Pair Of Distinct Points Research Articles

Metric hypergraphs and metric-line equivalences

In a metric space with distance function dist, we say that v is between u and w if u,v,w are pairwise distinct and dist(u,w)=dist(u,v)+dist(v,w). All triples {u,v,w} such that v is between u and w constitute a hypergraph induced by this metric space. In answer to a question posed in [3], we construct an infinite family of 3-uniform hypergraphs induced by no metric space.The line determined by a pair of distinct points x and y in a metric space is the set consisting of x, y, and all points z such that one of x,y,z is between the remaining two. Two pairs of distinct points are said to be equivalent if they determine the same line. The resulting equivalence relation is a metric-line equivalence. We construct an infinite family of genuinely distinct obstacles that prevent an abstract equivalence relation from being a metric-line equivalence.

On (βρn)-OS in Pythagorean Neutrosophic Topological Spaces

In this paper, we introduce a new set called Pythagorean neutrosophic beta-open set with this concept, and we introduce interior and closure of Pythagorean neutrosophic beta-open set in a Pythagorean neutrosophic topological spaces by utilizing beta-open set and we introduce the chii, ii, iiibeta-spaces and di, ii, iiibetaspaces from the pair of distinct points and we have derived the necessary and sufficient conditions by utilizing beta-open sets. We also go through some containment relations for interiors and closures of beta-open sets and studied some of their characteristics.

A characterization of uniquely representable graphs

The betweenness structure of a finite metric space $M = (X, d)$ is a pair $\mathcal{B}(M) = (X,\beta_M)$ where $\beta_M$ is the so-called betweenness relation of $M$ that consists of point triplets $(x, y, z)$ such that $d(x, z) = d(x, y) + d(y, z)$. The underlying graph of a betweenness structure $\mathcal{B} = (X,\beta)$ is the simple graph $G(\mathcal{B}) = (X, E)$ where the edges are pairs of distinct points with no third point between them. A connected graph $G$ is uniquely representable if there exists a unique metric betweenness structure with underlying graph $G$. It was implied by previous works that trees are uniquely representable. In this paper, we give a characterization of uniquely representable graphs by showing that they are exactly the block graphs. Further, we prove that two related classes of graphs coincide with the class of block graphs and the class of distance-hereditary graphs, respectively. We show that our results hold not only for metric but also for almost-metric betweenness structures.

Timelike Hilbert and Funk geometries

A timelike space is a Hausdorff topological space equipped with a partial order relation < and a distance function ρ satisfying a collection of axioms including a set of compatibility conditions between the partial order relation and the distance function. The distance function is defined only on a subset of the product of the space with itself that contains the diagonal, namely, ρ(x,y) is defined if and only if x<y or x=y. Distances between pairs of distinct points in a triple x,y,z, whenever these distances are defined, satisfy the so-called time inequality, which is a reverse triangle inequality ρ(x,y)+ρ(y,z)≤ρ(z,y).In the 1960s, Herbert Busemann developed an axiomatic theory of timelike spaces and of locally timelike spaces. His motivation comes from the geometry underlying the theory of relativity, and he tried to adapt to this setting his geometric theory of metric spaces, namely, his theory of G-spaces (geodesic spaces). The classical example he considers is the n-dimensional Lorentzian space. Two other interesting classes of examples of timelike spaces he introduced are the timelike analogues of the Funk and Hilbert geometries. In this paper, we investigate these two geometries, and in doing this, we introduce variants of them, in particular the timelike relative Funk and Hilbert geometries, in the Euclidean and spherical settings. We describe the Finsler infinitesimal structure of each of these geometries (with an appropriate notion of Finsler structure) and we display the interactions among the Euclidean and timelike spherical geometries. In particular, we characterize the de Sitter geometry as a special case of a timelike spherical Hilbert geometry.

Transitions and anti-integrable limits for multi-hole Sturmian systems and Denjoy counterexamples

Abstract For a Denjoy homeomorphism $f$ of the circle $S$, we call a pair of distinct points of the $\omega$-limit set $\omega (\,f)$ whose forward and backward orbits converge together a gap, and call an orbit of gaps a hole. In this paper, we generalize the Sturmian system of Morse and Hedlund and show that the dynamics of any Denjoy minimal set of finite number of holes is conjugate to a generalized Sturmian system. Moreover, for any Denjoy homeomorphism $f$ having a finite number of holes and for any transitive orientation-preserving homeomorphism $f_1$ of the circle with the same rotation number $\rho (\,f_1)$ as $\rho (\,f)$, we construct a family $f_\varepsilon$ of Denjoy homeomorphisms of rotation number $\rho (\,f)$ containing $f$ such that $(\omega (\,f_\varepsilon ), f_\varepsilon )$ is conjugate to $(\omega (\,f), f)$ for $0&lt;\varepsilon &lt;\tilde{\varepsilon }&lt;1$, but the number of holes changes at $\varepsilon =\tilde{\varepsilon }$, that $(\omega (\,f_\varepsilon ), f_\varepsilon )$ is conjugate to $(\omega (\,f_{\tilde{\varepsilon }}), f_{\tilde{\varepsilon }})$ for $\tilde{\varepsilon }\leqslant \varepsilon &lt;1$ but $\lim _{\varepsilon \nearrow 1}f_\varepsilon (t)=f_1(t)$ for any $t\in S$, and that $f_\varepsilon$ has a singular limit when $\varepsilon \searrow 0$. We show this singular limit is an anti-integrable limit (AI-limit) in the sense of Aubry. That is, the Denjoy minimal system reduces to a symbolic dynamical system. The AI-limit can be degenerate or nondegenerate. All transitions can be precisely described in terms of the generalized Sturmian systems.

Induced Ramsey-Type Results and Binary Predicates for Point Sets

Let $k$ and $p$ be positive integers and let $Q$ be a finite point set in general position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is a finite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$ there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order type and $\binom{Q'}{p}$ is monochromatic in $c$. Nešetřil and Valtr proved that for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They also proved that for every $k \ge 2$ and $p \ge 2$, there are point sets that are not $(k,p)$-Ramsey.As our main result, we introduce a new family of $(k,2)$-Ramsey point sets, extending a result of Nešetřil and Valtr. We then use this new result to show that for every $k$ there is a point set $P$ such that no function $\Gamma$ that maps ordered pairs of distinct points from $P$ to a set of size $k$ can satisfy the following "local consistency" property: if $\Gamma$ attains the same values on two ordered triples of points from $P$, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.

Distributionally chaotic systems of type 2 and rigidity

In this paper we deal with uniformly rigid systems obtained by a method introduced by Katznelson and Weiss and show that such systems never contain DC2 pairs. On the other hand, we introduce a modification of this technique that leads to a uniformly rigid system with DC2 pairs. We also show that every dynamical system contains a pair of distinct points which is not DC2.

On Removability Properties of $$\psi $$ ψ -Uniform Domains in Banach Spaces

Suppose that E is a real Banach space with dimension at least 2. The main aim of this paper is to show that a domain D in E is a \(\psi \)-uniform domain if and only if \(D\backslash P\) is a \(\psi _1\)-uniform domain, and D is a uniform domain if and only if \(D\backslash P\) also is a uniform domain, whenever P is a countable subset of D satisfying a quasihyperbolic separation condition. This condition requires that the quasihyperbolic distance with respect to D between each pair of distinct points in P has a lower bound greater than or equal to \(\frac{1}{2}\).

Proper holomorphic immersions in homotopy classes of maps from finitely connected planar domains into CxC*

Gromov, in his seminal 1989 paper on the Oka principle, proved that every continuous map from a Stein manifold into an elliptic manifold is homotopic to a holomorphic map. Previously we have shown that, given a continuous map X → C×C∗ from a finitely connected planar domain X without isolated boundary points, a stronger Oka property holds, namely that the map is homotopic to a proper holomorphic embedding. Here we show that every continuous map from a finitely connected planar domain, possibly with punctures, into C × C∗ is homotopic to a proper immersion that identifies at most countably many pairs of distinct points, and in most cases, only finitely many pairs. By examining situations in which the immersion is injective, we obtain a strong Oka property for embeddings of some classes of planar domains with isolated boundary points. It is not yet clear whether a strong Oka property for embeddings holds in general when the domain has isolated boundary points. We conclude with some observations on the existence of a null-homotopic proper holomorphic embedding C∗ → C× C∗.

Intra-orbit separation of dense orbits of dendrite maps

In this note, we will generalize the notion of separation index, which was introduced by Manjunath et al. (2006), to dendrite maps, and use the notion to characterize the intra-orbit separation for the orbits of continuous transitive dendrite maps. We will show: (i) For a dendrite map f, the separation index γ is greater than zero if and only if the set of fixed points of f is not an arc connected subspace. (ii) If the separation index γ of a dendrite map f is greater than zero, then for every 0<τ<γ and any pair of distinct points x and y on a dense orbit, {x,y} is a Li–Yorke pair of modulus τ.

Sharply 2-Transitive Linear Groups

A group Γ is sharply 2-transitive if it admits a faithful permutation representation that is transitive and free on pairs of distinct points. Conjecturally, for all such groups there exists a near-field N (i.e. a skew field that is distributive only from the left, see Definition 2) such that ⁠. This is well known in the finite case. We prove this conjecture when Γ<GLn(F) is a linear group, where F is any field with char(F)≠2 and that p−char(Γ)≠2 (see Definition 2.2).

Topological entropy zero and asymptotic pairs

Let T be a continuous map on a compact metric space (X, d). A pair of distinct points x, y ∈ X is asymptotic if lim n→∞ d(T n x, T n y) = 0. We prove the following four conditions to be equivalent: 1. h top(T) = 0; 2. (X, T) has a (topological) extension (Y,S) which has no asymptotic pairs; 3. (X, T) has a topological extension (Y ′, S′) via a factor map that collapses all asymptotic pairs; 4. (X, T) has a symbolic extension (i.e., with (Y ′, S′) being a subshift) via a map that collapses asymptotic pairs. The maximal factors (of a given system (X, T)) corresponding to the above properties do not need to coincide.

COMPUTING THE MAXIMUM DETOUR OF A PLANE GEOMETRIC GRAPH IN SUBQUADRATIC TIME

Let G be a plane geometric graph where each edge is a line segment. We consider the problem of computing the maximum detour of G , defined as the maximum over all pairs of distinct points (vertices as well as interior points of edges) p and q of G of the ratio between the distance between p and q in G and the Euclidean distance || pq || 2 . The fastest known algorithm for this problem has Θ ( n 2 ) running time where n is the number of vertices. We show how to obtain O ( n 3/2 (log n ) 3 ) expected running time. We also show that if G has bounded treewidth, its maximum detour can be computed in O ( n (log n ) 3 ) expected time.

Super-simple, Pan-orientable, and Pan-decomposable BIBDs with Block Size 4 and Related Structures

Let v, k, and μ be positive integers. A tournament T of order k, briefly k-tournament, is a directed graph on k vertices in which there is exactly one directed edge between any two vertices. A (v, k, λ = 2μ)-BIBD is called T-orientable if for each of its blocks B, it is possible to replace B by a copy of T on the set B so that every ordered pair of distinct points appears in exactly μ k-tournaments. A (v, k, λ = 2μ)-BIBD is called pan-orientable if it is T-orientable for every possible k-tournament T. In this paper, we continue the earlier investigations and complete the spectrum for (v, 4, λ = 2μ)-BIBDs which possess both the pan-orientable property and the pan-decomposable property first introduced by Granville et al. (Graphs Comb 5:57–61, 1989). For all μ, we are able to show that the necessary existence conditions are sufficient. When λ = 2 and v > 4, our designs are super-simple, that is they have no two blocks with more than two common points. One new corollary to this result is that there exists a (v, 4, 2)-BIBD which is both super-simple and directable for all v ≡ 1, 4 (mod 6), v > 4. Finally, we investigate the existence of pan-orientable, pan-decomposable (v, 4, λ = 2μ)-BIBDs with a pan-orientable, pan-decomposable (w, 4, λ = 2μ)-BIBD as a subdesign; here we obtain complete results for λ = 2, 4, but there remain several open cases for λ = 6 (mostly for v < 4w), and the case λ = 12 still has to be investigated.

Quasilinearization and curvature of Aleksandrov spaces

We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the quasi-inner product is defined by means of the quadrilateral cosine—a metric substitute for the angular measure between two directions at different points. Our main result states that a geodesically connected metric space \({\left(\mathcal{M},\rho\right)}\) is an Aleksandrov \({\Re_{0}}\) domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of distinct points in \({\mathcal{M}}\) . We also observe that a geodesically connected metric space \({\left(\mathcal{M},\rho\right)}\) is an \({\Re_{0}}\) domain if and only if, for every quadruple of points in \({\mathcal{M}}\) , the quadrilateral inequality (known as Euler’s inequality in \({\mathbb{R}^{2}}\)) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an \({\Re_{0}}\) domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.

Critical Values in a Long-range Percolationon Spaces Like Fractals

On a certain class of general discrete spaces including fractals, we consider a model in which each pair of distinct points is connected by a random bond. The main question we are concerned is whether a connected component consisting of infinitely many points exists or not. This depends on the choice of parameters in the connecting probabilities, and the aim of this paper is to find thresholds of the parameters

Limiting Distributions for the Maximum of a Symmetric Function on a Random Point Set

Let U1, U2, ... be a sequence of independent random points taking values in a measurable space (S, Σ) according to a common probability P and let \(h:S\times S\rightarrow\)R be a symmetric, Borel/ \(\Sigma\times\Sigma\)-measurable function. Let Hn = max{h(Ui,Uj): 1≤ i < j≤ n} denote the maximum h-value over pairs of distinct points from U1,U2,...,Un. Assumptions on the distribution of h(U1,·) are provided under which a continuous function of Hn converges in law to an extreme-value distribution upon suitable rescaling. The work is complementary to that appearing in Appel et al. (1999) J. Theor. Probab. 12, 27–47. on the almost-sure limiting behavior of Hn. In the first of two examples, the main result applied to the case of i.i.d. points distributed uniformly on the surface of a unit hypersphere in Rd provides the limiting distribution of the maximum pairwise distance (chord length) among the first n of the points. The second example exhibits the limiting distribution of the minimum pair-wise distance among the first n of i.i.d. uniform points in a compact subset of Rd.

The topology of surface mediatrices

Given a pair of distinct points p and q in a metric space with distance d, the mediatrix is the set of points x such that d ( x , p ) = d ( x , q ) . In this paper, we examine the topological structure of mediatrices in connected, compact, closed 2-manifolds whose distance function is inherited from a Riemannian metric. We determine that such mediatrices are, up to homeomorphism, finite, closed simplicial 1-complexes with an even number of incipient edges emanating from each vertex. Using this and results from [J.J.P. Veerman, J. Bernhard, Minimally separating sets, mediatrices and Brillouin spaces, Topology Appl., in press], we give the classification up to homeomorphism of mediatrices on genus 1 tori (and on projective planes) and outline a method which may possibly be used to classify mediatrices on higher-genus surfaces.

Partition of Triples of Order 6k+5 into 6k+3 Optimal Packings and One Packing of Size 8k+4

A (2,3)-packing on X is a pair (X,[InlineMediaObject not available: see fulltext.]), where [InlineMediaObject not available: see fulltext.] is a set of 3-subsets (called blocks) of X, such that any pair of distinct points from X occurs together in at most one block. For a (6k+5)-set X, an optimal partition of triples (denoted by OPT(6k+5)) is a set of 6k+3 optimal (2,3)-packings and a (2,3)-packing of size 8k+4 on X. Etzion conjectured that there exists an OPT(6k+5) for any positive integer k. In this paper, we construct such a system for any k?1. This complete solution is based on the known existence results of S(3,4,v)s by Hanani and that of special S(3,{4,6},6m)s by Mills. Partitionable candelabra systems also play an important role together with an OPT(11) and a holey OPT(11).

More results on perfect (3,3,6k + 4; 6k − 2)‐threshold schemes

AbstractA (2,3)‐packing on X is a pair (X,$\cal A$), where $\cal A$ is a set of 3‐subsets (called blocks) of X, such that any pair of distinct points from X occurs together in at most one block. Its leave is a graph (X,E) such that E consists of all the pairs which do not appear in any block of $\cal A$. In this article, we shall construct a set of 6k − 2 disjoint (2,3)‐packings of order 6k + 4 with K1,3 ∪ 3kK2 or G1 ∪ (3k − 1)K2 as their common leave for any integer k ≥ 1 with a few possible exceptions (G1 is a special graph of order 6). Such a system can be used to construct perfect threshold schemes as noted by Schellenberg and Stinson (1989). © 2006 Wiley Periodicals, Inc. J Combin Designs