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The concept of contraction mappings plays a significant role in mathematics, particularly in the study of fixed points and the existence of solutions for various equations. In this study, we described two types of enriched contractions: enriched F-contraction and enriched F'-contraction associated with u-fold averaged mapping, which are involved with Kirk's iterative technique with order u. The contractions extracted from this paper generalized and unified many previously common super contractions. Furthermore, u-fold averaged mappings can be seen as a more general form of both averaged mappings and double averaged mappings. Moreover, we demonstrated that the ufold averaged mapping with enriched contractions has a unique fixed point. Our work examined the necessary conditions for the u-fold averaged mapping and weak enriched contractions to have equal sets of fixed points. Additionally, we illustrated that an appropriate Kirk's iterative algorithm can effectively approximate a fixed point of a u-fold averaged mapping as well as the two enriched contractions. Also, we delved into the well-posedness, limit shadowing property, and Ulam-Hyers stability of the u-fold averaged mapping. Furthermore, we established necessary conditions that guaranteed the periodic point property for each of the illustrated strengthened contractions. To underscore the generality of our findings, we presented several examples that aligned with comparable results found in the existing literature. [ABSTRACT FROM AUTHOR]

This paper discusses the formation of math grammar rules for LATEX math equations. These rules are used to generate Abstract Syntax Tree (AST) which extracts structural information from mathematical expressions given in LATEX format. Later AST is used to generate XML structure of mathematical expressions that make mathematical expressions machine-readable in heterogeneous environments. A rule-based algorithm is also proposed that converts LATEX math expressions into Content MathML (CMML), which produces semantic enrichment in web documents. The rules for writing LATEX math equations are formulated and implemented as LATEX Math Grammar (LMG), which are used for generating AST. Further, AST is converted into XML structure which is used to generate CMML encoding. Initially, the conversion algorithm is tested on 20 equations used in an NTCIR-12 math competition, then the algorithm is tested on NTCIR-12 Wikipedia-MathIR and ArXiv data sets. The results show that our algorithm is capable of converting LATEX complex equations into CMML extensively as compared to the existing ones as well as its time efficiency is better than contemporary systems. [ABSTRACT FROM AUTHOR]