In this set of lecture notes we focus on the point-value representation obtained by looking at a particular set of points, the nth roots of unity. In the field of complex numbers C, there are exactly n different solutions to the equation x = 1. We call these solutions the n-th roots of unity. One of these roots of unity is ωn = cos(2π/n)+ i sin(2π/n), and this is called the principal nth root o...

The main result of this paper gives necessary and sufficient conditions for the ^-primary part S(K)P of the Schur group S (K) to be induced from S (F)p for any subfield F of K where K is contained in Q (εn), under the restriction that εp2 is not in K if p > 2 and n is odd if p~2, where εn is a primitive nth root of unity. Moreover we completely answer the question: "When is SiQiεn + ε-)) induce...

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F .

We consider the model with discrete innovations introduced in [6] and studied in [10, 11, 12, 13]. In this model, the state of the economy is determined by a set of operating technologies, a collection of funds corresponding to these technologies, and a set of known, but as yet not implemented technologies. To introduce a new technology, expenditures of the already utilized types of funds are r...

In this paper, a generalization of a group-magic graph is introduced and studied. Let R be a commutative ring with unity 1. A graph G = (V,E) is called R-ring-magic if there exists a labeling f : E → R−{0} such that the induced vertex labelings f : V → R, defined by f(v) = Σf(u, v) where (u, v) ∈ E, and f : V → R, defined by f(v) = Πf(u, v) where (u, v) ∈ E, are constant maps. General algebraic...

Enterprise systems are attractive exactly as they promise a stronger unity – integration, collaboration and standardization – across distinct and different organisational units of a business. However, empirical research on enterprise systems has documented convincingly how situated workarounds undermine the unity of enterprise systems through local thus different practices and adoptions. This p...

It is common knowledge that the Democratic Russia Movement has become the first political organization able to unite the democratic forces in Russia, and thus challenge in an efficient way the political monopoly of the Communist Party of the Soviet Union (CPSU). However, the political and economic crises in the country have affected the success of this large organization. Cracks in its unity ha...

In this paper we used the finite Fourier transformation to obtain the polar decomposition of the q-deformed boson algebra with q a root of unity. Since the q-deformed boson algebra was known [1,2], much work has been accomplished in this direction [3-8]. It is well known that this algebra has an infinite dimensional representation when the deformation parameter q is real, while it has a finite ...

We deal with the unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths a; b + m; c; a + m; b; c + m, where an equilateral triangle of side length m has been removed from the center. We give closed formulas for the plain enumeration and for a certain (?1)-enumeration of these lozenge tilings. In the case that a = b = c, we also provide closed formulas for certain...