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Locally polynomially bounded structures
Get PDFWe prove a theorem which provides a method for constructing points on varieties defined by
certain smooth functions. We require that the functions are definable in a definably complete
expansion of a real closed field and are locally definable in a fixed o-minimal and polynomially
bounded reduct.
As an application we show that in certain o-minimal structures definable functions are piecewise
implicitly defined over the basic functions in the in the language
Groups with an automorphism of prime order that is almost regular in the sense of rank
Get PDFLet be an automorphism of prime order of a finite group
, and let be the (Pr\"ufer) rank of the fixed-point subgroup
. It is proved that if is nilpotent, then there
exists a characteristic subgroup of nilpotency class bounded in
terms of such that the rank of is bounded in terms of
and .
For infinite (locally) nilpotent groups a similar result holds if the
group is torsion-free (due to Makarenko), or periodic, or finitely
generated; but examples show that these additional conditions cannot
be dropped, even for nilpotent groups.
As a corollary when is an arbitrary finite group, the
combination with the recent theorems of the author and Mazurov
gives characteristic subgroups such that
is nilpotent of class bounded in terms of , while the
ranks of and are bounded in terms of and (under
the additional unavoidable assumption that if is
insoluble); in general it is impossible to get rid of the
subgroup~. The inverse limit argument yields corresponding
consequences for locally finite groups
Some local definability theory for holomorphic functions
Get PDFLet F be a collection of holomorphic functions and let R(PR(F))
denote the reduct of the structure Ran to the ordered field operations
together with the set of proper restrictions (see below) of the
real and imaginary parts of all functions in F. We ask the question:
Which holomorphic functions are locally definable (ie have their real
and imaginary parts locally definable) in the structure R(PR(F))? It
is easy to see that the collection of all such functions is closed under
composition, partial differentiation, implicit definability (via the
Implicit Function Theorem in one dependent variable) and Schwarz
Reflection. We conjecture that this exhausts the possibilities and we
prove as much in the neighbourhood of generic points. More precisely,
we show that these four operations determine the natural pregeometry
associated with R(PR(F))-definable, holomorphic functions
A Preconditioned Newton Algorithm for the Nearest Correlation Matrix
Get PDFVarious methods have been developed for computing the correlation matrix nearest
in the Frobenius norm to a given matrix.
We focus on a quadratically convergent Newton algorithm recently derived by
Qi and Sun.
Various improvements to the efficiency and reliability of the
algorithm are introduced.
Several of these relate to the linear algebra:
the Newton equations are solved by minres instead of the conjugate gradient
method, as it more quickly satisfies the inexact Newton condition;
we apply a Jacobi preconditioner,
which can be computed efficiently even though
the coefficient matrix is not explicitly available;
an efficient choice of eigensolver is identified;
and a final scaling step is introduced
to ensure that the returned matrix has unit diagonal.
Potential difficulties caused by rounding errors in the Armijo
line search are avoided by altering the step selection strategy.
These
and other improvements lead to a significant speedup over the original
algorithm and allow the solution of problems of dimension a few
thousand in a few tens of minutes
Adaptive time-stepping for incompressible flow. Part I: scalar advection-diffusion
Get PDFEven the simplest advection-diffusion problems can exhibit multiple time scales. This means that robust variable step time integrators are a prerequisite if such problems are to be efficiently solved computationally. The performance of the second order Trapezoid Rule using an explicit Adams-Bashforth method for error control is assessed in this work. This combination is particularly well suited to long time integration of advection-dominated problems. Herein it is shown that a stabilized implementation of the Trapezoid Rule leads to a very effective integrator in other situations: specifically diffusion problems with rough initial data; and general advection-diffusion problems with different physical time scales governing the system evolution
Mathematical modelling of tumour acidity
Get PDFAcid-mediated tumour invasion is receiving increasing experimental and clinical attention. Previous models proposed to describe this phenomenon failed to capture key properties of the system, such as the existence of the benign steady state, or predicted incorrectly the size of the inter-tissue gap. Here we show that taking proper account of quiescence ameliorates these drawbacks as well as revealing novel behaviour. The simplicity of the model allows us to fully identify the key parameters controlling different aspects of behaviour
Cost-cautious designs for confirmatory bioassay
No full textConfirmatory bioassay experiments take place in late stages of the drug discovery processwhen
a small number of compounds have to be compared with respect to their properties. As the
cost of the observations may differ considerably, the design problem is well specified by the
cost of compound used rather than by the number of observations. We show that cost-efficient
designs can be constructed using useful properties of the minimum support designs. These designs
are particularly suited for studies where the parameters of the model to be estimated are
known with high accuracy prior to the experiment, although they prove to be robust against
typical inaccuracies of these values. When the parameters of the model can only be specified
with ranges of values or by a probability distribution, we use a Bayesian criterion of optimality
to construct the required designs. Typically, the number of their support points depends on
the prior knowledge for the model parameters. In all cases we recommend identifying a set of
designs with good statistical properties but different potential costs to choose from
Ranking the Importance of Boards of Directors
Get PDFWe measure the importance (centrality) of boards of directors using the PageRank algorithm from computational graph theory. PageRank is at the heart of the immensely successful Google web search engine and, we argue, can be naturally extended to social network settings. In this view, a board can be represented as part of an affiliation network or, in graph theoretic-terms, an undirected bipartite graph. But PageRank operates on directed graphs, so we develop a procedure to pass from an undirected bipartite graph to appropriately weighted, directed projections. Finally, we present the rankings of publicly traded US and UK firms using this method
Inverse problems in industry
Get PDFAn overview applications of inverse problems in industry and medicin