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Elementary properties of minimal and maximal points in Zariski spectra
Get PDFWe investigate connections between arithmetic properties of rings and topological properties
of their prime spectrum. Any property that the
prime spectrum of a ring may or may not have, defines the class of rings whose prime spectrum has the given property. We ask whether a class of rings defined in this way is axiomatizable in the model theoretic sense. Answers are provided for a variety of different properties of prime spectra, e.g., normality or complete normality, Hausdorffness of the space of
maximal points, compactness of the space of minimal points
On LP-models of arithmetic
Get PDFWe answer some problems set by Priest in [11] and [12], in particular refuting Priest’s
Conjecture that all LP-models of Th(N) essentially arise via congruence relations on classical models
of Th(N). We also show that the analogue of Priest’s Conjecture for I±0 + Exp implies the existence of
truth definitions for intervals [0, a] ⊂e M |= I±0 + Exp in any cut [0, a] ⊂e K ⊆e M closed under
successor and multiplication
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
Maximum Entropy Inference with Quantified Knowledge
Get PDFWe investigate uncertain reasoning with quantified sentences of the
predicate calculus treated as the limiting case of maximum entropy infer-
ence applied to finite domains
Structure sheaves of definable additive categories
Get PDFA 2-equivalence is described between the category of small abelian categories with exact functors and the category of definable additive categories with functors which commute with products and direct limits.
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)
A note on quantum chaology and gamma approximations to eigenvalue spacings for infinite random matrices
Get PDFQuantum counterparts of certain simple classical systems can exhibit chaotic behaviour
through the statistics of their energy levels. Gamma distributions do not
precisely model the various analytic systems discussed here, but some features may be useful in studies of qualitative generic properties in applications to data from real systems which manifestly seem to exhibit behaviour reminiscent of near-random processes.
We use known bounds on the distribution
function for eigenvalue spacings for the Gaussian orthogonal ensemble (GOE) and show that gamma distributions, which have an important uniqueness property, can yield an approximation similarly good, except near the origin, to that of the widely used Wigner surmise.
This has the advantage that then both the chaotic and non chaotic cases fit in the
information geometric framework of the manifold of gamma distributions, which has been
the subject of recent work on neighbourhoods of randomness for more general stochastic systems
An Interview with Gene Golub
Get PDFOn July 3, 2005 I interviewed
Gene Golub (1932--2007)
during a visit he made to The University of Manchester
to attend a workshop.
This document provides an edited transcript of the interview
The Zariski spectrum of the category of finitely presented modules
Get PDFA representation-theoretic description of the Zariski spectrum of a commutative noetherian ring is applied to more general categories, giving the "Gabriel-Zariski" spectrum. Applied to functor categories it gives a topology, the "rep-Zariski spectrum" on the set of indecomposable pure-injective modules. This topology is dual to Ziegler's topology on the same underlying set. Associated presheaves of rings and of small abelian categories are defined.
Examples of rep-Zariski spectra are computed.
Over commutative coherent rings it is shown that, although its underlying set might be larger, the Gabriel-Zariski spectrum is topologically equivalent to the Zariski spectrum
NLEVP: A Collection of Nonlinear Eigenvalue Problems
Get PDFWe describe a collection of nonlinear eigenvalue problems
that we provide in the form of a MATLAB toolbox.
The collection contains problems from models of real-life applications
as well as ones constructed specifically to have particular
properties.
A brief description is given of each problem and
the problems are classified according to their structural properties
Combinatorics of simple polytopes and differential equations
Get PDFSimple polytopes play important role in applications
of algebraic geometry to physics. They are also main
objects in toric topology.
There is a commutative associative ring P generated by simple polytopes. The ring P possesses a natural derivation d, which comes from the boundary operator. We shall describe a ring homomorphism from the ring P to the ring of polynomials Z[t,α] transforming the operator d to the partial derivative ∂/∂t.
This result opens way to a relation between polytopes
and differential equations. As it has turned out, certain
important series of polytopes (including some recently
discovered) lead to fundamental nonlinear differential
equations in partial derivative