We construct an example of a group
A typeII InAs/AlAs
 NSFPAR ID:
 10229067
 Publisher / Repository:
 Nature Publishing Group
 Date Published:
 Journal Name:
 Scientific Reports
 Volume:
 11
 Issue:
 1
 ISSN:
 20452322
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract for a finite abelian group$$G = \mathbb {Z}^2 \times G_0$$ $G={Z}^{2}\times {G}_{0}$ , a subset$$G_0$$ ${G}_{0}$E of , and two finite subsets$$G_0$$ ${G}_{0}$ of$$F_1,F_2$$ ${F}_{1},{F}_{2}$G , such that it is undecidable in ZFC whether can be tiled by translations of$$\mathbb {Z}^2\times E$$ ${Z}^{2}\times E$ . In particular, this implies that this tiling problem is$$F_1,F_2$$ ${F}_{1},{F}_{2}$aperiodic , in the sense that (in the standard universe of ZFC) there exist translational tilings ofE by the tiles , but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in$$F_1,F_2$$ ${F}_{1},{F}_{2}$ ). A similar construction also applies for$$\mathbb {Z}^2$$ ${Z}^{2}$ for sufficiently large$$G=\mathbb {Z}^d$$ $G={Z}^{d}$d . If one allows the group to be nonabelian, a variant of the construction produces an undecidable translational tiling with only one tile$$G_0$$ ${G}_{0}$F . The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles. 
Abstract Twodimensional electron systems subjected to high transverse magnetic fields can exhibit Fractional Quantum Hall Effects (FQHE). In the GaAs/AlGaAs 2D electron system, a double degeneracy of Landau levels due to electronspin, is removed by a small Zeeman spin splitting,
, comparable to the correlation energy. Then, a change of the Zeeman splitting relative to the correlation energy can lead to a reordering between spin polarized, partially polarized, and unpolarized many body ground states at a constant filling factor. We show here that tuning the spin energy can produce fractionally quantized Hall effect transitions that include both a change in$$g \mu _B B$$ $g{\mu}_{B}B$ for the$$\nu$$ $\nu $ minimum, e.g., from$$R_{xx}$$ ${R}_{\mathrm{xx}}$ to$$\nu = 11/7$$ $\nu =11/7$ , and a corresponding change in the$$\nu = 8/5$$ $\nu =8/5$ , e.g., from$$R_{xy}$$ ${R}_{\mathrm{xy}}$ to$$R_{xy}/R_{K} = (11/7)^{1}$$ ${R}_{\mathrm{xy}}/{R}_{K}={(11/7)}^{1}$ , with increasing tilt angle. Further, we exhibit a striking size dependence in the tilt angle interval for the vanishing of the$$R_{xy}/R_{K} = (8/5)^{1}$$ ${R}_{\mathrm{xy}}/{R}_{K}={(8/5)}^{1}$ and$$\nu = 4/3$$ $\nu =4/3$ resistance minima, including “avoided crossing” type lineshape characteristics, and observable shifts of$$\nu = 7/5$$ $\nu =7/5$ at the$$R_{xy}$$ ${R}_{\mathrm{xy}}$ minima the latter occurring for$$R_{xx}$$ ${R}_{\mathrm{xx}}$ and the 10/7. The results demonstrate both size dependence and the possibility, not just of competition between different spin polarized states at the same$$\nu = 4/3, 7/5$$ $\nu =4/3,7/5$ and$$\nu$$ $\nu $ , but also the tilt or Zeemanenergydependent crossover between distinct FQHE associated with different Hall resistances.$$R_{xy}$$ ${R}_{\mathrm{xy}}$ 
Abstract Let
be an elliptically fibered$$X\rightarrow {{\mathbb {P}}}^1$$ $X\to {P}^{1}$K 3 surface, admitting a sequence of Ricciflat metrics collapsing the fibers. Let$$\omega _{i}$$ ${\omega}_{i}$V be a holomorphicSU (n ) bundle overX , stable with respect to . Given the corresponding sequence$$\omega _i$$ ${\omega}_{i}$ of Hermitian–Yang–Mills connections on$$\Xi _i$$ ${\Xi}_{i}$V , we prove that, ifE is a generic fiber, the restricted sequence converges to a flat connection$$\Xi _i_{E}$$ ${\Xi}_{i}{}_{E}$ . Furthermore, if the restriction$$A_0$$ ${A}_{0}$ is of the form$$V_E$$ ${V}_{E}$ for$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j0)$$ ${\oplus}_{j=1}^{n}{O}_{E}({q}_{j}0)$n distinct points , then these points uniquely determine$$q_j\in E$$ ${q}_{j}\in E$ .$$A_0$$ ${A}_{0}$ 
Abstract The elliptic flow
of$$(v_2)$$ $\left({v}_{2}\right)$ mesons from beautyhadron decays (nonprompt$${\textrm{D}}^{0}$$ ${\text{D}}^{0}$ was measured in midcentral (30–50%) Pb–Pb collisions at a centreofmass energy per nucleon pair$${\textrm{D}}^{0})$$ ${\text{D}}^{0})$ TeV with the ALICE detector at the LHC. The$$\sqrt{s_{\textrm{NN}}} = 5.02$$ $\sqrt{{s}_{\text{NN}}}=5.02$ mesons were reconstructed at midrapidity$${\textrm{D}}^{0}$$ ${\text{D}}^{0}$ from their hadronic decay$$(y<0.8)$$ $\left(\righty<0.8)$ , in the transverse momentum interval$$\mathrm {D^0 \rightarrow K^\uppi ^+}$$ ${D}^{0}\to {K}^{}{\pi}^{+}$ GeV/$$2< p_{\textrm{T}} < 12$$ $2<{p}_{\text{T}}<12$c . The result indicates a positive for nonprompt$$v_2$$ ${v}_{2}$ mesons with a significance of 2.7$${{\textrm{D}}^{0}}$$ ${\text{D}}^{0}$ . The nonprompt$$\sigma $$ $\sigma $ meson$${{\textrm{D}}^{0}}$$ ${\text{D}}^{0}$ is lower than that of prompt nonstrange D mesons with 3.2$$v_2$$ ${v}_{2}$ significance in$$\sigma $$ $\sigma $ , and compatible with the$$2< p_\textrm{T} < 8~\textrm{GeV}/c$$ $2<{p}_{\text{T}}<8\phantom{\rule{0ex}{0ex}}\text{GeV}/c$ of beautydecay electrons. Theoretical calculations of beautyquark transport in a hydrodynamically expanding medium describe the measurement within uncertainties.$$v_2$$ ${v}_{2}$ 
Abstract We evaluate the
decay width from the perspective that the$$a_1(1260) \rightarrow \pi \sigma (f_0(500))$$ ${a}_{1}\left(1260\right)\to \pi \sigma \left({f}_{0}\left(500\right)\right)$ resonance is dynamically generated from the pseudoscalar–vector interaction and the$$a_1(1260)$$ ${a}_{1}\left(1260\right)$ arises from the pseudoscalar–pseudoscalar interaction. A triangle mechanism with$$\sigma $$ $\sigma $ followed by$$a_1(1260) \rightarrow \rho \pi $$ ${a}_{1}\left(1260\right)\to \rho \pi $ and a fusion of two pions within the loop to produce the$$\rho \rightarrow \pi \pi $$ $\rho \to \pi \pi $ provides the mechanism for this decay under these assumptions for the nature of the two resonances. We obtain widths of the order of 13–22 MeV. Present experimental results differ substantially from each other, suggesting that extra efforts should be devoted to the precise extraction of this important partial decay width, which should provide valuable information on the nature of the axial vector and scalar meson resonances and help clarify the role of the$$\sigma $$ $\sigma $ channel in recent lattice QCD calculations of the$$\pi \sigma $$ $\pi \sigma $ .$$a_1$$ ${a}_{1}$