We introduce and study the concept of $alpha $-semi short modules.Using this concept we extend some of the basic results of $alpha $-short modules to $alpha $-semi short modules.We observe that if $M$ is an $alpha $-semi short module then the dual perfect dimension of $M$ is $alpha $ or $alpha +1$.%In particular, if a semiprime ring $R$ is $alpha $-semi short as an $R$-module, then its Noetheri...

A lattice is called dual strongly perfect if both, the lattice and its dual, are strongly perfect. We show that there are no dual strongly perfect lattices of dimension 13 and 15.

The extremal 3-modular lattice [±G2(3)]14 with automorphism group C2 × G2(F3) is the unique dual strongly perfect lattice of dimension 14.

We determine the dimension and in some cases the minimum distance of primitive, narrow sense BCH codes of length n with small designed distance. We show that such a code contains its Euclidean dual code and, when the size of the field is a perfect square, also its Hermitian dual code. We establish two series of quantum error-correcting codes.

This paper examines the pure-strategy subgame-perfect equilibrium payoffs in discounted supergames with perfect monitoring. It is shown that the payoff sets are typically fractals unless they are full-dimensional, which may happen when the discount factors are large enough. More specifically, the equilibrium payoffs can be identified as subsets of self-affine sets or graph-directed self-affine ...

Let $Gamma$ be a group, $Gamma'$ a subgroup of $Gamma$ with finite index and $M$ be a $Gamma$-module. We show that $M$ is cotorsion if and only if it is cotorsion as a $Gamma'$-module. Using this result, we prove that the global cotorsion dimensions of rings $ZGamma$ and $ZGamma'$ are equal.