We discuss the problem of partitioning a set of points into two subsets with certain conditions on the diameters of the subsets and on their cardinalities. For example, we give an O(n log n) algorithm to find the smallest t such that the set can be split into two equal cardinality subsets each of which has diameter at most t. We also give an algorithm that takes two pairs of points (x, y) and (...

The transfinite diameter is a way of quantifying the size of compact sets in Euclidean space. This quantity is related to the Hausdorff dimension and the Lebesgue measure, but gives a slightly different perspective on the set than either of those do. In this paper, we introduce the transfinite diameter, and outline some attempts to calculate this quantity for three sets in R. For z1, z2, . . . ...

Let me recall what we proved about the Galton-Watson process, and one thing that I forgot to mention. In particular, I will be concerned with the case in which a cell splits into k children, each of which survives to split themselves with probability p = c/k. Theorem 5.2.1. Let Z be one plus the number of descendants of the first organism that survive to reproduce (not counting the first organi...

8 We present an algorithm that computes a (1 + ε)-approximation of the diameter of a weighted, 9 undirected planar graph of n vertices with non-negative edge lengths inO ( n logn ( logn+ (1/ε)5 )) 10 expected time, improving upon the O ( n ( (1/ε)4 log4 n+ 2O(1/ε) )) -time algorithm of Weimann 11 and Yuster [ICALP 2013]. Our algorithm makes two improvements over that result: first and 12 foremo...

Pupil diameter is enhanced in a variety of emotional contexts, including viewing pictures, listening to sounds, and during threat of shock. In this study, we investigated pupil diameter changes during emotional imagery. Participants imagined scenes describing pleasant, unpleasant, or neutral events while pupil diameter was continuously recorded. Second by second changes in pupil diameter were a...

We prove a formula for the Fekete-Leja transfinite diameter of the pullback of a set E ⊂ C by a regular polynomial map F , expressing it in terms of the resultant of the leading part of F and the transfinite diameter of E. We also establish the nonarchimedean analogue of this formula. A key step in the proof is a formula for the transfinite diameter of the filled Julia set of F .

Let V denote a compact (without boundary) connected Riemannian manifold of dimension n. We denote by K the sectional curvature of V and we set inf K = ird~ K(-r) where 9 runs over all tangent 2-planes in V. One calls V a manifold of non-negative curvature ff inf K>~0. This condition has the following geometric meaning. A n n-dimensional Riemannian manifold has non-negative curvature iff for eac...

We highlight intriguing analogies between the diameter of a polytope and the largest possible total curvature of the associated central path. We prove continuous analogues of the results of Holt and Klee, and Klee and Walkup: We construct a family of polytopes which attain the conjectured order of the largest curvature, and prove that the special case where the number of inequalities is twice t...