مولوی عبدالباری
افسوس ہے کہ دارالمصفین کے قدیم اور مخلص خدمت گذار مولوی عبدالباری صاحب ۳۰؍ جون کو وفات پاگئے، ان کی عمر ۹۰ سال سے متجاوز تھی، دارالمصنفین کے ابتدائی دور میں حضرت مولانا سید سلیمان ندویؒ نے انہیں تصحیح اور کتب خانہ کی نگرانی کے کام پر مامور کیا تھا جس کو دو برس پہلے تک وہ انجام دیتے رہے، دارالمصنفین کے عروج کا دور دیکھنے والے اب تنہا وہی رہ گئے تھے، ان کی تعلیم مدرسۃ الاصلاح سرائمیر میں ہوئی تھی اور وہ مولانا امین اصلاحی مدظلہ، کے ہم سبق تھے، دارالمصنفین سے وابستگی کی وجہ سے انہیں مضمون نگاری کا چسکہ لگ گیا تھا، ابو علی اثری اور ابو علی اعظمی کے نام سے مدۃالعمر اخباروں اور رسالوں میں مضامین لکھتے رہے، علامہ شبلیؒ کے بڑے مداح اور سیدصاحب کے نہایت عقیدت مند تھے، ان کا ذکر برابر لطف ولذت سے کرتے تھے ان پر اور مولانا ابو الکلام آزاد پر بے شمار مضامین لکھے، دونوں بزرگوں پر ان کے مضامین کے ایک ایک مجموعے ضیاء اﷲ کھوکھر صاحب (گوجرانوالہ، پاکستان) نے شایع کیا تھا، اپنی خودداری کی وجہ سے کسی کا منت کش ہونا گوارا نہیں کیا اور قناعت پسندی کی بنا پر ایک قلیل مشاہرہ پر پوری زندگی گذار دی، اﷲ تعالیٰ ان کی بشری لغزشوں کو معاف فرمائے اور جنت نعیم میں جگہ دے، آمین۔ (ضیاء الدین اصلاحی، جولائی ۱۹۹۳ء)
Hazrat Abdul Rehman (may Allah be pleased with him) belonged to Arab tribe of Quraish and was a close relative of Mohammad (peace be upon him). At the time of conquest of Makkah He (may Allah be pleased with him) entered the circle of Islam. He (may Allah be pleased with him) is counted among the companions of Muhammad (may Allah be pleased with him) who came to sub-continent specially Balochistan in order to preach for Islam and Jihad during the Khilafat of orthodox caliphs. He (may Allah be pleased with him) came to Balochistan twice for Jihad and conquests first during the Khilafat of Hazrat Usman (may Allah be pleased with him) and second time in the early era of Hazrat Muawia (may Allah be pleased with him). He (may Allah be pleased with him) played a vital role in the wars of Balochistan. He (may Allah be pleased with him) established Zehri his abode and capital after conquering Kalat, Khuazdar (Sajistan), Kachi, Gandhava, and Chaghi, and from here he expanded the series of his conquests till Kabul and Qandar. Besides this, he included many areas of sub-continent in the Islamic empire of conquered areas. His (may Allah be pleased with him) life is consists of great chapters of sincerity in deeds. Wisdom and valor, determination fearlessness, strife, hospitality, simplicity and patience. He (may Allah be pleased with him) is counted among the great generals of Islam had the honour to have carried the message of Holy faith in every corner of Balochistan in tough and unfavorable conditions and planted the flag of Islam in Balochistan forever.
The modular exponentiation is considered to be one of the renowned problems in number theory and is of paramount importance in the field of cryptography. Now a days many security systems are based on powerful cryptographic algorithms. Most of them are designed by using the exponentiation x k ≡ y (mod n) as in RSA, Diffie- Hellman key exchange, Pseudo-random number generators etc. For the last two decades, this problem is being studied by associating the power digraphs with modular exponentiation. For the fixed values of n and k, a power digraph G(n, k) is formed by taking Z n as the set of vertices and the directed edges (x, y) from x to y if x k ≡ y (mod n) for the vertices x and y. These digraphs make a novel connection between three disciplines of discrete mathematics namely number theory, graph theory and cryptography. The objective of this dissertation is to generalize the results on symmetry, heights, isolated fixed points, the number of components of a power digraph and the primality of Fermat numbers. To obtain the desired goal, a power digraph is decomposed into the direct product of smaller power digraphs by using the Chinese Remainder Theorem. The method of elimination is adopted to discard those values of n and k which do not provide desired results. During the entire course of research, the Carmichael lambda-function λ(n) is used for developing the relations between the properties of a power digraph and the parameters n, k. For any prime divisor p of n, the concept of equivalence classes has been used to discuss the symmetry of order p of G(n, k). The general rules to determine the heights are formulated by comparing the prime factorizations of k, λ(n) and the orders of vertices. Some necessary and sufficient conditions for the existence of symmetric power digraphs G(n, k), where n = p α q 1 q 2 · · · q m such that p, q i are distinct primes and α > 1, of order p are established. Explicit formulae for the determination of the heights of the vertices and components of a power digraph in terms of n, k, λ(n) and the orders of vertices are formulated. An expression for the number of vertices at a specific height is established. The power digraphs in which each vertex of indegree 0 of a certain subdigraph is at height q ≥ 1 are characterized. The necessary and sufficient conditions on n and k for a digraph to have at least one isolated fixed point are obtained. The work ends with the complete classification of the power digraphs with exactly two components.