We study the local equivalence problem for real-analytic (Cω) hypersurfaces M5⊂C3 that, in some holomorphic coordinates (z1,z2,w)∈C3 with w=u+ −1v, are rigid sense that their graphing functions u=F(z1,z2,z‾1,z‾2) independent of v. Specifically, we group Holrigid(M) biholomorphic transformations form (z1,z2,w)⟼(f1(z1,z2),f2(z1,z2),aw+g(z1,z2)), where a∈R∖{0} and D(f1,f2)/D(z1,z2)≠0, which preser...

We systematically investigated electronic evolutions of nonsymmorphic borophene with chemical environments that were realized by the ion exchange method. Electronic structures can be characterized topological Z2 invariant. Spectroscopic experiments and DFT calculations unveiled a sheet hydrogenated (borophane) is Dirac nodal loop semimetal (Z2=−1), while layered crystal YCrB4 an insulator (Z2=1...

We note that the Chinese remainder theorem implies that this group is isomorphic with the cylic group of order 30, Z30. 31 is prime so the only abelian group of order 31 is Z31. 32 = 2. So there are many possibilities corresponding to all of the ways we can write m1 +m2 + ...+mr = 5 with m1 ≥ m2 ≥ ... ≥ mr. The solutions are 5 = 5 which corresponds to Z32, 4 ≥ 1 giving Z16 ×Z2, 3 ≥ 2 giving Z8 ...