Approximate fixed points in metric spaces

In this article the concept of coupled best proximity point and cyclic contraction pair are introduced and then we study the existence and convergence of these points in metric spaces. We also establish new results on the existence and convergence in a uniformly convex Banach spaces. Furthermore, we give new results of coupled fixed points in metric spaces and give some illustrative examples. An open problems are also given at the end for further investigation.

The notion of complex valued metric spaces proved the common fixed point theorem that satisfies rational mapping of contraction. In the contraction mapping theory, several researchers demonstrated many fixed-point theorems, common fixed-point theorems and coupled fixed-point theorems by using complex valued metric spaces. The idea of b-metric spaces proved the fixed point theorem by the principle of contraction mapping. The notion of complex valued b-metric spaces, and this metric space was the generalization of complex valued metric spaces. They explained the fixed point theorem by using the rational contraction. In the metric spaces, we refer to this metric space as a quasi-metric space, the symmetric condition d(x, y) = d(y, x) is ignored. Metric space is a special kind of space that is quasi-metric. The Quasi metric spaces were discussed by many researchers. Banach introduced the theory of contraction mapping and proved the theorem of fixed points in metric spaces. We are now introducing the new notion of complex quasi b-metric spaces involving rational type contraction which proved the unique fixed point theorems with continuous as well as non-continuous functions. Illustrate this with example.

We investigate the connections between UC and UC$^*$ properties for ordered pairs of subsets $(A,B)$ in metric spaces, which are involved in the study of existence and uniqueness of best proximity points. We show that the UC property and the UC$^*$ property lead to one and the same corollaries, when iterated sequences, generated by cyclic maps, are investigated. We introduce some new notions: bounded UC (BUC) property and uniformly convex set about a function $\phi$. We prove that these new notions are generalizations of the UC property and that both of them are sufficient to ensure existence and uniqueness of best proximity points. We show that these two new notions are different from a uniform convexity and even from a strict convexity. If we consider the underlying space to be a Banach space, we find a sufficient condition which ensures that from the UC property follows the uniform convexity of the underlying Banach space. We illustrate the new notions with examples. We present an example of a cyclic contraction $T$ in a space, which is not even strictly convex and the ordered pair $(A,B)$ does not have the UC property, but has the BUC property and thus there is a unique best proximity point of $T$ in $A$.

Strong fairness and ultra metrics

Approximation algorithms for clustering points in metric spaces is a flourishing area of research, with much research effort spent on getting a better understanding of the approximation guarantees possible for many objective functions such as k-median, k-means, and min-sum clustering.This quest for better approximation algorithms is further fueled by the implicit hope that these better approximations also yield more accurate clusterings. E.g., for many problems such as clustering proteins by function, or clustering images by subject, there is some unknown correct clustering and the implicit hope is that approximately optimizing these objective functions will in fact produce a clustering that is close pointwise to the truth.In this paper, we show that if we make this implicit assumption explicit---that is, if we assume that any c-approximation to the given clustering objective φ is e-close to the target---then we can produce clusterings that are O(e)-close to the target, even for values c for which obtaining a c-approximation is NP-hard. In particular, for k-median and k-means objectives, we show that we can achieve this guarantee for any constant c > 1, and for the min-sum objective we can do this for any constant c > 2.Our results also highlight a surprising conceptual difference between assuming that the optimal solution to, say, the k-median objective is e-close to the target, and assuming that any approximately optimal solution is e-close to the target, even for approximation factor say c = 1.01. In the former case, the problem of finding a solution that is O(e)-close to the target remains computationally hard, and yet for the latter we have an efficient algorithm.

Hyperspherical neighbourhoods and pattern recognition using neural networks

In this paper, we modify the definition of some generalized proximal contractions and enumerate a list of equivalent conditions for various versions of generalized proximal contractions of non-self set-valued mappings on (ordered) metric spaces.By using the fixed point means, we establish the existence of best proximity points for mappings involving such contractions which extend and improve many existing related results, as well as, reveal that most of existing best proximity point theorems on metric spaces are in fact equivalent and immediate consequences of well-known fixed point theorems.Finally, some examples are given to support our results.

Cut points in metric spaces

The k nearest neighbors (kNN) is an algorithm for finding the closest k points in metric spaces. Due to its high computational costs, many parallel solutions have been proposed, including some implementations targeted at modern accelerators. However, most approaches assume relatively low dimensionality and dense data. Such conditions do not apply to textual datasets, which are known for their high dimensionality and sparsity. This work presents a fine-grained parallel algorithm that applies filtering technique based on most common important terms of the query document using an inverted index and its implementation on GPU. Our method improves the top k nearest neighbors search in textual datasets by up to 37x with a single GPU.

Approximation algorithms for clustering points in metric spaces is a flourishing area of research, with much research effort spent on getting a better understanding of the approximation guarantees possible for many objective functions such as k-median, k-means, and min-sum clustering. This quest for better approximation algorithms is further fueled by the implicit hope that these better approximations also yield more accurate clusterings. E.g., for many problems such as clustering proteins by function, or clustering images by subject, there is some unknown correct “target” clustering and the implicit hope is that approximately optimizing these objective functions will in fact produce a clustering that is close pointwise to the truth. In this paper, we show that if we make this implicit assumption explicit—that is, if we assume that any c-approximation to the given clustering objective Φ is ∊-close to the target—then we can produce clusterings that are O(∊)-close to the target, even for values c for which obtaining a c-approximation is NP-hard. In particular, for k-median and k-means objectives, we show that we can achieve this guarantee for any constant c > 1, and for the min-sum objective we can do this for any constant c > 2. Our results also highlight a surprising conceptual difference between assuming that the optimal solution to, say, the k-median objective is ∊-close to the target, and assuming that any approximately optimal solution is ∊-close to the target, even for approximation factor say c = 1.01. In the former case, the problem of finding a solution that is O(∊)-close to the target remains computationally hard, and yet for the latter we have an efficient algorithm.

Approximation of point of coincidence and common fixed points of quasi-contraction mappings using the Jungck iteration scheme

Two convergence principles with applications to fixed points in metric spaces

Nonparametric clustering algorithms, including mode-seeking, valley-seeking, and unimodal set algorithms, are capable of identifying generally shaped clusters of points in metric spaces. Most mode and valley-seeking algorithms, however, are iterative and the clusters obtained are dependent on the starting classification and the assumed number of clusters. In this paper, we present a noniterative, graph-theoretic approach to nonparametric cluster analysis. The resulting algorithm is governed by a single-scalar parameter, requires no starting classification, and is capable of determining the number of clusters. The resulting clusters are unimodal sets.

This article introduces a novel class of Reich-type contractions that meet the graph preservation criteria in the context of complete fuzzy metric spaces. The provided contraction condition is satisfied through various forms of contractive mappings via directed graphs in the literature. Our key result is the natural extension of fuzzy metric spaces to fuzzy metrics enriched with a graph, which adds the understanding of fixed points in metric spaces within the realm of graph structure. The findings are further supported by examples and applications.

A sequence (z 0,z 1,z 2,, ...,z n, z n+1) of points fromp=z 0 toq=z n+1 in a metric spaceX is said to besequentially equidistant ifd(z i−1,z i)=d(z i,z i+1) for 1≦i≦n. If there is path inX fromp toq (or if a certain weaker condition holds), then such a sequence exists, with all points distinct, for every choice ofn, while ifX is compact and connected, then such a sequence exists at least forn=2. An example is given of a dense connected subspaceS ofR m ,m≧2, and an uncountable dense subsetE disjoint fromS for which there is no sequentially equidistant sequence of distinct points (n ≧ 2) inS ∪E between any two points ofE. Techniques of dimension theory are utilized in the construction of these examples, as well as in the proofs of some of the positive results.