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....

Schinzel-Giedion syndrome (SGS) is a rare autosomal dominant disorder that results in facial dysmorphism, multiple congenital anomalies, and an increased risk of malignancy. Recently, using exome sequencing, de novo heterozygous mutations in the SETBP1 gene have been identified in patients with SGS. Most affected individuals do not survive after childhood because of the severity of this disorde...

We observe that five polynomial families have all of their roots on the unit circle. We prove the statements explicitly for four of the polynomial families. The polynomials have coefficients which involve Bernoulli numbers, Euler numbers, and the odd values of the Riemann zeta function. These polynomials are closely related to the Ramanujan polynomials, which were recently introduced by Murty, ...