S=1/2 Kagome antiferromagnets Cs2_2Cu3MF_3MF_{12}$ with M=Zr and Hf

Magnetization and specific heat measurements have been carried out on Cs$_2$Cu$_3$ZrF$_{12}$ and Cs$_2$Cu$_3$HfF$_{12}$ single crystals, in which Cu$^{2+}$ ions with spin-1/2 form a regular Kagom\'{e} lattice. The antiferromagnetic exchange interaction between neighboring Cu$^{2+}$ spins is $J/k_{\rm B}\simeq 360$ K and 540 K for Cs$_2$Cu$_3$ZrF$_{12}$ and Cs$_2$Cu$_3$HfF$_{12}$, respectively. Structural phase transitions were observed at $T_{\rm t}\simeq 210$ K and 175 K for Cs$_2$Cu$_3$ZrF$_{12}$ and Cs$_2$Cu$_3$HfF$_{12}$, respectively. The specific heat shows a small bend anomaly indicative of magnetic ordering at $T_\mathrm{N}= 23.5$ K and 24.5 K in Cs$_2$Cu$_3$ZrF$_{12}$ and Cs$_2$Cu$_3$HfF$_{12}$, respectively. Weak ferromagnetic behavior was observed below $T_\mathrm{N}$. This weak ferromagnetism should be ascribed to the antisymmetric interaction of the Dzyaloshinsky-Moriya type that are generally allowed in the Kagom\'{e} lattice.Comment: 6 pages, 4 figure. Conference proceeding of Highly Frustrated Magnetism 200

Boundary definition of a multiverse measure

We propose to regulate the infinities of eternal inflation by relating a late time cut-off in the bulk to a short distance cut-off on the future boundary. The light-cone time of an event is defined in terms of the volume of its future light-cone on the boundary. We seek an intrinsic definition of boundary volumes that makes no reference to bulk structures. This requires taming the fractal geometry of the future boundary, and lifting the ambiguity of the conformal factor. We propose to work in the conformal frame in which the boundary Ricci scalar is constant. We explore this proposal in the FRW approximation for bubble universes. Remarkably, we find that the future boundary becomes a round three-sphere, with smooth metric on all scales. Our cut-off yields the same relative probabilities as a previous proposal that defined boundary volumes by projection into the bulk along timelike geodesics. Moreover, it is equivalent to an ensemble of causal patches defined without reference to bulk geodesics. It thus yields a holographically motivated and phenomenologically successful measure for eternal inflation.Comment: 39 pages, 4 figures; v2: minor correction

Arithmeticity vs. non-linearity for irreducible lattices

We establish an arithmeticity vs. non-linearity alternative for irreducible lattices in suitable product groups, such as for instance products of topologically simple groups. This applies notably to a (large class of) Kac-Moody groups. The alternative relies on a CAT(0) superrigidity theorem, as we follow Margulis' reduction of arithmeticity to superrigidity.Comment: 11 page

The constraint equations for the Einstein-scalar field system on compact manifolds

We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for the scalar field, we are able to divide the set of free conformal data into subclasses depending on the possible signs for the coefficients of terms in the resulting Einstein-scalar field Lichnerowicz equation. For many of these subclasses we determine whether or not a solution exists. In contrast to other well studied field theories, there are certain cases, depending on the mean curvature and the potential of the scalar field, for which we are unable to resolve the question of existence of a solution. We consider this system in such generality so as to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed scalar curvature problem as special cases.Comment: Minor changes, final version. To appear: Classical and Quantum Gravit

Knaster's problem for (Z2)k(Z_2)^k-symmetric subsets of the sphere S2k−1S^{2^k-1}

We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-symmetric convex fans

Diffusion-Weighted Imaging and Diffusion Tensor Imaging of Asymptomatic Lumbar Disc Herniation

Diffusion-weighted imaging (DWI) and diffusion tensor imaging (DTI) were performed on a healthy 31-year-old man with asymptomatic lumbar disc herniation. Although the left S1 nerve root was obviously entrapped by a herniated mass, neither DWI nor DTI showed any significant findings for the nerve root. Decreased apparent diffusion coefficient (ADC) values and increased fractional anisotropy (FA) values were found. These results are contrary to those in previously published studies of symptomatic patients, in which a combination of increased ADC and decreased FA seem to have a relationship with nerve injury and subsequent symptoms, such as leg pain or palsy. Our results seen in an asymptomatic subject suggest that the compressed nerve with no injury, such as edema, demyelination, or persistent axonal injury, may be indicated by a combination of decreased ADC and increased FA. ADC and FA could therefore be potential tools to elucidate the pathomechanism of radiculopathy

Quantitative and qualitative relationship between microstructural factors and fatigue lives under load- And strain-controlled conditions of Ti-5Al-2Sn-2Zr-4Cr-4Mo (Ti-17) fabricated using a 1500-ton forging simulator

The fatigue lives of forged Ti-17 using a 1500-ton forging simulator subjected to different solution treatments and a common aging treatment were evaluated under both load- and strain-controlled conditions: high and low cycle fatigue lives, respectively. Then, the tensile properties and microstructures were also examined. Finally, the relationships among fatigue lives and the microstructural factors and tensile properties were examined. The microstructure after solution treatment at 1203 K, which is more than the β transus temperature, and aging treatment exhibits equiaxed prior β grains composed of fine acicular ¡. On the other hand, the microstructures after solution treatment at temperatures of 1063, 1123, and 1143 K, which are less than the β transus temperature, and aging treatment exhibit elongated prior β grains composed of two different microstructural feature regions, which are acicular α and fine spheroidal α phase regions. The 0.2% proof stress, σ₀.₂, and tensile strength, σB, increase with increasing solution treatment temperature up to 1143 K within the (α + β) region, but decrease with further increasing solution treatment temperature to 1203 K within the β region. The elongation (EL) and reduction of area (RA) decrease with increasing solution treatment temperature, and it becomes nearly 0% corresponding to a solution treatment temperature of 1203 K. The high cycle fatigue limit increases with increasing solution treatment temperature up to 1143 K, corresponding to the (α + β) region. However, it decreases with further increase in the solution treatment temperature to 1203 K in the β region. The fatigue ratio in high cycle fatigue life region is increasing with decreasing solution treatment temperature, namely increasing the volume fraction of the primary α phase, and it relates well qualitatively with the volume fraction of the primary α phase when the solution treatment temperature is less than the β transus temperature. The low cycle fatigue life increases with decreasing solution treatment temperature, namely increasing the volume fraction of the primary α phase. The low cycle fatigue life relates well quantitatively with the tensile true strain at breaking of the specimen and the volume fraction of the primary α phase for each total strain range of low cycle fatigue testing.Niinomi M., Akahori T., Nakai M., et al. Quantitative and qualitative relationship between microstructural factors and fatigue lives under load- And strain-controlled conditions of Ti-5Al-2Sn-2Zr-4Cr-4Mo (Ti-17) fabricated using a 1500-ton forging simulator. Materials Transactions 60, 1740 (2019); https://doi.org/10.2320/matertrans.ME201904

A compactness theorem for scalar-flat metrics on manifolds with boundary

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this set is compact for dimensions greater than or equal to 7 under the generic condition that the trace-free 2nd fundamental form of the boundary is nonzero everywhere.Comment: 49 pages. Final version, to appear in Calc. Var. Partial Differential Equation