المذهب الحنفي, نشأته, أطواره, استقراره, كتبه وطبقاته

In this article different era of ╓anaf┘ School of thought is discussed in which it is mentioned that ╓anaf┘ School got the general acceptance among people after the initial three stages of its Origination which lays the foundation of its Evolution and Promotion. First era is known as the era of “ Zahir ul Riw┐yah” renaissance starts from Hazrat Imam Ab┴ ╓an┘fa and moves on up to his disciple Hassan L┴lo’ (204 A.H.). There is no book found under the authorship of Imam Abu Han┘fa but his disciple R┐shid Al Mohammad has written various books. Some famous renowned books under the authorship of Im┐m Muhammad are Al J┐m‘ Alkab┘r (الكبري اجلامع, (Al J┐m‘ Al Sagh┘r (الصغري اجلامع, (Al Siyar Al kab┘r(الكبري السري, (Al Siyar Al Saghir(الصغري السري, (Al Mabs┴t(املبسوط (and Al Ziy┐d┐t(الزايدات.(Second era is known as Mas┐’l-e Naw┐dir. Books like Al-Kais┐niy┐t(كيسانيات َال, (AlH┐r┴niy┐t(اهلارونيات, (Al-Jurj┐niy┐t(رجانيات ُ اجل (and AlRuqiy┐t(ياتَّق َّ الر (are written under the authorship of Imam Muhammad during this era are consists of new topics that are not mentioned in the written books of earlier period.

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.