Primitive divisors on twists of the Fermat cubic

We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u3+v3=m, with m cube-free, all the terms beyond the first have a primive divisor

Primitive divisors of elliptic divisibility sequences

Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. As an application, we show that every term beyond the fourth of the Somos-4 sequence has a primitive divisor

Dynamical systems arising from elliptic curves

We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve whose topological entropy is given by the local canonical height. Also, a precise formula for the periodic points is given. There follows a discussion of how these local results may be glued together to give a map on the adelic curve. We are able to give a map whose entropy is the global canonical height and whose periodic points are counted asymptotically by the real division polynomial (although the archimedean component of the map is artificial). Finally, we set out a precise conjecture about the existence of elliptic dynamical systems and discuss a possible connection with mathematical physics

The elliptic curve discrete logarithm problem and equivalent hard problems for elliptic divisibility sequences

We define three hard problems in the theory of elliptic divisibility sequences (EDS Association, EDS Residue and EDS Discrete Log), each of which is solvable in sub-exponential time if and only if the elliptic curve discrete logarithm problem is solvable in sub-exponential time. We also relate the problem of EDS Association to the Tate pairing and the MOV, Frey-R\"{u}ck and Shipsey EDS attacks on the elliptic curve discrete logarithm problem in the cases where these apply.Comment: 18 pages; revised version includes some small mathematical corrections, reformatte

Euclidean Mahler measure and twisted links

If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily alternating, have bounded twist numbers, then both the Jones polynomials and a parametrization of the 2-variable Homflypt polynomials of the corresponding links have bounded Mahler measure.Comment: This is the version published by Algebraic & Geometric Topology on 7 April 200

Squirrelpox virus: assessing prevalence, transmission and environmental degradation

Red squirrels (Sciurus vulgaris) declined in Great Britain and Ireland during the last century, due to habitat loss and the introduction of grey squirrels (Sciurus carolinensis), which competitively exclude the red squirrel and act as a reservoir for squirrelpox virus (SQPV). The disease is generally fatal to red squirrels and their ecological replacement by grey squirrels is up to 25 times faster where the virus is present. We aimed to determine: (1) the seropositivity and prevalence of SQPV DNA in the invasive and native species at a regional scale; (2) possible SQPV transmission routes; and, (3) virus degradation rates under differing environmental conditions. Grey (n = 208) and red (n = 40) squirrel blood and tissues were sampled. Enzyme-linked immunosorbent assay (ELISA) and quantitative real-time polymerase chain reaction (qPCR) techniques established seropositivity and viral DNA presence, respectively. Overall 8% of squirrels sampled (both species combined) had evidence of SQPV DNA in their tissues and 22% were in possession of antibodies. SQPV prevalence in sampled red squirrels was 2.5%. Viral loads were typically low in grey squirrels by comparison to red squirrels. There was a trend for a greater number of positive samples in spring and summer than in winter. Possible transmission routes were identified through the presence of viral DNA in faeces (red squirrels only), urine and ectoparasites (both species). Virus degradation analyses suggested that, after 30 days of exposure to six combinations of environments, there were more intact virus particles in scabs kept in warm (25°C) and dry conditions than in cooler (5 and 15°C) or wet conditions. We conclude that SQPV is present at low prevalence in invasive grey squirrel populations with a lower prevalence in native red squirrels. Virus transmission could occur through urine especially during warm dry summer conditions but, more notably, via ectoparasites, which are shared by both species

Physiological responses during ascent to high altitude and the incidence of acute mountain sickness.

Acute mountain sickness (AMS) occurs when there is failure of acclimatisation to high altitude. The aim of this study was to describe the relationship between physiological variables and the incidence of AMS during ascent to 5300 m. A total of 332 lowland-dwelling volunteers followed an identical ascent profile on staggered treks. Self-reported symptoms of AMS were recorded daily using the Lake Louise score (mild 3-4; moderate-severe ≥5), alongside measurements of physiological variables (heart rate, respiratory rate (RR), peripheral oxygen saturation (SpO2 ) and blood pressure) before and after a standardised Xtreme Everest Step-Test (XEST). The overall occurrence of AMS among participants was 73.5% (23.2% mild, 50.3% moderate-severe). There was no difference in gender, age, previous AMS, weight or body mass index between participants who developed AMS and those who did not. Participants who had not previously ascended >5000 m were more likely to get moderate-to-severe AMS. Participants who suffered moderate-to-severe AMS had a lower resting SpO2 at 3500 m (88.5 vs. 89.6%, p = 0.02), while participants who suffered mild or moderate-to-severe AMS had a lower end-exercise SpO2 at 3500 m (82.2 vs. 83.8%, p = 0.027; 81.5 vs. 83.8%, p 5000 m (OR 2.740, p-value 0.003) predicted the development of moderate-to-severe AMS. The Xtreme Everest Step-Test offers a simple, reproducible field test to help predict AMS, albeit with relatively limited predictive precision

On the Isomorphic Description of Chiral Symmetry Breaking by Non-Unitary Lie Groups

It is well-known that chiral symmetry breaking ($\chi$SB) in QCD with $N_{f}=2$ light quark flavours can be described by orthogonal groups as $O(4) \to O(3)$, due to local isomorphisms. Here we discuss the question how specific this property is. We consider generalised forms of $\chi$SB involving an arbitrary number of light flavours of continuum or lattice fermions, in various representations. We search systematically for isomorphic descriptions by non-unitary, compact Lie groups. It turns out that there are a few alternative options in terms of orthogonal groups, while we did not find any description entirely based on symplectic or exceptional Lie groups. If we adapt such an alternative as the symmetry breaking pattern for a generalised Higgs mechanism, we may consider a Higgs particle composed of bound fermions and trace back the mass generation to $\chi$SB. In fact, some of the patterns that we encounter appear in technicolour models. In particular if one observes a Higgs mechanism that can be expressed in terms of orthogonal groups, we specify in which cases it could also represent some kind of $\chi$SB of techniquarks.Comment: 18 pages, to appear in Int. J. Mod. Phys.

Novel imaging tools for investigating the role of immune signalling in the brain

Abstract not availableJonathan Henry W. Jacobsen, Lindsay M. Parker, Arun V. Everest-Dass, Erik P. Schartner, Georgios Tsiminis, Vasiliki Staikopoulos, Mark R. Hutchinson, Sanam Mustaf