We study the Minkowski length L(P ) of a lattice polytope P , which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in P . The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in P , and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a...

A total of 78 orthodontic patients participated in a study of the degree of decalcification inhibition induced by daily treatment with 0.4% SnF2 gel. Control patients (n = 39) were managed exactly as were the test patients (n = 39) except that the latter were instructed to brush at bedtime each day with the gel. In the nongel users, 24 (64.1%) developed decalcification after 18-24 months bandin...

In all, 182 and 220 replies were received from patients in group A and B, respectively. Early complications included haematoma in five (2%) and two (0.9%) patients, respectively, and wound infection in 24 (13%) and 17 (7.7%). Eight patients who had scrotal pain even before vasectomy were excluded from the analysis (two in group A and six in group B). In group A, 25 of 180 (13.8%) had a new onse...

We investigate the structure of the Minkowski sum of standard simplices in R. In particular, we investigate the one-dimensional structure, the vertices, their degrees and the edges in the Minkowski sum polytope. 2000 MSC: 52B05, 52B11, 05C07.

Let A be a finite nonempty set of integers. An asymptotic estimate of several dilates sum size was obtained by Bukh. The unique known exact bound concerns the sum |A + k ·A|, where k is a prime and |A| is large. In its full generality, this bound is due to Cilleruelo, Serra and the first author. Let k be an odd prime and assume that |A| > 8kk. A corollary to our main result states that |2 · A +...

Recently, Eisenbrand, Pach, Rothvoß, and Sopher studied the function M(m, n), which is the largest cardinality of a convexly independent subset of the Minkowski sum of some planar point sets P and Q with |P | = m and |Q| = n. They proved that M(m, n) = O(m2/3n2/3+m+n), and asked whether a superlinear lower bound exists for M(n, n). In this note, we show that their upper bound is the best possib...

Article history: Available online 17 July 2008