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Implicit contractive mappings in modular metric and fuzzy metric spaces.

ABSTRACT: The notion of modular metric spaces being a natural generalization of classical modulars over linear spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, and Calderon-Lozanovskii spaces was recently introduced. In this paper we investigate the existence of fixed points of generalized α-admissible modular contractive mappings in modular metric spaces. As applications, we derive some new fixed point theorems in partially ordered modular metric spaces, Suzuki type fixed point theorems in modular metric spaces and new fixed point theorems for integral contractions. In last section, we develop an important relation between fuzzy metric and modular metric and deduce certain new fixed point results in triangular fuzzy metric spaces. Moreover, some examples are provided here to illustrate the usability of the obtained results.

SUBMITTER: Hussain N

PROVIDER: S-EPMC4070533 | biostudies-other | 2014

REPOSITORIES: biostudies-other

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Project description:The present paper explores nonnative (L2) phonological encoding of lexical entries and dissociates the difficulties associated with L2 phonological and phonolexical encoding by focusing on similarly sounding L2 words that are not differentiated by difficult phonological contrasts. We test two main claims of the fuzzy lexicon hypothesis: (1) L2 fuzzy phonolexical representations are not fully specified and lack details at both phonological and phonolexical levels of representation (Experiment 1); and (2) fuzzy phonolexical representations can lead to establishing incorrect form-to-meaning mappings (Experiment 2). The Russian-English Translation Judgment Task (Experiment 1, TJT) explores how the degree of phonolexical similarity between a word and its lexical competitor affects lexical access of Russian words. Words with smaller phonolexical distance (e.g., parent-parrot) show longer reaction times and lower accuracy compared to words with a larger phonolexical distance (e.g., parent-parchment) in lower-proficiency nonnative speakers, and, to a lesser degree, higher-proficiency speakers. This points to a lack of detail in nonnative phonolexical representations necessary for efficient lexical access. The Russian Pseudo-Semantic Priming task (Experiment 2, PSP) addresses the vulnerability of form-to-meaning mappings as a consequence of fuzzy phonolexical representations in L2. We primed the target with a word semantically related to its phonological competitor, or a potentially confusable word. The findings of Experiment 2 extend the results of Experiment 1 that, unlike native speakers, nonnative speakers do not properly encode phonolexical information. As a result, they are prone to access an incorrect lexical representation of a competitor word, as indicated by a slowdown in the judgments to confusable words. The study provides evidence that fuzzy phonolexical representations result in unfaithful form-to-meaning mappings, which lead to retrieval of incorrect semantic content. The results of the study are in line with existing research in support of less detailed L2 phonolexical representations, and extend the findings to show that the fuzziness of phonolexical representations can arise even when confusable words are not differentiated by difficult phonological contrasts.

Project description:BackgroundIn the post-genomic era, a central and overarching question in the analysis of protein-protein interaction networks continues to be whether biological characteristics and functions of proteins such as lethality, physiological malfunctions and malignancy are intimately linked to the topological role proteins play in the network as a mathematical structure. One of the key features that have implicitly been presumed is the existence of hubs, highly connected proteins considered to play a crucial role in biological networks. We explore the structure of protein interaction networks of a number of organisms as metric spaces and show that hubs are non randomly positioned and, from a distance point of view, centrally located.ResultsBy analysing how the human functional protein interaction network, the human signalling network, Saccharomyces cerevisiae, Arabidopsis thaliana and Escherichia coli protein-protein interaction networks from various databases are distributed as metric spaces, we found that proteins interact radially through a central node, high degree proteins coagulate in the centre of the network, and those far away from the centre have low degree. We further found that the distribution of proteins from the centre is in some hierarchy of importance and has biological significance.ConclusionsWe conclude that structurally, protein interaction networks are mathematical entities that share properties between organisms but not necessarily with other networks that follow power-law. We therefore conclude that (i) if there are hubs defined by degree, they are not distributed randomly; (ii) zones closest to the centre of the network are enriched for critically important proteins and are also functionally very specialised for specific 'house keeping' functions; (iii) proteins closest to the network centre are functionally less dispensable and may present good targets for therapy development; and (iv) network biology requires its own network theory modelled on actual biological evidence and that simply adopting theories from the social sciences may be misleading.

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Implicit contractive mappings in modular metric and fuzzy metric spaces.

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