یوں مہرباں ہوا ہے مہینہ رسولؐ کا
دل بن گیا ہے پھر سے مدینہ رسولؐ کا
خاکِ مدینہ پائی تو خوشبو خجل ہوئی
شامل ہے اِس میں پاک پسینہ رسولؐ کا
فدیہ ہو علم کا کہ مواخات کا عمل
تاریخ دیکھتی تھی قرینہ رسولؐ کا
قربان ہم تو دونوں کی آب و ہوا پہ ہیں
مکّہ خدا کا اور مدینہ رسولؐ کا
طوفان میں گھرا ہوں پہ مدحت زباں پہ ہے
یہ نعت بھی ہے ایک سفینہ رسولؐ کا
The most prominent and living name among the African biographers is Qazi Ayaz Mālkī. His book “Al-Shifā Ba Ta‘rīf-e-Ḥuqūq-e-Muṣṭafā” has an important place in the books of Sīrah and also has the status of the most popular book among all kinds of people. This book is also called Kitab-e-Shifā (meaning the one who revives hearts and heals from heart diseases). The reason for writing this book was that the people were becoming oblivious to the obligatory status and virtues of the Holy Prophet ﷺ and were becoming completely incapable of paying their dues. In these circumstances, the demand for preparation and compilation of a collection of rights and manāqibs related to it became intense and he compiled this magnificent book. On the one hand, it teaches love and respect for the Prophet (peace and blessings of Allah be upon him) and on the other hand, it mentions his rights and rewards for fulfilling them. The writing style of the book is simple and smooth as well as eloquent. The book is not so long that it would be too long for the reader to comprehend, nor is it so short that it would not be possible to get access to the concepts and demands due to its brevity. The people have made this book their favorite and have been studying it and the biographers have adopted Al-Shifā as an authoritative and reliable source. Because of its importance and usefulness, an introductory and analytical study of this book will be presented in this article.
Let J G denote the binomial edge ideal of a connected undirected graph G on n vertices. This is the ideal generated by the binomials x i y j −x j y i , 1 ≤ i < j ≤ n, in the polynomial ring S = K[x 1 , . . . , x n , y 1 , . . . , y n ] where {i, j} is an edge of G. Our aim in this thesis is to compute certain algebraic invariants like dimension, depth, system of parameters, regular sequence, Hilbert series and multiplicity of J G of some particular classes of binomial edge ideals of graphs. A large amount of information of an ideal is carried by its minimal free resolution. So we give information on the minimal free resolution on certain binomial edge ideals. We also give a complete description of the structure of the modules of deficiencies of binomial edge ideals of some classes of graphs. A generalization of the concept of a Cohen-Macaulay ring was introduced by S. Goto [7] under the name approximately Cohen-Macaulay. In this thesis we collect a few graphs G such that the associated ring S/J G is approximately Cohen-Macaulay. We also characterize all the trees that are approximately Cohen-Macaulay. As more generalized notion than approximately Cohen-Macaulay we also study se- quentially Cohen-Macaulay property for binomial edge ideals. We give a nice con- struction principle in this topic. ̃ on n vertices has the property that S/J̃ is a Cohen-Macaulay The complete graph G G domain with a 1-linear resolution. As one of the main results we clarify the structure of S/J K m,n , where K m,n denotes the complete bipartite graph.