Four-point Properties Research Articles

Segmented pseudometrics and four-point Fermat-Torricelli problems

We introduce segmented pseudometrics on a finite dimensional vector space as those having triangle equality at any intermediate point of any segment. Some example families are given. We characterize when a segmented (pseudo)metric is (semi)norm induced. We investigate the relation between a pseudometric being segmented and the concave four-point Fermat-Torricelli property, which states that the sum of the four pseudometric distances to the vertices of a triangle and to some inner point is always minimized at this inner point. These two properties are shown to be equivalent in dimension at most 2. For metrics in higher dimension the second property always implies the first, but not inversely, even when induced by a norm.

Query filtering using two-dimensional local embeddings

In high dimensional data sets, exact indexes are ineffective for proximity queries, and a sequential scan over the entire data set is unavoidable. Accepting this, here we present a new approach employing two-dimensional embeddings. Each database element is mapped to the XY plane using the four-point property. The caveat is that the mapping is local: in other words, each object is mapped using a different mapping.The idea is that each element of the data is associated with a pair of reference objects that is well-suited to filter that particular object, in cases where it is not relevant to a query. This maximises the probability of excluding that object from a search. At query time, a query is compared with a pool of reference objects which allow its mapping to all the planes used by data objects. Then, for each query/object pair, a lower bound of the actual distance is obtained. The technique can be applied to any metric space that possesses the four-point property, therefore including Euclidean, Cosine, Triangular, Jensen–Shannon, and Quadratic Form distances.Our experiments show that for all the data sets tested, of varying dimensionality, our approach can filter more objects than a standard metric indexing approach. For low dimensional data this does not make a good search mechanism in its own right, as it does not scale with the size of the data: that is, its cost is linear with respect to the data size. However, we also show that it can be added as a post-filter to other mechanisms, increasing efficiency with little extra cost in space or time. For high-dimensional data, we show related approximate techniques which, we believe, give the best known compromise for speeding up the essential sequential scan. The potential uses of our filtering technique include pure GPU searching, taking advantage of the tiny memory footprint of the mapping.

Supermetric search

Metric search is concerned with the efficient evaluation of queries in metric spaces. In general, a large space of objects is arranged in such a way that, when a further object is presented as a query, those objects most similar to the query can be efficiently found. Most mechanisms rely upon the triangle inequality property of the metric governing the space. The triangle inequality property is equivalent to a finite embedding property, which states that any three points of the space can be isometrically embedded in two-dimensional Euclidean space. In this paper, we examine a class of semimetric space which is finitely four-embeddable in three-dimensional Euclidean space. In mathematics this property has been extensively studied and is generally known as the four-point property. All spaces with the four-point property are metric spaces, but they also have some stronger geometric guarantees. We coin the term supermetric11This term has previously been used in the domains of particle physics and evolutionary biology as a pseudonym for the mathematical term ultra-metric, a concept of no interest in metric search; we believe our concept is of sufficient importance to the domain to justify its reuse with a different meaning.space as, in terms of metric search, they are significantly more tractable. Supermetric spaces include all those governed by Euclidean, Cosine,22For the correct formulation of Cosine distance, see Connor et al., 2016 [1] for details. Jensen–Shannon and Triangular distances, and are thus commonly used within many domains. In previous work we have given a generic mathematical basis for the supermetric property and shown how it can improve indexing performance for a given exact search structure. Here we present a full investigation into its use within a variety of different hyperplane partition indexing structures, and go on to show some more of its flexibility by examining a search structure whose partition and exclusion conditions are tailored, at each node, to suit the individual reference points and data set present there. Among the results given, we show a new best performance for exact search using a well-known benchmark.

Hilbert Exclusion

Most research into similarity search in metric spaces relies on the triangle inequality property. This property allows the space to be arranged according to relative distances to avoid searching some subspaces. We show that many common metric spaces, notably including those using Euclidean and Jensen-Shannon distances, also have a stronger property, sometimes called the four-point property: In essence, these spaces allow an isometric embedding of any four points in three-dimensional Euclidean space, as well as any three points in two-dimensional Euclidean space. In fact, we show that any space that is isometrically embeddable in Hilbert space has the stronger property. This property gives stronger geometric guarantees, and one in particular, which we name the Hilbert Exclusion property, allows any indexing mechanism which uses hyperplane partitioning to perform better. One outcome of this observation is that a number of state-of-the-art indexing mechanisms over high-dimensional spaces can be easily refined to give a significant increase in performance; furthermore, the improvement given is greater in higher dimensions. This therefore leads to a significant improvement in the cost of metric search in these spaces.

Study of divalent elements (Mg, Sr and Ba)-doped LaMnO3 nano-manganites

Doping of group-II divalent cations in LaMnO3 can produce advanced multi-functionalities in these nano-manganite materials according to recent theoretical approaches. An experimental investigation of Mg, Sr and Ba-doped LaMnO3 nano-structures has been made by synthesizing these compositions with fuel agent assistive sol–gel auto-combustion technique. Detailed structural studies were carried out for crystal structure, lattice parameters, crystallite size, density and strain in structure using the data obtained from X-ray diffraction (XRD). Structural parameters were largely affected and significantly varied in all compositions due to doping of group-II elements with different sized ionic radii. Morphological and compositional analyzes were carried out using scanning electron microscopy and energy dispersive X-ray spectroscopy, respectively. Electrical and magnetic properties were explored using four-point probe and physical property measurement setups and observed a significant change in various compositions. The substitution of group-II atoms with different sized ionic radii on the regular sites in structure provides a distinctive behavior in various properties.

Characteristics of Low Temperature Degradation Free ZTA for Artificial Joint

Low temperature degradation free Zirconia toughened alumina (ZTA) has been developed. It is reported that ZTA has higher mechanical strength compared to alumina due to the stress induced transformation and spontaneously transformation of zirconia phase on some ZTA have been occurred. For achieving the higher reliability of artificial joint prosthesis alternative to alumina and other ceramic materials, it is necessary to improve and validate the both mechanical characteristics and phase stability at the same time. We evaluated that microstructure, mechanical characteristics and phase stability of newly developed ZTA (BIOCERAM®AZUL). It was confirmed that four-point bending strength and weibull modulus were extreamly high, and ZTA has higher reliability. There were no significant changes and deterioration in four-point bending strength, crystal structure and wear property with and without accelerated aging test. Newly developed ZTA not only with high mechanical characteristics but also with phase stability could be quite useful as bearing materials in artificial joints for longer clinical use.

Intrinsic four-point properties

Many characterizations of euclidean spaces (real inner product spaces) among metric spaces have been based on euclidean four point embeddability properties. Related “intrinsic” four point properties have also been used to characterize euclidean or hyperbolic spaces among a suitable class of metric spaces. The present paper provides new characterizations of euclidean or hyperbolic spaces based on intrinsic four point properties which are related to known four point embedding properties.

Experimental Study on Bending Property of 8-Shape 3D Composite

The 8-shape 3D composites were fabricated by using 3D woven fabrics of glass fibers as the reinforcement and unsaturated polyester resin as the matrix with hand lay-up molding technique. In the fabric, two groups of binder yarns in warp direction, connecting top and bottom skins, form 8 shape. The four-point bending property was tested. The supporting methods, including warp- and weft-supporting, were analyzed according to the direction of 8 shape in fabrics in this study. Experimental results showed that the onset fractures of weft-supporting method were larger than that of warp-supporting method. Under the same displacement, the load increased with the addition of thickness for all supporting methods. Moreover, the selection of the supporting skins of 8-shape composite affected the testing results of the onset fracture and the Young’s modulus.

New intrinsic four point properties which characterize hyperbolic and euclidean spaces

Many characterizations of euclidean spaces (real inner product spaces) among metric spaces have been based on euclidean four-point embeddability properties. Related "intrinsic" four point properties have also been used to characterize euclidean or hyperbolic spaces among a suitable class of metric spaces. The present paper provides new characterizations of euclidean or hyperbolic spaces based on intrinsic four point properties which are related to known euclidean four point embedding properties, and in addition provides a new euclidean four-point embedding property which characterizes euclidean spaces among metric spaces. Application of this property in normed linear spaces provides an apparently new characterization of inner product spaces among normed linear spaces.

A new class of four-point properties which characterize euclidean spaces

Characterizations of generalized euclidean spaces by means of euclidean four-point properties.state that every metric space which is complete, and which contains a metric line joining each two of its points is a generalized euclidean space if and only if each quadruple from a certain class of quadruples of the space is congruent with a quadruple of points in a euclidean space. It is known that it suffices to consider only quadruples containing a linear triple, or quadruples in which one of the linear points is a metric midpoint of the other two. Another class of four-point properties involves quadruples which contain a linear triple and a point equidistant from two of the linear points. The present paper presents three characterizations of euclidean spaces based on four-point properties in which the embedded quadruples contain a linear triple and some three of the distances determined by the four points are equal.

Banach-Euclidean four-point properties

Banach-Euclidean four-point properties

Local Euclidean Four-Point Properties Which Characterize Inner-Product Spaces

Let $M$ be a complete, convex, externally convex metric space. We show $M$ is an inner-product space if and only if for each point $t$ of $M,M$ contains a sphere ${S_t}$ which has the euclidean queasy, feeble or weak four-point property.

Consequences of delayed lateral inhibition in the retina of Limulus II. theory of spatially uniform fields, assuming the four-point property

The theory developed in paper I of this series is here extended to enable discussion of the behavior of retinae which can exhibit a range of uniform excitation in which the response makes a transition from transient to sustained oscillation. Propositions proven in paper I about the existence and stability of periodic bursting behavior in the response to high levels of excitation are here broadly generalized.

Local Euclidean four-point properties which characterize inner-product spaces

Let M be a complete, convex, externally convex metric space. We show M is an inner-product space if and only if for each point t of M, M contains a sphere St which has the euclidean queasy, feeble or weak four-point property.

Metric transforms and the hyperbolic four-point property

The purpose of this paper is to show that any metric space is homeomorphic to a metric space with each quadruple of its points congruently imbeddable in three-dimensional hyperbolic space.

Metric Transforms and the Hyperbolic Four-Point Property

The purpose of this paper is to show that any metric space is homeomorphic to a metric space with each quadruple of its points congruently imbeddable in three-dimensional hyperbolic space.

On the Imbedding of Metric Sets in Euclidean Space

1. It is the purpose of this note to make a slight extension of results previously obtained by the writer t and to give a modification of Menger's general conditions for the imbedding of n points of a metric space in Euclidean space. With regard to the first topic it is proved on pp. 515-16 of the paper mentioned that a complete space, which is convex and externally convex and has the four-point property, has the n-point property for every integer n. We now proceed to show that the requirement of external convexity is needless. Using the notation of this proof, let T1T'1, To T'o, and ToL T0ol. If the line through a'o and a', meets T'0,, external convexity is not needed for the proof as given.: In the opposite case it is clear that there is a point u' in T'1 near enough to the centroid of T'oo so that: if ao' and a', are on the same side of E,e-2 and adou' is produced to meet T'0o in x', u'x' lies also in T'o; and, if a'o and a', are on opposite sides of E.n2, a'lu' cuts T'o0. In the congruence T1 T'1, let u u'. Then by the argument of p. 516 the points u, a1, a2, . . ., a. can be imbedded in En and the sign of the determinant D (u, a,, a2,' * * , a.) =,4 sign (1) f. Let us now suppose that sign D(ao, al, , a,) = sign((1)'. Since D is a continuous function of each variable and changes sign when ao is replaced by u, there is some point on the segment uao for which D (v, a1, a2, *, an) =O. Then the points v, a,, a2, . . ., a,, can be imbedded in En l so that To T'o in one of two ways: (1) v on the same side of En l as a'1; (2) v' on the opposite side. In the first case, since lies within T1, the congruence + a2 + + an v' + a'2 + + a'., which is a sub-congruence of + a1 + + a0 v'+ a'1 + + a*,,, defined in the preceding paragraph, is also a subcongruence of T1 T'1, which, includes To, T'O,.t If x x' in the congruence T1 T'1, we have a0 + + x a'o + v' + x' and (by the previous paragraph) + x + a1 v' + x' + a'1. Now by the four-point property

A note on the four-point property

A note on the four-point property