The band structure for a crystal generally consists of connected components in energy-momentum space, known as complexes. Here, we explore fundamental aspect regarding the maximal number bands that can be accommodated single complex. We show that, principle, complex have no finite upper bound certain space groups. This means infinitely many entangle together, forming pattern stable against symm...

The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring elementary, i.e. we can express with elementary ones in unique way. does not extend directly to semirings, but do have analogous results for some special example, the tropical, extended and supertropical semirings. These all fall into larger class upper-bound In this...