Using train tracks on a non-exceptional oriented surface S of finite type in a systematic way we give a proof that the complex of curves C(S) of S is a hyperbolic geodesic metric space. We also discuss the relation between the geometry of the complex of curves and the geometry of Teichmüller space.

Relations of convergence of real sequences and convergence of metric spaces are investigated. An abstract intermediate value theorem for two closed sets in the range is presented. At the end, it is proven that an arc connecting the west minimal point and the east maximal point in a simple closed curve must be identical to the upper arc or lower arc of the closed curve. provide the notation and ...

We call a comb a map f : I → [0,∞), where I is a compact interval, such that {f ≥ ε} is finite for any ε. A comb induces a (pseudo)-distance d̄f on {f = 0} defined by d̄f (s, t) = max(s∧t,s∨t) f . We describe the completion Ī of {f = 0} for this metric, which is a compact ultrametric space called comb metric space. Conversely, we prove that any compact, ultrametric space (U, d) without isolated p...

Considering the increasing interest in fuzzy theory and possible applications,the concept of fuzzy metric space concept has been introduced by severalauthors from different perspectives. This paper interprets the theory in termsof metrics evaluated on fuzzy numbers and defines a strong Hausdorff topology.We study interrelationships between this theory and other fuzzy theories suchas intuitionis...

We prove the norm identity kId + T k = 1 + kTk, which is known as the Daugavet equation, for operators T on C (S) not xing a copy of C (S), where S is a compact metric space without isolated points.

We prove the norm identity ‖Id + T ‖ = 1 + ‖T ‖, which is known as the Daugavet equation, for operators T on C(S) not fixing a copy of C(S), where S is a compact metric space without isolated points.

Here shape space is either the manifold of simple closed smooth unparameterized curves in R or is the orbifold of immersions from S to R modulo the group of diffeomorphisms of S. We investige several Riemannian metrics on shape space: L-metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two le...

In this paper, we discuss the existence and uniqueness of points of coincidence and common fixed points for a pair of graph preserving mappings in parametric $N_b$-metric spaces. As some consequences of this study, we obtain several important results in parametric $b$-metric spaces, parametric $S$-metric spaces and parametric $A$-metric spaces. Finally, we provide some illustrative examples to ...

The existence of fixed point in orthogonal metric spaces has been initiated by Eshaghi and et. al [7]. In this paper, we prove existence and uniqueness theorem of fixed point for mappings on $varepsilon$-connected orthogonal metric space. As a consequence of this, we obtain the existence and uniqueness of fixed point for analytic function of one complex variable. The paper concludes with some i...