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Goodwin's model is a cornerstone in the study of dynamical systems within macroeconomics, explaining the interaction between employment ratio and wage share in a closed economy. Analogous to predator-prey dynamics in mathematical economics, the Goodwin model, despite its simplicity, effectively captures the periodic behavior of state variables over specific time intervals. By relaxing the initial assumptions, the model can be adapted to account for more complex economic scenarios. In this article, we study a higher-dimensional extension of the Goodwin model that incorporates variable capacity utilization and capital coefficient alongside employment ratio and wage share. In particular instances, the wage share and employment rate equations decouple from the overall system. For these cases, by incorporating a delay effect in the Phillips curve, we demonstrate that while the equilibrium of the generalized system remains stable within certain parameter domains in the absence of delay, the introduction of delay can induce a Hopf bifurcation, leading to periodic oscillations. We analytically derive the critical delay parameter value that destabilizes the equilibrium point via a Hopf bifurcation., Comment: 30 pages, 12 figures

The objective of pansharpening and hypersharpening is to accurately combine a high-resolution panchromatic (PAN) image with a low-resolution multispectral (MS) or hyperspectral (HS) image, respectively. Unfolding fusion methods integrate the powerful representation capabilities of deep learning with the robustness of model-based approaches. These techniques involve unrolling the steps of the optimization scheme derived from the minimization of an energy into a deep learning framework, resulting in efficient and highly interpretable architectures. In this paper, we propose a model-based deep unfolded method for satellite image fusion. Our approach is based on a variational formulation that incorporates the classic observation model for MS/HS data, a high-frequency injection constraint based on the PAN image, and an arbitrary convex prior. For the unfolding stage, we introduce upsampling and downsampling layers that use geometric information encoded in the PAN image through residual networks. The backbone of our method is a multi-head attention residual network (MARNet), which replaces the proximity operator in the optimization scheme and combines multiple head attentions with residual learning to exploit image self-similarities via nonlocal operators defined in terms of patches. Additionally, we incorporate a post-processing module based on the MARNet architecture to further enhance the quality of the fused images. Experimental results on PRISMA, Quickbird, and WorldView2 datasets demonstrate the superior performance of our method and its ability to generalize across different sensor configurations and varying spatial and spectral resolutions. The source code will be available at https://github.com/TAMI-UIB/MARNet.

We present an experimental investigation of the magnetic ground state in Ba$_4$NbIr$_3$O$_{12}$, a fractional valent trimer iridate. X-ray absorption and photoemission spectroscopy show that the Ir valence lies between 3+ and 4+ while Nb is pentavalent. Combined dc/ac magnetization, specific heat, and muon spin rotation/relaxation ($\mu$SR) measurements reveal no magnetic phase transition down to 0.05~K. Despite a significant Weiss temperature ($\Theta_{\mathrm{W}} \sim -15$ to $-25$~K) indicating antiferromagnetic correlations, a quantum spin-liquid (QSL) phase emerges and persists down to 0.1~K. This state likely arises from geometric frustration in the edge-sharing equilateral triangle Ir network. Our $\mu$SR analysis reveals a two-component depolarization, arising from the coexistence of rapidly (90\%) and slowly (10\%) fluctuating Ir moments. Powder x-ray diffraction and Ir-L$_3$edge x-ray absorption fine structure spectroscopy identify ~8-10\% Nb/Ir site-exchange, reducing frustration within part of the Ir network, and likely leading to the faster muon spin relaxation, while the structurally ordered Ir ions remain highly geometrically frustrated, giving rise to the rapidly spin-fluctuating QSL ground state. At low temperatures, the magnetic specific heat varies as $\gamma T + \alpha T^2$, indicating gapless spinon excitations, and possible Dirac QSL features with linear spinon dispersion, respectively., Comment: 27 pages, 15 figures

Frustrated magnets offer a plethora of exotic magnetic ground states, including quantum spin liquids (QSLs), in which enhanced quantum fluctuations prevent a long-range magnetic ordering of the strongly correlated spins down to lowest temperature. Here we have investigated the trimer based mixed valence hexagonal rhodate Ba$_4$NbRh$_3$O$_{12}$ using a combination of dc and ac magnetization, electrical resistivity, specific heat, and muon spin rotation/relaxation ($\mu$SR) measurements. Despite the substantial antiferromagnetic exchange interactions, as evident from the Weiss temperature ($\theta_{\mathrm{W}}\sim -35$ to -45 K), among the Rh-local moments, neither long-range magnetic ordering nor spin-freezing is observed down to at least 50 mK, in ac-susceptibility, specific heat and ZF-$\mu$SR measurements (down to 0.26 K). We ascribe the absence of any magnetic transition to enhanced quantum fluctuations as a result of geometrical frustration arising out of the edge-sharing equilateral Rh-triangular network in the structure. Our longitudinal-field $\mu$SR result evidences persistent spin fluctuations down to 0.26~K, thus stabilizing a dynamic QSL ground state in Ba$_4$NbRh$_3$O$_{12}$. Furthermore, the magnetic specific heat ($C_{\mathrm{m}}$) data at low-$T$ reveal a significant $T$-linear contribution plus a quadratic $T$-dependence. A $T$-linear behavior is evocative of gapless spin excitations, while the $T^2$-term of $C_{\mathrm{m}}$ may indicate the Dirac QSL phenomenology of the spinon excitations with a linear dispersion., Comment: 21 pages, 11 figures

Over the past decades, libraries of stellar spectra have been used in a large variety of science cases, including as sources of reference spectra for a given object or a given spectral type. Despite the existence of large libraries and the increasing number of projects of large-scale spectral surveys, there is to date only one very high-resolution spectral library offering spectra from a few hundred objects from the southern hemisphere (UVES-POP) . We aim to extend the sample, offering a finer coverage of effective temperatures and surface gravity with a uniform collection of spectra obtained in the northern hemisphere. Between 2010 and 2020, we acquired several thousand echelle spectra of bright stars with the Mercator-HERMES spectrograph located in the Roque de Los Muchachos Observatory in La Palma, whose pipeline offers high-quality data reduction products. We have also developed methods to correct for the instrumental response in order to approach the true shape of the spectral continuum. Additionally, we have devised a normalisation process to provide a homogeneous normalisation of the full spectral range for most of the objects. We present a new spectral library consisting of 3256 spectra covering 2043 stars. It combines high signal-to-noise and high spectral resolution over the entire range of effective temperatures and luminosity classes. The spectra are presented in four versions: raw, corrected from the instrumental response, with and without correction from the atmospheric molecular absorption, and normalised (including the telluric correction)., Comment: 17 pages, 18 figures Preview and access to the library: https://www.royer.se/melchiors.html

CeRh$_2$As$_2$ is a new multiphase superconductor with strong suggestions for an additional itinerant multipolar ordered phase. The modeling of the low temperature properties of this heavy fermion compound requires a quartet Ce$^{3+}$ crystal-field ground state. Here we provide the evidence for the formation of such a quartet state using x-ray spectroscopy. Core-level photoelectron and x-ray absorption spectroscopy confirm the presence of Kondo hybridization in CeRh$_2$As$_2$. The temperature dependence of the linear dichroism unambiguously reveils the impact of Kondo physics for coupling the Kramer's doublets into an effective quasi-quartet. Non-resonant inelastic x-ray scattering data find that the $|\Gamma_7^- \rangle$ state with its lobes along the 110 direction of the tetragonal structure ($xy$ orientation) contributes most to the multi-orbital ground state of CeRh$_2$As$_2$., Comment: 8 pages, 7 figures

GaGeTe is a layered topological semimetal that has been recently found to exist in at least two different polytypes, $\alpha$-GaGeTe ($R\bar{3}m$) and $\beta$-GaGeTe ($P6_3 mc$). Here we report a joint experimental and theoretical study of the structural, vibrational, and electronic properties of these two polytypes at high pressure. Both polytypes show anisotropic compressibility and two phase transitions, above 7 and 15 GPa, respectively, as confirmed by XRD and Raman spectroscopy measurements. Although the nature of the high-pressure phases is not confirmed, comparison with other chalcogenides and total-energy calculations allow us to propose possible high-pressure phases for both polytypes with an increase in coordination for Ga and Ge atoms from 4 to 6. In particular, the simplification of the X-ray patterns for both polytypes above 15 GPa suggests a transition to a structure of relatively higher symmetry than the original one. This result is consistent with the rocksalt-like high-pressure phases observed in parent III-VI semiconductors, such as GaTe, GaSe, and InSe. Pressure-induced amorphization is observed upon pressure release. The electronic band structures of $\alpha$-GaGeTe and $\beta$-GaGeTe and their pressure dependence also show similarities to III-VI semiconductors, thus suggesting that the germanene-like sublayer induces a semimetallic character in both GaGeTe polytypes. Above 3 GPa, both polytypes lose their topological features, due to the opening of the direct band gap, while the reduction of the interlayer space increases the thermal conductivity at high pressure., Comment: Main manuscript: 47 pages, 12 figures, 4 tables; ESI: 20 pages, 12 figures, 9 tables