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The trace norm of a matrix is the sum of its singular values. This paper presents results on the minimum trace norm $\psi_{n}\left( m\right) $ of $\left( 0,1\right) $-matrices of size $n\times n$ with exactly $m$ ones. It is shown that: (1) if $n\geq2$ and $n

Let G denote a group and let W be an algebra over a commutative ring R. We will say that W is a G-graded twisted algebra (not necessarily commutative, neither associative) if there exists a G-grading W=\bigoplus_{g \in G}W_{g} where each summand W_{g} is a free rank one R -module, and W has no monomial zero divisors (for each pair of nonzero elements w_{a},w_{b} en W_{a} and W_{b} their product is not zero, w_{a}w_{b}\neq 0). It is also assumed that W has an identity element. In this article, methods of group cohomology are used to study the general problem of classification under graded isomorphisms. We give a full description of these algebras in the associative cases, for complex and real algebras. In the nonassociative case, an analogous result is obtained under a symmetry condition of the corresponding associative function of the algebra, and when the group providing the grading is finite cyclic., Comment: 15 pages, no figures