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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 m d ( X ) such that there is a set A of the cardinality m d ( X ) that resolves X. In this note we are going to determine the metric dimension of metric orbit spaces in some special cases and find an upper bound for a general case. This category contains a vast domain of topological spaces and topological manifolds.

In this paper, we will attempt to enforce the feeling that cone metric spaces are not real generalization of metric spaces, by a renorming of the Banach space. In specific, we can convert any strongly minihedral normal cone P to a normal cone with normal constant 1. Consequently, and due to this result, every cone metric space is really a metric space, and every theorem in metric spaces is valid for cone metric spaces automatically.

We investigate the recently introduced geometric quench protocol for fractional quantum Hall (FQH) states within the framework of exactly solvable quantum Hall matrix models. In the geometric quench protocol a FQH state is subjected to a sudden change in the ambient geometry, which introduces anisotropy into the system. We formulate this quench in the matrix models and then we solve exactly for the post-quench dynamics of the system and the quantum fidelity (Loschmidt echo) of the post-quench state. Next, we explain how to define a spin-2 collective variable $\hat{g}_{ab}(t)$ in the matrix models, and we show that for a weak quench (small anisotropy) the dynamics of $\hat{g}_{ab}(t)$ agrees with the dynamics of the intrinsic metric governed by the recently discussed bimetric theory of FQH states. We also find a modification of the bimetric theory such that the predictions of the modified bimetric theory agree with those of the matrix model for arbitrarily strong quenches. Finally, we introduce a class of higher-spin collective variables for the matrix model, which are related to generators of the $W_{\infty}$ algebra, and we show that the geometric quench induces nontrivial dynamics for these variables., 18 pages, 1 figure, v2: minor changes, accepted for publication in PRB

The Sierpinski gasket (also known as the Sierpinski triangle) is one of the fundamental models of self-similar sets. There have been many studies on different features of this set in the last decades. In this paper, initially we construct a dynamical system on the Sierpinski gasket by using expanding and folding maps. We then obtain a surprising shift map on the code set of the Sierpinski gasket, which represents the dynamical system, and we show that this dynamical system is chaotic on the code set of the Sierpinski gasket with respect to the intrinsic metric. Finally, we provide an algorithm to compute periodic points for this dynamical system.

We establish heat kernel upper bounds for a continuous-time random walk under unbounded conductances satisfying an integrability assumption, where we correct and extend recent results by the authors to a general class of speed measures. The resulting heat kernel estimates are governed by the intrinsic metric induced by the speed measure. We also provide a comparison result of this metric with the usual graph distance, which is optimal in the context of the random conductance model with ergodic conductances., Comment: 19 pages; accepted version, to appear in Electron. Commun. Probab

Fermat–Steiner problem consists in finding all points in a metric space Y such that the sum of distances from each of them to the points from some fixed finite subset A of Y is minimal. Such points are sometimes referred as geometric medians of A. This problem is investigated for the metric space \(Y=H(X)\) of compact subsets of a metric space X, endowed with the Hausdorff distance. For the case of a proper metric space X a description of all compacts \(K\in H(X)\) which the minimum is attained at is obtained. In particular, the Steiner minimal trees for three-element boundaries are described. We also construct a surprising example of a quite symmetric regular triangle in \(H(\mathbb {R}^2)\), such that all its shortest trees have no “natural” symmetry.

Given a connected graph G = ( V , E ) , a set of vertices S ⊂ V is an edge metric generator for G , if any two edges of G are identified by S by mean of distances to the vertices in S . Moreover, in a natural way, S is a mixed metric generator, if any two elements of G (vertices or edges) are identified by S by mean of distances. In this work we study the (edge and mixed) metric dimension of graphs.

In the note we discuss partial metric type structures and present their basic properties as well as their relationship with metric type structures. As an application, fixed point theorems for a Lipschitzian map on this structures will be presented.

In this work, the class of b-rectangular metric spaces is introduced, some fixed point results dealing with rational type contractions and almost generalized weakly contractive mappings are obtained. Certain examples are given to support the results.

The Weil–Petersson metric is a Hermitian metric originally defined on finite-dimensional Teichmuller spaces. Ahlfors proved that this metric is a Kahler metric and has some negative curvatures. Takhtajan and Teo showed that this result is also valid for the universal Teichmuller space equipped with a complex Hilbert manifold structure. In this paper, we stated that the Weil–Petersson metric can be also defined on a Hilbert manifold contained in the Teichmuller space of Fuchsian groups with Lehner’s condition, which we call the square integrable Teichmuller space, and proved that the results given by Ahlfors, Takhtajan, and Teo also hold in that case. Many parts of the proof were based on their ones. However, we needed more careful estimations in the infinite-dimensional case, which was achieved by two complex analytic characterizations of Lehner’s condition, by a certain integral equality for the partition of the upper half-plane by a Fuchisian group and by the invariant formula for the Bergman kernel.

We consider a weighted lattice Z d with conductance μ e = ∣ e ∣ − α . We show that the heat kernel of a variable speed random walk on it satisfies a two-sided Gaussian bound by using an intrinsic metric. We also show that when d = 2 and α ∈ ( − 1 , 0 ) , two independent random walks on such weighted lattice will collide infinitely many times while they are transient.

An approximation modulus, the $\mathrm{AM}$-modulus, for a family of paths is introduced in a metric measure space ${(X,d,\nu)}$. It is shown that a function of bounded variation in X is a BV function on almost every path with respect to the $\mathrm{AM}$-modulus. Properties of the $\mathrm{AM}$-modulus, its relations to the usual ${M_{1}}$-modulus and to the BV-capacities of condensers are studied. Only minimal assumptions on X and the measure ν are employed.

In this paper we discuss the topological properties of quasi-partialb-metric spaces. The notion of quasi-partialb-metric space was introduced and fixed point theorem and coupled fixed point theorem on this space were studied. Here the concept of quasi-partialb-metric topology is discussed and notion of product of quasi-partialb-metric spaces is also introduced.

In this paper we study smooth metric measure spaces with boundary via the Bakry–Émery curvature and the weighted mean curvature of the boundary. We establish the weighted Laplacian comparison theorems and the upper bound estimates of the distance from any point of the manifold to its boundary. As applications, we derive lower bound estimates for the first Dirichlet eigenvalue.

Developing A.D. Aleksandrov's ideas, the first-named author of this article proposed the following approach to study of rigidity problems for the boundary of a $C^0$-submanifold in a smooth Riemannian manifold: Let $Y_1$ be a 2-dimensional compact connected $C^0$-submanifold with nonempty boundary in a 2-dimensional smooth connected Riemannian manifold $(X,g)$ without boundary satisfying the condition $$\rho_{Y_1}(x,y) = \liminf_{x' \to x, y' \to y, x',y' \in \mathop{\rm Int} Y_1} \{[l(\gamma_{x',y',\mathop{\rm Int} Y_1})]\} < \infty,$$ if $x,y \in Y_1$. Here $\inf[l(\gamma_{x',y',\mathop{\rm Int} Y_1})]$ is the infimum of the length of smooth paths joining $x'$ and $y'$ in the interior $\mathop{\rm Int} Y_1$ of $Y_1$. In the present paper, we first establish that $\rho_{Y_1}$ is a metric on $Y_1$. Suppose further that $Y_1$ is strictly convex in the metric $\rho_{Y_1}$. Consider another 2-dimensional compact connected $C^0$-submanifold $Y_2$ of $X$ with boundary satisfying the condition $\rho_{Y_2}(x,y) < \infty$, $x,y \in Y_2$, and assume that $\partial Y_1$ and $\partial Y_2$ are isometric in the metrics $\rho_{Y_j}$, $j = 1,2$. There appears the following natural question: Under which additional conditions are the boundaries $\partial Y_1$ and $\partial Y_2$ of $Y_1$ and $Y_2$ isometric in the metric $\rho_X$ of the ambient manifold $X$? The paper is devoted to the detailed discussions of this question. In it, we in particular obtain a number of new results concerning the rigidity problems for the boundaries of $C^0$-submanifolds in a Riemannian manifold. The case of $\dim Y_j = \dim X = n$, $n > 2$, is also considered., Comment: arXiv admin note: substantial text overlap with arXiv:1305.6169

Let ( M , g ) be a Riemannian manifold and N a C 2 submanifold without boundary. If we multiply the metric g by the inverse of the squared distance to N , we obtain a new metric structure on M ∖ N called the condition metric. A question about the behavior of this new metric arises from the works of Beltran, Dedieu, Malajovich and Shub: is it true that for every geodesic segment in the condition metric its closest point to N is one of its endpoints? Previous works show that the answer to this question is positive (under some smoothness hypotheses) when M is the Euclidean space R n . Here we find that there is a strong relation between this property and the curvature of M . In particular, we prove that the answer is also positive if M has non-negative curvature and that this property does not hold when the curvature of M is strictly negative at some point.

The aim of our paper is to present new fixed point theorems under very general contractive conditions in generalized metric spaces which were recently introduced by Jleli and Samet in [Fixed Point Theory Appl. 2015 (2015), doi: 10.1186/s13663-015-0312-7 ]. Although these spaces are not endowed with a triangle inequality, these spaces extend some well known abstract metric spaces (for example, b-metric spaces, Hitzler–Seda metric spaces, modular spaces with the Fatou property, etc.). We handle several types of contractive conditions. The main theorems we present involve a reflexive and transitive binary relation that is not necessarily a partial order. We give a counterexample to a recent fixed point result of Jleli and Samet. Our results extend and unify recent results in the context of partially ordered abstract metric spaces.

In this paper we consider, discuss, improve and generalize recent fixed point results for mappings in b -metric, rectangular metric and bb-rectangular metric spaces established by Đukic et al. (2011), George and Rajagopalan (2013) and Roshan et al. (2015). Also, we prove a Geraghty type theorem in the setting of bb-metric spaces as well as a Boyd–Wong type theorem in the framework of bb-rectangular metric spaces, in both cases, without using Hausdorff assumption. One example is given to support the results.

In order to obtain tractable formal descriptions of poorly formalized problems within the context of the algebraic approach to pattern recognition, we develop methods for analyzing metric configurations. In this paper, using the concepts of ź-isomorphism and ź-completion of metric configurations, a system of criteria for assessing the properties of "generalized density" is obtained. The analysis of the density properties along the axes of a metric configuration allowed us to formulate methods for calculating the topological neighborhoods of points and for finding the "grains" of metric condensations. The theoretical results point to a new plethora of algorithms for searching metric condensations ź methods based on the "restoration" of the set (the condensation searched) using the data on the components of the projection of the set on the axes of the metric configuration. The only mandatory parameters of any algorithm of this family of algorithms are the metric itself and the distribution of ź, which characterizes the accuracy of the values of the metric.

In this article, we introduce the concept of ordered dualistic partial metric spaces and establish an order relation on quasi dualistic partial metric spaces. Later on, using this order relation, we prove fixed point theorems for single and multivalued mappings. We support our results with some illustrative examples.

The distortion of distances between points under maps is studied. We first prove a Schwarz-type lemma for quasiregular maps of the unit disk involving the visual angle metric. Then we investigate conversely the quasiconformality of a bilipschitz map with respect to the visual angle metric on convex domains. For the unit ball or half space, we prove that a bilipschitz map with respect to the visual angle metric is also bilipschitz with respect to the hyperbolic metric. We also obtain various inequalities relating the visual angle metric to other metrics such as the distance ratio metric and the quasihyperbolic metric., Comment: 14 pages, 3 figures

We introduce a scale-invariant version of the Cassinian metric and study its basic properties. We establish connections between the new metric and other well-known metrics such as, j-metric, \(\tilde{j}\)-metric, the half-apollonian metric, and the hyperbolic metric. We also show that the density of the new metric is the same as the density of the quasihyperbolic metric.

Coupled coincidence and fixed point problems have been in the focus of the research interest for last few years. The problem was introduced in fuzzy metric spaces only very recently. In this paper, we work out a weak coupled coincidence point theorem for a compatible pair of mappings in fuzzy metric spaces. As one of the corollaries we have a weak coupled fuzzy contraction mapping theorem. The space is endowed with a partial ordering. We use a combination of analytic and order theoretic concepts in the proof of our main theorem. The result is illustrated with an example.

We consider several generalized metric properties and study the relation between a space X satisfying such property and its n-fold symmetric product satisfying the same property.

In this paper we prove some conditions for equivalent Cauchy sequences in partial metric spaces. These conditions are necessary and sufficient for 0-equivalent 0-Cauchy sequences in partial metric spaces. Some examples are given to illustrate the observed results.

In the present paper we obtain new main metric invariants of finite metric spaces. These invariants can be used for classification of the finite metric spaces and their recognition.

We use the ideas in Samet et al. (Fixed Point Theory Appl 2013:5, 2013) and Vetro and Vetro (Topol Appl 164:125–137, 2014) and the notion of simulation function to establishing existence and uniqueness of fixed points in the setting of ordered metric and ordered partial metric spaces. As consequences of this study, we give a result of existence and uniqueness of a first-order periodic differential problem. From main theorems, we deduce several related fixed point results, in ordered metric and ordered partial metric spaces.

In this paper, we derive some implications and nonimplications among various kinds of convergence axioms derived in generalized metric space. Moreover, we prove an alternative proof for generalization of Banach contraction principle and illustrate this with a suitable example.

In this paper we prove a common fixed point theorem in fuzzy metric space using an implicit relation. 2010 AMS Classification: 47H10, 54H25

In the context of the algebraic approach to recognition of Yu.I. Zhuravlev's scientific school, metric analysis of feature descriptions is necessary to obtain adequate formulations for poorly formalized recognition/classification problems. Formalization of recognition problems is a cross-disciplinary issue between supervised machine learning and unsupervised machine learning. This work presents the results of the analysis of compact metric spaces arising during the formalization of recognition problems. Necessary and sufficient conditions of compactness of metric spaces over lattices of the sets of feature descriptions are analyzed, and approaches to the completion of the discrete metric spaces (completion by lattice expansion or completion by variation of estimate) are formulated. It is shown that the analysis of compactness of metric spaces may lead to some heuristic cluster criteria commonly used in cluster analysis. During the analysis of the properties of compactness, a key concept of a ?-network arises as a subset of points that allows one to estimate an arbitrary distance in an arbitrary metric configuration. The analysis of compactness properties and the conceptual apparatus introduced (?-networks, their quality functionals, the metric range condition, i- and ?-spectra, ?-neighborhood in a metric cone, ?-isomorphism of complete weighted graphs, etc.) allow one to apply the methods of functional analysis, probability theory, metric geometry, and graph theory to the analysis of poorly formalized problems of recognition and classification.

Ordered pairs form of a metric space (S,d), where d is the metric on a nonempty set S. Concept of partial metric space is a minimal generalization of a metric space where each x∈S,d(x,x) does not need to be zero, in other terms is known as non-self-distance. Axiom obtained from the generalization is following properties p(x,x)≤p(x,y) for every x,y∈S. The results of this paper are few studies in the form of definitions and theorems concerning continuity function on partial metric space.

A translation operator acting in a space with a diagonal metric is introduced to describe the motion of a particle in a quantum system. We show that the momentum operator and, as a consequence, the uncertainty relation now depend on the metric. It is also shown that, for any metric expanded up to second order, this formalism naturally leads to an extended uncertainty principle (EUP) with a minimum momentum dispersion. The Ehrenfest theorem is modified to include an additional term related to a tidal force arriving from the space curvature introduced by the metric. For one-dimensional systems, we show how to map a harmonic potential to an effective potential in Euclidean space using different metrics.

In this paper, we study the relations between the Thurston metric and the hyperbolic metric on a closed surface of genus . We show a rigidity result which says if there is an inequality between the marked length spectra of these two metrics, then they are isotopic. We obtain some inequalities on length comparisons between these metrics. Besides, we show certain distance distortions under conformal graftings, with respect to the metric, the length spectrum metric and Thurston`s asymmetric metrics.

This is an exposition of the theory of differentiable structures onmetric measure spaces, in the sense of Cheeger and Keith

Fixed point results for generalized weak contractions under w-distance are proved using discontinuous control functions that are more relaxed than functions used in related work for metric spaces. Later, we apply our theory to coupled coincidence point problems and existence of solution of Fredholm type integral equation. We present examples to justify our claims.

Based on the concept of truth degree for logical formulas, a pseudo-metric is constructed on the set of all classical propositional formulas, and a metric is naturally induced on the corresponding Lindenbaum algebra of Boolean type, which constitutes a metric space, called the classical logic metric space. We respectively study both lattice completions and metric completions of this space, and compare the two kinds of completions from the angle of lattice structure as well as metric structure. On one hand, it is proved that metric completions of the classical logic metric space are complete Boolean algebras, which also act as lattice completions of the Lindenbaum algebra. On the other hand, it is pointed out that the normal lattice completion of the Lindenbaum algebra constitutes a Boolean algebra as well, which is strictly smaller than metric completions of the classical logic metric space in the sense of order-embedding. Also, the normal lattice completion can be seen as a dense subspace of the metric completions in the sense of isometry.

We introduce new metric characteristics for nonrectifiable curves. They admit applications to the theory of boundary value problems for analytic functions. Using these characteristics, we in particular obtain some sharper conditions than those available for the solvability of the jump problem and the Riemann problem in domains with nonrectifiable boundaries.

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.