The dimension of any module over an algebra of affiliated operators U of a finite von Neumann algebra A is defined using a trace on A. All zero-dimensional U-modules constitute the torsion class of torsion theory (T,P). We show that every finitely generated U-module splits as the direct sum of torsion and torsion-free part. Moreover, we prove that the theory (T, P) coincides with the theory of ...

We investigate systems of equations and the first-order theory one-relator monoids. describe a family F monoids form 〈A|w=1〉 where for each monoid M in F, longstanding open problem decidability word with length constraints reduces to Diophantine (i.e. equations) M. achieve this result by finding an interpretation free monoid, using only together relations. It follows that has undecidable positi...

Let M be a torus bundle, i.e. an (n + 1)-dimensional manifold, which fibers over the circle with the fiber an n-dimensional torus. A covariant functor from M to the Cuntz-Krieger algebra (a C∗-algebra) is constructed. The functor maps homeomorphic torus bundles to the stably isomorphic Cuntz-Krieger algebras. It is shown that in general the K-theory of the Cuntz-Krieger algebra counts the torsi...

In the present article we discuss an approach to cohomological invariants of algebraic groups over fields characteristic zero based on Morava $K$-theories, which are generalized oriented cohomology theories in sense Levine--Morel. We show that second $K$-theory detects triviality Rost invariant and, more generally, relate and splitting motives. describe motives, compute some affine varieties,...

We first characterize $tau$-complemented modules with relative (pre)-covers. We also introduce an extending module relative to $tau$-pure submodules on a hereditary torsion theory $tau$ and give its relationship with $tau$-complemented modules.

We give some simple examples of mirror Calabi-Yau fourfolds in Type II string theory. These are realised as toroidal orbifolds. Motivated by the Strominger, Yau, Zaslow argument we give explicitly the mirror transformation which maps Type IIA/IIB on such a four-fold to Type IIA/IIB on the mirror. The mirrors are related by the inclusion/exclusion of discrete torsion. Implicit in the result is a...

Zero modes are an essential part of topological field theories, but they frequently also obstacle to the explicit evaluation associated path integrals. In order address this issue in case Ray-Singer Torsion, which appears various gauge we introduce a massive variant Torsion involves determinants twisted Laplacian with mass without zero modes. This has advantage allowing one explicitly keep trac...