Theory and applications of distance geometry

By L. M. Blumenthal. 1954

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On the Schoenberg Transformations in Data Analysis: Theory and Illustrations (2010, background): “The Menger’s curvature κ is deﬁned as the limit (Blumenthal 1953 p. 75) where ˜ A ijk is the area of the triangle ijk in the transformed space and ˜ d denotes the length of the corresponding sides.”
Properties of Classical and Quantum Jensen-Shannon Divergence (2008, methodology): “We refer to this book, to Berg, Christensen and Ressel [48], and to Blumenthal [41] for extensive introductions to the used results.”
Optimal contact definition for reconstruction of Contact Maps (2010, background): “Most of them develop around the common mathematical theory of distance geometry first applied to chemistry by Blumenthal [19].”
The geometry of graphs and some of its algorithmic applications (1994, background): “Remark 3.7. The case c -- 1, i.e., the characterization of metric spaces that isometrically embed in Euclidean space is classical (see [ 11 ]).”
The Hutchinson-Barnsley theory for infinite iterated function systems (2005, background): “Recall that (X,d) is said to be metrically convex or Menger convex (see, for example, [ 5 ]) if given x,y 2 X, x 6= y, there is a z 2 X such that x 6= z 6= y and d(x,y) = d(x,z)+d(z,y).”
Multivalued fractals in b-metric spaces (2010, background): “In a b-metric space (X, d) the following assertions hold: (i) a convergent sequence has a unique limit; (ii) each convergent sequence is Cauchy; (iii) (X, d→) is an L-space (see Fréchet [15], Blumenthal [7]); (iv) in general, a b-metric is not continuous.”
Sourcebook of parallel computing (2003, background): “For all possible pairs, the problem still can be solved in polynomial time [122, 233].”
Distance geometry and data science (2019, background): “of points and lines (Euclid) or point coordinates (Descartes). The axiomatic foundations of DG were rst laid out in full generality by Menger [135], and later organized and systematized by Blumenthal [31]. A metric space is a triplet (X;D;d), where X is an abstract set, D R +, and dis a binary relat”
The geometry of Minkowski spaces — A survey. Part I (2007, background): “Closely related is the subject of distance geometry, cf. the work of Menger and his school, summarized by Menger [138] and Blumenthal [29] (see also [30]), as well as the works of Alexandrov [2] and Rinow [167].”
Geometry without topology as a new conception of geometry (2001, background): “There is a generalization of metric geometry, known as distance geometry [9], which differs from the metric geometry in absence of constraint (1.”
Formation control using range-only measurements (2011, unknown): “The following theorem is a classical result on the Cayley–Menger determinant (Blumenthal, 1953).”