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Metamathematics of Elementary Mathematics
Get PDFWhy is mathematics so difficult? My talk will be devoted to hidden structures and concepts of elementary mathematics which frequently remain unnoticed but seriously influence students' perception of mathematics
A measure of the information content of EIT data
Get PDFWe ask: how many bits of information (in the Shannon sense) do we
get from a set of EIT measurements? Here, the term information in measurements (IM) is denotned as: the decrease in uncertainty about the contents of a medium, due to a set of measurements. This decrease in uncertainly is quantified by the change from the the inter-class model, q, denotned by the prior information, to the intra-class model, p, given by the measured data (corrupted by noise). IM is measured by the expected relative entropy (Kullback-Leibler divergence) between distributions q and
p, and corresponds to the channel capacity in an analogous communications system.
Based on a Gaussian model of the measurement noise, Σ_n, and a prior model of the image element covariances Σ_x, we calculate
IM= (1/2) Σ log_2([SNR]_i + 1), where [SNR]_i
is the signal to noise ratio for each independent measurement calculated from the prior and noise models. For an example, we consider saline tank measurements from a 16 electrode EIT system, with a 2 cm radius non-conductive target, and calculate IM= 179 bits. Temporal sequences of frames are considered, and formulae for IM as a function of temporal image element correlations are derived. We suggest that this measure may allow novel insights into questions such as distinguishability limits, optimal measurement schemes and data fusio
Stability and Convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains
No full textThe Method of Fundamental Solutions (MFS) is a popular tool to solve Laplace
and Helmholtz boundary value problems. Its main drawback is that it often leads
to ill-conditioned systems of equations. In this paper we investigate for the
interior Helmholtz problem
on analytic domains how the singularities (charge points) of the MFS basis
functions have to be chosen such that approximate solutions can be represented
by the MFS basis in a numerically stable way. For Helmholtz problems on the
unit disc we give a full analysis
which includes the high frequency (short wavelength) limit.
For more difficult and nonconvex
domains such as crescents we demonstrate how the
right choice of charge points is connected to how far
into the complex plane the solution of the boundary value problem can be
analytically continued, which in turn depends on both domain shape
and boundary data.
Using this we develop a recipe for locating charge points which
allows us to reach error norms of typically on a wide variety
of analytic domains.
At high frequencies of order only 3 points per wavelength are
needed, which compares very favorably to boundary integral methods
Evaluating Deformation Corrections in Electrical Impedance Tomography
Get PDFElectrical Impedance Tomography (EIT) uses
the difference in measurements between surface electrodes to
reconstruct an image of the conductivity of the contained
medium. However, changes in measurements result from
changes in internal conductivity and changes in the shape
of the medium relative to the electrode positions. Failure to
account for shape changes results in a conductivity image with
significant artifacts. Previous work to address shape changes
in EIT has shown that: a) theoretically, for an infinite number
of electrodes, non-conformal changes in boundary shapes and
electrode locations can be uniquely determined (Lionheart,
1998); and b) in some cases, conductivity and shape changes
can be recovered using a combined image reconstruction model
of both conductivity and shape changes (Soleimani et al, 2006).
This work has shown that the shape change problem can
be partially addressed. In this paper, we explore the limits
of compensation for boundary movement in EIT, using three
approaches: first, a theoretical model is developed to separate
a deformation vector field into conformal and non-conformal
components, from which the reconstruction limits may be
determined; next, finite element models are constructed from
which EIT measurements are simulated; finally, an experimental
phantom is constructed using a deformable gasket
and stainless steel electrodes in a saline medium, from which
boundary deformation measurements are acquired
Conjugate connections and differential equations on infinite dimensional manifolds
Get PDFOn a smooth manifold M, the vector bundle structures of the second order
tangent bundle, T^2M bijectively correspond to linear connections. In this
paper we classify such structures for those Frechet manifolds which can
be considered as projective limits of Banach manifolds. We investigate also
the relation between ordinary differential equations on Frechet spaces
and the linear connections on their trivial bundle; the methodology extends
to solve differential equations on those Frechet manifolds which are
obtained as projective limits of Banach manifolds. Such equations arise in
theoretical physics. We indicate an extension of the Earle and Eells foliation theorem
to the Frechet case
Tumour glycolysis: the many faces of HIF
Get PDFWe present a model for tumour metabolism that incorporates both microenvironmental (extracellular) and oncogenic (intracellular) influences. We explore the effects of the interaction between the hypoxic microenvironment and intracellular signalling on the glycolytic response of tumour tissue, finding that the glycolytic state is dependent on a delicately balanced interplay between the cellular hypoxic response, mediated by hypoxia-inducible factor-1α (HIF-1α), and growth-factor signalling cascades, which are frequently mutated in cancers. Our findings demonstrate the importance of considering both environmental and intracellular regulation when interpreting tumour metabolism for diagnostic or prognostic purposes. To illustrate this, we demonstrate the potential impact of this multi-factorial regulation on the kinetics of radiolabelled glucose analogues, used in positron emission tomography (PET)
Dynamics of a hybrid thermostat model with discrete sampling time control
Get PDFThe dynamics of a simple thermostat model is described. In the model the control system samples the temperature at regular but discrete time intervals rather than by continuous monitoring. The model exhibits quasi-periodic oscillations and banding, where the response falls into two or more bands of phase space representing either better or poorer control. A return circle map is derived which explains the observed dynamics. Some extensions of these results to the case where the flow is nonlinear are also given
Strangely Dispersed Minimal Sets in the Quasiperiodically Forced Arnold Circle Map
Get PDFWe study quasiperiodically forced circle endomorphisms, homotopic to the identity, and
show that under suitable conditions these exhibit uncountably many minimal sets with a
complicated structure, to which we refer to as ‘strangely dispersed’. Along the way, we
generalise some well-known results about circle endomorphisms to the uniquely ergodically
forced case. Namely, all rotation numbers in the rotation interval of a uniquely ergodically
forced circle endomorphism are realised on minimal sets, and if the rotation interval has
non-empty interior then the topological entropy is strictly positive. The results apply in
particular to the quasiperiodically forced Arnold circle map, which serves as a paradigm
example
The Steady Propagation of an Air Finger into a Rectangular Tube
Get PDFThe steady propagation of an air finger into a fluid-filled tube of uniform rectangular
cross-section is investigated. This paper is primarily focused on the influence of the
aspect ratio, α, on the flow properties, but the effects of a transverse gravitational field
are also considered. The three-dimensional interfacial problem is solved numerically
using the object-oriented multi-physics finite-element library oomph-lib and the
results agree with our previous experimental results (de Lo´ zar et al. Phys. Rev. Lett.
vol. 99, 2007, article 234501) to within the ±1% experimental error.
At a fixed capillary number Ca (ratio of viscous to surface-tension forces) the pressure
drops across the finger tip and relative finger widths decrease with increasing α.
The dependence of the wet fraction m (the relative quantity of liquid that remains on
the tube walls after the propagation of the finger) is more complicated: m decreases
with increasing α for low Ca but it increases with α at high Ca. Our results also
indicate that the system is approximately quasi-two-dimensional for α 8, when we
obtain quantitative agreement with McLean & Saffman’s two-dimensional model for
the relative finger width as a function of the governing parameter 1/B =12α2Ca.
The action of gravity causes an increase in the pressure drops, finger widths and wet
fractions at fixed capillary number. In particular, when the Bond number (ratio of
gravitational to surface-tension forces) is greater than one the finger lifts off the bottom
wall of the tube leading to dramatic increases in the finger width and wet fraction at a
given Ca.
For α 3 a previously unobserved flow regime has been identified in which a
small recirculation flow is situated in front of the finger tip, shielding it from any
contaminants in the flow. In addition, for α 2 the capillary number, Cac, above
which global recirculation flows disappear has been observed to follow the simple
empirical law: Ca2/3
c α =1.21
Definable additive categories: purity and model theory
Get PDFDefinable additive categories and their model theory are the topic of this paper. We begin with background and preliminary results on additive categories. Then definable subcategories, their properties and the morphisms between them are investigated, as are certain associated topological spaces (``spectra"). It was in the model theory of modules that these categories were first considered and model theory provides some of the tools for exploring them. Some general model-theoretic background is presented, then various aspects of the model theory of definable categories are considered