Search

Given an n -point metric space (M , d) , metric 1 -median asks for a point p ∈ M minimizing ∑ x ∈ M d (p , x). We show that for each computable function f : Z + → Z + satisfying f (n) = ω (1) , metric 1 -median has a deterministic, o (n) -query, o (f (n) ⋅ log ⁡ n) -approximation and nonadaptive algorithm. Previously, no deterministic o (n) -query o (n) -approximation algorithms are known for metric 1 -median. On the negative side, we prove each deterministic O (n) -query algorithm for metric 1 -median to be not (δ log ⁡ n) -approximate for a sufficiently small constant δ > 0. We also refute the existence of deterministic o (n) -query O (log ⁡ n) -approximation algorithms. [ABSTRACT FROM AUTHOR]

We show that the standard partition of unity subordinate to an open cover of a metric space has Lipschitz constant max ⁡ (1 , M − 1) / L , where L is the Lebesgue number and M is the multiplicity of the cover. If the metric space satisfies the approximate midpoint property, as length spaces do, then the upper bound improves to (M − 1) / (2 L). These Lipschitz estimates are optimal. We also address the Lipschitz analysis of ℓ p -generalizations of the standard partition of unity, their partial sums, and their categorical products. Lastly, we characterize metric spaces with Assouad–Nagata dimension n as exactly those metric spaces for which every Lebesgue cover admits an open refinement with multiplicity n + 1 while reducing the Lebesgue number by at most a constant factor. [ABSTRACT FROM AUTHOR]

• We propose the novel concept of metric hulls. • We define the covered operator using mutual distances only. • We evaluate the proposal on synthetic and real-life data. • Our method achieves online response times. • We show that a few hull objects provide high coverage in comparison to the related work. [Display omitted] Similarity searching has become widely utilized in many online services processing unstructured and complex data, e.g., Google Images. Metric spaces are often applied to model and organize such data by their mutual similarity. As top-k queries provide only a local view on data, a data analyst must pose multiple requests to observe the entire dataset. Thus, group-by operators for metric data have been proposed. These operators identify groups by respecting a given similarity constraint and produce a set of objects per group. The analyst can then tediously browse these sets directly, but representative members may provide better insight. In this paper, we focus on concise representations of metric datasets. We propose a novel concept of a metric hull which encompasses a given set by selecting a few objects. Testing an object to be part of the set is then made much faster. We verify this concept on synthetic Euclidean data and real-life image and text datasets and show its effectiveness and scalability. The metric hulls provide much faster and more compact representations when compared with commonly used ball representations. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study a new dispersive order in an arbitrary metric space. More precisely, given two random variables on a metric space, the order decides which one has a more concentrated distribution. It is motivated by the wish to be able to make new comparisons that the currently defined orderings cannot make. Related to this problem, several statistical parameters of a random variable, giving different kinds of information about the distribution, are studied. It is proved that the new dispersive order also compares some of these parameters. Special attention is paid to Riemannian manifolds, a natural class of metric spaces. Here, the new order behaves in a distinguished way: the differentiability axioms allow obtaining several equations which facilitate the decision. The principal properties of the order are presented, as well as several applications. Finally, the independence of this order from those already known is proved. • A new dispersive order for an arbitrary metric space. • Distinguished properties in Riemannian manifolds. • Better behaviour than other dispersive stochastic orders. [ABSTRACT FROM AUTHOR]

In this paper, we equip the partition lattice of the positive integers with a metric structure, and study the properties of this metric. We show, that this new space is complete, compact and separable. We also investigate how continuous functions look like in this structure. [ABSTRACT FROM AUTHOR]

The 'the' in the title hides a subtlety. A metric space induces not one but four topologies - by means of open sets, closed sets, closure, and interior - they just so happen to coincide. The agreement between these four structures arising from a metric function d : X × X → 0 , ∞ is due to a combination of the metric axioms and the lattice structure of 0 , ∞. Further motivation materializes from Lawvere's observation from 1973 to the effect that a (slightly generalized) metric space is a category enriched in 0 , ∞. Metric spaces taking values in structures other than 0 , ∞ are relevant for generalizations of metric spaces and find a natural home in Lawvere's categorical setting. In particular, in recent years quantales emerged as structures occupying an important niche in between 0 , ∞ and arbitrary monoidal categories. Since a category enriched in a quantale Q is the same thing as a metric space taking values in Q one may ask whether such a thing belongs to algebra or geometry. Further, does the quadruplet of topologies associated to a Q -valued space/category still consist of identical siblings? We propose a litmus test for the geometricity of Q -valued spaces as we investigate these issues. [ABSTRACT FROM AUTHOR]

In this article, we propose a generalization of both b -metric and dislocated metric, namely, dislocated extended b -metric space. After getting some fixed point results, we suggest a relatively simple solution for a Volterra integral equation by using the technique of fixed point in the setting of dislocated extended b -metric space. [ABSTRACT FROM AUTHOR]

Limit of sequences is a basic concept in mathematical analysis. In this paper we study it in more details using another basic concept of analysis, measure on sets of positive integers. A key role in our considerations is played by the concept of a degree of convergence of a given sequence to a given point with respect to a particular measure on the set of positive integers, as a number in interval [ 0 , 1 ]. We study its properties depending on properties of the chosen measure. It appears that standard limits and their known generalizations (convergence with respect to a filter or ideal) are extremal special cases in our approach. [ABSTRACT FROM AUTHOR]

Logical systems containing knowledge and know-how modalities have been investigated in several recent works. Independently, epistemic modal logics in which every knowledge modality is labeled with a degree of uncertainty have been proposed. This article combines these two research lines by introducing a bimodal logic containing knowledge and know-how modalities, both labeled with a degree of uncertainty. The main technical results are soundness, completeness, and incompleteness of the proposed logical system with respect to two classes of semantics. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a Lipschitz version of the Bishop–Phelps–Bollobás property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map F and a pair of points at which F almost attains its norm by a Lipschitz map G and a pair of points such that G strongly attains its norm at the new pair of points. We first show that if M is a finite pointed metric space and Y is a finite-dimensional Banach space, then the pair (M , Y) has the Lip-BPB property, and that both finiteness assumptions are needed. Next, we show that if M is a uniformly Gromov concave pointed metric space (i.e. the molecules of M form a set of uniformly strongly exposed points), then (M , Y) has the Lip-BPB property for every Banach space Y. We further prove that this is the case for finite concave metric spaces, ultrametric spaces, and Hölder metric spaces. The extension of the Lip-BPB property from (M , R) to some Banach spaces Y and some results for compact Lipschitz maps are also discussed. [ABSTRACT FROM AUTHOR]

Permutation-based Indexing (PBI) approaches have been proven to be particularly effective for conducting large-scale approximate metric searching. These methods rely on the idea of transforming the original metric objects into permutation representations, which can be efficiently indexed using data structures such as inverted files. The standard conceptualization of permutation associated with a metric object involves only the use of object distances and their relative orders from a set of anchors called pivots. In this paper, we generalized this definition in order to enlarge the class of permutation representations that can be used by PBI approaches. In particular, we introduced the concept of permutation induced by a space transformation and a sorting function, and we investigated which properties these transformations should possess to produce permutations that are effective for metric search. Furthermore, as a practical outcome, we defined a new type of permutation representation that is calculated using distances from pairs of pivots. This proposed technique allowed us to produce longer permutations than traditional ones for the same number of object-pivot distance calculations. The advantage lies in the fact that when longer permutations are employed, the use of inverted files built on permutation prefixes leads to greater efficiency in the search phase. • Permutation-based Indexing methods are effective for large-scale metric search. • Definition of permutations induced by a space transformation. • Study of space transformation properties to induce good permutations for metric search. • Novel permutations calculated using both object-pivot and pivot-pivot distances. [ABSTRACT FROM AUTHOR]

This paper develops the so-called Weighted Energy-Dissipation (WED) variational approach for the analysis of gradient flows in metric spaces. This focuses on the minimization of the parameter-dependent global-in-time functional of trajectories I ε [ u ] = ∫ 0 ∞ e − t / ε (1 2 | u ′ | 2 (t) + 1 ε ϕ (u (t))) d t , featuring the weighted sum of energetic and dissipative terms. As the parameter ε is sent to 0, the minimizers u ε of such functionals converge, up to subsequences, to curves of maximal slope driven by the functional ϕ. This delivers a new and general variational approximation procedure, hence a new existence proof, for metric gradient flows. In addition, it provides a novel perspective towards relaxation. [ABSTRACT FROM AUTHOR]

Abstract Let X be a uniform space and U (X) the linear space of real-valued uniformly continuous functions on X. Our main objective is to give a number of properties characterizing the fact that U (X) is stable under pointwise product in case X is a metric space. Some of these characterizations hold in much more general circumstances. [ABSTRACT FROM AUTHOR]

Abstract 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 supermetric 1 1 This 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, 2 2 For 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. [ABSTRACT FROM AUTHOR]

Abstract We show that the category of intrinsic metric spaces and Lipschitz maps is coreflective in the category of all rectifiably connected metric spaces and Lipschitz maps. We also introduce a class of spaces called rectifiably chain-connected metric spaces, and for such spaces we define the concept of a weakly intrinsic distance. We then prove that the category of rectifiably connected metric spaces with weakly intrinsic distance is coreflective in the category of all rectifiably connected metric spaces and Lipschitz maps. [ABSTRACT FROM AUTHOR]

Abstract Let P be a path graph of n vertices embedded in a metric space. We consider the problem of adding a new edge to P such that the diameter of the resulting graph is minimized. Previously (in ICALP 2015) the problem was solved in O (n log 3 ⁡ n) time. In this paper, based on new observations and different algorithmic techniques, we present an O (n log ⁡ n) time algorithm. [ABSTRACT FROM AUTHOR]

Abstract We construct Jordan arcs of prescribed conformal dimension which are "minimal for conformal dimension," meaning the Hausdorff and conformal dimensions are equal. These curves are used to design fractal rugs, similar to Rickman's rug, that are also minimal for conformal dimension. These fractal rugs could potentially settle a standing conjecture regarding the existence of metric spaces of prescribed topological conformal dimension. [ABSTRACT FROM AUTHOR]

We introduce a new class of multidistances, they are called S -generalized multidistances, defined by means of a t-conorm S. The main properties of them are analyzed and some examples are given. We have also established some relations between a multidistance and an S -generalized multidistance through the additive generator of the t-conorm. In this context, we study how to combine a collection of ({ S k } k = 1 m)-generalized multidistances in order to obtain a new S -generalized multidistance. In addition, we prove that properties of the S -generalized multidistances are preserved through the aggregation process. Finally, we give some characterizations of the aggregation functions of a family of ({ S k } k = 1 m)-generalized multidistances based on another aggregation function and the additive generators of the involved t-conorms. [ABSTRACT FROM AUTHOR]

This work is a companion reproducible paper of a previous paper (Antol et al., 2021) in which we presented an alternative to the traditional paradigm of similarity searching in metric spaces called the Learned Metric Index. Inspired by the advance in learned indexing of structured data, we used machine learning models to replace index pivots, thus posing similarity search as a classification problem. This implementation proved to be more than competitive with the conventional methods in terms of speed and recall, proving the concept as viable. The aim of this publication is to make our source code, datasets, and experiments publicly available. For this purpose, we create a collection of Python3 software libraries, YAML reproducible experiment files, and JSON ground-truth files, all bundled in a Docker image – the Learned Metric Index Framework (LMIF) – which can be run using any Docker-compatible operating system on a CPU with Advanced vector extensions (AVX). We introduce a reproducibility protocol for our experiments using LMIF and provide a closer look at the experimental process. We introduce new experimental results by running the reproducibility protocol introduced herein and discussing the differences with the results reported in our primary work (Antol et al., 2021). Finally, we make an argument that these results can be considered weakly reproducible (in both of the performance metrics), since they point to the same conclusions derived in the primary paper. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of learning nearest-prototype classifiers in any finite distance space; that is, in any finite set equipped with a distance function. An important advantage of a distance space over a metric space is that the triangle inequality need not be satisfied, which makes our results potentially very useful in practice. We consider a family of binary classifiers for learning nearest-prototype classification on distance spaces, building on the concept of large-width learning which we introduced and studied in earlier works. Nearest-prototype is a more general version of the ubiquitous nearest-neighbor classifier: a prototype may or may not be a sample point. One advantage in the approach taken in this paper is that the error bounds depend on a ‘width’ parameter, which can be sample-dependent and thereby yield a tighter bound. [ABSTRACT FROM AUTHOR]

It is proved that a metric space is sober, as an approach space, if and only if it is Smyth complete. [ABSTRACT FROM AUTHOR]

Let ( X , d ) be a complete, bounded, metric space. For a nonempty, closed subset A of X denote by C ⁎ ( A × A ) the set of all continuous, bounded, real-valued functions on A × A . Denote by C † = ⋃ { C ⁎ ( A × A ) | A is a nonempty closed subset of X } the set of all partial, continuous and bounded functions. We prove that there exists a linear, regular extension operator from C † endowed with the topology of convergence in the Hausdorff distance of graphs of partial functions to the space C ⁎ ( X × X ) with the topology of uniform convergence on compact sets. The constructed extension operator preserves constant functions, pseudometrics, metrics and admissible metrics. For a fixed, nonempty, closed subset A of X the restricted extension operator from C ⁎ ( A × A ) to C ⁎ ( X × X ) is continuous with respect to the topologies of pointwise convergence, uniform convergence on compact sets and uniform convergence considered on both C ⁎ ( A × A ) and C ⁎ ( X × X ) . [ABSTRACT FROM AUTHOR]

In many practical stochastic dynamic optimization problems with countable states, the optimal policy possesses certain structural properties. For example, the (s , S) policy in inventory control, the well-known c μ -rule and the recently discovered c / μ -rule (Xia et al. (2022)) in scheduling of queues. A presumption of such results is that an optimal stationary policy exists. There are many research works regarding to the existence of optimal stationary policies of Markov decision processes with countable state spaces (see, e.g., Bertsekas (2012); Hernández-Lerma and Lasserre (1996); Puterman (1994); Sennott (1999)). However, these conditions are usually not easy to verify in such optimization problems. In this paper, we study the optimization of long-run average of continuous-time Markov decision processes with countable state spaces. We provide an intuitive approach to prove the existence of an optimal stationary policy. The approach is simply based on compactness of the policy space, with a special designed metric, and the continuity of the long-run average in the space. Our method is capable to handle cost functions unbounded from both above and below, which makes a complementary contribution to the literature work where the cost function is unbounded from only one side. Examples are provided to illustrate the application of our main results. [ABSTRACT FROM AUTHOR]

It is shown that every non-trivial subgroup with one generator p of the multiplicative group of positive numbers R + is the stabilizer of some cloud in the Gromov–Hausdorff class. Namely, this is the stabilizer both of the cloud of the geometric progression X p ⊂ R + and of the cloud of the discrete hedgehog X p ˆ over this progression. It is proved that the center Z [ X p ˆ ] consists of a single complete space. It is proved that the center Z [ [ 0 , ∞) ] (of the half-line cloud) also consists of a single complete space, but the half-line cloud stabilizer is the maximum possible – the whole group of similarities R +. It is shown that the stabilizer of any cloud is the union of isometric stabilizers of spaces from the center of this cloud. [ABSTRACT FROM AUTHOR]

We consider the proper class of all metric spaces endowed with the Gromov–Hausdorff distance. Its maximal subclasses, consisting of spaces at finite distance from each other, we call clouds. Multiplying all distances in a metric space by the same positive real number, we obtain a similarity transformation of the Gromov–Hausdorff class. In our previous work, we observed that with such a transformation, some clouds can jump to others. To characterize the phenomenon, we studied the stabilizers of the similarity action. In this paper, we prove that every cloud with a nontrivial stabilizer has a center, i.e., a metric space for which all similarities from the stabilizer generate a new space at zero distance. Moreover, the center is unique modulo zero distance. The proof is based on the cloud completeness theorem. [ABSTRACT FROM AUTHOR]

We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions. [ABSTRACT FROM AUTHOR]

We show that, unlike minima of superharmonic functions which are again superharmonic, the same property fails for Q -quasisuperminimizers. More precisely, if u i is a Q i -quasisuperminimizer, i = 1 , 2 , where 1 < Q 1 ≤ Q 2 , then u = min { u 1 , u 2 } is a Q -quasisuperminimizer, but there is an increase in the optimal quasisuperminimizing constant Q . We provide the first examples of this phenomenon, i.e. that Q > Q 2 . In addition to lower bounds for the optimal quasisuperminimizing constant of u we also improve upon the earlier upper bounds due to Kinnunen and Martio. Moreover, our lower and upper bounds turn out to be quite close. We also study a similar phenomenon in pasting lemmas for quasisuperminimizers, where Q = Q 1 Q 2 turns out to be optimal, and provide results on exact quasiminimizing constants of piecewise linear functions on the real line, which can serve as approximations of more general quasiminimizers. [ABSTRACT FROM AUTHOR]

We investigate the geometry of the family M of isometry classes of compact metric spaces, endowed with the Gromov–Hausdorff metric. We show that sufficiently small neighborhoods of generic finite spaces in the subspace of all finite metric spaces with the same number of points are isometric to some neighborhoods in the space R ∞ N , i.e., in the space R N with the norm ‖ ( x 1 , … , x N ) ‖ = max i ⁡ | x i | . As a corollary, we get that each finite metric space can be isometrically embedded into M in such a way that its image belongs to a subspace consisting of all finite metric spaces with the same number k of points. If the initial space has n points, then one can take k as the least possible integer with n ≤ k ( k − 1 ) / 2 . [ABSTRACT FROM AUTHOR]

In this paper we discuss the following Tkachuk's question in the sense of ideal convergence [16,18] : Is any Tychonoff connected sequential space a quotient image of a connected metric space? It is proved that let I be an ideal on the set N then a topological space X is an I -connected space with an I - csf -network if and only if X is a continuous I -covering image of a connected metric space. It follows that a topological space X is a connected I -sequential space with an I - csf -network if and only if X is a quotient I -covering image of a connected metric space. [ABSTRACT FROM AUTHOR]

The term Stone-type duality often refers to a dual equivalence between a category of lattices or other partially ordered structures on one side and a category of topological structures on the other. This paper is part of a larger endeavour that aims to extend a web of Stone-type dualities from ordered to metric structures and, more generally, to quantale-enriched categories. In particular, we improve our previous work and show how certain duality results for categories of [ 0 , 1 ] -enriched Priestley spaces and [ 0 , 1 ] -enriched relations can be restricted to functions. In a broader context, we investigate the category of quantale-enriched Priestley spaces and continuous functors, with emphasis on those properties which identify the algebraic nature of the dual of this category. [ABSTRACT FROM AUTHOR]

Let X be a zero-dimensional metric space and X ′ its derived set. We prove the following assertions: (1) the space C k ( X , 2 ) is an Ascoli space iff C k ( X , 2 ) is k R -space iff either X is locally compact or X is not locally compact but X ′ is compact, (2) C k ( X , 2 ) is a k -space iff either X is a topological sum of a Polish locally compact space and a discrete space or X is not locally compact but X ′ is compact, (3) C k ( X , 2 ) is a sequential space iff X is a Polish space and either X is locally compact or X is not locally compact but X ′ is compact, (4) C k ( X , 2 ) is a Fréchet–Urysohn space iff C k ( X , 2 ) is a Polish space iff X is a Polish locally compact space, (5) the space C k ( X , 2 ) is normal iff X ′ is separable, (6) C k ( X , 2 ) has countable tightness iff X is separable. In cases (1)–(3) we obtain also a topological and algebraic structure of C k ( X , 2 ) . [ABSTRACT FROM AUTHOR]

Let K be an uncountable metric compact space. It is well known that C ( K ) is isometrically universal for the separable Banach spaces, but the continuous functions that compose the isometric image of finite dimensional spaces are typically far from being Lipschitz. We prove that the possibility of embedding Euclidean spaces R n ↪ C ( K ) in such a way that the image in C ( K ) is made of Lipschitz functions is tightly related to the dimension (topological or Hausdorff) of K . [ABSTRACT FROM AUTHOR]

Connectedness, path connectedness, and uniform connectedness are well-known concepts. In the traditional presentation of these concepts there is a substantial difference between connectedness and the other two notions, namely connectedness is defined as the absence of disconnectedness, while path connectedness and uniform connectedness are defined in terms of connecting paths and connecting chains, respectively. In compact metric spaces uniform connectedness and connectedness are well-known to coincide, thus the apparent conceptual difference between the two notions disappears. Connectedness in topological spaces can also be defined in terms of chains governed by open coverings in a manner that is more reminiscent of path connectedness. We present a unifying metric formalism for connectedness, which encompasses both connectedness of topological spaces and uniform connectedness of uniform spaces, and which further extends to a hierarchy of notions of connectedness. [ABSTRACT FROM AUTHOR]

We continue our investigation on those (partially) ordered metric spaces ( X , m , ≤ ) for which there exists a T 0 -quasi-metric d on X such that sup ⁡ { d , d − 1 } = m and for any x , y ∈ X we have that x ≤ y if and only if d ( x , y ) = 0 . In particular such partially ordered metric spaces are determined (in the sense of Nachbin) by a T 0 -quasi-uniformity with a countable base. We show that the compatibility conditions obtained in an earlier study between partial order and metric on the investigated spaces X become more transparent when additional compatibility conditions between metric and partial order with appropriate algebraic operations on X are assumed to hold. [ABSTRACT FROM AUTHOR]

We study adaptive data-dependent dimensionality reduction in the context of supervised learning in general metric spaces. Our main statistical contribution is a generalization bound for Lipschitz functions in metric spaces that are doubling, or nearly doubling. On the algorithmic front, we describe an analogue of PCA for metric spaces: namely an efficient procedure that approximates the data's intrinsic dimension, which is often much lower than the ambient dimension. Our approach thus leverages the dual benefits of low dimensionality: (1) more efficient algorithms, e.g., for proximity search, and (2) more optimistic generalization bounds. [ABSTRACT FROM AUTHOR]

In the first half of the paper, we are concerned with the problems of existence (and searching) of coincidence points and the common preimage of a closed subset (in particular, a common root) in the case of a finite system of mappings of one metric space to another one. The second half of the paper is devoted to fiberwise variants of Arutyunov's theorem on coincidence points. Obtaining the main results of the paper is based on the use of the class of almost exactly ( α , β ) -search functionals that is wider than Fomenko's class of ( α , β ) -search functionals. [ABSTRACT FROM AUTHOR]

We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach is not dependent on function spaces and therefore applies equally well to functions on metric spaces as to operator algebras. In particular, we consider analogues of Dirichlet and obstacle problems, as well as first eigenvalue problems, and formulate conditions for the existence of solutions and their uniqueness. Moreover, we investigate to what extent a lattice structure may be introduced on (ordered) Banach spaces via a norm-minimizing variational problem. A multitude of examples is provided to illustrate the versatility of our approach. [ABSTRACT FROM AUTHOR]

We prove that a homeomorphism of a semi-locally connected compact metric space is equicontinuous if and only if it satisfies the following property (CW): the distance between the iterates of a given point and a given subcontinuum not containing that point is bounded away from zero. This equivalence is false for general compact metric spaces. We also prove that a homeomorphism with the property (CW) satisfies that the set of automorphic points contains those points where the space is not semi-locally connected. [ABSTRACT FROM AUTHOR]

Fixed point theory is very wide field, which have many applications in different areas. The concept of altering distance to find fixed point results have been explored by many authors. In the present paper, we achieve unique common fixed point results for two self-mappings using a generalization of altering distance and a rational type contractive condition. An example has also been given in support of our result. [ABSTRACT FROM AUTHOR]

We give a deterministic O ( h n 1 + 1 / h ) -time ( 2 h ) -approximation nonadaptive algorithm for 1-median selection in n -point metric spaces, where h ∈ Z + ∖ { 1 } is arbitrary. Our proof generalizes that of Chang [2] . [ABSTRACT FROM AUTHOR]

It is well known that by Szpilrajn's Theorem each partial order on a set can be extended to a linear order. In this paper we study variants of that theorem that hold for T 0 -quasi-metrics instead of partial orders. More specifically, given a metric space ( X , m ) , we call a T 0 -quasi-metric d defined on the set X m -splitting provided that its symmetrization d s = d ∨ d − 1 is equal to m . For a given metric space ( X , m ) we obtain results about T 0 -quasi-metrics on X that are minimal among the collection of all m -splitting T 0 -quasi-metrics on X . We are also interested in the maximal possible specialization orders of m -splitting T 0 -quasi-metrics. [ABSTRACT FROM AUTHOR]

For each point V , a subset of R 3 , we define a distance on the one skeleton of curve complex for each point and prove that (1) for each point in V with all positive entries, the one skeleton of curve complex under this distance is a metric space and δ -hyperbolic for some δ ∈ R + ; (2) for each point in V with at least one non-positive entry, the diameter of vertices of curve complex under this distance is finite. [ABSTRACT FROM AUTHOR]

Recently, permutation based indexes have attracted interest in the area of similarity search. The basic idea of permutation based indexes is that data objects are represented as appropriately generated permutations of a set of pivots (or reference objects). Similarity queries are executed by searching for data objects whose permutation representation is similar to that of the query, following the assumption that similar objects are represented by similar permutations of the pivots. In the context of permutation-based indexing, most authors propose to select pivots randomly from the data set, given that traditional pivot selection techniques do not reveal better performance. However, to the best of our knowledge, no rigorous comparison has been performed yet. In this paper we compare five pivot selection techniques on three permutation-based similarity access methods. Among those, we propose a novel technique specifically designed for permutations. Two significant observations emerge from our tests. First, random selection is always outperformed by at least one of the tested techniques. Second, there is no technique that is universally the best for all permutation-based access methods; rather different techniques are optimal for different methods. This indicates that the pivot selection technique should be considered as an integrating and relevant part of any permutation-based access method. [ABSTRACT FROM AUTHOR]

Besides integers and short character strings, current applications require storing photos, videos, genetic sequences, multidimensional time series, large texts and several other types of “complex” data elements. Comparing them by equality is a fundamental concept for the Relational Model, the data model currently employed in most Database Management Systems. However, evaluating the equality of complex elements is usually senseless, since exact match almost never happens. Instead, it is much more significant to evaluate their similarity. Consequently, the set theoretical concept that a set cannot include the same element twice also becomes senseless for data “sets” that must be handled by similarity, suggesting that a new, equivalent definition must take place for them. Targeting this problem, this paper introduces the new concept of “similarity sets”, or SimSets for short. Specifically, our main contributions are: (i) we highlight the central properties of SimSets; (ii) we develop the basic algorithm required to create SimSets from metric datasets, ensuring that they always meet the required properties; (iii) we formally define unary and binary operations over SimSets; and (iv) we develop an efficient algorithm to perform these operations. To validate our proposals, the SimSet concept and the tools developed for them were employed over both real world and synthetic datasets. We report results on the adequacy, stability, speed and scalability of our algorithms with regard to increasing data sizes. [ABSTRACT FROM AUTHOR]

Sparse spatial selection (SSS) allows insertions of new database objects and dynamically promotes some of the new objects as pivots. In this paper, we argue that SSS has fundamental problems that result in poor query performance for clustered or otherwise skewed distributions. Real datasets have often been observed to show such characteristics. We show that SSS has been optimized to work for a symmetrical, balanced distribution and for a specific radius value. Our main contribution is offering a new pivot promotion scheme that can perform robustly for clustered or skewed distributions. We show that our new indexing scheme performs significantly better than tree-based dynamic structures while having lower insertion costs. [ABSTRACT FROM AUTHOR]

Due to its possible applications, Fixed Point Theory has become one of the most useful branches of Nonlinear Analysis. In a very recent paper, Khojasteh et al. introduced the notion of simulation function in order to express different contractivity conditions in a unified way, and they obtained some fixed point results. In this paper, we slightly modify their notion of simulation function and we investigate the existence and uniqueness of coincidence points of two nonlinear operators using this kind of control functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of coincidence points for nonlinear mappings of any number of arguments under a weak (ψ,φ) -contractive condition. Our results generalize, extend and unify several classical and very recent related results in the literature (see Aydi et al. (2011), Berinde (2010), Gnana-Bhaskar and Lakshmikantham (2006), Berzig and Samet (2012), Borcut and Berinde (2012), Choudhury et al. (2011), Karapınar and Luong (2012), Lakshmikantham and Ćirić (2009), Luong and Thuan (2011), Roldán (2012)). [ABSTRACT FROM AUTHOR]

For a metric space ( X , d ) , a subset A resolves ( X , d ) if each point x ∈ X is uniquely determined by the distances d ( x , a ) for a ∈ A . Also the metric dimension of ( X , d ) is the smallest cardinality md ( X ) such that there is a set A of the cardinality md ( X ) that resolves X . In this work we are going to compute md ( X ) for the case that X is an n -dimensional geometric space. This category includes a vast domain of Riemannian manifolds. [ABSTRACT FROM AUTHOR]