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Graphical Metric Spaces and Fixed Point Theorems
Published in Dhananjay Gopal, Praveen Agarwal, Poom Kumam, Metric Structures and Fixed Point Theory, 2021
In this chapter, an approach to fixed point theory via graphical metric is discussed. In view of the fact that the class of contractions in graphical metric spaces is larger than that in metric spaces, we see that the fixed point results in graphical metric spaces can be applied to a larger class of existence problems which use the fixed point methods, e.g., existence of a solution of an integral equation. As well as, apart from the topology of usual metric spaces, the topology generated by graphical metric spaces is non-Hausdorff which can be considered a useful tool in theories where non-Hausdorffness occurs, e.g., in algebraic geometry, in representation theory, in domain theory, in computer applications etc.
Final Thoughts
Published in Pascal Hitzler, Anthony Seda, Mathematical Aspects of Logic Programming Semantics, 2016
Domain theory,2 based on order continuity of semantic operators, is the dominant theory underlying the denotational semantics of programming languages. However, an alternative tradition in the semantics of programming languages to that using domains is an approach based on the use of metric spaces, as already mentioned in Chapter 4.3 A reconciliation of these two approaches is of obvious interest for the theory of programming semantics, and a considerable body of work has been done on this very topic,4 resulting in the area of quantitative domain theory. Indeed, the Rutten-Smyth theorem arose out of precisely these considerations.
A Neuro-Symbolic Hybrid Intelligent Architecture with Applications
Published in Lakhmi Jain, Anna Maria Fanelli, Recent Advances in Artificial Neural Networks, 2000
Before training the initial connectionist architecture, a fuzzy subsystem incorporating a coarse coding scheme is used to discretize the input parameters into multi-interval inputs with initial mean and variance for each interval. During the training phase of the connectionist architecture, an Augmented Backpropagation Algorithm (ABA) with momentum term is used to refine the discretization parameters and thus enhance the domain parameters. Therefore, the connectionist architecture can improve the efficiency of the domain theory and incorporate it in its topology.
An induction theorem and Ekeland's variational principle in partial metric spaces with applications
Published in Optimization, 2020
Metric spaces have been proved to be crucial underlying spaces for wide ranges of important studies in continuous mathematics in general. For variational analysis, optimization and related areas, regularity properties, various types of continuity, approximations, stability/sensitivity, well-posedness, fixed points, etc., have been extensively investigated in metric spaces, because such underlying spaces are suitable for developments of deeper and more concrete results than a general topological setting. Moreover, during the last several decades a number of notions of generalized metric spaces have been proposed as underlying spaces to develop the aforementioned topics to meet the variety of practical problems. The reader is referred to recent contributions, e.g. [1] for a survey of generalized metric spaces with a focus on fixed-point results, [2–7] for results related to the Ekeland variational principle (EVP) on some (generalized) metric settings with applications. The concept of a partial metric space (PMS) and its first basic properties have their roots in [8]. Unlike in a metric space, the self-distance of a point in a partial metric space is not necessarily zero. Hence, a PMS has been shown to have important applications, e.g. in areas of computer science like domain theory, semantics, and programming language (see [9–16]). There have been also applications of PMSs in different areas of mathematics such as integral equations [17], fixed-point theorems for single-valued maps and applications [18,19], fixed-point theorems for set-valued maps [17,18,20,21], and Ekeland's variational principle [22,23].