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یہ عشق میں نہ سوچ تو کہ کیا نہیں ملا

یہ عشق میں نہ سوچ، تُجھے کیا نہیں ملا
ہے کر لیا، تو خاک میں اپنی جبیں ملا

ہم راہ دیکھتے ہی رہے جس کی عمر بھر
آیا وہ شہر میں بھی تو ہم سے نہیں ملا

بچپن میں دل کی بستی میں رہتے تھے کتنے لوگ
دیکھا شباب میں تو فقط اک مکیں ملا

اگلے جہاں کے عہد پہ ہم کو دیا ہے ٹال
کم بخت ہم کو وہ تو بلا کا ذہیں ملا

کہتے رہے تھے یار جسے ہم تمام عمر
اک دن عدو کی بزم میں وہ نازنیں ملا

گر یاں دیا نہ تُو نے تو نہ لوں گا حشر میں
یارب اسے اگر ہے ملانا، یہیں ملا

بر صغیر میں علم حدیث کی نشر و اشاعت اور محدثین کرام کی خدمات کا علمی جائزہ Analytical study of the propagation of Ḥadīth-learning and the services of Muḥaddithīn in the Subcontinent

Ḥadīth learning has a very high status in Islamic education after the Exegesis of the Holy Qurān (Tafsīr). Oddly enough, Non-Arab (‘Ajam) has a more important and key role than Arabs in the service of this knowledge. In this regard, the subcontinent is also being remained in the forefront. It started when Muhammad bin Qasim strengthened the Arab government in Sindh. Sheikh Abdul Haq Muhaddith Dehalvi is one of the prominent names in the teaching and writing of ḥadīth which travels gradually and reaches the family of Shah Waliullah, which is undoubtedly the pinnacle of ḥadīth knowledge in the subcontinent. The types of hadith that have been worked on, in connection with the service of ḥadīth, include the text of ḥadīth, commentaries, as well as summaries of ḥadīth and others. The entire Islamic world has benefited from all these teaching and writing services. The subcontinent, especially Pakistan, is gaining a central position in the Islamic world, which is the result of the propagation of Islam and understanding of the Qurān and Ḥadīth. In this regard, it is imperative to re-evaluate the services of the historians of this land. The said article will not only provide an introductory review of the Muḥaddith in the subcontinent and their authorship as well as a compilation but will also highlight the broadcasting and writing services of the hadith learning.

Combinatorial Generation Algorithms for Subgraphs

The field of computer science has greatly benefitted from the advances in combina- torial algorithms in the last few decades. This is because the advent of high speed computers has made it possible to generate lists of combinatorial objects in a practi- cal amount of time. In areas such as genome science and data mining the problems are often vaguely defined, and researchers have to look for meaningful information in huge datasets. In this thesis, efficient generation algorithms for subgraphs such as bicliques and paths in cliques are developed. Cliques and bicliques are used to model various real-world problems encountered in bio-informatics, data mining and networks. We consider two variations of bicliques: pseudo-bicliques and c-isolated bicliques. Pseudo- bicliques relax the rigid connectivity requirement of bicliques to cater for missing and noisy data. On the other hand, the c-isolated bicliques enforce a restriction on the external connectivity of the vertices in a biclique to model cohesive communities. This thesis presents an algorithm based on reverse search to list all pseudo-bicliques in a graph G. The algorithm takes linear time on average to generate each pseudo- biclique. On the other hand, our generation algorithm for c-isolated bicliques exploit underlying properties of an isolated biclique to trim the input graph. Furthermore, the algorithm deploys the vertex cover enumeration algorithm based on fixed point tractability (FTP) and lists all isolated bicliques in linear time, in the case where c is constant. The performance of the proposed algorithms is evaluated on random graphs and real-world problems. The results are quite promising and confirm our theoretical findings. In this research work, we also explore another combinatorial object called a clique. A constant amortized time algorithm is proposed to generate all spanning paths and all paths in a clique in minimal change order (an ordering in which successive elements differ in a small way)
Asian Research Index Whatsapp Chanel
Asian Research Index Whatsapp Chanel

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