The ferroelectric and cubic phases in BaTiO_3 ferroelectrics are also antiferroelectric

Using quantum mechanics (QM, Density Functional Theory) we show that all four phases of barium titanate (BaTiO3) have local Ti distortions toward (an octahedral face). The stable rhombohedral phase has all distortions in phase (ferroelectric, FE), whereas higher temperature phases have antiferroelectric coupling (AFE) in one, two, or three dimensions (orthorhombic, tetragonal, cubic). This FE–AFE model from QM explains such puzzling aspects of these systems as the allowed Raman excitation observed for the cubic phase, the distortions toward observed in the cubic phase using x-ray fine structure, the small transition entropies, the heavily damped soft phonon modes, and the strong diffuse x-ray scattering in all but the rhombohedral phase. In addition, we expect to see additional weak Bragg peaks at the face centers of the reciprocal lattice for the cubic phase. Similar FE–AFE descriptions are expected to occur for other FE materials. Accounting for this FE–AFE nature of these phases is expected to be important in accurately simulating the domain wall structures, energetics, and dynamics, which in turn may lead to the design of improved materials

On hypergraph Lagrangians

It is conjectured by Frankl and F\"uredi that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform hypergraphs with $m$ edges in \cite{FF}. Motzkin and Straus' theorem confirms this conjecture when $r=2$. For $r=3$, it is shown by Talbot in \cite{T} that this conjecture is true when $m$ is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for $r$-uniform hypergraphs. As an implication of this connection, we prove that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform graphs with $t$ vertices and $m$ edges satisfying ${t-1\choose r}\leq m \leq {t-1\choose r}+ {t-2\choose r-1}-[(2r-6)\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\choose r-2}-1)$ for $r\geq 4.$Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1312.7529, arXiv:1211.7057, arXiv:1211.6508, arXiv:1311.140