شیخ ابن تیمیہ کےاصولی تفردات کا تجزیاتی مطالعہ An Analytical Study of the Ibn-e-Teymiyah’s Usooli Distinctions

The very existence of theology was a prelude to the competition of philosophy and the support of religion. Other religions, for the sake of legitimacy of their religion, have been raising objections to Islam in all respects. The mixture of nations, non-Arab influences, the negligent rhetoric of scholars and shaykhs, and the erroneous beliefs and practices of the people were tarnishing the clean face of Islam. In the time of Ibn Taymiyyah, there was an emphasis on theology. Theologians specialize in their knowledge and art of rationalism and logic. Their method consisted of reasoning and inference, rational and logical proofs. In contrast to them, the Hanbalis were holding the line of reasoning from the apparent meaning of verses and hadiths and texts. In the case of debates and debates, the weight of the theologians would be heavy and the opponent would be considered ignorant. Ibn Taymiyyah paid homage to the authenticity and supremacy of the Qur'an and Sunnah in such a way that he made critical commentary and analysis on philosophy and Greek logic and reasoned weaknesses and errors. One-half or two-thirds of his total writing services on the subject of speech and logic. This shows that Ibn Taymiyyah was feeling the need of defending Islam in this regard because in his time these sciences and arts had become very popular and common practice. The far-reaching effects of his thoughts and opinions in this regard have been felt in every age. The issues in which he has differed are the result of his research, jurisprudence and ijtihad, and constant reflection. The basis and foundation of which is the Quran and Sunnah, the interaction of companions and speculation. In these Ijtihad issues of hiss, there is a collection of arguments and proofs related to the Quran and Sunnah. In this article, an introductory and analytical study of Ibn Taymiyyah's fundamental differences will be presented.

Combinatorial and Arithmetic Study of Binomial Edge Ideal

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.