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It is smooth everywhere except the z-axis. A remarkable fact is that all smooth quasiregular maps are local homeomorphisms. Even more remarkable is that every quasiregular local homeomorphism Rn → Rn, where n ≥ 3, is a homeomorphism (this is a theorem of Vladimir Zorich). This explains why in the definition of quasiregular maps it is not reasonable to restrict oneself to smooth maps: all smooth quasiregular maps of Rn to itself are quasiconformal.
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Also the density matrix renormalization group (DMRG) method, next to QMC, is a very reliable method, with an expanding community of users and an increasing number of applications to physical systems. When the first quantum computer is plugged in and functioning, the perspectives for the field of computational physics will look rather promising, but until that day one has to restrict oneself to the mundane tools offered by classical computers. While experimental physicists are putting a lot of effort in trying to build the first quantum computer, theoretical physicists are searching, in the field of quantum information theory (QIT), for genuine quantum algorithms, appropriate for problems that would perform badly when trying to be solved on a classical computer, but pretty fast and successful on a quantum one. The search for such algorithms is still going, the best-known (and almost the only ones found) being the Shor's algorithm, for factoring large numbers, and Grover's search algorithm.
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