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Let u be a word over the positive integers. Motivated in part by a question from representation theory, we study the centralizer set of u which is C(u) = {w | uw is Knuth-equivalent to wu}. In particular, we give various necessary conditions for w to be in C(u). We also characterize C(u) when u has few letters, when it has a single repeated entry, or when it is a certain type of decreasing sequence. We consider c_{n,m}(u), the number of w in C(u) of length n with max w at most m. We prove that for |u| = 1 the value of this function depends only on the relative sizes of u and m and not on their actual values. And for various u we use Stanley's theory of poset partitions to show that, for fixed n, c_{n,m}(u) is a polynomial in m with certain degree and leading coefficient. We end with various conjectures and directions for further research.

Skyrmions are particle-like vortices of magnetization with non-trivial topology, which are usually stabilized by Dzyaloshinskii-Moriya interactions (DMI) in noncentrosymmetric bulk materials. Exceptions are centrosymmetric Gd- and Eu-based skyrmion-lattice (SkL) hosts with net-zero DMI, where both the SkL stabilization mechanisms and magnetic ground states remain controversial. We address these by investigating both static and dynamic spin properties of the centrosymmetric SkL host Gd$_2$PdSi$_3$ using muon spectroscopy ($\mu$SR). We find that spin fluctuations in its non-coplanar SkL phase are highly anisotropic, implying that spin anisotropy plays a prominent role in stabilizing this phase. We also observe strongly-anisotropic spin dynamics in the ground-state (IC-1) incommensurate magnetic phase of the material, indicating that it is a meron-like multi-$q$ structure. In contrast, the higher-field, coplanar IC-2 phase is found to be single-$q$ with nearly-isotropic spin dynamics., Comment: Main text: 7 pages, 3 figures. Supplemental Material: 6 pages, 5 figures

Eu$_5$In$_2$Sb$_6$ is a member of a family of orthorhombic nonsymmorphic rare-earth intermetallics that combines large localized magnetic moments and itinerant exchange with a low carrier density and perpendicular glide planes. This may result in special topological crystalline (wallpaper fermion) or axion insulating phases. Recent studies of Eu$_5$In$_2$Sb$_6$ single crystals have revealed colossal negative magnetoresistance and multiple magnetic phase transitions. Here, we clarify this ordering process using neutron scattering, resonant elastic X-ray scattering, muon spin-rotation, and magnetometry. The nonsymmorphic and multisite character of Eu$_5$In$_2$Sb$_6$ results in coplanar noncollinear magnetic structure with an Ising-like net magnetization along the $a$ axis. A reordering transition, attributable to competing ferro- and antiferromagnetic couplings, manifests as the onset of a second commensurate Fourier component. In the absence of spatially resolved probes, the experimental evidence for this low-temperature state can be interpreted either as an unusual double-$q$ structure or in a phase separation scenario. The net magnetization produces variable anisotropic hysteretic effects which also couple to charge transport. The implied potential for functional domain physics and topological transport suggests that this structural family may be a promising platform to implement concepts of topological antiferromagnetic spintronics.

Background: Asymmetric fission is known to occur in two regions, the actinides and sub-lead, and is dependent on the fissioning system excitation energy. Experimental evidence in the sub-lead region show that this mode is surprisingly persistent with increasing energy and its origin is not fully understood. Purpose: To experimentally study the fusion-fission reaction of $^{215}$Fr at moderate excitation energy and determine previously unknown independent fission yields and other properties. Method: The compound nucleus was formed in the reaction $^{18}$O + $^{197}$Au. The prompt gamma-rays emitted during the reaction were measured with the high efficiency and high granularity $\nu$-ball2 spectrometer. Independent fission yields of even-even nuclei were determined by detecting triple-gamma cascades in the fission fragments. Results: The observed yields, although dominated by a symmetric peak, show maxima for heavy fragment of $Z \approx 54-56$, which is consistent with the known results in the actinide region but unexpected for the nuclide of interest, and at the studied excitation energy. Conclusions: The mode of asymmetric fission is present even at relatively high excitation energies in the system studied. This observation matches experimental findings in the sub-lead region, contrary to the actinides, and so far there is no well-developed explanation of this phenomenon.

The chemical enrichment of dust and metals are vital processes in constraining the star formation history of the universe. Previously, the dust masses of high-redshift star-forming galaxies have been determined through their far-infrared continuum, however, equivalent, and potentially simpler, approaches to determining the metal masses have yet to be explored at $z\gtrsim 2$. Here, we present a new method of inferring the metal mass in the interstellar medium (ISM) of galaxies out to $z\approx 8$, using the far-infrared [CII]$-158\mu$m emission line as a proxy. We calibrated the [CII]-to-$M_{\rm Z,ISM}$ conversion factor based on a benchmark observational sample at $z\approx 0$, in addition to gamma-ray burst sightlines at $z>2$ and cosmological hydrodynamical simulations of galaxies at $z\approx 0$ and $z\approx 6$. We found a universal scaling across redshifts of $\log (M_{\rm Z,ISM}/M_\odot) = \log (L_{\rm [CII]}/L_\odot) - 0.45,$ with a 0.4 dex scatter, which is constant over more than two orders of magnitude in metallicity. We applied this scaling to recent surveys for [CII] in galaxies at $z\gtrsim 2$ and determined the fraction of metals retained in the gas-phase ISM, $M_{\rm Z,ISM} / M_\star$, as a function of redshift showing that an increasing fraction of metals reside in the ISM of galaxies at higher redshifts. We place further constraints on the cosmic metal mass density in the ISM ($\Omega_{\rm Z,ISM}$) at $z\approx 5$ and $\approx 7$, yielding $\Omega_{\rm Z,ISM} = 6.6^{+13}_{-4.3}\times 10^{-7}\,M_\odot\, {\rm Mpc}^{-3}$ ($z\approx 5$) and $\Omega_{\rm Z,ISM} = 2.0^{+3.5}_{-1.3}\times 10^{-7}\,M_\odot\, {\rm Mpc}^{-3}$ ($z\approx 7$). These results are consistent with the expected metal yields from the integrated star formation history at the respective redshifts. This suggests that the majority of metals produced at $z\gtrsim 5$ are confined to the ISM of galaxies., Comment: Accepted in A&A, abstract abridged

The chemical enrichment of dust and metals in the interstellar medium (ISM) of galaxies throughout cosmic time is one of the key driving processes of galaxy evolution. Here we study the evolution of the gas-phase metallicities, dust-to-gas (DTG), and dust-to-metal (DTM) ratios of 36 star-forming galaxies at $1.7 < z < 6.3$ probed by gamma-ray bursts (GRBs). We compile all GRB-selected galaxies with intermediate (R=7000) to high (R>40,000) resolution spectroscopic data for which at least one refractory (e.g. Fe) and one volatile (e.g. S or Zn) element have been detected at S/N>3. This is to ensure that accurate abundances and dust depletion patterns can be obtained. We first derive the redshift evolution of the dust-corrected, absorption-line based gas-phase metallicity [M/H]$_{\rm tot}$ in these galaxies, for which we determine a linear relation with redshift ${\rm [M/H]_{tot}}(z) = (-0.21\pm 0.04)z -(0.47\pm 0.14)$. We then examine the DTG and DTM ratios as a function of redshift and through three orders of magnitude in metallicity, quantifying the relative dust abundance both through the direct line-of-sight visual extinction $A_V$ and the derived depletion level. We use a novel method to derive the DTG and DTM mass ratios for each GRB sightline, summing up the mass of all the depleted elements in the dust-phase. We find that the DTG and DTM mass ratios are both strongly correlated with the gas-phase metallicity and show a mild evolution with redshift as well. While these results are subject to a variety of caveats related to the physical environments and the narrow pencil-beam sightlines through the ISM probed by the GRBs, they provide strong implications for studies of dust masses to infer the gas and metal content of high-redshift galaxies, and particularly demonstrate the large offset from the average Galactic value in the low-metallicity, high-redshift regime., Comment: Accepted in A&A

The assembly and build-up of neutral atomic hydrogen (HI) in galaxies is one of the most fundamental processes in galaxy formation and evolution. Studying this process directly in the early universe is hindered by the weakness of the hyperfine 21-cm HI line transition, impeding direct detections and measurements of the HI gas masses ($M_{\rm HI}$). Here we present a new method to infer $M_{\rm HI}$ of high-redshift galaxies using neutral, atomic oxygen as a proxy. Specifically, we derive metallicity-dependent conversion factors relating the far-infrared [OI]-$63\mu$m and [OI]-$145\mu$m emission line luminosities and $M_{\rm HI}$ in star-forming galaxies at $z\approx 2-6$ using gamma-ray bursts (GRBs) as probes. We substantiate these results by observations of galaxies at $z\approx 0$ with direct measurements of $M_{\rm HI}$ and [OI]-$63\mu$m and [OI]-$145\mu$m in addition to hydrodynamical simulations at similar epochs. We find that the [OI]$_{\rm 63\mu m}$-to-HI and [OI]$_{\rm 145\mu m}$-to-HI conversion factors universally appears to be anti-correlated with the gas-phase metallicity. The high-redshift GRB measurements further predict a mean ratio of $L_{\rm [OI]-63\mu m} / L_{\rm [OI]-145\mu m}=1.55\pm 0.12$ and reveal generally less excited [CII]. The $z \approx 0$ galaxy sample also shows systematically higher $\beta_{\rm [OI]-63\mu m}$ and $\beta_{\rm [OI]-145\mu m}$ conversion factors than the GRB sample, indicating either suppressed [OI] emission in local galaxies or more extended, diffuse HI gas reservoirs traced by the HI 21-cm. Finally, we apply these empirical calibrations to the few high-redshift detections of [OI]-$63\mu$m and [OI]-$145\mu$m line transitions from the literature and further discuss the applicability of these conversion factors to probe the HI gas content in the dense, star-forming ISM of galaxies at $z\gtrsim 6$, well into the epoch of reionization., Comment: Submitted to A&A. Comments welcome

We present a joint analysis of the galaxy S04590 at $z=8.496$ based on NIRSpec, NIRCam, and NIRISS observations obtained through as part of Early Release Observations programme of the James Webb Space Telescope (JWST) and the far-infrared [CII]-$158\mu$m emission line detected by dedicated Atacama Large Millimeter/submillimeter Array (ALMA) observations. We determine the physical properties of S04590 from modelling of the spectral energy distribution (SED) and through the redshifted optical nebular emission lines detected with JWST/NIRSpec. The best-fit SED model reveals a low-mass ($M_\star = 10^{7.2}-10^{8}\,M_{\odot}$) galaxy with a low oxygen abundance of $12+\log{\rm (O/H)} = 7.16^{+0.10}_{-0.12}$ derived from the strong nebular and auroral emission lines. Assuming that [CII] effectively traces the interstellar medium (ISM), we estimate the total gas mass of the galaxy to be $M_{\rm gas} = (8.0\pm 4.0)\times 10^{8}\,M_\odot$ based on the luminosity and spatial extent of [CII]. This yields an exceptionally high gas fraction, $f_{\rm gas} = M_{\rm gas}/(M_{\rm gas} + M_\star) \gtrsim 90\%$, though still consistent within the range expected for its low metallicity. We further derive the metal mass of the galaxy based on the gas mass and gas-phase metallicity, which we find to be consistent with the expected metal production from Type II supernovae. Finally, we make the first constraints on the dust-to-gas (DTG) and dust-to-metals (DTM) ratios of galaxies in the epoch of reionization at $z\gtrsim 6$, showing overall low mass ratios of logDGT $<-3.8$ and logDTM $<-0.5$, though consistent with local scaling relations and in particular the local metal-poor galaxy I Zwicky 18. Our analysis highlights the synergy between ALMA and JWST in characterizing the gas, metal, and stellar content of the first generation of galaxies., Comment: Accepted for publication in ApJL

Skyrmion-hosting multilayer stacks are promising avenues for applications, although little is known about the depth dependence of the magnetism. We address this by reporting the results of circular dichroic resonant elastic x-ray scattering (CD-REXS), micromagnetic simulations, and low-energy muon-spin rotation (LE-$\mu^+$SR) measurements on a stack comprising [Ta/CoFeB/MgO]$_{16}$/Ta on a Si substrate. Energy-dependent CD-REXS shows a continuous, monotonic evolution of the domain-wall helicity angle with incident energy, consistent with a three-dimensional hybrid domain-wall-like structure that changes from N\'eel-like near the surface to Bloch-like deeper within the sample. LE-$\mu^+$SR reveals that the magnetic field distribution in the trilayers near the surface of the stack is distinct from that in trilayers deeper within the sample. Our micromagnetic simulations support a quantitative analysis of the $\mu^+$SR results. By increasing the applied magnetic field, we find a reduction in the volume occupied by domain walls at all depths, consistent with a crossover into a region dominated by skyrmions above approximately 180 mT., Comment: 9 pages, 4 figures

Mechanical straining of skyrmion hosting materials has previously demonstrated increased phase stability through the expansion of the skyrmion equilibrium pocket. Additionally, metastable skyrmions can be generated via rapid field-cooling to form significant skyrmion populations at low temperatures. Using small-angle x-ray scattering and x-ray holographic imaging on a thermally strained 200 nm thick FeGe lamella, we observe temperature-induced strain effects on the structure and metastability of the skyrmion lattice. We find that in this sample orientation (H || [1 1 0]) with no strain, metastable skyrmions produced by field cooling through the equilibrium skyrmion pocket vanish from the sample upon dropping below the well known helical reorientation temperature. However, when strain is applied along [110] axis, and this procedure is repeated, a substantial volume fraction of metastable skyrmions persist upon cooling below this temperature down to 100 K. Additionally, we observe a large number of skyrmions retained after a complete magnetic field polarity reversal, implying that the metastable energy barrier protecting skyrmions from decay is enhanced.

The $\beta$ decays of the ground state (gs) and isomeric state (m) of $^{96}$Y have been studied with the total absorption $\gamma$-ray spectroscopy technique at the Ion Guide Isotope Separator On-Line facility. The separation of the 8$^{+}$ isomeric state from the 0$^{-}$ ground state was achieved thanks to the purification capabilities of the JYFLTRAP double Penning trap system. The $\beta$-intensity distributions of both decays have been independently determined. In the analyses the de-excitation of the 1581.6 keV level in $^{96}$Zr, in which conversion electron emission competes with pair production, has been carefully considered and found to have significant impact on the $\beta$-detector efficiency, influencing the $\beta$-intensity distribution obtained. Our results for $^{96\text{gs}}$Y (0$^+$) confirm the large ground state to ground state $\beta$-intensity probability, although a slightly larger value than reported in previous studies was obtained, amounting to $96.6_{-2.1}^{+0.3}\%$ of the total $\beta$ intensity. Given that the decay of $^{96\text{gs}}$Y is the second most important contributor to the reactor antineutrino spectrum between 5 and 7 MeV, the impact of the present results on reactor antineutrino summation calculations has been evaluated. In the decay of $^{96\text{m}}$Y (8$^{+}$), previously undetected $\beta$ intensity in transitions to states above 6 MeV has been observed. This shows the importance of total absorption $\gamma$-ray spectroscopy measurements of $\beta$ decays with highly fragmented de-excitation patterns. $^{96\text{m}}$Y (8$^{+}$) is a major contributor to reactor decay heat in uranium-plutonium and thorium-uranium fuels around 10 s after fission pulses, and the newly measured average $\beta$ and $\gamma$ energies differ significantly from the previous values in evaluated databases (...)

We present magnetization and muon-spin spectroscopy measurements of N\'{e}el skyrmion-host GaV$_4$S$_8$ under the application of hydrostatic pressures up to $P=2.29$ GPa. Our results suggest that the magnetic phase diagram is altered with pressure via a reduction in the crossover temperature from the cycloidal (C) to ferromagnetic-like state with increasing $P$, such that, by 2.29 GPa, the C state appears to persist down to the lowest measured temperatures. With the aid of micromagnetic simulations, we propose that the driving mechanism behind this change is a reduction in the magnetic anisotropy of the system, and suggest that this could lead to an increase in stability of the skyrmion lattice., Comment: 9 pages, 6 figures