A theory of graded manifolds can be viewed as a generalization differential geometry smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also $\mathbb{Z}$-graded variables either commute or anticommute, according their degree. To obtain consistent global description manifolds, resorts sheaves commutative associative algebras secon...

Additive manufacturing has emerged as a promising approach for fabricating graded refractive index structures that control the electromagnetic response of radio frequency (RF) devices. However, current 3D printing methods cannot produce continuous gradients from multiple materials. Here, low-loss dielectrics via active mixing nanocomposite inks composed block copolymers and oxide nanoparticles ...

We introduce Morita and Rickard equivalences over a group graded G-algebra between block extensions. A consequence of such is that Späth's central order relation holds two corresponding character triples.

Let k be a field, G an abelian group with a bicharacter, A a colour algebra; i.e., an associative graded k-algebra with identity, C a graded A-coring that is projective as a right A-module, C∗ the graded dual ring of C and Λ a left graded C-comodule that is finitely generated as a graded right C∗-module. We give necessary and sufficient conditions for projectivity and flatness of a graded modul...