This article investigates the evolution of hog spatial market integration in China based on a minimal spanning tree by using provincial hog price data. The empirical results show that hog spatial market integration in China has increased gradually and reached a high stable level after 2012. Hog spatial market integration underwent a structural break in April 2007, after which hog market integration was greatly strengthened. Moreover, the market power of hog markets in eastern China and central China is increasing, and Shandong is a price setter, whereas hog markets in southwestern, northeastern, and northern China are price followers.

The primary dendrite arm spacing and its distribution at the solid–liquid interface has been examined in directionally solidified Sn–36 at.% Ni peritectic alloys under constant temperature gradient in a range of growth rates (2–200 μm/s). Statistical analysis of the primary dendrite arm spacing on transverse sections has been carried out using the minimum spanning tree and Voronoi polygon. The frequency distribution of the number of nearest neighbors determined by the Voronoi polygon suggested that the arrangement of dendrites at the solid–liquid interface could be visualized as hexagonal tessellation. The primary dendrite arm spacing determined by the conventional area counting method and minimum spanning tree all decreased with increasing growth rate, and a range of primary dendrite spacing was present during solidification. The range first increased with increasing growth rate, but when the growth rate exceeded 20 μm/s, it turned to decrease, which can be attributed to disorder induced by growth rate and interdendritic convection.

For n independent, identically distributed uniform points in [0, 1]d, d ≥ 2, let Ln be the total distance from the origin to all the minimal points under the coordinatewise partial order (this is also the total length of the rooted edges of a minimal directed spanning tree on the given random points). For d ≥ 3, we establish the asymptotics of the mean and the variance of Ln, and show that Ln satisfies a central limit theorem, unlike in the case d = 2.

We study the asymptotic properties of a minimal spanning tree formed by n points uniformly distributed in the unit square, where the minimality is amongst all rooted spanning trees with a direction of growth. We show that the number of branches from the root of this tree, the total length of these branches, and the length of the longest branch each converges weakly. This model is related to the study of record values in the theory of extreme-value statistics and this relation is used to obtain our results. The results also hold when the tree is formed from a Poisson point process of intensity n in the unit square.

Let Xi : i ≥ 1 be i.i.d. points in ℝd, d ≥ 2, and let Tn be a minimal spanning tree on X1,…,Xn. Let L(X1,…,Xn) be the length of Tn and for each strictly positive integer α let N(X1,…,Xn;α) be the number of vertices of degree α in Tn. If the common distribution satisfies certain regularity conditions, then we prove central limit theorems for L(X1,…,Xn) and N(X1,…,Xn;α). We also study the rate of convergence for EL(X1,…,Xn).

Let n points be placed independently in ν-dimensional space according to the standard ν-dimensional normal distribution. Let Mn be the longest edge-length of the minimal spanning tree on these points; equivalently let Mn be the infimum of those r such that the union of balls of radius r/2 centred at the points is connected. We show that the distribution of (2 log n)1/2Mn - bn converges weakly to the Gumbel (double exponential) distribution, where bn are explicit constants with bn ~ (ν - 1)log log n. We also show the same result holds if Mn is the longest edge-length for the nearest neighbour graph on the points.

The numerical taxonomic analysis of encrinurine trilobites by Temple and Tripp (1979) is extended to 28 described Chinese species, 35 other Chinese samples, and 13 non-Chinese species, in addition to 56 species analysed previously. The 132 entities are coded for 34 cranidial and pygidial attributes, and are ordinated on the eigenvectors of the interattribute correlation matrix for 32 attributes. Chinese material falls into four clusters informally compared with Coronocephalus, Kailia, Encrinuroides and Encrinurus. The Encrinurus-like Chinese species occur mainly in Northern China and have links with non-Chinese species.

British and Irish species of the trilobite family Trinucleidae are investigated by numerical taxonomie methods. Forty-five attributes are defined, relating mainlyto the dorsal surface of the cephalon, and several of them representing new concepts. Of the 106 described species from the British Isles, 31 are fully codable on the basis of topo types. An analysis is based on these species, together with 28 partly topotypically codable species and samples of 9 other species (undescribed, non-topotypic or non-British). The relations of the species are depicted in an ordination derived by principal components analysis of the interattribute correlation matrix and with superimposed minimal spanning tree. On the basis of the ordination most of the analysed species can be grouped into two subfamilies, Trinucleinae and Cryptolithinae (including Marrolithinae); thersites is excluded from Reedolithinae. Trinucleinae and Cryptolithinae are strongly differentiated on the y1 axis of the ordination. The two subfamilies show parallel evolution during the Ordovician in general cranidial attributes (e.g. increasing length, declination of fringe). In Trinucleinae the fringe also shows evolutionary changes (e.g. increases in numbers and regularity of arcs); in Cryptolithinae the fringe evolves very little. Ordination of the attributes is by non-linear mapping based on a distance matrix derived from the interattribute correlation matrix by the transformation (tanh-1|r|)-1. Lectotypes of 12 species are here selected: albidus Reed, ceryx Lamont, gibbsii Salter, girvanensis Lamont, hibernicus Reed, inconstans Whittard, persulcatus Reed, praeterita Reed, rhodesi Whittard, sortita Reed, subradiatus Reed and thersites Salter.