We employ min-max techniques to show that the unit ball in $mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
We study the Yamabe flow on a Riemannian manifold of dimension $mgeq3$ minus a closed submanifold of dimension $n$ and prove that there exists an instantaneously complete solution if and only if $n>frac{m-2}{2}$. In the remaining cases $0leq nleqfrac
{m-2}{2}$ including the borderline case, we show that the removability of the $n$-dimensional singularity is necessarily preserved along the Yamabe flow. In particular, the flow must remain geodesically incomplete as long as it exists. This is contrasted with the two-dimensional case, where instantaneously complete solutions always exist.
We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension $mgeq3$ starting from any smooth, conformally hyperbolic initial metric. We do not require initial completeness or curvature bounds. With th
e same methods, we show rigidity of hyperbolic space under the Yamabe flow.
We prove global existence of Yamabe flows on non-compact manifolds $M$ of dimension $mgeq3$ under the assumption that the initial metric $g_0=u_0g_M$ is conformally equivalent to a complete background metric $g_M$ of bounded, non-positive scalar curv
ature and positive Yamabe invariant with conformal factor $u_0$ bounded from above and below. We do not require initial curvature bounds. In particular, the scalar curvature of $(M,g_0)$ can be unbounded from above and below without growth condition.
We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension $mgeq3$. The initial metric is assumed to be conformally hyperbolic with conformal factor and scalar curvature bounded from above. We do not
require initial completeness or bounds on the Ricci curvature. If the initial data are rotationally symmetric, the solution is proven to be unique in the class of instantaneously complete, rotationally symmetric Yamabe flows.
New classes of distance-constrained structures are introduced, namely string-node nets and meshes, a mesh being a string-node net for which the nodes are dense in the strings. Various construction schemes are given including the minimal extension of
a (countable) line segment net by an arbitrary countable scaling group. A linear mesh has strings that are straight lines and nodes given by the intersection points of these lines. Classes of linear meshes, such as the regular linear meshes in R^2 and R^3, are defined and classified. Contrasting flexing and rigidity properties are obtained for some fundamental meshes in the plane with respect to noncrossing motions in the space of smooth meshes.
We compare the absolute gain photometric calibration of the Planck/HFI and Herschel/SPIRE instruments on diffuse emission. The absolute calibration of HFI and SPIRE each relies on planet flux measurements and comparison with theoretical far-infrared
emission models of planetary atmospheres. We measure the photometric cross calibration between the instruments at two overlapping bands, 545 GHz / 500 $mu$m and 857 GHz / 350 $mu$m. The SPIRE maps used have been processed in the Herschel Interactive Processing Environment (Version 12) and the HFI data are from the 2015 Public Data Release 2. For our study we used 15 large fields observed with SPIRE, which cover a total of about 120 deg^2. We have selected these fields carefully to provide high signal-to-noise ratio, avoid residual systematics in the SPIRE maps, and span a wide range of surface brightness. The HFI maps are bandpass-corrected to match the emission observed by the SPIRE bandpasses. The SPIRE maps are convolved to match the HFI beam and put on a common pixel grid. We measure the cross-calibration relative gain between the instruments using two methods in each field, pixel-to-pixel correlation and angular power spectrum measurements. The SPIRE / HFI relative gains are 1.047 ($pm$ 0.0069) and 1.003 ($pm$ 0.0080) at 545 and 857 GHz, respectively, indicating very good agreement between the instruments. These relative gains deviate from unity by much less than the uncertainty of the absolute extended emission calibration, which is about 6.4% and 9.5% for HFI and SPIRE, respectively, but the deviations are comparable to the values 1.4% and 5.5% for HFI and SPIRE if the uncertainty from models of the common calibrator can be discounted. Of the 5.5% uncertainty for SPIRE, 4% arises from the uncertainty of the effective beam solid angle, which impacts the adopted SPIRE point source to extended source unit conversion factor (Abridged)
We report on the sensitivity of SPIRE photometers on the Herschel Space Observatory. Specifically, we measure the confusion noise from observations taken during the Science Demonstration Phase of the Herschel Multi-tiered Extragalactic Survey. Confus
ion noise is defined to be the spatial variation of the sky intensity in the limit of infinite integration time, and is found to be consistent among the different fields in our survey at the level of 5.8, 6.3 and 6.8 mJy/beam at 250, 350 and 500 microns, respectively. These results, together with the measured instrument noise, may be used to estimate the integration time required for confusion-limited maps, and provide a noise estimate for maps obtained by SPIRE.
A new excitation is observed at 201 meV in the doped-hole ladder cuprate Sr$_{14}$Cu$_{24}$O$_{41}$, using ultraviolet resonance Raman scattering with incident light at 3.7 eV polarized along the direction of the rungs. The excitation is found to be
of charge nature, with a temperature independent excitation energy, and can be understood via an intra-ladder pair-breaking process. The intensity tracks closely the order parameter of the charge density wave in the ladder (CDW$_L$), but persists above the CDW$_L$ transition temperature ($T_{CDW_L}$), indicating a strong local pairing above $T_{CDW_L}$. The 201 meV excitation vanishes in La$_{6}$Ca$_{8}$Cu$_{24}$O$_{41+delta}$, and La$_{5}$Ca$_{9}$Cu$_{24}$O$_{41}$ which are samples with no holes in the ladders. Our results suggest that the doped holes in the ladder are composite bosons consisting of paired holons that order below $T_{CDW}$.
We investigate the time evolution of the entropy for a paradigmatic conservative dynamical system, the standard map, for different values of its controlling parameter $a$. When the phase space is sufficiently ``chaotic (i.e., for large $|a|$), we rep
roduce previous results. For small values of $|a|$, when the phase space becomes an intricate structure with the coexistence of chaotic and regular regions, an anomalous regime emerges. We characterize this anomalous regime with the generalized nonextensive entropy, and we observe that for values of $a$ approaching zero, it lasts for an increasingly large time. This scenario displays a striking analogy with recent observations made in isolated classical long-range $N$-body Hamiltonians, where, for a large class of initial conditions, a metastable state (whose duration diverges with $1/Nto 0$) is observed before it crosses over to the usual, Boltzmann-Gibbs regime.
