The pentagram integrals for Poncelet families

The pentagram map is now known to be a discrete integrable system. We show that the integrals for the pentagram map are constant along Poncelet families. That is, if $P_1$ and $P_2$ are two polygons in the same Poncelet family, and $f$ is a monodromy invariant for the pentagram map, then $f\left(P_1\right)=f\left(P_2\right)$. Our proof combines complex analysis with an analysis of the geometry of a degenerating sequence of Poncelet polygons.     

Numerically anticanonical divisors on Kato surfaces

The compact curves of an intermediate Kato surface $S$ form a basis of $H^2(S,\mathbb{Q})$. We present a way to compute the associated rational coefficients of the first Chern class $c_1\left(S\right)$. We get in particular a simple geometric obstruction for $c_1\left(S\right)$ to be an integral class, or equivalently index $\left(S\right)=1$. In the final part we discuss relations with some recent work of Dloussky (2011) and Oeljeklaus and Toma (2009).     

Lattice dynamics and phase transition of NiTi alloy

We have performed detailed first-principles calculations to investigate the structural and lattice dynamical properties of NiTi alloy. The calculated static structures consist well with the experimental data and other theoretical results. With quasi harmonic approximation, the phase boundary between B19′ and BCO phases can be described as a five order polynomial $T=100?89.28P+296.75P^2?717.94P^3+734.62P^4?274.25P^5$. The change of vibrational entropy is $0.07k_B$/atom at the transition temperature 100 K under zero pressure.     

Pseudo anti-commuting and Ricci soliton real hypersurfaces in complex two-plane Grassmannians

In this paper, first we introduce a new notion of pseudo anti-commuting for real hypersurfaces in complex two-plane Grassmannians $G_2\left({\mathbb{C}}^{m+2}\right)$ and prove a complete classification theorem, which gives a shrinking Ricci soliton with potential Reeb flow on Hopf real hypersurfaces and a tube over a totally real totally geodesic $\mathbb{Q}{P}^n$, $m=2n$ in $G_2\left({\mathbb{C}}^{m+2}\right)$.     

Pressure dependence of negative thermal expansion in Zr2(WO4)(PO4)2

Negative thermal expansion materials can experience significant stresses when they are used in composites. Under ambient conditions Zr₂(WO₄)(PO₄)₂ displays anisotropic negative thermal expansion (NTE) (${\alpha }_v=?14.0\left(10\right)\times 10^{?6}{K}^{?1}$, ${\alpha }_a=?7.9\left(5\right)\times 10^{?6}{K}^{?1}$, ${\alpha }_b=2.5\left(5\right)\times 10^{?6}{K}^{?1}$, ${\alpha }_c=?8.7\left(2\right)\times 10^{?6}{K}^{?1}$ at 0 GPa). The effect of hydrostatic pressure on its thermal expansion characteristics was investigated by neutron diffraction between 300 and 60 K at pressures up to 0.3 GPa. No phase transitions were observed in the pressure and temperature range examined. The material was found to have a bulk modulus, B₀, of 61.3(8) GPa at ambient temperature, and unlike some other NTE materials, pressure had no detectable effect on thermal expansion (${\alpha }_v=?14.2\left(8\right)\times 10^{?6}{K}^{?1}$, ${\alpha }_a=?7.9\left(3\right)\times 10^{?6}{K}^{?1}$, ${\alpha }_b=2.9\left(5\right)\times 10^{?6}{K}^{?1}$, ${\alpha }_c=?9.2\left(2\right)\times 10^{?6}{K}^{?1}$ at 0.3 GPa).     

p(x) and Tc(x) characteristics in the Y 1−b(Ca)bBa2Cu3O6+x cuprate family

A model has been proposed to calculate the $p\left(x\right)$ and $T_c\left(x\right)$ dependences in the Y _1?b(Ca)_bBa₂Cu₃O_6+x high-$T_c$ cuprate family and applied to $b=0$, $b=0.1$, and $b=0.2$ cases, for which experimental data exist in the literature. The results obtained imply that the Ca efficiency to provide holes is independent of the basal plane oxygen concentration, which is consistent with a view that electrons from CuO₂ layers would go primarily to Ca since it is twice closer than oxygen (in addition, the chain oxygen is screened by a layer made up of Ba and O(4) ions). It is shown that, in fully oxygenized compounds ($x=1$) the average efficiency, $\chi$, of a chain oxygen to attract an electron from the two nearby layers is reduced by the Ca insertion, though not because the charge transfer mechanism is in itself weakened by Ca, but because a part of electrons that are otherwise available in CuO${}_2$ layers has already been removed by the substitution of Y ³⁺ with Ca²⁺. It has been found that the $b$-dependence of the average oxygen doping efficiency can be fairly accurately described by the following relation: $\chi \left(b\right)=0.39\times (1?0.78b)$. The calculated $p\left(x\right)$ and $T_c\left(x\right)$ dependences are in very good agreement with the available experimental data.     

Compressibility of hexagonal selenium to 45 kbar

Compression data to 45 kbar have been obtained for hexagonal selenium by static methods in a piston-cylinder apparatus. Cylindrical samples (1 cm dia. and 1·2 cm length) of polycrystalline hexagonal selenium, with a bulk density close to the X-ray density, were prepared by pressing the samples at 7 kbar and 170°C. The compression data are described by a third degree polynomial,

$?\left(\frac{\Delta V}{V_O}\right)=?64·14\times {10}^{?4}P+109·9\times {10}^{?6}{P}^2?96·77\times {10}^{?8}{P}^3$

,where P is pressure in kbar. The zero pressure bulk modulus and the pressure derivative of the bulk modulus are 156 ± 10 kbar and 4·33 respectively.     

Monopoles on the Bryant–Salamon G2-manifolds

$G_2$-monopoles are solutions to gauge theoretical equations on noncompact $7$-manifolds of $G_2$ holonomy. We shall study this equation on the $3$ Bryant–Salamon manifolds. We construct examples of $G_2$-monopoles on two of these manifolds, namely the total space of the bundle of anti-self-dual two forms over the ${\mathbb{S}}^4$ and ${\mathbb{C}\mathbb{P}}^2$. These are the first nontrivial examples of $G_2$-monopoles.Associated with each monopole there is a parameter $m\in {\mathbb{R}}^{+}$, known as the mass of the monopole. We prove that under a symmetry assumption, for each given $m\in {\mathbb{R}}^{+}$ there is a unique monopole with mass $m$. We also find explicit irreducible $G_2$-instantons on ${\Lambda }_{-}^2\left({\mathbb{S}}^4\right)$ and on ${\Lambda }_{-}^2\left({\mathbb{C}\mathbb{P}}^2\right)$.The third Bryant–Salamon $G_2$-metric lives on the spinor bundle over the $3$-sphere. In this case we produce a vanishing theorem for monopoles.