The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the “boundary knot” that occurs around the singular set, the set of points at which the embedding is not lo...

‎we call a ring $r$ right generalized semiperfect if every simple right $r$-module is an epimorphic image of a flat right $r$-module with small kernel‎, ‎that is‎, ‎every simple right $r$-module has a flat $b$-cover‎. ‎we give some properties of such rings along with examples‎. ‎we introduce flat strong covers as flat covers which are also flat $b$-covers and give characterizations of $a$-perfe...

Throughout this paper, k is a field, R is an algebra over k, and H is a Hopf algebra over k. We say that R# σ H is the crossed product of R and H if R# σ H becomes an algebra over k by multiplication: (a#h)(b#g) = h,g a(h 1 · b)σ(h 2 , g 1)#h 3 g 2 Let lpd(R M), lid(R M) and lf d(R M) denote the left projective dimension, left injective dimension and left flat dimension of left R-module M, resp...

where k is the first Betti number b^M), T is a flat riemannian λ -torus, M~ is a compact connected Ricci-flat (n — λ;)-manifold, and Ψ is a finite group of fixed point free isometries of T x M' of a certain sort (Theorem 4.1). This extends Calabi's result on the structure of compact euclidean space forms ([7] see [20, p. 125]) from flat manifolds to Ricci-flat manifolds. We use it to essentiall...

A Riemannian manifold is said to be Kähler if the holonomy group is contained in U(n). It is quaternion Kähler if the holonomy group is contained in Sp(n)Sp(1). It is known that quaternion Kähler manifolds of dimension ≥ 8 are Einstein, so the scalar curvature s splits these manifolds according to whether s > 0, s = 0 or s < 0. Ricci flat quaternion Kähler manifolds include hyperkähler manifold...

Abstract The Hessian of the entropy function can be thought of as a metric tensor on state space. In the context of thermodynamical fluctuation theory Ruppeiner has argued that the Riemannian geometry of this metric gives insight into the underlying statistical mechanical system; the claim is supported by numerous examples. We study these geometries for some families of black holes and find tha...

We study the three-dimensional topological insulators in the continuum by coupling spin-1/2 fermions to the Aharonov-Casher SU(2) gauge field. They exhibit flat Landau levels in which orbital angular momentum and spin are coupled with a fixed helicity. The three-dimensional lowest Landau level wave functions exhibit the quaternionic analyticity as a generalization of the complex analyticity of ...

The aim of this paper is to show that the supercover of an m-flat (i.e. a Euclidean affine subspace of dimension m) in Euclidean n-space is a discrete analytical object. The supercover of a Euclidean object F is a discrete object consisting of all the voxels that intersect F . A discrete analytical object is a set of discrete points that is defined by a finite set of inequalities. A method to d...