موجودہ وبائی ماحول میں عسل کی طبی افادیت جدیدسائنس اور اسلامی تعلیمات کی روشنی میں

 Honey is an unparalleled treasure of medicinal properties. Honey has the ability to strengthen the human body's immune system. Therefore, the use of honey and other natural foods is extremely beneficial and effective in current epidemic environment. Honey is one of the most appreciated and valued natural blessing of Allah for human being. Medicinal importance of honey hase been described in Holy Quran. Holy prophet (SAW) has advised to use honey because honey provides body instant energy. There are several varieties of honey are found in different areas of the world. Honey has been taken for research because of its nutritional popularity and medicinal quality. Honey is high in important nutrients of benefits and uses. Honey contains carbohydrates, vitamins, minerals and a significant amount of fiber. Nutrients of honey can improve digestive system by preventing constipation. Immunity system can be improved by using honey in epidemic environment. Honey is a source of fructose, which does not harm patients of diabetes. The fructose makes a better substitute for sugar because of the fiber, nutrients and antioxidants. Honey is comprised of several minerals including phosphorus, iron, potassium, calcium and magnesium. Honey has very small amount of protein and fats. This research paper analyzes ingredients and minerals of honey in Islamic perspective. The research work highlights medical benefits of honey in the light of modern science and Islamic teachings in the current epidemic environment.

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.