‎By the Mordell‎- ‎Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎This paper studies the rank of the family Epq:y2=x3-pqx of elliptic curves‎, ‎where p and q are distinct primes‎. ‎We give infinite families of elliptic curves of the form y2=x3-pqx with rank two‎, ‎three and four‎, ‎assuming a conjecture of Schinzel ...

We prove that for certain families of semi-algebraic convex bodies in R3, the convex hull of n disjoint bodies has O(nλs(n)) features, where s is a constant depending on the family: λs(n) is the maximum length of order-s Davenport-Schinzel sequences with n letters. The argument is based on an apparently new idea of ‘compact families’ of convex bodies or discs, and of ‘crossing content’ and ‘foo...

A finite sequence u = a1a2 . . . ap of some symbols is contained in another sequence v = b1b2 . . . bq if there is a subsequence bi1bi2 . . . bip of v which can be identified, after an injective renaming of symbols, with u. We say that u = a1a2 . . . ap is k-regular if i − j ≥ k whenever ai = aj , i > j. We denote further by |u| the length p of u and by ‖u‖ the number of different symbols in u....