between any two coordinates ( i) and ( i). Hence any choice of Γ k ij in one coordinate chart suffices to define the connection in another. Let T k ij := Γkij − Γkji be the torsion tensor of the connection. The case when torsion is zero (torsion-free) is of particular interest in geometry. In Hermann Weyl’s terms [11], the very existence of inertia systems in the Universe warrants that its line...

We will answer a question raised by Emmanuel Dror Farjoun concerning the existence of torsion-free abelian groups G such that for any ordered pair of pure elements there is a unique automorphism mapping the first element onto the second one. We will show the existence of such a group of cardinality λ for any successor cardinal λ = μ+ with μ = μ0.

A classical theorem of Fritz John allows one to describe a convex body, up to constants, as an ellipsoid. In this article we establish similar descriptions for generalized (i.e. multidimensional) arithmetic progressions in terms of proper (i.e. collision-free) generalized arithmetic progressions, in both torsion-free and torsion settings. We also obtain a similar characterization of iterated su...

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results...

The residual torsion-free nilpotence of the commutator subgroup a knot group has played key role in studying bi-orderability groups. A technique developed by Mayland provides sufficient condition for to be residually-torsion-free nilpotent using work Baumslag. In this paper, we apply Mayland's several genus one pretzel knots and family with arbitrarily high genus. As result, obtain large number...

We study algebraic properties of categories of Merotopic, Nearness, and Filter Algebras. We show that the category of filter torsion free abelian groups is an epireflective subcategory of the category of filter abelian groups. The forgetful functor from the category of filter rings to filter monoids is essentially algebraic and the forgetful functor from the category of filter groups to the cat...

The author is reporting a case of torsion of gallbladder. The patient used to have episodes of intermittent torsion and there was no acute presentation. Patient was presented with abdominal swelling which was misdiagnosed as a mesenteric cyst. Distended gallbladder, free floating type, tortuous cystic artery with the mesentry were factors responsible for torsion. Gallbladder exhibited 180 degre...

In this article we study the space of leftand bi-invariant orderings on a torsion-free nilpotent group G. We will show that generally the set of such orderings is equipped with a faithful action of the automorphism group of G. We prove a result which allows us to establish the same conclusion when G is assumed to be merely residually torsion-free nilpotent. In particular, we obtain faithful act...

Recently, Bismut[ 2] proved a new local index theorem for Dirac operators associated with certain non-torsion-free connections. In this paper, we discuss the odd dimensional analogue. Here the object is the 17-function introduced by Atiyah, Patodi and Singer[l]. And we prove the regularity property of the ~7-function of Dirac operators associated with the appropriate non-torsion-free connection...

Let M be the moduli space of generalized parabolic bundles (GPBs) of rank r and degree d on a smooth curve X . Let ML̄ be the closure of its subset consisting of GPBs with fixed determinant L̄. We define a moduli functor for which ML̄ is the coarse moduli scheme. Using the correspondence between GPBs on X and torsion-free sheaves on a nodal curve Y of which X is a desingularization, we show that M...