Cubic Curves, Finite Geometry and Cryptography

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a `shared secret' related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.Comment: This is a version of our article to appear in Acta Applicandae Mathematicae. In this version, we have corrected a sentence in the third paragraph. The final publication is available at springerlink.com at http://www.springerlink.com/content/xh85647871215644

Is the Shroud of Turin in Relation to the Old Jerusalem Historical Earthquake?

Phillips and Hedges suggested, in the scientific magazine Nature (1989), that neutron radiation could be liable of a wrong radiocarbon dating, while proton radiation could be responsible of the Shroud body image formation. On the other hand, no plausible physical reason has been proposed so far to explain the radiation source origin, and its effects on the linen fibres. However, some recent studies, carried out by the first author and his Team at the Laboratory of Fracture Mechanics of the Politecnico di Torino, found that it is possible to generate neutron emissions from very brittle rock specimens in compression through piezonuclear fission reactions. Analogously, neutron flux increments, in correspondence to seismic activity, should be a result of the same reactions. A group of Russian scientists measured a neutron flux exceeding the background level by three orders of magnitude in correspondence to rather appreciable earthquakes (4th degree in Richter Scale). The authors consider the possibility that neutron emissions by earthquakes could have induced the image formation on Shroud linen fibres, trough thermal neutron capture by Nitrogen nuclei, and provided a wrong radiocarbon dating due to an increment in C(14,6)content. Let us consider that, although the calculated integral flux of 10^13 neutrons per square centimetre is 10 times greater than the cancer therapy dose, nevertheless it is100 times smaller than the lethal dose.Comment: 13 pages, 1 figur

Pan-genome analysis highlights the role of structural variation in the evolution and environmental adaptation of Asian honeybees.

The Asian honeybee, Apis cerana, is an ecologically and economically important pollinator. Mapping its genetic variation is key to understanding population-level health, histories and potential capacities to respond to environmental changes. However, most efforts to date were focused on single nucleotide polymorphisms (SNPs) based on a single reference genome, thereby ignoring larger scale genomic variation. We employed long-read sequencing technologies to generate a chromosome-scale reference genome for the ancestral group of A. cerana. Integrating this with 525 resequencing data sets, we constructed the first pan-genome of A. cerana, encompassing almost the entire gene content. We found that 31.32% of genes in the pan-genome were variably present across populations, providing a broad gene pool for environmental adaptation. We identified and characterized structural variations (SVs) and found that they were not closely linked with SNP distributions; however, the formation of SVs was closely associated with transposable elements. Furthermore, phylogenetic analysis using SVs revealed a novel A. cerana ecological group not recoverable from the SNP data. Performing environmental association analysis identified a total of 44 SVs likely to be associated with environmental adaptation. Verification and analysis of one of these, a 330 bp deletion in the Atpalpha gene, indicated that this SV may promote the cold adaptation of A. cerana by altering gene expression. Taken together, our study demonstrates the feasibility and utility of applying pan-genome approaches to map and explore genetic feature variations of honeybee populations, and in particular to examine the role of SVs in the evolution and environmental adaptation of A. cerana

Global patterns of diapycnal mixing from measurements of the turbulent dissipation rate

The authors present inferences of diapycnal diffusivity from a compilation of over 5200 microstructure profiles. As microstructure observations are sparse, these are supplemented with indirect measurements of mixing obtained from (i) Thorpe-scale overturns from moored profilers, a finescale parameterization applied to (ii) shipboard observations of upper-ocean shear, (iii) strain as measured by profiling floats, and (iv) shear and strain from full-depth lowered acoustic Doppler current profilers (LADCP) and CTD profiles. Vertical profiles of the turbulent dissipation rate are bottom enhanced over rough topography and abrupt, isolated ridges. The geography of depth-integrated dissipation rate shows spatial variability related to internal wave generation, suggesting one direct energy pathway to turbulence. The global-averaged diapycnal diffusivity below 1000-m depth is O(10?4) m2 s?1 and above 1000-m depth is O(10?5) m2 s?1. The compiled microstructure observations sample a wide range of internal wave power inputs and topographic roughness, providing a dataset with which to estimate a representative global-averaged dissipation rate and diffusivity. However, there is strong regional variability in the ratio between local internal wave generation and local dissipation. In some regions, the depth-integrated dissipation rate is comparable to the estimated power input into the local internal wave field. In a few cases, more internal wave power is dissipated than locally generated, suggesting remote internal wave sources. However, at most locations the total power lost through turbulent dissipation is less than the input into the local internal wave field. This suggests dissipation elsewhere, such as continental margins

Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.Comment: 39 page

Amicable pairs and aliquot cycles for elliptic curves

An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good reduction for E satisfying #E(F_p) = q and #E(F_q) = p. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We show that there exist elliptic curves with arbitrarily long aliqout cycles, but that CM elliptic curves (with j not 0) have no aliqout cycles of length greater than two. We give conjectural formulas for the frequency of amicable pairs. For CM curves, the derivation of precise conjectural formulas involves a detailed analysis of the values of the Grossencharacter evaluated at a prime ideal P in End(E) having the property that #E(F_P) is prime. This is especially intricate for the family of curves with j = 0.Comment: 53 page