In this note, we calculate all the basic invariants of the number field K = Q(3 √ 5, ω), where ω = (−1 + √ −3)/2 is a primitive cube root of unity. Here is the notation for the fields and Galois groups to be used. Let

The quintuple identity has a long history and, as Berndt [5] points out, it is difficult to assign priority to it. It seems that a proof of the identity was first published in H. A. Schwartz’s book in 1893 [19]. Watson gave a proof in 1929 in his work on the RogersRamanujan continued fractions [20]. Since then, various proofs have appeared. To name a few, Carlitz and Subbarao gave a simple proo...

Hall-Littlewood functions indexed by rectangular partitions, specialized at primitive roots of unity, can be expressed as plethysms. We propose a combinatorial proof of this formula using Schilling’s bijection between ribbon tableaux and ribbon rigged configurations.

We must mention that the assumption that we are dealing with a Cartan matrix of finite type and a root of unity appears only at the very end (see Chapter 4). We need these assumptions in order to compare our representations with the conventional definition of the category C. All previous results are valid in more general assumptions. In particular a Cartan matrix could be arbitrary and a deform...

the main purpose of this study is to comparative of two alternative approaches capabilities in the field of religious education pathology. salafism and fundamentalism is moving toward quasi education in religious education; but religious traditionalism and post revivalism can be adopted an alternative approach to religious education. critical analysis and interpretation of concept method used i...

because of using principles such as principality, unity, gradation of existence and concomitance between existence and unity, along with an exhaustive understanding of ibn sina's philosophical principle about our knowledge of unity and multiplicity, mulla sadra believed new philosophical-aesthetical principles such as principality, unity and gradation of beauty, though he didn't use t...

unity is one of the most important social issues in the holy religion of islam. unity makes possible entanglement from dependencies, success in settling social issues, and firmness in confronting conspiracies and invasions. since islam is a religion of kindness and brotherhood, it invites its followers to engage in being united, which will, however, never be achieved without resolving conflicts...

in order to investigate the effect of irrigation intervals and harvesting time on yield and yield components of anise (pimpinella anisum l.) a field experiment was conducted at the agricultural research station, university of zanjan in 2010. for this purpose a spilt-plot experiment based on complete randomized block design with three replications was used. three irrigation intervals (every 7, 1...

in this paper, we try to determine the status of the relation between dahr world and matrial world. the conclusion shows the subjective duality and objective unity of dahr and material entities. two interpretations of this duality can be suggested. in the first interpretation, this unity, and the duality supposed, are explained on the basis of different intellectual cognition of immaterial and ...

A k-element subset D of a finite multiplicative group G of order v is called a (v, k, λ, μ)partial difference set in G (PDS) provided that the multiset of ‘differences’ {d1d −1 2 | d1, d2 ∈ D, d1 6= d2} contains each nonidentity element of D exactly λ times and each nonidentity element in G\D exactly μ times. See [11] for background on partial difference sets. We will limit our attention to abe...