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We introduce and study a new family of pseudo-Riemannian metrics on the anti-de Sitter three-space . These metrics will be called “of Kaluza-Klein type” , as they are induced in a natural way by the corresponding metrics defined on the tangent sphere bundle . For any choice of three real parameters , the pseudo-Riemannian manifold is homogeneous. Moreover, we shall introduce and study some natural almost contact and paracontact structures , compatible with , such that is a homogeneous almost contact (respectively, paracontact) metric structure. These structures will be then used to show the existence of a three-parameter family of homogeneous metric mixed 3-structures on the anti-de Sitter three-space.

Whole system metrics are valuable tools for use in systems science and engineering. Entropy metrics are defined, developed, and demonstrated in this paper. Based on classical systems engineering methods and practices, these entropy metrics indicate the degree of order/disorder in any given system. A physical-entropy based metric and an information-based entropy metric are aligned with the two primary components of a system: objects and relationships. The physical-entropy based metric is called a relational score, and the information-based metric is called an object score. A subsystem score, based on the relational score and object score, is also developed and presented in this paper. A well-defined process, using these metrics, is used to evaluate the reduction of entropy and complexity associated with any specific system. The metrics and processes developed in this work have a prose component, a graphic component, and a mathematical component. These three components are aligned with the systems science techniques developed by John N. Warfield.

We generalize the following classical result of Fubini to pseudo-Riemannian metrics: if three essentially different metrics on an (n ≥ 3)-dimensional manifold M share the same unparametrized geodesics, and two of them (say, g and g) are strictly nonproportional (that is, the minimal polynomial of the g-self-adjoint (1, 1)-tensor defined by g coincides with the characteristic polynomial) at least at one point, then they have constant sectional curvature.

We study the Lipschitz metric on Teichmuller space (defined by Thurston) and compare it with the Teichmuller metric. We show that in the thin part of Teichmuller space the Lipschitz metric is approximated up to bounded additive distortion by the sup metric on a product of lower-dimensional spaces (similar to the Teichmuller metric as shown by Minsky), and in the thick part, the two metrics are comparable within additive error. However, these metrics are not comparable in general; we construct a sequence of pairs of points in Teichmuller space whose distances approach zero in the Lipschitz metric while they approach infinity in the Teichmuller metric., 22 Pages

We study in a constructive manner some problems of topology related to the set Irr of irrational reals. The constructive approach requires a strong notion of an irrational number; constructively, a real number is irrational if it is clearly different from any rational number. We show that the set Irr is one-to-one with the set Dfc of infinite developments in continued fraction (dfc). We define two extensions of Irr, one, called Dfc1, is the set of dfc of rationals and irrationals preserving for each rational one dfc, the other, called Dfc2, is the set of dfc of rationals and irrationals preserving for each rational its two dfc. We introduce six natural distances over Irr wich we denote by dfc0, dfc1, dfc2, d, dmir and dcut. We show that only the four distances dfco, dfc1, d and dmir among the six make Irr a complete metric space. The last distances define in Irr the same topology in a constructive sens. We study further the set Dfc1 in which we show that the irrationals constitute a closed subset. Finally, we make a particular study of the completion Dfc2 of Dfc for the two equivalent metrics dfc2 and dcut.

We show that the embedding of metric spaces into the l 1 -grid Z 2 can be characterized inessentially the same fashion as in the case of the l 1 -plane R 2 . In particular, a metric space can beembedded into Z 2 iff every subspace with at most 6 points is embeddable. Moreover, if such an embed-ding exists, it can be constructed in polynomial time (for finite spaces). q 1998 John Wiley & Sons, Inc.Networks 32: 127–132, 1998 The rectilinear metric (alias l 1 -metric) is probably the for the embedding in the gridZ 2 (‘‘digital plane’’) en-simplest distance measure in R n . This explains why in dowed with the rectilinear distance (alias city block met-many cases rectilinear spaces are selected as host spaces ric) (see Fig. 1). Again, as in [1],wecantake advantagefor embedding a given metric space; we refer to Hubert of such notions as ‘‘d-split’’ and ‘‘totally decomposableet al. [7] for an application to multidimensional scaling metrics,’’ introduced and investigated in the case of finitewhere one aims at producing a visual display of a given metric spaces by Bandelt and Dress [2].data set and to Deza and Laurent [6] for other applica- Let (X, d) be a metric space. We will say that (X, d)tions. Surprisingly, the problem to characterize the metric satisfies theparity condition if d(u, £) /d(£, w) /d(w,subspaces of the rectilinear space of a given dimension u) is an even integer for any u, £, w ˆX (cf. [10] forseems to be much more difficult than is the analogous a straightforward metric characterization of bipartiteproblem for Euclidean spaces, where suitable criteria are graphs). To every pair S †{A, B} of nonempty subsetsgiven by the classical results of Menger and Schoenberg of X we associate the isolation index a

In this paper we develop some connections between the partial metrics of Matthews and the topological aspects of domain theory. We do this by introducing the valuation spaces, which are a special class of partial metric spaces. We develop the natural duality of partial metrics and propose that a natural context in which to view a partial metric space is as a bitopological space. We then see that successive conditions on a valuation can ensure that the pmetric topology is first of all order consistent (with the underlying poset), then equivalent to the Scott topology, and finally that the induced metric topology is equivalent to the patch topology.

Generalized metric spaces are a common generalization of preorders and ordinary metric spaces. Every generalized metric space can be isometrically embedded in a complete function space by means of a metric version of the categorical Yoneda embedding. This simple fact gives naturally rise to: 1. a topology for generalized metric spaces which for arbitrary preorders corresponds to the Alexandroff topology and for ordinary metric spaces reduces to the e-ball topology; 2. a topology for algebraic generalized metric spaces generalizing both the Scott topology for algebraic complete partial orders and the e-ball topology for metric spaces.

The backgrounds of the FCM metric, which was presented at Color 77, are discussed. Analysis of color-discrimination data [threshold as well as color-difference scaling data] and interpretation in terms of present conceptions on zonal processing of primary stimulations leads to principles that have to be respected by any metric. The FCM metric is a mathematical approximation of these principles. This metric presents a color-order system in L a b rectangular coordinates, which links up with the Munsell System. However, it is not a uniform color space. Difference scaling is handled by the FCM color-difference metric. The CIELAB formula is in serious conflict with several of the principles developed.