and Applied Analysis 3 Remark 2.2. For each t0 ∈ T \ {maxT}, the single-point set {t0} is Δ-measurable, and its Δmeasure is given by μΔ {t0} σ t0 − t0 μ t0 . 2.2 Obviously, E1 ⊂ A does not have any right-scattered points. For a set E ⊂ T, define the Lebesgue Δ-integral of f over E by ∫ Ef t Δt and let f ∈ LT E,R see 8 . Lemma 2.3 see 8 . Let f : a, b T → R. f̃ : a, b → R is the extension of f to...

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...

Given a graph H = (U,E) and connectivity requirements r = {r(u, v) : u, v ∈ R ⊆ U}, we say that H satisfies r if it contains r(u, v) pairwise internally-disjoint uv-paths for all u, v ∈ R. We consider the Survivable Network with Minimum Number of Steiner Points (SN-MSP) problem: given a finite set V of points in a normed space (M, ‖·‖) and connectivity requirements, find a minimum size set S ⊂ ...

Given a set P of n points in R, a typical clustering problem asks for partitioning P into a family of k subsets {P1, . . . , Pk}, so that a certain objective function is minimized. The objective function depends on the application and varies. A typical clustering problem is the so-called centered clustering problem in which a representative point ci (or a q-flat) is chosen for each cluster Pi a...

We propose a scalable technique called Seeded Clustering that allows us to maintain R-tree indices by bulk insertion while keeping pace with high data arrival rates. Our approach uses a seed tree, which is copied from the top k levels of a target R-tree, to classify input data objects into clusters. We then build an Rtree for each of the clusters and insert the input R-trees into the target R-t...

where B : E x E —• E' is a bilinear map and E' is a vector space. (E may be identified with the Zariski tangent space to Q at 0.) Let V be an algebraic variety and x € V be a point. We say that V is quadratic at x if the analytic germ of V at x is equivalent to the germ of a quadratic cone at 0. Let T be a finitely generated group and G a k-algebraic group. We identify G with its set of k-point...

2 Hierarchic Clustering 1 2.1 Description of Classes . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Numbers of Classes . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Clustering and Ordination . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Reordering a Dendrogram . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Minimum Spanning Tree . . . . . . . . . . . . . . . . . . . . ....