After publication, the authors realized that an antibody and siRNAs used in the paper were incorrect. Specifically, the antibody reported as Cdh1 in Fig. 5 E was instead an antibody against CDH1 (Cadherin). In addition, the three siRNAs in Fig. 5 E reported as siCdh1#1, siCdh1#2, and siCdh1#3 were also inadvertently synthesized against CDH1 (Cadherin). Upon discovering the error, the authors ge...

We present an algorithm for computing Minkowski sums among surfaces of revolution and surfaces of linear extrusion, generated by slope-monotone closed curves. The special structure of these simple surfaces allows the process of normal matching between two surfaces to be expressed as an explicit equation. Based on this insight, we also present an efficient algorithm for computing the distance be...

We show that for polytopes P1, P2, . . . , Pr ⊂ R, each having ni ≥ d + 1 vertices, the Minkowski sum P1 + P2 + · · · + Pr cannot achieve the maximum of ∏ i ni vertices if r ≥ d. This complements a recent result of Fukuda & Weibel (2006), who show that this is possible for up to d − 1 summands. The result is obtained by combining methods from discrete geometry (Gale transforms) and topological ...

It is known that in the Minkowski sum of r polytopes in dimension d, with r < d, the number of vertices of the sum can potentially be as high as the product of the number of vertices in each summand [2]. However, the number of vertices for sums of more polytopes was unknown so far. In this paper, we study sums of polytopes in general orientations, and show a linear relation between the number o...

We consider the problem of traveling the contour of the set of all points that are within distance 1 of a connected planar curve arrangement P, forming an embedding of the graph G. We show that if the overall length of P is L, there is a closed roundtrip that visits all points of the contour and has length no longer than 2L + 2n. This result carries over in a more general setting: if R is a com...

With the developing technology and increasing urbanization, many problems have started to arise in use of energy resources our natural environment. Energy-saving constitutes main purpose any proposed smart system. In addition, environmental effects should be taken into consideration. an outdoor lighting, along with energy-efficient bulbs, lighting done right method way. addition these, intellig...

submitted at the Oberwolfach Conference “Combinatorial Convexity and Algebraic Geometry” 26.10–01.11, 1997 Throughout, we fix the notation M := Z and MR := R . Given convex lattice polytopes P, P ′ ⊂ MR, we have (M ∩ P ) + (M ∩ P ) ⊂ M ∩ (P + P ), where P + P ′ is the Minkowski sum of P and P , while the left hand side means {m+m | m ∈ M ∩ P,m ∈ M ∩ P }. Problem 1 For convex lattice polytopes P...

In this paper we have shall generalize Shearer’s entropy inequality and its recent extensions by Madiman and Tetali, and shall apply projection inequalities to deduce extensions of some of the inequalities concerning sums of sets of integers proved recently by Gyarmati, Matolcsi and Ruzsa. We shall also discuss projection and entropy inequalities and their connections.

We present a tight bound on the exact maximum complexity of Minkowski sums of polytopes in R. In particular, we prove that the maximum number of facets of the Minkowski sum of k polytopes with m1,m2, . . . ,mk facets respectively is bounded from above by