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...

If A is a matrix with no negative real eigenvalues and all zero eigenvalues of A are semisimple, the principal pth root of A can be computed by Newton’s method or Halley’s method, with a preprocessing procedure if necessary. We prove a new convergence result for Newton’s method, and discover an interesting property of Newton’s method and Halley’s method in terms of series expansions. We explain...

Abstract. We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function ζ(s), this is expected to be an accurate description for the horizontal distribution of the zeros of ζ (s) to the right of ...

Newton’s method for the inverse matrix pth root, A−1/p, has the attraction that it involves only matrix multiplication. We show that if the starting matrix is cI for c ∈ R then the iteration converges quadratically to A−1/p if the eigenvalues of A lie in a wedge-shaped convex set containing the disc { z : |z−cp| < cp }. We derive an optimal choice of c for the case where A has real, positive ei...

Abstract. The worst-case residual norms of the GMRES method for linear algebraic systems [3] can, in case of a normal matrix, be characterized by a min-max approximation problem on the matrix eigenvalues. In [2] we derive a lower bound on this min-max value (worst-case residual norm) for each step of the GMRES iteration. We conjecture that the lower bound and the min-max value agree up to a fac...

We propose that the Baxter Q-operator for the spin-1/2 XXZ quantum spin chain is given by the j → ∞ limit of the transfer matrix with spin-j (i.e., (2j+1)-dimensional) auxiliary space. Applying this observation to the open chain with general (nondiagonal) integrable boundary terms, we obtain from the fusion hierarchy the T -Q relation for generic values (i.e. not roots of unity) of the bulk ani...

Root decay after timber cutting can lead to slope failure. In situ measurements of soil with tree roots showed that soil strength increased linearly as root biomass increased. Forests clear-felled 3 years earlier contained about one-third of the root biomass of oldgrowth forests. Nearly all of the roots < 2 mm in diameter were gone from 7-year-old logged areas while about 30 percent of the < 17...

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...

Philip Gibbs Diophantine m-tuples with property D(n), for n an integer, are sets of m positive integers such that the product of any two of them plus n is a square. Triples and quadruples with this property can be classed as regular or irregular according to whether they satisfy certain polynomial identities. Given any such m-tuple, a symmetric integer matrix can be formed with the elements of ...

AIMS AND OBJECTIVES Obtaining root coverage has become an important part of periodontal therapy. The aims of this studyare to evaluate the clinical efficacy of acellular dermal matrix graft in the coverage of denuded roots and also to examine the change in the width of keratinized gingiva. MATERIALS AND METHODS A total of 20 sites with more than or equal to 2 mm of recession depth were taken ...