Let $A$ be a nite dimensional $k-$algebra and $R$ be a locally bounded category such that $R rightarrow R/G = A$ is a Galois covering dened by the action of a torsion-free group of automorphisms of $R$. Following [30], we provide criteria on the convex subcategories of a strongly simply connected category R in order to be a cycle- nite category and describe the module category of $A$. We p...

In this article, we formalize in Mizar [7] the definition of “torsion part” of Z-module and its properties. We show Z-module generated by the field of rational numbers as an example of torsion-free non free Z-modules. We also formalize the rank-nullity theorem over finite-rank free Z-modules (previously formalized in [1]). Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lo...

We show that if G is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group G∗, such that G is a subgroup of G∗ but G is not quasiconvex in G∗ . We also prove that any non-elementary subgroup of a torsion-free word hyperbolic group G contains a free group of rank two which is malnormal and quasiconvex in G.

The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsion-free groups is proved. In a special case, this generalized version become a multivariable variant of this theorem. For unimodular equations over torsion free groups, we prove an analogue of Magnus...

Adding two generators and one arbitrary relator to a nontrivial torsion-free group, we always obtain an SQ-universal group. In the course of the proof of this theorem, we obtain some other results of independent interest. For instance, adding one generator and one relator in which the exponent sum of the additional generator is one to a free product of two nontrivial torsion-free groups, we als...

We consider the Dirac equation in flat Minkowski 3–space and rewrite it as the Maxwell equation in Minkowski 4–space with torsion. The torsion tensor is defined as the dual of the electromagnetic vector potential. Our model clearly distinguishes the electron and the positron without resorting to “negative frequencies”: we produce a real scalar invariant (charge) which indicates whether we are l...

We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for n p2, the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion-free part of the group is free abelia...

A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolds. All conformal deformations sending the standard flat torsion-free quaternionic contac...

In a pointed category with kernels and cokernels, we characterize torsion–free classes in terms of their closure under extensions. They are also described as indexed reflections. We obtain a corresponding characterization of torsion classes by formal categorical dualization.

The study of modules over a finite von Neumann algebra A can be advanced by the use of torsion theories. In this work, some torsion theories for A are presented, compared and studied. In particular, we prove that the torsion theory (T,P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for A. Using torsion theories, we describe the inj...