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Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice U⊕E8(−1) depends only on the group but not on the K3 surface. For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces.

While the equation above is correct, it is not very useful for incremental updates since the equation above updates the checksum C, rather than the 1’s complement of the checksum, ̃C, which is the value stored in the checksum field. In addition, it suffers because the notation does not clearly specify that all arithmetic, including the unary negation, must be performed one’s complement, and so i...

When the degree of intersections A∩B of events A,B is unknown arises a problem: how to evaluate the probability P(A ∩ B) and P(A ∩B) from P(A) and P(B). To treat related problems two models of valuation: interval and paired probabilities are proposed. It is shown that the valuation corresponding to the set operations ∩ (intersection), ∪ (union) and ∼ (complement) can be described by the truth f...

An efficient chiral squaramide-catalysed enantioselective Michael addition of pyrazolin-5-ones to 3-nitro-2H-chromenes for the synthesis of chiral heterocyclic systems containing both chroman and pyrazolone derivatives has been developed. This reaction afforded the desired products in high to excellent yields (up to 98%) with high enantioselectivities (up to 96%) and excellent diastereoselectiv...

An efficient chiral squaramide-catalysed enantioselective Michael addition of pyrazolin-5-ones to nitroalkenes has been developed. This reaction afforded the chiral pyrazol-3-ol derivatives in high to excellent yields (up to >99%) with high enantioselectivities (up to 94% ee) for most substrates under very low catalyst loading (0.25 mol%). This catalytic asymmetric reaction provides valuable an...

This paper describes a simple means for embedding a checksum within a FITS header, providing a mechanism for later verifying the integrity of the FITS le. The method uses ASCII coded 32-bit 1's complement arithmetic to force the checksum of each FITS HDU to zero. This technique requires no extension of the FITS standard to be used by any given project, in fact the technique may be used to zero ...

The second author’s ω, ∆, χ conjecture proposes that every graph satisties χ ≤ d 1 2 (∆ + 1 + ω)e. In this paper we prove that the conjecture holds for all claw-free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way we discuss a stronger local conjecture, and prove that it holds for claw-free graphs with a three-colourable complement. To prove our results ...