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The tan theta theorem with relaxed conditions
Get PDFThe Davis-Kahan tan theta theorem bounds the tangent of the angles between an approximate and an
exact invariant subspace of a Hermitian matrix. When applicable, it gives a sharper bound than the
sin theta theorem. However, the tan theta theorem requires more restrictive conditions on the spectrums,
demanding that the entire approximate eigenvalues (Ritz values) lie above (or below) the set of
exact eigenvalues corresponding to the orthogonal complement of the invariant subspace. In this
paper we show that the conditions of the tan theta theorem can be relaxed, in that the same bound holds
even when the Ritz values lie both below and above the exact eigenvalues, but not vice versa
Frobenius groups of automorphisms and their fixed points
Get PDFSuppose that a finite group admits a Frobenius group of
automorphisms with kernel and complement such that
the fixed-point subgroup of is trivial: . In this
situation various properties of are shown to be close to the
corresponding properties of . By using Clifford's theorem
it is proved that the order is bounded in terms
of and , the rank of is bounded in terms
of and the rank of , and
that is nilpotent if is nilpotent. Lie ring methods
are used for bounding the exponent
and the nilpotency class
of in the case of metacyclic . The exponent of is
bounded in terms of and the exponent of by using Lazard's
Lie algebra associated with the Jennings--Zassenhaus
filtration and its connection with powerful subgroups. The
nilpotency class of is bounded in terms of and the
nilpotency class of by considering Lie rings with a
finite cyclic grading satisfying a certain `selective nilpotency'
condition. The latter technique also yields similar results
bounding the nilpotency class of Lie rings and algebras with a
metacyclic Frobenius group of automorphisms, with corollaries for
connected Lie groups and torsion-free locally nilpotent groups
with such groups of automorphisms. Examples show that such nilpotency results
are no longer true for non-metacyclic Frobenius groups of automorphisms
Reducing the Influence of Tiny Normwise Relative Errors on Performance Profiles
Get PDFIt is a widespread but little-noticed phenomenon that the
normwise relative error
of vectors and of floating point numbers of the same precision,
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 normwise relative 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
Mildness and the density of rational points on certain transcendental curves
No full textWe use a result due to Rolin, Speissegger, and Wilkie to show that definable sets in certain o-minimal structures admit definable parameterizations by mild maps. We then use this parameterization to prove a result on the density of rational points on curves defined by restricted Pfaffian functions
Single Scan Parameterization of Space-Variant Point Spread Functions in Image Space via a Printed Array: Impact for two PET/CT Scanners
Get PDFIncorporation of a resolution model during statistical image reconstruction often produces images of improved resolution and signal-to-noise ratio. A novel and practical methodology to rapidly and accurately determine the overall emission and detection blurring component of the system matrix using a printed point source array within a custom-made Perspex phantom is presented. The array was scanned at different positions and orientations within the field of view (FOV) to examine the feasibility of extrapolating the measured point source blurring to other locations in the FOV and the robustness of measurements from a single point source array scan. We measured the spatially-variant image-based blurring on two PET/CT scanners, the B-Hi-Rez and the TruePoint TrueV. These measured spatially-variant kernels and the spatially-invariant kernel at the FOV centre were then incorporated within an ordinary Poisson ordered subset expectation maximization (OP-OSEM) algorithm and compared to the manufacturer's implementation using projection space resolution modelling (RM). Comparisons were based on a point source array, the NEMA IEC image quality phantom, the Cologne resolution phantom and two clinical studies (carbon-11 labelled anti-sense oligonucleotide [11C]-ASO and fluorine-18 labelled fluoro-l-thymidine [18F]-FLT). Robust and accurate measurements of spatially-variant image blurring were successfully obtained from a single scan. Spatially-variant resolution modelling resulted in notable resolution improvements away from the centre of the FOV. Comparison between spatially-variant image-space methods and the projection-space approach (the first such report, using a range of studies) demonstrated very similar performance with our image-based implementation producing slightly better contrast recovery (CR) for the same level of image roughness (IR). These results demonstrate that image-based resolution modelling within reconstruction is a valid alternative to projection-based modelling, and that, when using the proposed practical methodology, the necessary resolution measurements can be obtained from a single scan. This approach avoids the relatively time-consuming and involved procedures previously proposed in the literature
Micro-chaos in Relay Feedback Systems with Bang-Bang Control and Digital Sampling
No full textWe investigate a class of linear relay feedback systems with bang-bang control and
with the control input applied at discrete time instances. Using a third order system as a
representative example we show that stable oscillations with so-called sliding motion, with
sliding present in continuous time system, loose the sliding segment of evolution, but do not
loose their stability if the open loop system is stable. We then carry on our investigations
and consider a situation when stable self-sustained oscillations are generated with the unstable
open loop system. In the latter case a transition from a stable limit cycle to micro-chaotic
oscillations occurs. The presence of micro-chaotic oscillations is shown by considering a linearised
map that maps a small neighbourhood of initial conditions back to itself. Using this map the
presence of the positive Lyapunov exponent is shown. The largest Lyapunov exponent is then
calculated numerically for an open set of sampling times, and it is shown that it is positive. The
boundedness of the attractor is ensured for sufficiently small sampling times; with the sampling
time tending to zero these switchings become faster and they turn into sliding motion. It is
the presence of the underlying sliding evolution that ensures the boundedness of the chaotic
attractor. Our finding implies that what may be considered as noise in systems with digital
control should actually be termed as micro-chaotic behaviour. This information may be helpful
in designing digital control systems where any element contributing to what appears as noise
should be suppressed
Differential formulations of Maxwell's equations in ansiotropic materials
No full textWe consider a differential-form formulation of Maxwell's equations for anisotropic and bianisotropic medi
Generating the Pfaffian closure with total Pfaffian functions
No full textGiven an ominimal
expansion R of the real field, we show that the
structure obtained from R by iterating the operation of adding all total Pfaffian
functions over R defines the same sets as the Pfaffian closure of
Mildness and the density of rational points on certain transcendental curves
No full textWe use a result due to Rolin, Speissegger, and Wilkie to show that definable sets in certain o-minimal structures admit definable parameterizations by mild maps. We then use this parameterization to prove a result on the density of rational points on curves defined by restricted Pfaffian functions
Some illustrations of information geometry in biology and physics
Get PDFMany real processes have stochastic features which
seem to be representable in some intuitive sense as `close to Poisson', `nearly random', `nearly uniform'
or with binary variables `nearly independent'. Each of those particular reference states,
defined by an equation, is unstable in the formal sense,
but it is passed through or hovered
about by the observed process. Information geometry gives
precise meaning for nearness and neighbourhood in a state space of processes, naturally quantifying
proximity of a process to a particular state via an information
theoretic metric structure on smoothly parametrized families of
probability density functions. We illustrate some aspects of the
methodology through case studies: inhomogeneous statistical evolutionary rate processes
for epidemics, amino acid spacings along protein chains,
constrained disordering of crystals, distinguishing nearby signal distributions and testing pseudorandom number generators