On the Closing Lemma problem for vector fields of bounded type on the torus

We investigate the open Closing Lemma problem for vector fields on the 2-dimensional torus. Under the assumption of bounded type rotation number, the $C^r$ Closing Lemma is verified for smooth vector fields that are area-preserving at all saddle points. Namely, given such a $C^r$ vector field $X$, $r\geq 4$, with a non-trivially recurrent point $p$, there exists a vector field $Y$ arbitrarily near to $X$ in the $C^r$ topology and obtained from $X$ by a twist perturbation, such that $p$ is a periodic point of $Y$. The proof relies on a new result in 1-dimensional dynamics on the non-existence of semi-wandering intervals of smooth maps of the circle.Comment: 11 pages, 1 figur

A semi-invertible Oseledets Theorem with applications to transfer operator cocycles

Oseledets' celebrated Multiplicative Ergodic Theorem (MET) is concerned with the exponential growth rates of vectors under the action of a linear cocycle on R^d. When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise splitting of R^d into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets' MET only yields the existence of a filtration of subspaces, the elements of which contain all vectors that grow no faster than exponential rates given by the Lyapunov exponents. The authors recently demonstrated that a splitting over R^d is guaranteed even without the invertibility assumption on the linear actions. Motivated by applications of the MET to cocycles of (non-invertible) transfer operators arising from random dynamical systems, we demonstrate the existence of an Oseledets splitting for cocycles of quasi-compact non-invertible linear operators on Banach spaces.Comment: 26 page

Coherent sets for nonautonomous dynamical systems

We describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems. In the autonomous setting, such objects are variously known as almost-invariant sets, metastable sets, persistent patterns, or strange eigenmodes, and have proved to be important in a variety of applications. In this current work, we explain how to extend existing autonomous approaches to the nonautonomous setting. We call the new time-dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent. The new methods are illustrated via detailed examples in both discrete and continuous time

‘Not Just Another Anonymous Spot’: Government Support for Memory Institutions in Prince Edward Island and Wales

In view of the important role that libraries, archives and museums – collectively referred to here as memory institutions – can play in regenerating communities that have faced economic and social difficulties, this essay examines government support for this sector in Prince Edward Island (PEI) through a comparative study with Wales. A historical sketch of the development of government administrative and planning capacity in this field in both Wales and PEI is offered, followed by a comparison of current funding commitments by each government. The study finds that relative funding levels are about the same in both jurisdictions, but that the Welsh government can offer an instructive example to the government of PEI through the former’s commitment to planning for museums, archives and libraries, and for cultural and heritage activity as whole

Complete spatial characterization of an optical wavefront using a variable-separation pinhole pair

We present a technique for measuring the transverse spatial properties of an optical wavefront. Intensity and phase profiles are recovered by analysis of a series of interference patterns produced by the combination of a scanning X-shaped slit and a static horizontal slit; the spatial coherence may be found from the same data. We demonstrate the technique by characterizing high harmonic radiation generated in a gas cell, however the method could be extended to a wide variety of light sources.Comment: 4 pages, 3 figures, 1 tabl

Mapping PAMP Responses and Disease Resistance in Brassicas

The first layer of active defence in plants is based on the perception of pathogen-associated molecular patterns (PAMPs) leading to PAMP-triggered immunity (PTI). PTI is increasingly being investigated in crop plants, where it may have potential to provide durable disease resistance in the field. Limiting this work, however, is an absence of reliable bioassays to investigate PAMP responses in some species. Presented here is a series of methods to investigate PTI in Brassica napus. The assays allow measuring early cell signalling responses, gene expression changes, cell wall reinforcement, metabolome changes and scoring PAMP-Induced resistance. Illumina-based RNA sequencing analysis produced a genome-wide survey of transcriptional changes upon PAMP treatment seen in both the A and C genomes of the allotetraploid B. napus. Using these assays substantial variation in PAMP-responsiveness was observed amongst elite varieties of B. napus. Taking this further, a genome wide association study (GWAS) of the flg22 and elf18 triggered oxidative bursts, resistance to Pseudomonas syringae and Botrytis cinerea was carried out in a population of B. napus. A substantial number of molecular markers, covering both sequence and expression variation, have been identified as having significant association with these four traits. QTL mapping of the flg22 triggered oxidative burst in the ADxGD double haploid cross identified a major quantitative trait loci (QTL) on C9 of B. oleracea. mRNA-seq of the parents led to a non-synonymous single nucleotide polymorphism (SNP) list and enabled fine mapping through the addition of KASPar markers to the original map. This produced a relatively small list of candidate genes including CYLIC NUCLEOTIDE GATED ION CHANNEL 4 (CNGC4) also known as DEFENCE NO DEATH 2 (DND2). An insertion in Arabidopsis thaliana DND2 showed phenotypic difference in the oxidative burst between the insertion line and Col-0, potentially indicating a role for the gene in regulating early PTI

Quantum Computing with an 'Always On' Heisenberg Interaction

Many promising ideas for quantum computing demand the experimental ability to directly switch 'on' and 'off' a physical coupling between the component qubits. This is typically the key difficulty in implementation, and precludes quantum computation in generic solid state systems, where interactions between the constituents are 'always on'. Here we show that quantum computation is possible in strongly coupled (Heisenberg) systems even when the interaction cannot be controlled. The modest ability of 'tuning' the transition energies of individual qubits proves to be sufficient, with a suitable encoding of the logical qubits, to generate universal quantum gates. Furthermore, by tuning the qubits collectively we provide a scheme with exceptional experimental simplicity: computations are controlled via a single 'switch' of only six settings. Our schemes are applicable to a wide range of physical implementations, from excitons and spins in quantum dots through to bulk magnets.Comment: 4 pages, 3 figs, 2 column format. To appear in PR