Decomposability of an algebraic structure into the union of its sub-structures goes back to G. Scorza's Theorem of 1926 for groups. An analogue of this theorem for rings has been recently studied by A. Lucchini in 2012. On the study of this problem for non-group semigroups, the first attempt is due to Clifford's work of 1961 for the regular semigroups. Since then, N.P. Mukherjee in 1972 studied...

Some important examples and explications of Grothendieck duality which should have been included in Chapter 5 of [C1] were overlooked. Several such topics are treated here. One question which can be asked about the trace map H(X,ΩX/k) → k for a proper smooth scheme X of pure dimension n over a field k is how it relates to the trace map for a smooth divisor. Generalizing a little, one can ask wh...

We show how certain diffeomorphism-invariant functionals on differential forms in dimensions 6, 7 and 8 generate in a natural way special geometrical structures in these dimensions: metrics of holonomy G2 and Spin(7), metrics with weak holonomy SU(3) and G2, and a new and unexplored example in dimension 8. The general formalism becomes a practical tool for calculating homogeneous or cohomogenei...

Modeling Cloth from Examples

Developing the relationships with students during teaching and learning was by method. As a result, approaches made significant contribution to activities. Learning examples that are not one example of media-based methodology. The media in our problem process because component will give an effect on student outcomes messages can be delivered according objectives through media. To maximize effec...