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نظم و ضبط

نظم وضبط
تنظیم بڑی چیز ہے اے اہلِ گلستاں
بکھرے ہوئے تنکوں کو نشیمن نہیں کہتے
کائنات جن اٹل اصولوں پر چل رہی ہے ان میں زیادہ اہم قانون نظم و ضبط کا ہے۔ کائنات کی لامتناہی اور بے کراں وسعتوں میں کوئی معمولی سی شے یا کوئی حقیر ساذرّہ بھی اس اصول سے مستثنیٰ نہیں ، نظام ِقدرت کے تحت مختلف مخلوقات کو جو ذمہ داریاں تفویض کی گئی ہیں وہ انہیں بطریقہ احسن انجام دے رہی ہیں۔ نظامِ شمسی کے اندر مختلف ستاروں اور سیاروں کی حرکت اور گردش کا جو قانون خالقِ کائنات نے مقرر کیا ہے۔ اس پر صحیح طریقے سے عملدرآمد ہو رہا ہے۔ زمین اپنے مقررہ ضابطے کے تحت محو گردش ہے۔ اسی طرح فضاؤں ، بادلوں کا سفر، در یائوں اور نالوں کی روانی، طغیانی، پہاڑوں کی آتش فشانی اور سمندر کے مدّو جزر وغیرہ کے جواصول اس حکیم مطلق نے مقرر کیے ہیں یہ سب چیزیں انہی قوانین کے مطابق کام کر رہی ہیں۔
قدرت کا تکوینی نظام (جونظم و ضبط کی بنیاد پر قائم ہے) انسان کی نظم و ضبط کی اہمیت کو سمجھنے اور اس پر عمل پیرا ہونے کی دعوت دے رہا ہے۔ کائنات کی مختلف اشیاء ہمیں یہ سکھاتی ہیں کہ قواعد وضوابط کی خلاف ورزی بہر حال نقصان دہ ہے۔ پانی کی مثال ہی لیجئے اگر پانی کی روانی کے لیے شہر کی حدود مقرر نہ کر دی جائیں تو وہ کناروں کو توڑ کر اِدھر اُدھر پھیل کر تباہی مچادیتا ہے۔ لہذا اسے نظم وضبط کے حصار میں رکھنا انتہائی ضروری ہے اگر ہم غور کریں تو یہ بات سمجھ لینا کچھ مشکل نہیں کہ نظم و ضبط اپنانے ہی سے سکھ اور سکون حاصل ہوتا ہے۔
دہر میں عیش دوام آئیں کی پابندی سے ہے
موج کی آزادیاں سامانِ...

Jugni, Dhola and Mahiya: Comparing

Among the amazing variety of forms of poetic expression by the folk of the Punjab region, this essay has selected three genres: mahiya, dhola and jugni. The study is meant to compare these three genres of Punjabi folklore, in their evolution, structure, expression and themes. The study finds that the three genres are very old in time origin and tracing their exact origins in history is impossible, only few hints are available. Their structures are variable, as mahiya has a fixed structure, dhola has rather loose structure giving more freedom to the singer-poet, and jugni has a specific meter in certain lines, but it has freedom to repeat some lines for perfect expression of the melody. The structures in fact follow the tunes, distinct for each genre. Three genres have many themes common, but jugni has spirituality as dominant theme, dhola has expression of love as dominant them and mahiya has now become quite inclusive, but it originated as expression of love and it still retains that character in its core. The folk heart of Punjab has endeared these three genres so much that these are appreciated far and wide in original tunes, but new experiments of tunes and themes are also underway. Being a true mirror of simple unsophisticated villagers these folk songs would lose popularity if these villagers become sophisticated hence the need for their preservation is highlighted in this study.

Power Digraphs in Number Theory

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.
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