We provide an explicit classification of all simple $\mathfrak{sl}_2$-modules that are torsion free of rank $1$ over the Cartan subalgebra. We also establish a similar result for the first Weyl algebra and for the Lie superalgebra $\mathfrak{osp}(1|2)$.

Spin precession experiments in lateral spin devices are a powerful tool for probing the spin transport properties of materials. These experiments can be quantitatively described using the Bloch diffusion equation, which offers a practical framework for modeling spin-related phenomena. In this work, we present calculations of the spin density across heterogeneous, diffusive spintronic devices. The modeled devices feature spin transport channels that include both isotropic and in-plane anisotropic spin relaxation regions. We analyze how different geometric configurations and spin transport parameters influence the lineshape of spin precession signals under magnetic fields applied in different orientations and compare with experimental observations. Our results introduce a theoretical framework for interpreting spin transport measurements in lateral graphene spin devices. The framework is especially relevant when the graphene is partially proximitized by other two-dimensional materials, where proximity-induced spin-orbit coupling leads to anisotropic spin relaxation.

The flexural strength variability of α-SiC based ceramics at elevated temperatures creates the need for an Integrated Computational Materials Engineering (ICME) framework that relates the strength of a specimen directly to its manufacturing process. To create this ICME framework a model must first be developed which establishes a relationship between the chemical vapor infiltration (CVI) process and parameters, the resulting mesoscale pores, and the overall macroscale flexural strength. Here a nonlinear single pore model of CVI is developed used in conjunction with a four-way coupled themo-mechanical damage model. The individual components of the model are tested and a sample system under a four-point bending test is explored. Results indicate that specimens with an initial porosity greater than 30% require temperatures below 1273 K to maintain structural integrity, while those with initial porosities less than 30% are temperature-independent, allowing for optimization of the CVI processing time without compromising strength.

The progression of chronic diseases often follows highly variable trajectories, and the underlying factors remain poorly understood. Standard mixed-effects models typically represent inter-patient differences as random deviations around a common reference, which may obscure meaningful subgroups. We propose a probabilistic mixture extension of a mixed effects model, the Disease Course Mapping model, to identify distinct disease progression subtypes within a population. The mixture structure is introduced at the latent individual parameters, enabling clustering based on both temporal and spatial variability in disease trajectories. We evaluated the model through simulation studies to assess classification performance and parameter recovery. Classification accuracy exceeded 90% in simpler scenarios and remained above 80% in the most complex case, with particularly high recall and precision for fast-progressing clusters. Compared to a post hoc classification approach, the proposed model yielded more accurate parameter estimates, smaller biases, lower root mean squared errors, and reduced uncertainty. It also correctly recovered the true three-cluster structure in 93% of the simulations. Finally, we applied the model to a longitudinal cohort of CADASIL patients, identifying two clinically meaningful clusters, differentiating patients with early versus late onset and fast versus slow progression, with clear spatial patterns across motor and memory scores. Overall, this probabilistic mixture framework offers a robust, interpretable approach for clustering patients based on spatiotemporal disease dynamics.

Small-scale vortical motions in the upper solar atmosphere are abundant and occupy about 2.8% of the photosphere at any given time. Although considerable work has focused on the detection and analysis of individual solar vortices, the interconnected and multi-scale behaviour of these coherent structures remains largely unexplored. We present a methodology for studying this behaviour through vortex interactions, to improve our understanding of how small- and large-scale photospheric flows contribute to energy transfer into the upper solar atmosphere and to the driving of solar activity. We represent vortices as a network of interacting structures. We apply a community detection algorithm to derive an optimal reduced network composed of highly interconnected vortex groups. From the interaction patterns and group structure, we define three roles within each community: peripheral, connector and hub. We then track both vortex communities and their member vortices from the photosphere into the chromosphere and across their lifetimes. On average, vortices assigned to these roles persist to greater heights in the chromosphere and have longer lifetimes than unclassified vortices. This shows that community detection can identify vortices with greater dynamical influence on the upper solar atmosphere. We also find that 32% to 58.6% of vortex communities exhibit global periodic behaviour following a helical path. This collective vortical motion may indicate an enhanced mechanism for wave excitation. Solar vortical community detection, therefore, offers a new framework for studying solar vortices and a new perspective on the importance of collective vortex dynamics.

We present near-infrared (NIR) observations of the 2021 eruption of the recurrent nova RS Ophiuchi. The dataset provides both pre- and post-eruption perspectives on the eruption, as well as NIR spectra at high cadence. The spectrum obtained in 2020 June (14.3 years after the 2006 eruption, and 428.1 days before the 2021 eruption), is that of the red giant secondary, on which are superimposed several emission lines which most likely arise in the red giant wind. Spectra obtained during the eruption consist of emission (including coronal) lines, superimposed on a bremsstrahlung continuum at 8900K. The temperature of the coronal gas is estimated to be $10^{6.0}$K on day 11.7, and $10^{5.9}$K on day 31.7. The high cadence observations, obtained on day 31.7 of the eruption, provide no conclusive evidence for rapid ($<\sim1$~minute) variations in the HeI 1.0833$\,μ$m line. Data obtained about one year after the eruption show that there may have been changes in the spectral type of the secondary after the 2021 eruption.

We present an alternative measurement of the Baryon Acoustic Oscillation (BAO) using Legendre multipole representation of the Ly$α$ forest correlation functions from the second data release (DR2) of the Dark Energy Spectroscopic Instrument survey. Compressing the auto- and cross-correlation functions into Legendre multipoles yields a positive-definite covariance matrix without any smoothing -- unlike the baseline DR2 analysis -- thanks to a significantly reduced data vector size. We introduce the statistical corrections required to debias the finite-sample covariance matrix estimate and demonstrate that monopole and quadrupole terms for both auto- and cross-correlations can be used even when the correlation functions are distorted by continuum errors and contaminated by metals. This formalism has slightly diminished the constraining power of the BAO scale, while considerably weakening constraints on nuisance parameters. We measure the isotropic BAO scale with $0.96\%$ precision at $z_\mathrm{eff}=2.35$, the Hubble parameter $H(z_\mathrm{eff})=(238.7\pm3.4)~(147.09~\mathrm{Mpc}/r_d) ~\mathrm{km~s}^{-1}~\text{Mpc}^{-1}$, and the transverse comoving distance $D_M(z_\mathrm{eff})=(5.79 \pm 0.10)~(r_d/147.09~\mathrm{Mpc})$~Gpc for a given value of the sound horizon ($r_d$). Our BAO results are entirely consistent with the baseline DR2 analysis.

This paper studies maintenance optimization for a two-component system under mixed observability. Component~$U_1$ is fully monitored, whereas component~$U_2$ is only partially observable due to sensing limitations. The system exhibits unidirectional positive degradation dependence, in which the health state of component~$U_1$ influences the degradation process of component~$U_2$, but not vice versa. We propose a novel framework for modeling and optimizing maintenance decisions for such systems using a partially observable Markov decision process (POMDP). Under mild conditions, we analytically establish structural properties of the optimal maintenance policy. Baum-Welch algorithm with multiple sample paths is developed to estimate the unknown system parameters in the context of a covariate-dependent Hidden Markov Model. %from observational data with multiple trajectories. Numerical experiments demonstrate the effectiveness of the proposed parameter estimation and the maintenance policy. Across 64 instances, we show that it consistently outperforms classical threshold-based policies. Specifically, when the degradation of component $U_1$ is faster, it achieves maximal cost reductions of up to approximately $6\%$

Strong gravitational lensing provides an independent and powerful probe of cosmic expansion by directly linking observables to cosmological distances. Upcoming surveys such as LSST will discover large number of galaxy-galaxy strong lensing systems, offering a new route to precise cosmological constraints. In this paper, we propose a Fisher-like sensitivity factor to map how the cosmological information of strong-lensing distances changes across the lens-source redshift plane. Applying such factor to the distance ratio $D_{ls}/D_s$, the time-delay distance $D_{Δt}$, and the double-source-plane ratio, we determine the ``sensitivity valleys'' where an observable becomes insensitive to a given parameter. The realistically simulated LSST lens population, which largely lies outside the distance-ratio valleys, covers the most sensitive region for $(w_0,w_a)$ parameter space. We then develop a new hierarchical framework, which could calibrate the redshift evolution of lens mass-density slopes and constrain cosmological parameters simultaneously. Focusing on the LSST mock data, we demonstrate that ignoring mass-profile evolution can bias $Ω_m$ by up to $\sim 10σ$, while modeling the lens evolution could perfectly recovers the fiducial cosmology and yield stringent cosmological constraints (e.g., $ΔΩ_m \simeq 0.01$ and $Δw \simeq 0.1$ for $\sim 10^4$ lenses).

This paper develops a comprehensive Markov-based framework for modelling reservoir behaviour and assessing key performance measures such as reliability and resilience. We first formulate a stochastic model for a finite-capacity dam, analysing its long-term storage dynamics under both independent and identically distributed inflows, following the Moran model, and correlated inflows represented by an ergodic Markov chain in the Lloyd formulation. For this finite case, we establish stationary water balance relations and derive asymptotic results, including a central limit theorem for storage levels. The analysis is then extended to an infinite-capacity reservoir, for which normal limit distributions and analogous long-term properties are obtained. A continuous-state formulation is also introduced to represent reservoirs with continuous inflow processes, generalizing the discrete-state framework. On this basis, we define and evaluate reliability and resilience metrics within the proposed Markovian context. The applicability of the methodology is demonstrated through a real-world case study of the Quiebrajano dam, illustrating how the developed models can support efficient and sustainable reservoir management under hydrological uncertainty.

We introduce inference methods for score decompositions, which partition scoring functions for predictive assessment into three interpretable components: miscalibration, discrimination, and uncertainty. Our estimation and inference relies on a linear recalibration of the forecasts, which is applicable to general multi-step ahead point forecasts such as means and quantiles due to its validity for both smooth and non-smooth scoring functions. This approach ensures desirable finite-sample properties, enables asymptotic inference, and establishes a direct connection to the classical Mincer-Zarnowitz regression. The resulting inference framework facilitates tests for equal forecast calibration or discrimination, which yield three key advantages. They enhance the information content of predictive ability tests by decomposing scores, deliver higher statistical power in certain scenarios, and formally connect scoring-function-based evaluation to traditional calibration tests, such as financial backtests. Applications demonstrate the method's utility. We find that for survey inflation forecasts, discrimination abilities can differ significantly even when overall predictive ability does not. In an application to financial risk models, our tests provide deeper insights into the calibration and information content of volatility and Value-at-Risk forecasts. By disentangling forecast accuracy from backtest performance, the method exposes critical shortcomings in current banking regulation.

Parity-odd transport is a central signature of chiral fluids, yet analytical predictions are sparse. Here, we introduce a minimal two-dimensional hard-disk gas in which chirality arises solely from a collision-induced transverse impulse. Motivated by granular spinners, collisions are dissipative and inject orbital angular momentum through a fixed tangential ``kick'' at contact. Starting from a Boltzmann-Enskog description, we derive nonlinear hydrodynamic equations for density, momentum, and temperature, and show that chirality generates an antisymmetric homogeneous stress corresponding to a nonzero torque density. In the dilute limit, a Chapman-Enskog expansion yields analytical predictions for transport coefficients, including odd viscosity, odd thermal conductivity, and odd self-diffusivity, in good agreement with numerical simulations. This minimal kinetic model can serve as a foundation for systematic coarse-graining of chiral fluids and as a tractable benchmark for gaining insight into odd transport across a broader class of chiral systems.

The interfacial diffusion associated with finite volume method (FVM) discretizations of multiphase flows creates the need for an interface sharpening mechanism. Such solutions for structured quadrilateral grids are well documented, but various engineering applications require mesh designs specific to the irregular geometry of the physical system it is modeling. Therefore this study casts interface sharpening as an ant-idiffusive volumetric body force whose calculation procedure is generalizable to an arbitrarily constructed grid. The force magnitude is derived at cell centers as a function of the local compressible flow characteristics and the geometry of the cell neighborhood. The flow model uses an AUSM+up based method for flux evaluation and imposes a stiffened equation of state onto each of the fluids in order to close the linear system and extract auxiliary variables. Validation tests show good agreement with the Young-Laplace condition whereby the interface converges to the analytical solution corresponding to a balance between a pressure jump and interfacial forces. Further results show the recovery of a circle starting from a shape with highly variational curvature through the combined effects of surface tension and interface sharpening. Lastly shear-driven droplet pinchoff results show good agreement with droplet shapes provided by the surrounding literature at various Weber-Ohnesorge number combinations.

We present a novel method for the tomographic reconstruction of the bias-weighted mean electron pressure $\langle bP_e \rangle$ and star formation rate density $\langle bρ_{\mathrm{SFR}} \rangle$, by simultaneously modelling the contribution from the thermal Sunyaev-Zel'dovich (tSZ) effect and the Cosmic Infrared Background (CIB) to the cross-correlation between photometric galaxy samples and multi-frequency Cosmic Microwave Background (CMB) maps. The resulting measurements are independent of the galaxy clustering properties and robust against cross-contamination between tSZ and CIB. Applying this method to publicly available data, we reconstruct the cosmic evolution of $\langle bP_e \rangle$ and $\langle bρ_{\mathrm{SFR}} \rangle$ out to $z\sim1$, making our measurements publicly available. Our measurements of both quantities are broadly compatible with predictions from the fiducial FLAMINGO hydrodynamical simulation, although we observe a lower gas pressure at low redshifts, in agreement with other measurements.

We characterize the extreme and exposed points of the unit ball (with respect to the $L^1$-norm) in the shift-invariant space generated by the Gaussian function, as well as in the quasi shift-invariant space generated by the hyperbolic secant.

Wave-optics effects in strongly lensed gravitational waves (GWs) provide a new interferometric probe of dark matter substructure. We compute the full diffraction integral for GWs propagating through statistically generated cold dark matter subhalo populations and quantify the resulting frequency-dependent amplification in the Laser Interferometer Space Antenna (LISA) band. We show that realistic galaxy-scale lenses generically produce percent-level amplitude and phase distortions in strongly magnified images, primarily induced by subhalos in the mass range $10^4$-$10^7\,M_{\odot}$. These signatures arise naturally within the standard cold dark matter paradigm and should be detectable in high signal-to-noise LISA events. Strongly lensed GWs thus offer a direct and complementary window on dark matter structure at subgalactic mass scales inaccessible to electromagnetic measurements.

Since 2019, three scintillator detectors of the EEE collaboration have been continuously measuring cosmic muon rates at 78.9°N at the Ny-Ålesund Research Station (Svalbard). The resulting six-year time series reveals a pronounced annual modulation, driven primarily by seasonal atmospheric variations. Utilizing routine radiosonde profiles collected above the same site, we applied several established techniques --along with a tailored analysis approach-- to investigate the relationship between muon rate and atmospheric temperature. The temperature-corrected muon-rates are analysed using the Lomb-Scargle periodogram technique in order to investigate the presence of remaining periodic structures. Finally, the temperature corrections coefficients of our analysis are compared with measurements in other stations located at lower latitudes.

Lorentz invariance is a fundamental symmetry underlying both the Standard Model of particle physics and General Relativity. Testing its validity provides a direct means of searching for new physics emerging near the Planck scale. A search for isotropic Lorentz invariance violation with 1.4 years of atmospheric neutrino data collected by a partial configuration of the KM3NeT/ORCA detector comprising six detection units is presented. No evidence for such violation is found; thus, competitive limits are set on a subset of isotropic Lorentz invariance violating coefficients, which complement and extend existing experimental constraints.

We develop a quantum kinetic theory for QCD, which incorporates all leading order collision terms. At lowest order in gradient expansion, it reproduces the spin-averaged Boltzmann equation with both elastic and inelastic collisions. At next order in gradient expansion, the solution to the quantum kinetic equations give spin polarization of on-shell quarks and gluons in quark-gluon plasma when the gradients are of hydrodynamic ones. A power counting in the coupling shows the spin polarization behaves differently in vortical and non-vortical gradients: the former is free of collisional contribution to leading order, while the latter contains a collisional contribution at parametrically the same order as the free theory counterpart. We also find the inelastic collision in a spin basis provides a possible mechanism for conversion between spin and orbital angular momentum.

Two-dimensional particle-in-cell simulations with a realistic mass ratio reveal the generation mechanisms of the out-of-plane magnetic field in magnetic islands/flux ropes during magnetic reconnection. In the absence of an initial guide field, reconnection produces a large electron temperature anisotropy (around 4.5) inside magnetic islands that drives the Weibel instability. Strong out-of-plane magnetic fields (Bz/B0 around 0.4, greatly exceeding the Hall field) with a regular bipolar structure grow inside islands. A space-time analysis reveals a one-to-one correspondence between the temperature anisotropy and the development of the Weibel magnetic field. The instability relaxes the anisotropy, but island merging leads to anisotropy reemergence and re-excitation. In the presence of a strong ambient guide field (Bg/B0 = 0.5), the electron outflow from the X-point deflects along the separatrices and forms a circular current loop wrapping the flux ropes. This flux-rope separatrix current generates an out-ofplane magnetic field that reinforces the ambient guide field, reaching Bz/B0 around 1.4. The current can, in some cases, drive the electron Kelvin-Helmholtz instability, which produces electron vortices and strengthens the magnetic field. Mergers significantly broaden the islands and further strengthen the field. These self-generated out-of-plane magnetic fields scatter electrons and reduce their temperature anisotropy, which can potentially affect electron heating via Fermi reflection. The simulation results are supported by spacecraft observations suggesting that ambient guide fields can be enhanced within flux ropes in Earth's magnetotail.

Accurate estimation of Marginal Emission Factors (MEFs) is critical for evaluating the decarbonization potential of low-carbon technologies and demand-side management. However, canonical methodologies, predominantly relying on linear regression and differencing techniques, fail to capture the structural non-linearities inherent in the merit order, i.e. the marginal technology setting electricity prices. Utilizing Markov switching autoregressive models with exogenous regressors (MS-ARX) and hourly US data (2019-2025), we identify distinct, mutually exclusive regimes governed by fuel-price dynamics. We find that linear models overestimate abatement potential by masking the dichotomy between a gas-driven and coal-driven marginal system. Furthermore, using robust structural break detection, we link regime instability to a specific structural shift in natural gas pricing in May 2022. Our results indicate that post-2022, the grid has transitioned into a correction phase where the coal-driven regime is less persistent but highly volatile, necessitating state-dependent policy metrics rather than static annual averages.

In this paper we introduce conformal heat flow of (extrinsic) biharmonic maps on $4$-manifold, simply called bi-conformal heat flow (bi-CHF), and study its properties. Similar to other CHF of harmonic maps and regularized $n$-harmonic maps, (CHF and regularized $n$-CHF respectively), we obtain global smoothness and no finite time singularity.

A symbolic determinant under rank-one restriction computes a polynomial of the form $\det(A_0+A_1y_1+\ldots+A_ny_n)$, where $A_0,A_1,\ldots,A_n$ are square matrices over a field $\mathbb{F}$ and $rank(A_i)=1$ for each $i\in[n]$. This class of polynomials has been studied extensively, since the work of Edmonds (1967), in the context of linear matroids, matching, matrix completion and polynomial identity testing. We study the following learning problem for this class: Given black-box access to an $n$-variate polynomial $f=\det(A_0+A_1y_1+ \ldots+A_ny_n)$, where $A_0,A_1,\ldots,A_n$ are unknown square matrices over $\mathbb{F}$ and rank$(A_i)=1$ for each $i\in[n]$, find a square matrix $B_0$ and rank-one square matrices $B_1,\ldots,B_n$ over $\mathbb{F}$ such that $f=\det(B_0+B_1y_1+\ldots+B_ny_n)$. In this work, we give a randomized poly(n) time algorithm to solve this problem. As the above-mentioned class is known to be equivalent to the class of read-once determinants (RODs), we will refer to the problem as learning RODs. The algorithm for learning RODs is obtained by connecting with a well-known open problem in linear algebra, namely the Principal Minor Assignment Problem (PMAP), which asks to find (if possible) a matrix having prescribed principal minors. PMAP has also been studied in machine learning to learn the kernel matrix of a determinantal point process. Here, we study a natural black-box version of PMAP: Given black-box access to an $n$-variate polynomial $f = \det(A + Y)$, where $A \in \mathbb{F}^{n \times n}$ is unknown and $Y = diag(y_1,\ldots,y_n)$, find a $B\in\mathbb{F}^{n\times n}$ such that $f=det(B+Y)$. We show that black-box PMAP can be solved in randomized poly(n) time, and further, it is randomized polynomial-time equivalent to learning RODs. We resolve black-box PMAP by investigating a property of dense matrices that we call the rank-one extension property.

Frustrated kagome magnets provide a fertile platform for unconventional collective quantum phenomena, yet the role of lattice distortion in reorganizing magnetic degrees of freedom and controlling low-energy physics remains poorly understood. Here we report a rare realization of dimensional reduction in the distorted kagome material $\mathrm{YCa_3(CrO)_3(BO_3)_4}$, combining thermodynamic experiments with first-principles calculations and large-scale Monte Carlo simulations. Magnetic susceptibility and specific heat show no signatures of spin freezing or long-range magnetic order down to $65~\mathrm{mK}$ despite strong antiferromagnetic interactions. Instead, the susceptibility exhibits a broad maximum characteristic of quasi-one-dimensional spin correlations, while the magnetic specific heat follows a robust power law $C_{\mathrm{mag}}\sim T^2$ over more than a decade in temperature that remains unchanged in applied magnetic fields. This field-independent scaling rules out impurity or conventional magnon contributions and points to a collective low-energy excitation spectrum governed by frustration and local constraints. We show that a strongly hierarchical exchange network reorganizes the system into local antiferromagnetic dimers and weakly coupled spin chains, with frustrated inter-unit couplings suppressing three-dimensional order to ultralow temperatures. Our results demonstrate how a hierarchy of competing exchange interactions can reorganize a frustrated three-dimensional magnet into effectively lower-dimensional correlated units, stabilizing extended regimes of quantum-disordered behavior in realistic materials.

Experimentation is central to modern digital businesses, but many operational decisions cannot be randomized at the user level. In such cases, cluster-level experiments, where clusters are usually geographic, come to the rescue. However, such experiments often suffer from low power due to persistent cluster heterogeneity, strong seasonality, and autocorrelated outcome metrics, as well as common shocks that move many clusters simultaneously. On an example of airline pricing - where policies are typically applied at the route level and thus the A/B test unit of analysis is a route - we study switchback designs to remedy these problems. In switchback designs, each cluster (route in our case) alternates between treatment and control on a fixed schedule, creating within-route contrasts that mitigate time-invariant heterogeneity and reduce sensitivity to low-frequency noise. We provide a unified Two-Way Fixed Effects interpretation of switchback experiments that makes the identifying variation explicit after partialling out route and time effects, clarifying how switching cadence interacts with temporal dependence to determine precision. Empirically, we evaluate weekly and daily switchback cadences using calibrated synthetic regimes and operational airline data from ancillary pricing. In our evaluations, switchbacks decrease standard errors by up to 67%, with daily switching yielding the largest gains over short horizons and weekly switching offering a strong and simpler-to-operationalize alternative.

We study the occurrence of long-lived metastable states in the 4f$^{13}$5d6s electron configuration of Yb$^+$. By optical pumping of a single trapped ion on the $^2F^\text{o}_{7/2}\rightarrow (7/2,0)_{7/2}$ transition at 377.5 nm, we prepare a wide range of metastable electronic states. We use a co-trapped control ion to sympathetically cool the spectroscopy ion, allowing us to accurately time its subsequent decay. We record a strong decay signal corresponding to a lifetime of 0.92(8) s, a weaker decay signal with lifetime 9.8(+2.9, -2.0) s, and find evidence for a much longer lifetime, $>$ 30 s. We identify the metastable states with these lifetimes qualitatively, and corroborate our results with atomic structure calculations that support the observed lifetimes and decay paths. These long-lived states provide new opportunities in qubit and qudit state detection and optical clocks.

We highlight recent progress in the study of artificial flat band systems with a threefold focus. First, we discuss single-particle flat band physics, which has advanced through the design of various flat band generators. These generators rely on the classification of flat bands in terms of compact localized states - their fundamental building blocks. A related development is the complete real-space description of flat band projectors. Next, we review studies on perturbations of flat bands, which provide new insights into the effects of disorder and, more importantly, the intricate interplay between many-body interactions and flat band physics. Finally, we survey the growing number of experimental realizations of flat bands across diverse physical platforms.

Producing reliable estimates of health and demographic indicators at fine areal scales is crucial for examining heterogeneity and supporting localized health policy. However, many surveys release outcomes only at coarser administrative levels, thereby limiting their relevance for decision-making. We propose a fully Bayesian, single-stage spatial modeling framework for area-level disaggregation that generates fine-scale estimates of indicators directly from coarsely aggregated survey data. By defining a latent spatial process at the target resolution and linking it to observed outcomes through an aggregation step, the framework adopts small-area estimation techniques while incorporating covariates and delivering coherent uncertainty quantification. The proposed methods are implemented with inlabru to achieve computational efficiency. We evaluate performance through a simulation study of general fertility rates in Kenya to demonstrate the models' ability to recover fine-scale variation across diverse data-generating scenarios. We further apply the framework to two national surveys to produce district-level fertility estimates from the 2022 Kenya Demographic and Health Survey and, more importantly, district-level indicators for unpaid care and domestic work and mass media usage from the 2021 Kenya Time Use Survey.

Droplet microfluidics combined with optical detection has become a powerful approach for high-throughput single-cell assays, but these systems often face limited sensitivity and signal heterogeneity due to optical and geometrical constraints. We investigate how key operating parameters influence the performance of a droplet-based optomicrofluidic platform. Experiments examine optical interactions between guided light and aqueous droplets containing fluorescent (FL) particles flowing in oil. Geometrical optics simulations model light-droplet interactions, while FL simulations quantify signal variations caused by particle size and position. Two refracted signals are observed experimentally: a droplet-refracted signal (DRS) that scales with droplet diameter and a particle-refracted signal (PRS) produced by light interaction with encapsulated particles. Both experiments and simulations show that PRS becomes prominent when the particle-to-droplet size ratio $D^*_\text{p}$ lies between 0.23-0.33, enabling label-free detection. Particles near the droplet center ($r^*_\text{p} < 0.4$) display reduced angular dependence and more uniform FL signals. Simulations further show that FL intensity increases with $D^*_\text{p}$, rising sharply from 0.33 to 0.5 and more gradually up to 0.66. Additionally, reducing the oil layer thickness enhances fluorescence by minimizing optical losses at the droplet-channel interface. These results demonstrate that controlling $D^*_\text{p}$, particle position, and oil layer thickness improves FL strength and uniformity, providing a framework for optimizing droplet-based fluorescence detection in microflow cytometry and single-cell assays.

All utility-scale quantum computers will require some form of Quantum Error Correction in which logical qubits are encoded in a larger number of physical qubits. One promising encoding is known as the colour code which has broad applicability across all qubit types and can decisively reduce the overhead of certain logical operations when compared to other two-dimensional topological codes such as the surface code. However, whereas the surface code decoding problem can be solved exactly in polynomial time by finding minimum weight matchings in a graph, prior to this work, it was not known whether exact and efficient colour code decoding was possible. Optimism in this area, stemming from the colour code's significant structure and well understood similarities to the surface code, fanned this uncertainty. In this paper we resolve this, proving that exact decoding of the colour code is NP-hard -- that is, there does not exist a polynomial time algorithm unless P=NP. This highlights a notable contrast to some of the colour code's key competitors, such as the surface code, and motivates continued work in the narrower space of heuristic and approximate algorithms for fast, accurate and scalable colour code decoding.