arXiv subjects

John Armstrong

Publications and source records attributed to John Armstrong.

47 records · Page 3Linked to original sources

arXiv2016

Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples

We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite dimensional Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric and on a family of normal mixtures. We compare this method to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and we compare the L2 mixture projection filter with a particle method with the same number of parameters, using the Levy metric. We prove that for a simple choice of the mixture manifold the L2 mixture projection filter coincides with a Galerkin method, whereas for more general mixture manifolds the equivalence does not hold and the L2 mixture filter is more general. We study particular systems that may illustrate the advantages of this new filter over other algorithms when comparing outputs with the optimal filter. We finally consider a specific software design that is suited for a numerically efficient implementation of this filter and provide numerical examples.

math.PR↗

arXiv2014

Twistor Topology of the Fermat Cubic

We describe topologically the discriminant locus of a smooth cubic surface in the complex projective space ${\mathbb{CP}}^3$ that contains 5 fibres of the projection ${\mathbb{CP}}^3 \longrightarrow S^4$.

math.DG↗

arXiv2014

The twistor discriminant locus of the Fermat cubic

We consider the discriminant locus of the Fermat cubic under the twistor fibration $CP^3 \longrightarrow S^4$. We show that it has a conformal symmetry group of order $72$ and use this to identify its topology.

math.DG↗

arXiv2014

Superhabitable Worlds

To be habitable, a world (planet or moon) does not need to be located in the stellar habitable zone (HZ), and worlds in the HZ are not necessarily habitable. Here, we illustrate how tidal heating can render terrestrial or icy worlds habitable beyond the stellar HZ. Scientists have developed a language that neglects the possible existence of worlds that offer more benign environments to life than Earth does. We call these objects "superhabitable" and discuss in which contexts this term could be used, that is to say, which worlds tend to be more habitable than Earth. In an appendix, we show why the principle of mediocracy cannot be used to logically explain why Earth should be a particularly habitable planet or why other inhabited worlds should be Earth-like. Superhabitable worlds must be considered for future follow-up observations of signs of extraterrestrial life. Considering a range of physical effects, we conclude that they will tend to be slightly older and more massive than Earth and that their host stars will likely be K dwarfs. This makes Alpha Centauri B, member of the closest stellar system to the Sun that is supposed to host an Earth-mass planet, an ideal target for searches of a superhabitable world.

astro-ph.EP↗

arXiv2013

Curvature of Hessian Manfiolds

We prove that, in dimensions greater than 2, the generic metric is not a Hessian metric and find a curvature condition on Hessian metrics in dimensions greater than 3. In particular we prove that the forms used to define the Pontryagin classes in terms of the curvature vanish on a Hessian manifold. By contrast all analytic Riemannian 2-metrics are Hessian metrics.

math.DG↗

arXiv2012

Twistor lines on cubic surfaces

It is shown that there exist non-singular cubic surfaces in CP^3 containing 5 twistor lines. This is the maximum number of twistor fibres that a non-singular cubic can contain. Cubic surfaces in CP^3 with 5 twistor lines are classified up to transformations preserving the conformal structure of S^4.

math.DG↗

arXiv2008

Categorifying Coloring Numbers

Coloring numbers are one of the simplest combinatorial invariants of knots and links to describe. And with Joyce's introduction of quandles, we can understand them more algebraically. But can we extend these invariants to tangles -- knots and links with free ends? Indeed we can, once we categorify. Starting from the definition of coloring numbers, we will categorify them and establish this extension to tangles. Then, decategorifying will leave us with matrix representations of the monoidal category of tangles.

math.GT↗

arXiv2006

Functors Extending the Kauffman Bracket

In this article we extend evaluations of the Kauffman bracket on regular isotopy classes of knots and links to a variety of functors defined on the category of framed tangles. We show that many such functors exist, and that they correspond up to equivalence to bilinear forms on free, finitely-generated modules over commutative rings.

math.GT↗

arXiv2005

The Extension of Knot Groups to Tangles

The extension of the knot group $π_1(S^3\setminus K)$ to the category of tangles is introduced via a new category-theoretic construction. Through this presentation, a new avenue of proof for results about knot groups is opened.

math.AT↗

arXiv2002

Local models and integrability of certain almost Kahler 4-manifolds

We classify, up to a local isometry, all non-Kahler almost Kahler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kahler 4-manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure.

math.DG↗

arXiv2001

Local rigidity of certain classes of almost Kahler 4-manifolds

We show that any non-Kahler, almost Kahler 4-manifold for which both the Ricci and the Weyl curvatures have the same algebraic symmetries as they have for a Kahler metric is locally isometric to the (only) proper 3-symmetric 4-dimensional space described by O. Kowalski.

math.DG↗