheet runoff, driven by the AMOC, lead to a naturally arising oscillator, in which the rapid warmings come
about because the Arctic Ocean is starved of freshwater. The changing size of the ice sheets, as well as changes
in the background climate, would have aected the magnitude and extent of runoff, which altered the period and
magnitude of individual cycles. We suggest that this may provide a simple explanation for the absence of the
events during interglacials and around the time of glacial maxima