غریبوں کی معاشی کفالت میں صحابیات رضی اﷲ عنھن کا کردار

Islamic cardinal precepts include seeking benefaction and patronage of other Muslims. Generally, Islamic history is replete with such munificent and generous personages as were ever geared up to serving humanity, yet the golden era of Islamic history excelled in profusion of such noble persons. When we review the aspect of serving mankind in the life history of the reverend Sahaba (the holy prophet’s disciples), we come to know that they had highly remarkable passions for helping, beneficing, supporting and sustaining their relatives, friends as well as the common run. The level of their charity was so great that they felt felicity and prostrated to Allah in gratitude even after giving away their all possessions and belongings for Allah’s sake only. In fact, the motivation behind this generosity was the attainment of pleasing Allah and the obedience and submission to the holy prophet. They had no iota of worldly voracity or avarice. Therefore, they were ever elated over giving away their assets.

On Generation of Non-Equivalent Combinatorial Objects

The rapid growth in fields of computational biology, data mining and combinatorial chemistry results in an increased demand of combinatorial algorithms which produce exhaustive lists of combinatorial objects especially those objects which are symmetric under some equivalence relation. In this thesis we develop efficient algorithms for generation of bracelets with fixed density, bracelets with fixed content, and non- isomorphic unicyclic graphs. All of the above mentioned algorithms list only one representative object from each equivalence class. A bracelet is said to be of fixed density, if number of occurrences of the symbol 0 is fixed. The algorithm for generation of bracelets with fixed density generates a complete set of bracelets with fixed density of arbitrary base in lexicographic order. A simple mapping technique is used to prove that our algorithm works in constant amortized time. Bracelets with fixed content are those in which number of occurrences of each symbol is fixed. We devise an efficient algorithm to list bracelets with fixed content in reverse lexicographic order. Again, by using an injective mapping we prove that the algorithm works in constant amortized time with the condition that number of occurrences of the largest symbol is maximum. Our analysis also gives a simpler alternate proof for the original bracelet algorithm presented in “Generating bracelets in constant amortized time” by J. Sawada. Moreover, an enumeration formula for bracelets with fixed content is obtained. Listing of non-isomorphic graphs is known to be computationally very hard. However, when we restrict the graphs to have exactly one cycle, the problem can be efficiently solved in constant amortized time. In this thesis, we give a CAT algorithm to list all unlabeled non-isomorphic unicyclic graphs.