We have seen a number of loop transformations, but they all have been different, needing different analysis and implementation. However, a closer look reveals that the previous list of loop transformations (permutation, reversal, skewing) all follow a general pattern of linear loop transformations. Each of those transformations (and combinations and many others) can be represented by unimodular...

Continuing the thrust of our recent work, but with an important new idea, we find a cutoff regularization of the determinant of a scalar particle in a classical Euclidean gravitational field. The field is assumed asymptotically flat, and the regularization is diffeomorphism-invariant under coordinate changes that are the identity at infinity. The scalar field is expanded in term of variables th...

Using the language of h-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group, Fell GL n , from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra sln. We apply the generalized FRST construction and obtain an h-bialgebroid Fell M n . Natural analogs of the exterior algebra and ...

A standing conjecture in L-cohomology is that every finite CW complex X is of L-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class G of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L-acyclic, we also prove that the L-...

Using the language of h-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group, Fell(GL(n)), from Felder’s elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra sln. First we apply the generalized FRST construction and obtain a bialgebroid Fell(M(n)). Then, analogues of the exterior al...

Group G hemolytic streptococcal cell walls which have been treated with trypsin are composed of a group-specific polysaccharide moiety and a mucopeptide matrix. The mucopeptide contains N-acetylglucosamine, N-acetylmuramic acid, alanine, glutamic acid, lysine, and glycine, a composition similar to that of other groups of streptococci. The Group G carbohydrate is composed of rhamnose, N-acetylga...

In the Riemannian geometry of quantum computation, the quantum evolution is described in terms of the special unitary group of n-qubit unitary operators with unit determinant. To elaborate on several aspects of the methodology, the Riemannian curvature, geodesic equation, Jacobi equation, and lifted Jacobi equation on the group manifold are explicitly derived. This is important for investigatio...

We study the cancellation property of projective modules rank $2$ with a trivial determinant over Noetherian rings dimension $\leq 4$. If $R$ is smooth affine algebra $4$ an algebraically closed field $k$ such that $6 \in k^{\times}$, then we prove stably free $R$-modules are if and only Hermitian $K$-theory group $\tilde{V}_{SL} (R)$ trivial.