It is shown that a sufficiently pure mesoscopic metallic ring will exhibit a magnetic phase transition to a low-temperature state with a persistent orbital current in zero external field (to a flux-phase state). The transition temperature and the spontaneous persistent current depend strongly on the geometry of the ring and of the Fermi surface.

The possibility of making a flux qubit on a nonsuperconducting mesoscopic ballistic quasi-one-dimensional ring is discussed. We showed that such a ring can be effectively reduced to a two-state system with two external control parameters. The two states carry opposite persistent currents and are coupled by tunneling, which leads to a quantum superposition of states. The qubit states can be manipulated by resonant microwave pulses. The flux state of the sample can be measured by a superconducting quantum interference device magnetometer. Two or more qubits can be coupled by the flux the circulating currents generate. The problem of decoherence is also discussed.

Nash equilibria and correlated equilibria of classical and quantum games are investigated in the context of their Pareto efficiency. The examples of the prisoner’s dilemma, battle of the sexes and the game of chicken are studied. Correlated equilibria usually improve Nash equilibria of games but require a trusted correlation device susceptible to manipulation. The quantum extension of these games in the Eisert–Wilkens–Lewenstein formalism and the Frąckiewicz–Pykacz parameterization is analyzed. It is shown that the Nash equilibria of these games in quantum mixed Pauli strategies are closer to Pareto optimal results than their classical counter-parts. The relationship of mixed Pauli strategies equilibria and correlated equilibria is also studied.

An infinite series of (3,6) cages is defined by trivalent carbon polyhedra composed of hexagonal and four triangular rings. A zone-folding construction is applied to the graphene band structure to yield explicit expressions for the π-molecular orbitals, energies, and symmetries of the cages that depend only on four indices m, n, p, and q. Leapfrog members of the series $(m-n=0\mathrm{}$ mod 3 and $p-q=0\mathrm{}$ mod 3) have closed shells in a neutral form with two filled nonbonding orbitals; all others have closed shells as dications. Quantum chemical calculations on ${\mathrm{C}}_{12},$ ${\mathrm{C}}_{48},$ and ${\mathrm{C}}_{52}^{2+}$ confirm this result. Embedding relationships are proved for the spectra of (3,6) cages related by inflation transformations corresponding to stretching and rotation of the polyhedral net.