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Permutation groups of finite Morley rank
No full textThe paper bounds the Morley rank of a definably primitive permutation group of finite Morley rank in terms of the rank of the set on which it acts
Universal connection and curvature for statistical manifold geometry
Get PDFStatistical manifolds are representations of smooth families of
probability density functions
that allow differential geometric methods to be applied to
problems in stochastic processes, mathematical statistics and
information theory. It is common to have to consider a number of
linear connections on a given statistical manifold and so it is
important to know the corresponding universal connection and
curvature; then all linear connections and their curvatures are
pullbacks. An important class of statistical manifolds is that
arising from the exponential families and one particular family is
that of gamma distributions, which we showed recently to have
important uniqueness properties in stochastic processes. Here we
provide formulae for universal connections and curvatures on
exponential families and give an explicit example for the manifold
of gamma distributions
Symplectic group actions and covering spaces
Get PDFFor symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action is closed, the natural projection from the latter to the former is a symplectic covering. At the same time we give a classification of all Hamiltonian coverings of a given symplectic group action. The main properties of the lifting of a group action to a cover are studied
Twists of symmetric bundles
No full textWe establish comparison results between the Hasse--Witt invariants of a symmetric bundle over a scheme and the invariants of one of its twists . For general twists we describe the difference between and up to terms of degree . Next we consider a special kind of twist, which has been studied by A. Fröhlich. This arises from twisting by a cocycle obtained from an orthogonal representation. A simple important example of this twisting procedure is the bilinear trace form of an étale algebra, which is obtained by twisting the standard/sum-of-squares form by the orthogonal representation attached to the algebra. We show how to explicitly describe the twist for representations arising from very general tame actions. This involves the 'square root of the inverse different' which Serre, Esnault, Kahn, Viehweg and ourselves had studied before. For torsors we show that, in our geometric set-up, Jardine's generalization of Fröhlich's formula holds. Namely let be a torsor with quotient , let be a symmetric bundle over , let be an orthogonal representation and let be the corresponding twist of . Then we verify up to degree that the formula holds. Here and are respectively the spinor invariant and the Stiefel--Whitney class of . The case of genuinely tamely ramified actions is geometrically more involved and leads us to introduce an invariant of ramification, which in a sense gives a decomposition in terms of representations of the inertia groups of the invariant introduced by Serre for curves.The comparison result in the tamely ramified case proceeds by reduction to the case of a torsor. The reduction is carried out by means of a partial normalisation procedure, which we had introduced in a previous paper. An important lemma of Esnault, Kahn and Viehweg allows us to express the difference between the invariants of bundles before and after the normalisation procedure in terms of Chern classes of certain sub-bundles. As noted elsewhere, this result can be best understood in the context of symmetric complexes and their invariants. Our results are new even for bundles over curves and they allow us to weaken the regularity assumptions that we had to impose in previous work of ours
Neighbourhoods of independence and associated geometry in manifolds of bivariate Gaussians and Freund distributions
No full textWe provide explicit information geometric tubular neighbourhoods containing all bivariate
processes sufficiently close to the cases of independent Poisson or Gaussian processes.
This is achieved via affine immersions of the 4-manifold of Freund bivariate distributions
and of the 5-manifold of bivariate Gaussians. We provide also the alpha-geometry for both
manifolds. The Central Limit Theorem makes our neighbourhoods of independence limiting cases
for a wide range of bivariate processes; the topological character of the results makes
them stable under small perturbations, which is important for applications
A Note on Binary Inductive Logic
Get PDFWe consider the problem of induction over languages containing binary
relations and outline a way of interpreting and constructing a class of
probability functions on the sentences of such a language. Some principles
of inductive reasoning satisfied by these probability functions are discussed,
leading in turn to a representation theorem for a more general class of
probability functions satisfying these principles
The Solution of S exp(S) = A is Not Always the Lambert W Function of A
Get PDFWe study the solutions of the matrix equation .
Our motivation comes from the study of systems of delay differential equations
, which occur in some models of practical
interest, especially in mathematical biology. This paper
concentrates on the distinction between \emph{evaluating a matrix
function} and \emph{solving a matrix equation}.
In particular,
it shows that the matrix Lambert function evaluated at the
matrix does not represent all possible solutions of . These results can easily be extended to more general matrix
equations
Instability of a viscous interface under horizontal oscillation
No full textThe linear stability of superposed layers of viscous, immiscible fluids of different densities subject to horizontal oscillations, is analyzed with a spectral collocation method and Floquet theory. We focus on counterflowing layers, which arise when the horizontal volume-flux is conserved, resulting in a streamwise pressure gradient. This model has been shown to accurately predict the onset of the frozen wave observed experimentally [E. Talib, S. V. Jalikop, and A. Juel, J. Fluid Mech. 584, 45 (2007)]. The numerical method enables us to gain new insights into the Kelvin–Helmholtz (KH) mode usually associated with the frozen wave, and the harmonic modes of the parametric-resonant instability, by resolving the flow for an exhaustive range of vibrational to viscous forces ratios and viscosity contrasts. We show that the viscous model is essential to accurately predict the onset of each mode of instability. We characterize the evolution of the neutral curves from the multiple modes of the parametric-resonant instability to the single frozen wave mode encountered in the limit of practical flows. We find that either the KH or the first resonant mode may persist when the fluid parameters are varied toward this limit. Interestingly, these two modes exhibit opposite dependencies on the viscosity contrast, which are understood by examining the eigenmodes near the interface. ©2007 American Institute of Physic
Topological Wiener-Wintner ergodic theorems via non-abelian Lie group extensions
Get PDFWe generalize a series of topological Wiener–Wintner ergodic theorems due to Walters to the context of group extensions of measure-preserving transformations where the group is a non-abelian compact Lie group. Applications to random ergodic theorems for a shift map are given