Narrow Results

This paper attempts an extensive study on metric space and calculus under Type 2 interval uncertainty. Type 2 interval generalizes interval uncertainty considering both ends of the interval number to be imprecise. Type 2 interval philosophy was introduced in the literature with optimization perspectives. We prioritize the study of Type 2 interval-ruled dynamical systems. The concerns necessitate an extensive introduction of metric space and calculus for Type 2 interval-valued functions. We investigate several fundamental properties of metric space in the contemporary of Type 2 interval setting. After significant findings in differential calculus using generalized Hukuhara difference of Type 2 interval numbers, a detailed and novel manifestation of integral calculus including Riemann and Lebesgue senses is also done in this paper. We also provide hints for possible mathematical modelings of real-world scenarios using Type 2 interval-ruled uncertain decision realm.

Let L ≥ 1, ε > 0 be real numbers, (M, d) be a finite metric space and (N, ρ) be a metric space. A query to a metric space consists of a pair of points and asks for the distance between these points. We study the number of queries to metric spaces (M, d) and (N, ρ) needed to decide whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into N, ρ). When (M, d) is ∊-far from being L-bilipschitz embeddable into (N, ρ), we allow an o(1) probability of error (i.e., returning the wrong answer "L-bilipschitz embeddable"). However, no error is allowed when (M, d) is L-bilipschitz embeddable into (N, ρ). That is, algorithms with only one-sided errors are studied in this paper. When |M| ≤ |N| are both finite, we give an upper bound of on the number of queries for determining with one-sided error whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into (N, ρ). For the special case of finite |M| = |N|, the above upper bound evaluates to . We also prove a lower bound of Ω(|N|^3/2) for the special case when |M| = |N| are finite and L = 1, which coincides with testing isometry between finite metric spaces. For finite |M| = |N|, the upper and lower bounds thus match up to a multiplicative factor of at most , which depends only sublogarithmically in |N|. We also investigate the case when (N, ρ) is not necessarily finite. Our results are based on techniques developed in an earlier work on testing graph isomorphism.

We consider a local move, denoted by $\lambda$, on knot diagrams or virtual knot diagrams.If two (virtual) knots $K_1$ and $K_2$ are transformed into each other by a finite sequence of $\lambda$ moves, the $\lambda$ distance between $K_1$ and $K_2$ is the minimum number of times of $\lambda$ moves needed to transform $K_1$ into $K_2$. By ${\mathrm{\Gamma }}_{\lambda }(K)$, we denote the set of all (virtual) knots which can be transformed into a (virtual) knot $K$ by $\lambda$ moves. A geodesic graph for ${\mathrm{\Gamma }}_{\lambda }(K)$ is the graph which satisfies the following: The vertex set consists of (virtual) knots in ${\mathrm{\Gamma }}_{\lambda }(K)$ and for any two vertices $K_1$ and $K_2$, the distance on the graph from $K_1$ to $K_2$ coincides with the $\lambda$ distance between $K_1$ and $K_2$. When we consider virtual knots and a crossing change as a local move $\lambda$, we show that the $N$-dimensional lattice graph for any given natural number $N$ and any tree are geodesic graphs for ${\mathrm{\Gamma }}_{\lambda }(K)$.

The large intersection classes in real Euclidean spaces have been introduced by Falconer in 1985. First, we generalize Falconer’s large intersection classes to compact metric spaces and show that these classes are closed under countable intersections. Next, we prove that the limsup sets generated by rectangles in a compact metric space enjoy this intersection property under certain full Hausdorff measure assumptions. We also apply our results to high-dimensional Diophantine approximation on fractals.

We classify countable metrically homogeneous graphs of diameter 3.

A G-process is briefly a process ([A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, Vol. 182 (Springer, 2013)], [C. M. Dafermos, An invariance principle for compact processes, J. Differential Equations9 (1971) 239–252], [P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176 (Amer. Math. Soc., 2011)]) for which the role of evolution parameter is played by a general topological group $G$. These processes are broad enough to include the $G$-actions (characterized as autonomous $G$-processes) and the two-parameter flows (where $G=ℝ$). We endow the space of $G$-processes with a natural group structure. We introduce the notions of orbit, pseudo-orbit and shadowing property for $G$-processes and analyze the relationship with the $G$-processes group structure. We study the equicontinuous $G$-processes and use it to construct nonautonomous $G$-processes with the shadowing property. We study the global solutions of the $G$-processes and the corresponding global shadowing property. We study the expansivity (global and pullback) of the $G$-processes. We prove that there are nonautonomous expansive $G$-processes and characterize the existence of expansive equicontinuous $G$-processes. We define the topological stability for $G$-processes and prove that every expansive $G$-process with the shadowing property is topologically stable. Examples of nonautonomous topologically stable $G$-processes are given.

In recent years, Kider and Hussain introduced the concept of standard fuzzy metric space and proved some of its key properties. In this paper, we extend this notion to the intuitionistic fuzzy setting. Furthermore, a number of properties related to this notion are studied such as compactness and completeness properties defined in standard intuitionistic fuzzy metric space. Moreover, the Edelstein contraction theorem on this space is established.

Mendel and Naor's definition of metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz is equivalent to an ultrametric space having infimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov–Hausdorff limits, and use these facts to establish a partial converse of the main result.

Some new theorems on random coincidence points and random fixed points for weakly compatible mappings in convex separable complete metric spaces have been established. These results generalize some recent well known comparable results in the literature.

Hasegawa, Kawasaki and Kobayashi [Fixed point theorems for contractively widely more generalized hybrid mappings in metric spaces, to appear in Linear and Nonlinear Anal.] introduced the concept of contractively widely more generalized hybrid mappings in a metric space. On the other hand, Bogin [A generalization of a fixed point theorem of Goebel, Kirk and Shimi, Canad. Math. Bull.19 (1976) 7–12] showed a fixed point theorem. However, Bogin’s result is not included in our results. In this paper, we consider new sufficient conditions as to cover the Bogin’s fixed point theorem for contractively widely more generalized hybrid mappings to have a fixed point.

In this paper, we are interested in obtaining fixed point theorems by keeping the orthogonal completeness of the orthogonal metric space and replacing the $\perp$-contraction condition in theorems by another slightly modified conditions. The paper contains an example illustrating our results.

The purpose of this paper is to present some sufficient conditions for the existence of solutions for nonlinear matrix equations and the second-order two-point boundary value problems for nonlinear ordinary differential equations via fixed point theorem. To do this, we introduce two classes of mappings called the Suzuki–Geraghty-$(\alpha ,\beta )$-$F$-contraction mappings and generalize Suzuki-type $(\alpha ,\beta )$-F-contraction mappings and establish some fixed point results for these classes of mappings. The results obtained in this work complements, extends, generalizes, and improves some known results in the literature.

In this paper, we consider $(3,2)$-metric spaces and prove several fixed point theorems for self-mappings on $(3,2)$-metric spaces satisfying some conditions similar to contractions in ordinary metric spaces.

The Steiner tree problem is a well known network optimization problem which asks for a connected minimum network (called a Steiner minimum tree) spanning a given point set N. In the original Steiner tree problem the given points lie in the Euclidean plane or space, and the problem has many variants in different applications now. Recently a new type of Steiner minimum tree, probability Steiner minimum tree, is introduced by the authors in the study of phylogenies. A Steiner tree is a probability Steiner tree if all points in the tree are probability vectors in a vector space. The points in a Steiner minimum tree (or a probability Steiner tree) that are not in the given point set are called Steiner points (or probability Steiner points respectively). In this paper we investigate the properties of Steiner points and probability Steiner points, and derive the formulae for computing Steiner points and probability Steiner points in ℓ₁- and ℓ₂-metric spaces. Moreover, we show by an example that the length of a probability Steiner tree on 3 points and the probability Steiner point in the tree are smooth functions with respect to p in d-space.

Let T be a nonempty subset of ℝ, X a metric space with metric d and X^T the set of all functions mapping T into X. Given ε > 0 and f ∈ X^T, we denote by N(ε, f, T) the least upper bound of those n ∈ ∕, for which there exist numbers s₁, …, s_n, t₁, …, t_n from T such that s₁ < t₁ ≤ s₂ < t₂ ≤ … < s_n < t_n and d(f(s_i), f(t_i)) > ε for all i = 1,…, n (N(ε, f, T) = 0 if there are no such n's). The following pointwise selection principle is proved: If a sequence of functions is such that the closure in X of the sequence is compact for each t ∈ T and lim sup_j→∞N(ε, f_j, T) < ∞ for all ε > 0, then contains a subsequence, converging pointwise on T to a function f ∈ X^T, such that N(ε, f, T) < ∞ for all ε > 0. We establish several variants of this result for functions with values in a metric semigroup and reflexive separable Banach space as well as for the weak pointwise and almost everywhere convergence of extracted subsequences, and comment on the necessity of conditions in the selection principles. We show that many Helly-type pointwise selection principles are consequences of our results, which can be applied to sequences of non-regulated functions, and compare them with recent results by Chistyakov [J. Math. Anal. Appl. 310 (2005) 609–625] and Chistyakov and Maniscalco [J. Math. Anal. Appl. 341 (2008) 613–625].