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Dynamics of poles with variable strengths and optical analogies (early version)
No full textDynamics of point vortices is generalized to complex, variable strengths (poles), and several exact solutions with optical analogues, notably Snell's law, are given
On varieties of representations of finite groups
Get PDFThis is a manuscript last dated 30 June 1994; it was mostly written during the first author's visit to Eindhoven in December 1992. This approach to study of finite subgroups of simple algebraic groups could still be of some interest
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
Renormalization for the boundary of chaos in piecewise monotonic maps with a single discontinuity
Get PDFMonotonic maps with a single discontinuity arise in a variety of situations. We describe the infinite sets of periods for such maps on the boundary of chaos; this gives a sense of the routes to chaos in such maps. The description involves an explicit subshift of finite type which describes the sequences of different renormalizations possible in these maps
Border collision bifurcations, snap-back repellers and chaos
Get PDFThe normal form for codimension one border collision bifurcations of fixed points of discrete time piecewise smooth dynamical systems is considered in the unstable case. We show that in appropriate parameter regions there is a snap-back repeller immediately after the bifurcation, and hence that the bifurcation creates chaos. Although the chaotic solutions are repellers they may explain observations, and this is illustrated through an example
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
An Improved Arc Algorithm for Detecting Definite Hermitian Pairs
Get PDFA 25-year old and somewhat neglected algorithm of Crawford and Moon
attempts to determine
whether a given Hermitian matrix pair is definite by exploring the
range of the function
,
which is a subset of the unit circle.
We revisit the algorithm and show that
with suitable modifications and careful attention to implementation
details it provides a reliable and efficient means of testing definiteness.
A clearer derivation of the basic algorithm is given
that emphasizes an arc expansion viewpoint
and makes no assumptions about the definiteness of the pair.
Convergence of the algorithm is proved for all ), definite or not.
It is shown that proper handling of three details of the algorithm is
crucial to the efficiency and reliability:
how the midpoint of an arc is computed,
whether shrinkage of an arc is permitted,
and how directions of negative curvature are computed.
For the latter, several variants of
Cholesky factorization with complete pivoting are explored and the benefits of
pivoting demonstrated.
The overall cost of our improved algorithm is typically just a few Cholesky
factorizations.
Applications of the algorithm are described to testing the hyperbolicity of a
Hermitian quadratic matrix polynomial,
constructing conjugate gradient methods for sparse linear systems in saddle point form,
and computing the Crawford number of the pair via a quasiconvex univariate
minimization problem
Structure sheaves of definable additive categories
Get PDF2-equivalences are described between the category of small abelian categories with exact functors, the category of definable additive categories with functors which commute with products and direct limits and the category of locally
coherent Grothendieck categories with "coherent" morphisms.
There is a comparison, for definable additive categories, between the presheaf of finite-type localisations and the presheaf of localisations of associated functor categories.
The image of the free abelian category in Mod-R is described and related to special bases of the Ziegler and rep-Zariski spectra restricted to the set of indecomposable injectives. In the coherent case there is a particularly nice form (which is essentially elimination of imaginaries in the model-theoretic sense)
On Davis-Januszkiewicz Homotopy Types II; completion and globalisation
Get PDFFor any finite simplicial complex K, Davis and Januszkiewicz
have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction, which they showed to be homotopy equivalent to the original examples. It is therefore natural to investigate the extent to which the homotopy type of a space X is determined by such a cohomology ring. Having analysed this problem
rationally in Part I, we here consider it prime by prime, and utilise Lannes' T functor and Bousfield-Kan type obstruction theory to study the p-completion of X. We find the situation to be more subtle than for rationalisation, and confirm the uniqueness of the completion whenever K is a join of skeleta of simplices. We apply our results to the global problem by appealing to Sullivan's arithmetic square, and deduce integral uniqueness whenever the
Stanley-Reisner algebra is a complete intersection
On -representability of countable structures over real numbers, complex numbers and quaternions
Get PDFWe study Σ-definability of countable models over hereditarily finite superstructures over the field ℝ of reals, the field ℂ of complex numbers, and over the skew field ℍ of quaternions. In particular, it is shown that each at most countable structure of a finite signature, which is Σ-definable over HF(ℝ) with at most countable equivalence classes and without parameters, has a computable isomorphic copy. Moreover, if we lift the requirement on the cardinalities of the classes in a definition then such a model can have an arbitrary hyperarithmetical complexity, but it will be hyperarithmetical in any case. Also it is proved that any countable structure Σ-definable over HF(ℂ), possibly with parameters, has a computable isomorphic copy and that being Σ-definable over HF(H) is equivalent to being Σ-definable over HF(ℝ)