السلام فی القرآن الكريم

It goes without saying that peace has been a hot issue in the past; it is, still, a vital topic of discussion today, and it seems to continue to draw the attention of people in the future. We witness wars and destructions in several parts of the world. This has led to killings of millions of people, left innumerable number of families broken and displaced millions of people. We need to address it earnestly. The author of this dissertation has specified this topic with reference to the Holy Qur’ān, because being the book of Allāh Almighty, it means to us as the final constitution, the ultimate torchlight for our guidance, and a great blessing to the whole world. Peace means to be free from the calamities in this world and in the hereafter. Usually, peace is used to imply an opposition to war and violence between the nations. Peace is a virtue and it means absence of evil, and, therefore, it is always praiseworthy. The author of this paper deals with this topic by dividing it into six sections. These are, ‘The Peace’ as one of the divine names of Allāh Almighty; Peace equals good; Peace is praiseworthy; Peace in the sense of conciliation and security; Peace in the sense of customary Islamic salutation; and Refutation of the objection that Islām was spread by sword.

Stanley Depth and Sequentially Cohen- Macaulay Lexsegment Ideals

In the first chapter we give some basic definitions from commutative algebra. We give some results obtained in recent years for the Stanley depth of multigraded S-modules, where S = K[x1 , . . . , xn ] is a polynomial ring in n indeteminantes with coefficients in a field K. We also give some results regarding the progress towards the Stanley’s conjecture. In the second chapter, we show that if I ⊂ J be monomial ideals of a polynomial algebra S over a field. Then the Stanley depth of J/I is smaller or equal to the √ √ Stanley depth of J/ I. We give also an upper bound for the Stanley depth of the intersection of two primary monomial ideals Q, Q , which is reached if Q, Q √ √ are irreducible, ht(Q + Q ) is odd and Q, Q have no common variables. These results are proved in my paper [23]. In the third chapter, we give different bounds for the Stanley depth of a monomial ideal I of a polynomial algebra S over a field K. For example we show that the Stanley depth of I is less than or equal to the Stanley depth of any prime ideal associated to S/I. Also we show that the Stanley’s conjecture holds for I and S/I when the associated prime ideals of S/I are generated by disjoint sets of variables. These results are proved in my paper [24]. In the forth chapter, we give an upper bound for the Stanley depth of the edge ideal I of a k-partite complete graph and show that Stanley’s conjecture holds for I. Also we give an upper bound for the Stanley depth of the edge ideal of an s-uniform complete bipartite hypergraph. In this chapter we also give an upper bound for the Stanley depth of the edge ideal of a complete k-partite hypergraph and as an application we give an upper bound for the Stanley depth of a monomial ideal in a polynomial ring S. We give a lower and an upper bound for the cyclic module S/I associated to the complete k-partite hypergraph. These results are proved in our papers [26] and [27]. In the fifth chapter, the associated primes of an arbitrary lexsegment ideal I ⊂ S are determined. As application it is shown that S/I is a pretty clean module, therefore, S/I is sequentially Cohen-Macaulay and satisfies the Stanley’s conjecture. These results are proved in my paper [25].