Computing the Cassels–Tate pairing on the 3-Selmer group of an elliptic curve

We extend the method of Cassels for computing the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent.

Mersenne numbers

These notes have been issued on a small scale in 1983 and 1987 and on request at other times. This issue follows two items of news. First, WaIter Colquitt and Luther Welsh found the 'missed' Mersenne prime M110503 and advanced the frontier of complete Mp-testing to 139,267. In so doing, they terminated Slowinski's significant string of four consecutive Mersenne primes. Secondly, a team of five established a non-Mersenne number as the largest known prime. This result terminated the 1952-89 reign of Mersenne primes. All the original Mersenne numbers with p < 258 were factorised some time ago. The Sandia Laboratories team of Davis, Holdridge & Simmons with some little assistance from a CRAY machine cracked M211 in 1983 and M251 in 1984. They contributed their results to the 'Cunningham Project', care of Sam Wagstaff. That project is now moving apace thanks to developments in technology, factorisation and primality testing. New levels of computer power and new computer architectures motivated by the open-ended promise of parallelism are now available. Once again, the suppliers may be offering free buildings with the computer. However, the Sandia '84 CRAY-l implementation of the quadratic-sieve method is now outpowered by the number-field sieve technique. This is deployed on either purpose-built hardware or large syndicates, even distributed world-wide, of collaborating standard processors. New factorisation techniques of both special and general applicability have been defined and deployed. The elliptic-curve method finds large factors with helpful properties while the number-field sieve approach is breaking down composites with over one hundred digits. The material is updated on an occasional basis to follow the latest developments in primality-testing large Mp and factorising smaller Mp; all dates derive from the published literature or referenced private communications. Minor corrections, additions and changes merely advance the issue number after the decimal point. The reader is invited to report any errors and omissions that have escaped the proof-reading, to answer the unresolved questions noted and to suggest additional material associated with this subject.

Transcendental Brauer groups of products of CM elliptic curves

Let L be a number field and let E/L be an elliptic curve with complex multiplication by the ring of integers O_K of an imaginary quadratic field K. We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface ExE. The results for the odd order torsion also apply to the Brauer group of the K3 surface Kum(ExE). We describe explicitly the elliptic curves E/Q with complex multiplication by O_K such that the Brauer group of ExE contains a transcendental element of odd order. We show that such an element gives rise to a Brauer-Manin obstruction to weak approximation on Kum(ExE), while there is no obstruction coming from the algebraic part of the Brauer group.

Shadow lines in the arithmetic of elliptic curves

Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K)\otimes Q_p, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q)\otimes Qp. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We conclude by illustrating these algorithms in a collection of examples.

Law of large numbers for super-brownian motions with a single point source

We investigate the super-Brownian motion with a single point source in dimensions 2 and 3 as constructed by Fleischmann and Mueller in 2004. Using analytic facts we derive the long time behavior of the mean in dimension 2 and 3 thereby complementing previous work of Fleischmann, Mueller and Vogt. Using spectral theory and martingale arguments we prove a version of the strong law of large numbers for the two dimensional superprocess with a single point source and finite variance.

Finding the point on Bezier Curves with the normal vector passing an external point

Two algorithms for finding the point on non-rational/rational Bezier curves of which the normal vector passes through a given external point are presented. The algorithms are based on Bezier curves generation algorithms of de Casteljau's algorithm for non-rational Bezier curve or Farin's recursion for rational Bezier curve, respectively. Orthogonal projections from the external point are used to guide the directional search used in the proposed iterative algorithms. Using Lyapunov's method, it is shown that each algorithm is able to converge to a local minimum for each case of non-rational/rational Bezier curves. It is also shown that on convergence the distance between the point on curves to the external point reaches a local minimum for both approaches. Illustrative examples are included to demonstrate the effectiveness of the proposed approaches.

Reduced complexity set-point following in self-tuning control

An external input signal is incorporated into a self-tuning controller which, although it is based on a CARMA system model, employs a state-space framework for control law calculations. Steady-state set point following can then be accomplished even when only a recursive least squares parameter estimation scheme is used, despite the fact that the disturbance affecting the system may well be coloured.

In search of the aim of the free movement of persons provisions: has the Court of Justice missed the point?

The Court of Justice has, over the years, often been vilified for exceeding the limits of its jurisdiction by interpreting the provisions of Community legislation in a way not seem originally envisaged by its drafters. A recent example of this approach was a cluster of cases in the context of the free movement of workers and the freedom of establishment (Ritter-Coulais and its progeny), where the Court included within the scope of those provisions situations which, arguably, did not present a sufficient link with their (economic) aim. In particular, in that case law the Court accepted that the mere exercise of free movement for the purpose of taking up residence in the territory of another Member State whilst continuing to exercise an economic activity in the State of origin, suffices for bringing a Member State national within the scope of Articles 39 and 43 EC. It is argued that the most plausible explanation for this approach is that the Court now wishes to re-read the economic fundamental freedoms in such a way as to include within their scope all economically active Union citizens, irrespective of whether their situation presents a sufficient link with the exercise of an economic activity in a cross-border context. It is suggested that this approach is problematic for a number of reasons. It is, therefore, concluded that the Court should revert to its orthodox approach, according to which only situations that involve Union citizens who have moved between Member States for the purpose of taking up an economic activity should be included within the scope of the market freedoms.

From symmetry breaking to Poisson Point Process in 2D Voronoi Tessellations: the generic nature of hexagons

We bridge the properties of the regular triangular, square, and hexagonal honeycomb Voronoi tessellations of the plane to the Poisson-Voronoi case, thus analyzing in a common framework symmetry breaking processes and the approach to uniform random distributions of tessellation-generating points. We resort to ensemble simulations of tessellations generated by points whose regular positions are perturbed through a Gaussian noise, whose variance is given by the parameter α2 times the square of the inverse of the average density of points. We analyze the number of sides, the area, and the perimeter of the Voronoi cells. For all valuesα >0, hexagons constitute the most common class of cells, and 2-parameter gamma distributions provide an efficient description of the statistical properties of the analyzed geometrical characteristics. The introduction of noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α = 0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise withα <0.12. For all tessellations and for small values of α, we observe a linear dependence on α of the ensemble mean of the standard deviation of the area and perimeter of the cells. Already for a moderate amount of Gaussian noise (α >0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α >2, results converge to those of Poisson-Voronoi tessellations. The geometrical properties of n-sided cells change with α until the Poisson- Voronoi limit is reached for α > 2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established. This law allows for an easy link to the Lewis law for areas and agrees with exact asymptotic results. Finally, for α >1, the ensemble mean of the cells area and perimeter restricted to the hexagonal cells agree remarkably well with the full ensemble mean; this reinforces the idea that hexagons, beyond their ubiquitous numerical prominence, can be interpreted as typical polygons in 2D Voronoi tessellations.

The importance of attractive three-point interaction in enantioselective surface chemistry: stereospecific adsorption of serine on the intrinsically chiral Cu{531} surface

Both enantiomers of serine adsorb on the intrinsically chiral Cu{531} surface in two different adsorption geometries, depending on the coverage. At saturation, substrate bonds are formed through the two oxygen atoms of the carboxylate group and the amino group (μ3 coordination), whereas at lower coverage, an additional bond is formed through the deprotonated β−OH group (μ4 coordination). The latter adsorption geometry involves substrate bonds through three side groups of the chiral center, respectively, which leads to significantly larger enantiomeric differences in adsorption geometries and energies compared to the μ3 coordination, which involves only two side groups. This relatively simple model system demonstrates, in direct comparison, that attractive interactions of three side groups with the substrate are much more effective in inducing strong enantiomeric differences in heterogeneous chiral catalyst systems than hydrogen bonds or repulsive interactions.

Comparison of different boundary layer surface schemes using single point micrometeorological field data

 In the last decade, a vast number of land surface schemes has been designed for use in global climate models, atmospheric weather prediction, mesoscale numerical models, ecological models, and models of global changes. Since land surface schemes are designed for different purposes they have various levels of complexity in the treatment of bare soil processes, vegetation, and soil water movement. This paper is a contribution to a little group of papers dealing with intercomparison of differently designed and oriented land surface schemes. For that purpose we have chosen three schemes for classification: i) global climate models, BATS (Dickinson et al., 1986; Dickinson et al., 1992); ii) mesoscale and ecological models, LEAF (Lee, 1992) and iii) mesoscale models, LAPS (Mihailović, 1996; Mihailović and Kallos, 1997; Mihailović et al., 1999) according to the Shao et al. (1995) classification. These schemes were compared using surface fluxes and leaf temperature outputs obtained by time integrations of data sets derived from the micrometeorological measurements above a maize field at an experimental site in De Sinderhoeve (The Netherlands) for 18 August, 8 September, and 4 October 1988. Finally, comparison of the schemes was supported applying a simple statistical analysis on the surface flux outputs.

Comparison of different boundary layer surface schemes using single point micrometeorological field data

 In the last decade, a vast number of land surface schemes has been designed for use in global climate models, atmospheric weather prediction, mesoscale numerical models, ecological models, and models of global changes. Since land surface schemes are designed for different purposes they have various levels of complexity in the treatment of bare soil processes, vegetation, and soil water movement. This paper is a contribution to a little group of papers dealing with intercomparison of differently designed and oriented land surface schemes. For that purpose we have chosen three schemes for classification: i) global climate models, BATS (Dickinson et al., 1986; Dickinson et al., 1992); ii) mesoscale and ecological models, LEAF (Lee, 1992) and iii) mesoscale models, LAPS (Mihailović, 1996; Mihailović and Kallos, 1997; Mihailović et al., 1999) according to the Shao et al. (1995) classification. These schemes were compared using surface fluxes and leaf temperature outputs obtained by time integrations of data sets derived from the micrometeorological measurements above a maize field at an experimental site in De Sinderhoeve (The Netherlands) for 18 August, 8 September, and 4 October 1988. Finally, comparison of the schemes was supported applying a simple statistical analysis on the surface flux outputs.

Endocannabinoids, FOXO and the metabolic syndrome: redox, function and tipping point--the view from two systems

The endocannabinoid system (ECS) was only 'discovered' in the 1990s. Since then, many new ligands have been identified, as well as many new intracellular targets--ranging from the PPARs, to mitochondria, to lipid rafts. It was thought that blocking the CB-1 receptor might reverse obesity and the metabolic syndrome. This was based on the idea that the ECS was dysfunctional in these conditions. This has met with limited success. The reason may be that the ECS is a homeostatic system, which integrates energy seeking and storage behaviour with resistance to oxidative stress. It could be viewed as having thrifty actions. Thriftiness is an innate property of life, which is programmed to a set point by both environment and genetics, resulting in an epigenotype perfectly adapted to its environment. This thrifty set point can be modulated by hormetic stimuli, such as exercise, cold and plant micronutrients. We have proposed that the physiological and protective insulin resistance that underlies thriftiness encapsulates something called 'redox thriftiness', whereby insulin resistance is determined by the ability to resist oxidative stress. Modern man has removed most hormetic stimuli and replaced them with a calorific sedentary lifestyle, leading to increased risk of metabolic inflexibility. We suggest that there is a tipping point where lipotoxicity in adipose and hepatic cells induces mild inflammation, which switches thrifty insulin resistance to inflammation-driven insulin resistance. To understand this, we propose that the metabolic syndrome could be seen from the viewpoint of the ECS, the mitochondrion and the FOXO group of transcription factors. FOXO has many thrifty actions, including increasing insulin resistance and appetite, suppressing oxidative stress and shifting the organism towards using fatty acids. In concert with factors such as PGC-1, they also modify mitochondrial function and biogenesis. Hence, the ECS and FOXO may interact at many points; one of which may be via intracellular redox signalling. As cannabinoids have been shown to modulate reactive oxygen species production, it is possible that they can upregulate anti-oxidant defences. This suggests they may have an 'endohormetic' signalling function. The tipping point into the metabolic syndrome may be the result of a chronic lack of hormetic stimuli (in particular, physical activity), and thus, stimulus for PGC-1, with a resultant reduction in mitochondrial function and a reduced lipid capacitance. This, in the context of a positive calorie environment, will result in increased visceral adipose tissue volume, abnormal ectopic fat content and systemic inflammation. This would worsen the inflammatory-driven pathological insulin resistance and inability to deal with lipids. The resultant oxidative stress may therefore drive a compensatory anti-oxidative response epitomised by the ECS and FOXO. Thus, although blocking the ECS (e.g. via rimonabant) may induce temporary weight loss, it may compromise long-term stress resistance. Clues about how to modulate the system more safely are emerging from observations that some polyphenols, such as resveratrol and possibly, some phytocannabinoids, can modulate mitochondrial function and might improve resistance to a modern lifestyle.

Trans-floating-point arithmetic removes nine quadrillion redundancies from 64-bit IEEE 754 floating-point arithmetic

IEEE 754 floating-point arithmetic is widely used in modern, general-purpose computers. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity. Modifying the IEEE arithmetic so that it uses transreal arithmetic has a number of advantages. It removes one redundant binade from IEEE floating-point objects, doubling the numerical precision of the arithmetic. It removes eight redundant, relational,floating-point operations and removes the redundant total order operation. It replaces the non-reflexive, floating-point, equality operator with a reflexive equality operator and it indicates that some of the exceptions may be removed as redundant { subject to issues of backward compatibility and transient future compatibility as programmers migrate to the transreal paradigm.