A Planar 3-Convex Set is Indeed a Union of Six Convex Sets

Suppose S is a planar set. Two points $a,b$ in S  see each other via S if $[a,b]$ is included in S . F. Valentine proved in 1957 that if S is closed, and if for every three points of S, at least two see each other via S, then S is a union of three convex sets. The pentagonal star shows that the number three is the best possible. We drop the condition that S is closed and show that S is a union of (at most) six convex sets. The number six is best possible.     

Magnetic excitations in Cu6 and Mn6 hexagons embedded in D3d-symmetric polyoxotungstates

In this article we reconsider the discussion of the magnetic measurements for the two novel polyoxotungstates, (n-BuNH(3))(12)[(CuCl)(6)(AsW(9)O(33))(2)].6H(2)O and (n-BuNH(3))(12)[(MnCl)(6)(SbW(9)O(33))(2)].6H(2)O, which have been synthesized and characterized by Yamase et al. (Inorg.Chem. 2006, 45, 7698). Analysis of the magnetic susceptibility and magnetization for Cu(6)(12+) and Mn(6)(12+) hexagons based on the exact diagonalization of isotropic exchange Hamiltonian shows that the best-fit first-neighbor coupling parameters are J = 35 and 0.55 cm(-1), respectively, while the second-neighbor interactions are very small. These values exceed considerably those obtained by Yamase et al. (J = 8.82 and 0.14 cm(-1)) on the basis of the Kambe-Van Vleck formula that is inappropriate for six-membered rings. We also got perfect fits to the experimental data for the field dependence of magnetization at 1.8 K. The results imply the importance of axial anisotropy, which is shown to be especially pronounced for the Mn(6)(12+) cluster. We discuss also the symmetry assignments of exchange multiplets to the exact SGamma terms (full spin, S, and irreducible representation, Gamma, of the point group) and correlate the results with the selection rules for the anisotropic magnetic contributions. The antisymmetric exchange is shown to appear in orbitally degenerate multiplets as a first-order perturbation and gives rise to an easy axis of magnetization along the C(6) axis. Evaluation of the Zeeman levels shows that the field applied in the plane of the hexagon fully reduces the effect of the antisymmetric exchange.     

Bell inequalities with no quantum violation and unextendable product bases

The strength of classical correlations is subject to certain constraints, commonly known as Bell inequalities. Violation of these inequalities is the manifestation of nonlocality-displayed, in particular, by quantum mechanics, meaning that quantum mechanics can outperform classical physics at tasks associated with such Bell inequalities. Interestingly, however, there exist situations in which this is not the case. We associate an intriguing class of bound entangled states, constructed from unextendable product bases with a wide family of tasks, for which (i) quantum correlations do not outperform the classical ones but (ii) there exist supraquantum nonsignaling correlations that do provide an advantage.     

The Maximum Number of Edges in Geometric Graphs with Pairwise Virtually Avoiding Edges

Let G be a geometric graph on n vertices that are not necessarily in general position. Assume that no line passing through one edge of G meets the relative interior of another edge. We show that in this case the number of edges in G is at most 2n?3.     

Dipolar molecules in optical lattices

We study the extended Bose-Hubbard model describing an ultracold gas of dipolar molecules in an optical lattice, taking into account all on-site and nearest-neighbor interactions, including occupation-dependent tunneling and pair tunneling terms. Using exact diagonalization and the multiscale entanglement renormalization ansatz, we show that these terms can destroy insulating phases and lead to novel quantum phases. These considerable changes of the phase diagram have to be taken into account in upcoming experiments with dipolar molecules.