عظیم شخصیات کے اوصاف وخصائص: سیرت طیبہ کی روشنی میں

Great lineage, good manners, trustworthiness, sincerity, courage and bravery, obedience, generosity, modesty, humility, thankfulness, absolute trust in Allah, education and upbringing, affection and compassion towards the elderly, the youth, slaves and animals, the regard for truth, the endurance and patience etc. Are such features that at them neither was the match for the Holy Master (saw) born, nor will any such person be ever born. In addition, let alone friends, his enemies were helpless to hold him as-Sadiq (the Truthful) and al-Ameen (the Trustworthy) due to these and many other great attributes; and they would say: Muhammad is not a liar at all, but we can’t believe in him as a prophet sent by Allah. Moreover, even a non-Muslim or polytheist has never criticized his character to this day, nor will anybody do till the judgment day. Whenever the greatest personalities of the world were listed or somebody committed to writing on the said topic, the Holy Prophet (saw) remained at the top of the list. Drawing of the swords of father and son against each other, the change of the standard of friendship and enmity, fighting of master and his slave against each other, chanting the slogan of Ahad-Ahad (the only One, the only One) on the blazing sand, giving priority to martyrdom over the utterance of a word of unbelief, taking a husband away from his wife and separating a mother from her infant, deforming the condition after a blasphemous blow on the blessed face, forced migration to Habsha (Abyssinia) at times, and laying siege to Shb-e-Abi Talib (the valley of Abi Talib) at times.

Some Applications of Graph Transformations in Computational Algorithms and Group Automata

In my thesis some of the techniques of graph transformation and its applications are introduced. Some of the basic graph transformations are edge deletion, edge contraction, vertex splitting, inner dual of a planar graph and vertex deletion. Inner dual of a graph does not maintain the orientation of the edges in the original graph. To keep the orientation, various methods and techniques are used. Here the technique called He-Matrix is used. This is designed for any hexagonal system. Here the orientation of the edges is represented by using edge weights. These weights can be 1, 2 or 3 for any edge depending upon its orientation. When a hexagonal system is rotated through angles which are multiple of 60 degrees a new graph is obtained. Considering the edge weights, the inner dual of these graphs may be different. This implies that the different inner duals can be compared on different basis. In this thesis two different problems are solved related to these inner duals. The first problem is to find the orientation where the minimum spanning tree is the smallest. Different theorems and algorithms related to this topic are given. Also two linear time algorithms are presented. The first one finds the required orientation without computing the minimum spanning tree in any of the direction while the second solves the minimum spanning tree problem in linear time, in any of the given orientation. The second problem related to the Inner dualist is finding the orientation that gives the smallest shortest path between any two given points. Again some theorems are proved and a linear time algorithm is discussed that determines the orientation and finds the required shortest path is presented. The next problem is an application of graph transformation and network flows. Here a postman problem is discussed. This postman is working in an office with some constraints. These constraints are analogous to the working of a router, so the results from this part can be applied to finding faulty routers. The information available is the total number of letters sent and received by each person. The objective is to find some bounds on the total number of letters that the postman has lost. This number can be used to measure the performance of our postman. This can also be used to compare the performance of different routers, and also indicate the routers that are faulty. Two different linear time algorithms are given that solve this problem correctly. Moreover, a few theoretical results are also given. The concept of graph transformation in the area of teaching methodologies is also applied. Group theory is among the most difficult courses in undergraduate mathematics. The difficulty in learning arises from the fact that this course is abstract in nature and no pictorial representation or visualization is used in teaching this subject. Groups can be represented in various ways, and one of the representations is group automata. This is very similar to finite automata. Here a group is first transformed into an automaton, which is also abstract in nature. Representation for automata is available in the form of state diagrams and they can be viewed as a labeled graph. Here, the group axioms can be studied easily, and identity element and inverse of any element can be visualized. Also the closure property and the associative law can be understood with the help of such graphs. After the axioms, some of the fundamental theorems in group theory are proved in this model, and it is shown how group transformations can be used to present a proof using graphs, of otherwise abstract concepts and theorems. One such example is analogous to the technique where an equivalent minimum state deterministic finite automata is found for any given deterministic finite automata.