In this paper, we prove a common fixed point theorem for a quadruple of mappings by using an implicit relation [6] and property (E.A) [1] under weak compatibility. Our theorem improves and generalizes the main Theorems of Popa [6] and Aamri and Moutawakil [1] .Various examples verify the importance of weak compatibility and property (E.A) in the existence of common fixed point and examples are also given to the implicit relation and to validate our main Theorem. We also show that property (E.A) and Meir-Keeler type contractive condition are independent to each other. .
Cell-free DNA(cfDNA) methylation profiling is considered promising and potentially reliable for liquid biopsy to study progress of diseases and develop reliable and consistent diagnostic and prognostic biomarkers. There are several different mechanisms responsible for the release of cfDNA in blood plasma, and henceforth it can provide information regarding dynamic changes in the human body. Due to the fragmented nature, low concentration of cfDNA, and high background noise, there are several challenges in its analysis for regular use in diagnosis of cancer. Such challenges in the analysis of the methylation profile of cfDNA are further aggravated due to heterogeneity, biomarker sensitivity, platform biases, and batch effects. This review delineates the origin of cfDNA methylation, its profiling, and associated computational problems in analysis for diagnosis. Here we also contemplate upon the multi-marker approach to handle the scenario of cancer heterogeneity and explore the utility of markers for 5hmC based cfDNA methylation pattern. Further, we provide a critical overview of deconvolution and machine learning methods for cfDNA methylation analysis. Our review of current methods reveals the potential for further improvement in analysis strategies for detecting early cancer using cfDNA methylation.
Abstract. Various common fixed point theorems have been proved for one or two pairs of mappings using either (CLR) property ([44]), or by taking one of the range-subspace closed. In this paper, we introduce the notion of (CLCS)-property i.e., "common limit converging in the range sub-space". Using this property, we prove common fixed point theorems for two pairs of weakly compatible mappings in complex valued b-metric spaces satisfying a collection of contractive conditions. Our notion is meaningful and valid because the required common fixed point will always lie on the range-subspace of the mapping-pair. We give some examples to show that if a mapping pair (f, g) on a closed complex valued b-metric space X satisfy the (CLR f ) property, then it is also (CLRg), and vice-versa.
Varma [1,2] entropy has attracted attention for a new class of non-linear integer programming problems that arise during the course of discussion. Our focus in this communication is to explore the techniques of dynamic programming. This process requires splitting any optimization event into a finite number of subcomponents for any occurrence of a finite generalized problem. The capacity plan should be partitioned in such a way that the expression can be optimized.
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