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We study ancient solutions of polynomial growth to both continuous-time and discrete-time heat equations on graphs with unbounded Laplacians. We generalize Colding and Minicozzi's theorem [CM19] on manifolds, and the result [Hua19] on graphs with normalized Laplacians to the setting of graphs with unbounded Laplacians: For a graph admitting an intrinsic metric, which has polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the dimension of harmonic functions with the same growth up to some factor., 15 pages. arXiv admin note: text overlap with arXiv:1903.02411

We obtain functional inequalities for functions which are metric-preserving with respect to one of the following intrinsic metrics in a canonical plane domain: hyperbolic metric or some restrictions of the triangular ratio metric, respectively, of a Barrlund metric. The subadditivity turns out to be an essential property, being possessed by every function that is metric-preserving with respect to the hyperbolic metric and also by the composition with some specific function of every function that is metric-preserving with respect to some restriction of the triangular ratio metric or of a Barrlund metric. We partially answer an open question, proving that the hyperbolic arctangent is metric-preserving with respect to the restrictions of the triangular ratio metric on the unit disk to radial segments and to circles centered at origin.

In the paper weak isometries in directed groups are investigated. It is proved that for every weak isometryfin a directed groupGthe relationf(UL(x,y) ∩LU(x,y)) =UL(f(x),f(y)) ∩LU(f(x),f(y)) is valid for eachx,y∈G. The notions of an orthogonality of two elements and of a subgroup symmetry in directed groups are introduced and it is shown that each weak isometry in a 2-isolated directed group or in an abelian directed group is a composition of a subgroup symmetry and a right translation. It is also proved that stable weak isometries in a 2-isolated abelian directed groupGare directly related to subdirect decompositions of the subgroupG2= {2x;x∈G} ofG.

The intrinsic metric formula on the code set of the Sierpinski Gasket is explicitly given in Definition 1.1. In this paper, we obtain the intrinsic metric formula on the code set of the Sierpinski tetrahedron and we investigate some geometrical properties of this structure. Moreover, we define a dynamical system on the Sierpinski tetrahedron and then we give an algorithm to compute the periodic points of this dynamical system. Finally, we show that this dynamical system is both Devaney chaotic and Li-Yorke chaotic.

We define metrics on Culler-Vogtmann space, which are an analogue of the Teichmuller metric and are constructed using stretching factors. In fact the metrics we study are related, one being a symmetrised version of the other. We investigate the basic properties of these metrics, showing the advantages and pathologies of both choices. We show how to compute stretching factors between marked metric graphs in an easy way and we discuss the behaviour of stretching factors under iterations of automorphisms. We study metric properties of folding paths, showing that they are geodesic for the non-symmetric metric and, if they do not enter the thin part of Outer space, quasi-geodesic for the symmetric metric., changelog v1 -> v2: Added an example, suggested by Bert Wiest and Thierry Coulbois showing that the symmetric metric is not geodesic. Minor changes. Biblio updated

Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere $X$ is a quasisphere if and only if $X$ is linearly locally connected and carries a weak metric doubling measure, i.e., a measure that deforms the metric on $X$ without much shrinking., Comment: 15 pages

We prove the following localized version of a classical ellipsoid characterization: Let $B\subset\mathbb R^3$ be convex body with a smooth strictly convex boundary and 0 in the interior, and suppose that there is an open set of planes through 0 such that all sections of $B$ by these planes are linearly equivalent. Then all these sections are ellipses and the corresponding part of $B$ is a part of an ellipsoid. We apply this to differential geometry of Finsler surfaces in normed spaces and show that in certain cases the intrinsic metric of a surface imposes restrictions on its extrinsic geometry similar to implications of Gauss' Theorema Egregium. As a corollary we construct 2-dimensional Finsler metrics that do not admit local isometric embeddings to dimension~3., Comment: v2: minor clarification and cleanup

A result showed by M. Gursky in [4] ensures that any metric g on the 4-dimensional sphere S4 satisfying Ricg=3g and injg(S4)≥π34 is isometric to the round metric. In this note, we prove that there exists a universal number i0 such that any metric g on the 4-dimensional sphere S4 satisfying Ricg=3g and injg(S4)≥π34−i0 is isometric to the round metric. Moreover, there exists a universal e0>0 such that any metric g on the 4-dimensional sphere S4 with nonnegative sectional curvature, Ricg=3g and 89π2−e0≤Volg(S4) is isometric to the round metric. This last result slightly improves a rigidity theorem also proved in [4].

Far East Journal of Mathematical Sciences (FJMS), Volume 102, Number 5, 2017, Pages 965-978

A pseudo-Finsleroid metric function F of the two-axes structure that involves the vertical axis and the horizontal axis is proposed assuming constancy of the curvature of indicatrix. The curvature is negative and the signature of the Finslerian metric tensor is exactly $$(+-\cdots )$$ . The function F endows the tangent space with the geometry which possesses many interesting Finslerian properties. The use of the angle representation is the underlying method which has been conveniently and successfully applied. The appearance of the positive-definite Finsleroid metric function in the horizontal sections of the tangent space is established.

We explore the connection between the geometries generated by logarithmic oscillations and the class of metric spaces satisfying the Gromov hyperbolicity condition. We investigate the most fundamental examples, inspired from classical geometries, e.g. the Euclidean distance on the infinite strip or Hilbert’s distance on the unit disk. We continue our study with the Barbilian’s distance, which historically appeared as a natural extension of a model of hyperbolic geometry. We introduce and investigate a new metric, called the stabilizing metric. In a natural development, we explore a class of extensions of this distance which, under some analytic conditions, produces infinitely many new examples of Gromov hyperbolic metric spaces. Using similar procedures, we construct Vuorinen’s stabilizing metric and its extensions, and we discuss their Gromov hyperbolicity.

We use λ-sequences in this article to derive common fixed points for a family of self-mappings defined on a complete G-metric space. We imitate some existing techniques in our proofs and show that the tools employed can be used at a larger scale. These results generalize well known results in the literature.

In this work, we will prove some fixed point results for the class of (G,?)-contractions on vector metric spaces endowed with a graph. Our results extend and unify many known results for (G,?)- contractions on metric spaces with a graph and for ?-contractions on vector metric spaces. We apply our results to obtain an existence theorem for the solution of an integral equation.

In this paper, we study Ishikawa iterative scheme with error terms for a finite family of Iasymptotically quasi-nonexpansive mappings in a convex metric space. We established strong convergence theorems and their applications for the proposed algorithms in a convex metric space. Our theorems improve and extend the corresponding known results in Banach spaces.

[EN] In this paper we introduce and study the concept of strong convergence in fuzzy metric spaces (X, M,*) in the sense of George and Veeramani. This concept is related with the condition Lambda M-t > 0(x, y, t) > 0, which frequently is required or missing in this context. Among other results we characterize the class of s-fuzzy metrics by the strong convergence defined here and we solve partially the question of finding explicitly a compatible metric with a given fuzzy metric., Valentn Gregori acknowledges the support of the Spanish Ministry of Economy and Competitiveness under Grant MTM2015-64373-P (MINECO/FEDER, UE).

The aim of this article is to study common fixed point theorems in generalized metric spaces and obtain sufficient conditions for the existence of C-class function on Khan type common fixed points of a pair of mappings satisfying generalized contraction involving rational expressions. Also, some examples are obtained to support the obtained result. Also obtained results further generalize, extend and unify already existing results in literature.

We study sets with at most two-valued metric projection in Banach spaces. We show that a two-dimensional Banach space is smooth if and only if every point of the convex hull of an arbitrary closed set with at most two-valued metric projection lies on a segment with endpoints in that set.

Some classes of metric spaces satisfying properties stronger than completeness but weaker than compactness have been studied by many authors over the years. One such significant family consists of those metric spaces on which every real-valued continuous function is uniformly continuous, which are widely known as Atsuji spaces or UC spaces. Recently in 2014, two new kinds of complete metric spaces are introduced, namely Bourbaki-complete and cofinally Bourbaki-complete metric spaces, whose idea has come from some new classes of sequences acting as generalizations of Cauchy sequences. Our major goal is to give several new equivalent conditions for metric spaces whose completions are one of the aforesaid spaces, especially in terms of some functions, sequences and geometric functionals.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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 this paper, we introduce a new condition namely: condition (W.C.C.) and utilize the same to prove a Suzuki type unique common fixed point theorem for two hybrid pairs of mappings in partial metric spaces employing the partial Hausdorff metric which generalizes several known results of the existing literature proved in metric and partial metric spaces.

In this paper, we prove a Meir–Keeler type coupled fixed point results in metric spaces. We make an application of our result to obtain a corresponding result in partial metric spaces. The latter are generalizations of metric spaces having a T0-topology in general and admitting non-zero measures of self distances. It is only under special circumstances that the results obtained in metric spaces can be extended to partial metric spaces. Here, we show that the result we obtain in metric spaces can be applied to obtain similar fixed point result in partial metric spaces. Two illustrative examples are given one each for the metric spaces and partial metric spaces.

In this work, the authors introduce the concept of (p, q)-quasi-contraction mapping in a cone metric space. We prove the existence and uniqueness of a fixed point for a (p, q)-quasi-contraction mapping in a complete cone metric space. The results of this paper generalize and unify further fixed point theorems for quasi-contraction, convex contraction mappings and two-sided convex contraction of order 2.

In this paper we prove a fixed point theorem on a fuzzy set defining a new class of fuzzy metric space as structure fuzzy metric space.

It is proved that there exists a metric on a Cantor set such that any finite metric space whose diameter does not exceed 1 and the number of points does not exceed n can be isometrically embedded into it. It is also proved that for any m, n ∈ N there exists a Cantor set in Rm that isometrically contains all finite metric spaces which can be embedded into Rm, contain at most n points, and have the diameter at most 1. The latter result is proved for a wide class of metrics on Rm and, in particular, for the Euclidean metric.

In this paper we introduce a property and use this property to prove some common fixed point theorems in b-metric space. We also give some fixed point results on b-metric spaces endowed with an arbitrary binary relation which can be regarded as consequences of our main results. As applications, we applying our result to prove the existence of a common solution for the following system of integral equations: x ( t ) = ∫ a b K 1 ( t , r , x ( r ) ) d r , x ( t ) = ∫ a b K 2 ( t , r , x ( r ) ) d r , $$\matrix {x (t) = \int \limits_a^b {{K_1}} (t, r, x(r))dr, & & x(t) = \int \limits_a^b {{K_2}}(t, r, x(r))dr,} $$ where a, b ∈ ℝ with a < b, x ∈ C[a, b] (the set of continuous real functions defined on [a, b] ⊆ ℝ) and K 1, K 2 : [a, b] × [a, b] × ℝ → ℝ are given mappings. Finally, an example is also given in order to illustrate the effectiveness of such result.

We introduce the notion of a parametricS-metric space as generalization of a parametric metric space. Using some expansive mappings, we prove a fixed-point theorem on a parametricS-metric space. It is important to obtain new fixed-point theorems on a parametricS-metric space because there exist some parametricS-metrics which are not generated by any parametric metric. We expect that many mathematicians will study various fixed-point theorems using new expansive mappings (or contractive mappings) in a parametricS-metric space.

In this paper we introduce a new fuzzy contraction mapping and prove that such mappings have fixed point in complete fuzzy metric spaces. We give an illustrative example. The result generalizes some existing results.

We consider infinite graphs and the associated energy forms. We show that a graph is canonically compactifiable (i.e. all functions of finite energy are bounded) if and only if the underlying set is totally bounded with respect to any finite measure intrinsic metric. Furthermore, we show that a graph is canonically compactifiable if and only if the space of functions of finite energy is an algebra. These results answer questions in a recent work of Georgakopoulos, Haeseler, Keller, Lenz, Wojciechowski., Comment: This is a small note solving two problems posed in "Graphs of finite measure" (arXiv:1309.3501)

We first investigate on the asymptotics of the Kolmogorov metric entropy and nonlinear n -widths of approximation spaces on some function classes on manifolds and quasi-metric measure spaces. Secondly, we develop constructive algorithms to represent those functions within a prescribed accuracy. The constructions can be based on either spectral information or scattered samples of the target function. Our algorithmic scheme is asymptotically optimal in the sense of nonlinear n -widths and asymptotically optimal up to a logarithmic factor with respect to the metric entropy.