Smallest cyclically covering subspaces of F-q(n), and lower bounds in Isbell's conjecture

For a prime power q and a positive integer n, we say a subspace U of Fqn is cyclically covering if the union of the cyclic shifts of U is equal to Fqn. We investigate the problem of determining the minimum possible dimension of a cyclically covering subspace of Fqn. (This is a natural generalisation of a problem posed in 1991 by the first author.) We prove several upper and lower bounds, and for each fixed q, we answer the question completely for infinitely many values of n (which take the form of certain geometric series). Our results imply lower bounds for a well-known conjecture of Isbell, and a generalisation theoreof, supplementing lower bounds due to Spiga. We also consider the analogous problem for general representations of groups. We use arguments from combinatorics, representation theory and finite field theory.PostprintPeer reviewe

Can we predict the duration of an interglacial?

Differences in the duration of interglacials have long been apparent in palaeoclimate records of the Late and Middle Pleistocene. However, a systematic evaluation of such differences has been hampered by the lack of a metric that can be applied consistently through time and by difficulties in separating the local from the global component in various proxies. This, in turn, means that a theoretical framework with predictive power for interglacial duration has remained elusive. Here we propose that the interval between the terminal oscillation of the bipolar seesaw and three thousand years (kyr) before its first major reactivation provides an estimate that approximates the length of the sea-level highstand, a measure of interglacial duration. We apply this concept to interglacials of the last 800 kyr by using a recently-constructed record of interhemispheric variability. The onset of interglacials occurs within 2 kyr of the boreal summer insolation maximum/precession minimum and is consistent with the canonical view of Milankovitch forcing pacing the broad timing of interglacials. Glacial inception always takes place when obliquity is decreasing and never after the obliquity minimum. The phasing of precession and obliquity appears to influence the persistence of interglacial conditions over one or two insolation peaks, leading to shorter (~ 13 kyr) and longer (~ 28 kyr) interglacials. Glacial inception occurs approximately 10 kyr after peak interglacial conditions in temperature and CO2, representing a characteristic timescale of interglacial decline. Second-order differences in duration may be a function of stochasticity in the climate system, or small variations in background climate state and the magnitude of feedbacks and mechanisms contributing to glacial inception, and as such, difficult to predict. On the other hand, the broad duration of an interglacial may be determined by the phasing of astronomical parameters and the history of insolation, rather than the instantaneous forcing strength at inception

Intercellular ultrafast Ca(2+) wave in vascular smooth muscle cells: numerical and experimental study.

Vascular smooth muscle cells exhibit intercellular Ca(2+) waves in response to local mechanical or KCl stimulation. Recently, a new type of intercellular Ca(2+) wave was observed in vitro in a linear arrangement of smooth muscle cells. The intercellular wave was denominated ultrafast Ca(2+) wave and it was suggested to be the result of the interplay between membrane potential and Ca(2+) dynamics which depended on influx of extracellular Ca(2+), cell membrane depolarization and its intercel- lular propagation. In the present study we measured experimentally the conduction velocity of the membrane depolarization and performed simulations of the ultrafast Ca(2+) wave along coupled smooth muscle cells. Numerical results reproduced a wide spectrum of experimental observations, including Ca(2+) wave velocity, electrotonic membrane depolarization along the network, effects of inhibitors and independence of the Ca(2+) wave speed on the intracellular stores. The numerical data also provided new physiological insights suggesting ranges of crucial model parameters that may be altered experimentally and that could significantly affect wave kinetics allowing the modulation of the wave characteristics experimentally. Numerical and experimental results supported the hypothesis that the propagation of membrane depolarization acts as an intercellular messenger mediating intercellular ultrafast Ca(2+) waves in smooth muscle cells

Instanton bundles on Fano threefolds

We introduce the notion of an instanton bundle on a Fano threefold of index 2. For such bundles we give an analogue of a monadic description and discuss the curve of jumping lines. The cases of threefolds of degree 5 and 4 are considered in a greater detail.Comment: 31 page, to appear in CEJ

Hypercontractivity on the qq-Araki-Woods algebras

Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the $q$-Ornstein-Uhlenbeck semigroup on the $q$-deformation of the free group algebra. In this note, we look for an extension of this result to the type III situation, that is for the $q$-Araki-Woods algebras. We show that hypercontractivity from $L^p$ to $L^2$ can occur if and only if the generator of the deformation is bounded.Comment: 17 page

Atmospheric methane, record from greenland ice core over the last 1000 years

The atmospheric methane concentration in ancient times can be reconstructed by analysing air entrapped in bubbles of polar ice sheets. We present results from an ice core from Central Greenland (Eurocore) covering the last 1000 years. We observe variations of about 70 ppbv around the mean pre-industrial level, which is confirmed at about 700 ppbv on a global average. According to our data, the beginning of the anthropogenic methane increase can be set between 1750 and 1800. Changes in the oxidizing capacity of the atmosphere may contribute significantly to the pre-industrial methane concentration variations, but changes in methane emissions probably play a dominant role. Since methane release depends on a host of influences it is difficult to specify clearly the reasons for these emission changes. Methane concentrations correlate only partially with proxy-data of climatic factors which influence the wetland release (the main source in pre-industrial times). A good correlation between our data and a population record from China suggests that man may already have influenced the CH4-cycle significantly before industrialisation

The integral monodromy of hyperelliptic and trielliptic curves

We compute the \integ/\ell and \integ_\ell monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the \integ/\ell monodromy of the moduli space of hyperelliptic curves of genus $g$ is the symplectic group \sp_{2g}(\integ/\ell). We prove that the \integ/\ell monodromy of the moduli space of trielliptic curves with signature $(r,s)$ is the special unitary group \su_{(r,s)}(\integ/\ell\tensor\integ[\zeta_3])

Regulator constants and the parity conjecture

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-infinity Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is semistable at 2 and 3, K/Q is abelian and K^\infty is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of Gal(K^\infty/Q). We also give analogous results when K/Q is non-abelian, the base field is not Q and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their "regulator constants", and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.Comment: 50 pages; minor corrections; final version, to appear in Invent. Mat