Let f(d) be the smallest number so that every set in Rd of diameter 1 can be partitioned into f(d) sets of diameter smaller than 1. Borsuk's conjecture was that f(d)=d+\ . We prove that f(d)>(\.2)^ for large d.

Given a tree T, let q(T) be the minimum number of distinct eigenvalues in symmetric matrix whose underlying graph is T. It well known that q(T)≥d(T)+1, where d(T) diameter and T said to diminimal if q(T)=d(T)+1. In this paper, we present families trees any fixed diameter. Our proof constructive, allowing us compute, for d these families, M with spectrum has exactly d+1 eigenvalues.

Regular digraphs of degree d > 1, diameter k > 1 and order N(d, k) = d+· · ·+dk will be called almost Moore (d, k)-digraphs. So far, the problem of their existence has only been solved when d = 2, 3 or k = 2, 3. In this paper we prove that almost Moore digraphs of diameter 4 do not exist for any degree d.

Effects of 2 pellet diameters for piglets pre- and postweaning on feed intake, BW, and feed-related behavior were studied in 3 experiments. In Exp. 1, 19 litters were provided with pellets of 2 and 12 mm in diameter in a choice-feeding setup from d 4 of lactation onward. From d 4 to 18, piglets preferred the 12-mm diameter pellet over the 2-mm diameter pellet (519 vs. 168 g/pen; P < 0.001). In ...

We study connection networks in which certain pairs of nodes have to be connected by k edge-disjoint paths, and study bounds for the minimal sum of lengths of such k paths. We define the related notions of totalk-distance for a pair of nodes and totalk-diameter of a connection network, and study the value TDk(d) which is the maximal such totalk-diameter of a network with diameter d. These notio...

A classic result is that the number of distinct eigenvalues of a graph of diameter d is at least d+1. More generally, let G be a graph with n vertices, and let S(G) denote the set of all symmetric n by n matrices, A = [aij ], with the property that for off-diagonal entries, aij is nonzero if and only if G has an edge from i to j. The traditional proof of the classic result extends to A ∈ S(G). ...

in this study the effects of plant growth regulators including; ga3 (0 and 50 mg/l) and ethephon (0 and 500 mg/l) were studied on the quality properties of two grape cultivars (perlette and yaghuti). at the harvest time, some physicochemical characteristics such as fruit cluster weight, fruit diameter, length, volume and l/d ratio, flesh firmness, fresh weight, ta, ph, tss and fruit color were ...

PURPOSE To compare diameter-based glenoid bone loss quantification with a true geometric calculation for the area of a circular segment. METHODS By use of Maxima 12.01.0 mathematics modeling software (Macysma, Boston, MA), the diameter-based glenoid bone loss equation (% Bone loss = [Defect width (w)/Inferior glenoid circle diameter (D)] × 100%) was compared with a true geometric calculation ...

For any d ≥ 5 and k ≥ 3 we construct a family of Cayley graphs of degree d, diameter k, and order at least k((d−3)/3)k. By comparison with other available results in this area we show that, for all sufficiently large d and k, our family gives the current largest known Cayley graphs of degree d and diameter k.