We compute liftings of the Nichols algebra of a Yetter-Drinfeld module of Cartan type B 2 subject to the small restriction that the diagonal elements of the braiding matrix are primitive nth roots of 1 with odd n = 5. As well, we compute the lift-ings of a Nichols algebra of Cartan type A 2 if the diagonal elements of the braiding matrix are cube roots of 1; this case was not completely covered...

We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group Eτ,η(sl2) at the discrete coupling constants: 2Nη = m1 + im2τ , where N,m1 and m2 are integers. Then we show that...

We compute liftings of the Nichols algebra of a Yetter-Drinfeld module of Cartan type B2 subject to the small restriction that the diagonal elements of the braiding matrix are primitive nth roots of 1 with odd n 6= 5. As well, we compute the liftings of a Nichols algebra of Cartan type A2 if the diagonal elements of the braiding matrix are cube roots of 1; this case was not completely covered i...

In this paper we study the class of differential operators T = Pk j=1 QjD j with polynomial coefficients Qj in one complex variable satisfying the condition degQj ≤ j with equality for at least one j. We show that if degQk < k then the root with the largest modulus of the nth degree eigenpolynomial pn of T tends to infinity when n → ∞, as opposed to the case when degQk = k, which we have treate...

in this paper, the variational iteration method for solving nth-order fuzzy integro differential equations (nth-fide) is proposed. in fact the problem is changed to the system of ordinary fuzzy integro-differential equations and then fuzzy solution of nth-fide is obtained. some examples show the efficiency of the proposed method.

The algebra B of bicomplex numbers is viewed as a complexification the Archimedean f-algebra hyperbolic D. This lattice-theoretic approach allows us to establish new properties so-called D-norms. In particular, we show that D-norms generate same topology in B. We develop D-trigonometric form number which leads geometric interpretation nth roots terms polyhedral tori. use concepts developed, par...

Let C0 X,X be the set of all continuous self-mappings on a topological space X. Let f ∈ C0 X,X . For integer n ≥ 0, define the nth iterate of f by f f ◦ fn−1 and f0 id, where id denotes the identity mapping on X and ◦ denotes the composition of mappings. An equation with iteration as its main operation is simply called an iterative equation. Founded on the problem of iterative roots, the proble...

We study sums of the form ∑ ζ R(ζ), where R is a rational function and the sum is over all nth roots of unity ζ (often with ζ = 1 excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit formula for ∏ ζ(1− xR(ζ)). Multisection ca...

ABSTRACT. For parameters n, l and r, we consider the problem of maximizing the determinant of an l × l Vandermonde matrix V, with nodes selected from the set Ωn of nth roots of unity, but avoiding a forbidden subset R of size r. An asymptotically tight lower bound is given for the expected value of ln |detV | in case the nodes are selected uniformly at random from Ωn/R. We apply our result to g...

Our goal in this section will be to collect some standard examples of finite groups. The main emphasis will be on realizing them as subgroups of GL(n,R) or GL(n,C). Cyclic groups: For a natural number n, let Z/nZ be the standard cyclic group of order n. We denote its elements by 0, 1, . . . , n − 1, and 1 is a generator. Another model for the cyclic group of order n is the nth roots of unity, o...