سندھ کے عظیم روحانی پاگارہ خاندان کی تاریخ کا تحقیقی جائزہ

Allah Almighty, with his power, created the sky, earth and whole universe and provided all the necessities.  Therefore, the work of preaching Islam is conferred upon the Ummah of Holy Prophet (P.B.U.H). There are the people who had been taught the teachings of Islam directly by Holy Prophet and they preached Islam in the same way as they were taught. They practiced all the teachings in their lives. Sahaba R.A, for preaching Islam, went to the east, west, north, south, Hind, Sindh and the corners of the world. As a result of their efforts, almost half of the world got the message of Islam. Tabaeen consisted of many people, in the light of Quran and Sunnah, conveyed the message of Islam throughout the world. In the same way, another group of people named Tab Tabaeen became prepared to spread the universal message of Islam. They didn't ignore any sacrifice to uplift the spirit of Islam.  With their struggle, many great people like saints, jurist, reader and interpreter of Quran and Hadith born. These people tried their best to convey the message of Islam through written and spoken discourse. Religious books and places were introduced. That process of education, after many years, is still continuing. Presently, many religious places are constructed to preach spiritual guidance in which many great scholars and saints are engaged. History is witness that opponents went to extent possible level to aloof Muslims from teaching, preaching and guiding of lslam. Not only were this but the Muslims intended to end also. In these trials and tribulations, saints took responsibility on their shoulders and continued their struggle on the directed path of oneness, Prophet hood, Ahle Bait and Sahabas against the evil forces. Among such great Family of Pir Pagara is one whose line is continuing in Hind and Sindh. Insha Allah (If God wills) it'll continue till the last day. There is his great religious and social contribution. To convey, his valiant services towards people, research on this personality is dire need of the hour. In history, this research will be advantageous for the researchers to come.

Subalgebra Bases in Local Rings and its Efficient Computation in Polynomial Rings

In Chapter 1, there are given some necessary definitions and results about monomial orderings, Standard basis and Sagbi basis in polynomial ring over the field along with a description on the Gr ̈obner walk algorithm and Gr ̈obner basis under composition. In Chapter 2 we develop a theory of subalgebra basis analogous to Standard basis for ideals in polynomial rings over a field. We call this basis Sasbi Basis, standing for Subalgebra Analogue to Standard Basis for Ideals. Sasbi bases may be infinite. In this chapter we consider subalgebras admitting a finite Sasbi basis and give algorithms to compute them. Sasbi basis theory is given in my paper [22]. In Chapter 3, we present an algorithm which converts the Sagbi basis with respect to one ordering to the Sagbi basis with respect to another ordering, under the as- sumption that the subalgebra admits a finite Sagbi basis with respect to all monomial orderings. We called it Sagbi walk algorithm. Sagbi Walk algorithm is given in my paper [20]. Composition is an operation of replacing variables in a polynomial by other poly- nomials. In Chapter 4, we study the behavior of Sagbi basis under composition. Some related results are from my paper [21].