After a historical discussion of Einstein’s 1907 principle of equivalence, a homogeneous gravitational field in Minkowski spacetime is constructed. It is pointed out that the reference frames in gravitational theory can be understood as spaces with a flat connection and torsion defined through teleparallelism. This kind of torsion was introduced by Einstein in 1928. The concept of torsion is di...

If EQ(M,N) has positive rank, then there are infinitely many primitive integer solutions to (1); but if EQ(M,N) has rank 0, then there may be a non-trivial solution. Such a solution exists if and only if the torsion group is Z2×Z8 or Z2×Z6.We classify all such cases, thereby reducing Euler’s problem to a question of ranks. In some cases, the ranks of quadratic twists of EQ(M,N) are described by...

Suppose M is a compact connected odd-dimensional manifold with boundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the L2-topological torsion of M and the L2-analytic torsion of the Riemannian manifold M are equal. In particular, the L2-topological torsion of M is proportional to the hyperbolic volume of M , with a constant of proportionality w...

From local class-field theory and higher ramification theory one gets a classification up to conjugacy of the torsion elements of arbitrary order in the Nottingham group over a finite field, in terms of continuous characters on the multiplicative group of principal units in the ring of formal power series over the field. An essential part of this classification is to partition the torsion eleme...

‎let $r$ be a domain with quotiont field $k$‎, ‎and‎ ‎let $n$ be a submodule of an $r$-module $m$‎. ‎we say that $n$ is‎ ‎powerful (strongly primary) if $x,yin k$ and‎ ‎$xymsubseteq n$‎, ‎then $xin r$ or $yin r$ ($xmsubseteq n$‎ ‎or $y^nmsubseteq n$ for some $ngeq1$)‎. ‎we show that a submodule‎ ‎with either of these properties is comparable to every prime‎ ‎submodule of $m$‎, ‎also we show tha...