We develop tools to study Picard groups of quotients of ring spectra by a finitely generated ideal, which we use to show that $\mathrm{Pic}(\mathrm{E}_n/I) = \mathbb{Z}/2$, where $\mathrm{E}_n$ is a Lubin--Tate theory and $I$ is an ideal generated by suitable powers of a regular sequence. We apply this to obtain spectral sequences computing Picard groups of $\mathrm{K}(n)$-local generalized Moore algebras, and make some preliminary computations including the height $1$ case.

When AI systems allow human-like communication, they elicit increasingly complex relational responses. Knowledge workers face a particular challenge: They approach these systems as tools while interacting with them in ways that resemble human social interaction. To understand the relational contexts that arise when humans engage with anthropomorphic conversational agents, we need to expand existing human-computer interaction frameworks. Through three workshops with qualitative researchers, we found that the fundamental ontological and relational ambiguities inherent in anthropomorphic conversational agents make it difficult for individuals to maintain consistent relational stances toward them. Our findings indicate that people's articulated positioning toward such agents often differs from the relational dynamics that occur during interactions. We propose the concept of relational dissonance to help researchers, designers, and policymakers recognize the resulting tensions in the development, deployment, and governance of anthropomorphic conversational agents and address the need for relational transparency.

Casselgren, Markstörm, and Pham conjectured that any precolored dis\-tan\-ce-2 matching in the $d$-dimensional cube $Q_d$ with at most $d$ colors can be extended to a proper $d$-edge-coloring. In this paper, we prove this conjecture and some related theorems. Especially, our result establishes that if $G$ is a bipartite graph, then a precolored distance-2 matching in the Cartesian product $H = G \mathbin{\Box} K_{2m}$ with at most $χ'(H) = Δ(H) = Δ(G) + 2m - 1$ colors can be extended to an edge-coloring using at most $χ'(H)$ colors. As another generalization, we establish a similar result for the Cartesian product $G \mathbin{\Box} K_{1,m}$.

Dynamical quantum phase transitions (DQPTs) are a class of non-equilibrium phase transitions that occur in many-body quantum systems during real-time evolution, rather than through parameter tuning as in conventional phase transitions. This paper presents an exact analytical approach to studying DQPTs by combining complex dynamics with the real-space renormalization group (RG). RG transformations are interpreted as iterated maps on the complex plane, establishing a connection between DQPTs and the Julia set, the boundary separating the basins of attraction of the stable fixed points. This framework is applied to a quantum quench in the one-dimensional transverse field Ising model, where we examine the sensitivity of DQPTs to variations in boundary conditions. We show that altering the topology of the spin chain can suppress DQPTs and provide a qualitative explanation based on quantum speed limits.

We study Krylov complexity in Schrödinger field theory in the grand canonical ensemble with chemical potential $μ$, with an emphasis on the qualitatively new features that arise for $μ>0$. In this regime the fermionic Wightman power spectrum is effectively single-sided and sharply truncated at $ω=μ$, which induces a crossover in the Lanczos coefficients {and signals a dynamical transition from a bulk-dominated regime to a spectral-edge-dominated regime}: $b_n$ displays a two-stage linear growth (from an early-time slope $π/β$ to an asymptotic slope $2/β$), while $a_n$ bends from near-zero values to a linear descent with slope $-4/β$. We provide analytic support for the resulting complexity growth from three complementary viewpoints: (i) using an $SL(2,\mathbb{R})$ algebraic construction matched to the asymptotic Lanczos data, we show that the late-time Krylov complexity must grow quadratically, $K(t)\propto t^{2}$; (ii) by analyzing engineered Wightman spectra with controlled decay and truncation, we identify single-sided exponential decay as the key spectral feature responsible for the quadratic asymptotics, while an approximately even two-sided exponential spectrum explains the early-time $K(t)\sim\sinh^{2}(πt/β)$ behavior at large $μ$; (iii) we formulate the problem in terms of orthogonal polynomials and estimate the crossover scale separating the early- and late-stage regimes. Overall, our results help clarify the role of chemical potential and spectral truncation in shaping operator growth and Krylov complexity in this non-relativistic quantum field theory setting.

Peer review remains the central quality-control mechanism of science, yet its ability to fulfill this role is increasingly strained. Empirical studies document serious shortcomings: long publication delays, escalating reviewer burden concentrated on a small minority of scholars, inconsistent quality and low inter-reviewer agreement, and systematic biases by gender, language, and institutional prestige. Decades of human-centered reforms have yielded only marginal improvements. Meanwhile, artificial intelligence, especially large language models (LLMs), is being piloted across the peer-review pipeline by journals, funders, and individual reviewers. Early studies suggest that AI assistance can produce reviews comparable in quality to humans, accelerate reviewer selection and feedback, and reduce certain biases, but also raise distinctive concerns about hallucination, confidentiality, gaming, novelty recognition, and loss of trust. In this paper, we map the aims and persistent failure modes of peer review to specific LLM applications and systematically analyze the objections they raise alongside safeguards that could make their use acceptable. Drawing on emerging evidence, we show that targeted, supervised LLM assistance can plausibly improve error detection, timeliness, and reviewer workload without displacing human judgment. We highlight advanced architectures, including fine-tuned, retrieval-augmented, and multi-agent systems, that may enable more reliable, auditable, and interdisciplinary review. We argue that ethical and practical considerations are not peripheral but constitutive: the legitimacy of AI-assisted peer review depends on governance choices as much as technical capacity. The path forward is neither uncritical adoption nor reflexive rejection, but carefully scoped pilots with explicit evaluation metrics, transparency, and accountability.

In order to fully utilize the technological potential of unconventional superconductors, an enhanced understanding of the superconducting mechanism is necessary. In the best performing superconductors, the cuprates, superconductivity is intimately linked with magnetism, although the details of this coupling remain elusive. In search of clarity in the magnetism-superconductivity relationship, we focus on the electron-doped cuprate $\mathrm{Nd}_{1.85}\mathrm{Ce}_{0.15}\mathrm{CuO}_{4-δ}$ (NCCO). NCCO has an antiferromagnetic ground state when synthesized, and only becomes superconducting after a reductive annealing process. This makes NCCO an ideal template to study how the magnetism differs in the superconducting and non-superconducting state, while keeping the material template as constant as possible. Using neutron spectroscopy, we reveal that the as-grown crystal exhibits a large spin pseudogap in the magnetic fluctuation spectrum. Upon annealing, defects that are introduced by the commonly employed synthesis method are removed and the spin pseudogap is significantly reduced. While the spin pseudogap in the annealed sample is likely an effect of superconductivity, we argue that the spin pseudogap in the as-grown sample is caused by the absence of long-wavelength spin waves. The defects in as-grown NCCO thus play the dual role of suppressing both superconductivity and low-energy spin waves, highlighting a potential connection between these two phenomena.

Quantum key distribution (QKD) using entangled photon sources (EPS) is a cornerstone of secure communication. Despite rapid advances in QKD, conventional protocols still employ beam splitters (BSs) for passive random basis selection. However, BSs intrinsically suffer from photon loss, imperfect splitting ratios, and polarization dependence, limiting the key rate, increasing the quantum bit error rate (QBER), and constraining scalability, particularly over long distances. By contrast, EPSs based on spontaneous parametric down-conversion (SPDC) intrinsically exhibit quantum randomness in spatial and spectral degrees of freedom, offering a natural replacement for BS-based basis selection. Here, we demonstrate a proof-of-concept QKD scheme that exploits the intrinsic spatial randomness of SPDC without employing beam splitters. The annular SPDC emission ring is divided into four spatial sections, effectively generating two independent EPSs whose photon pairs are distributed to Alice and Bob. Crucially, the measurement basis is not predetermined but is assigned after photon detection by exploiting intrinsic detector timing jitter, thereby concealing the basis information from a potential eavesdropper. This post-detection basis assignment emulates stochastic basis choice while avoiding BS-induced losses and bias. Experimentally, our scheme achieves a 6.4-fold enhancement in sifted key rate, a consistently reduced QBER, and a near-ideal encoding balance between linear and rectilinear bases. Furthermore, the need for four spatial channels can be avoided by employing wavelength demultiplexing to generate two EPSs at distinct wavelength pairs. Harnessing intrinsic spatial/spectral randomness thus enables robust, bias-free, high-rate, and low-QBER QKD, offering a scalable pathway for next-generation quantum networks.

This paper investigates a coupled two-qubits heat engine fueled by generalized measurements of the spin components and using a single heat reservoir as sink. Our model extends the proposal of Yi and coworkers [Phys. Rev. E {\bf 96}, 022108 (2017)] where the role of a hot reservoir in a four-stroke cycle was replaced by a quantum measurement apparatus, the other steps being two quantum adiabatic strokes and thermalization with a cold reservoir. We propose a five-stroke cycle, where an ergotropy extracting stroke is introduced following the measurement stroke, and study the effect of measurements of different spin components on the performance of the machine. For measurements along z-z directions, we find two possible occupation distributions that yield an active state and the ergotropic stroke improves the performance of the engine over the four-stroke cycle. Further, the three-stroke engine ( {without the adiabatic strokes}) yields the same performance as the five-stroke engine. For arbitrary working medium and non-selective measurements, we prove that the total work output of a five-stroke engine is equal to the sum of the work outputs of the corresponding four-stroke and three-stroke engines. For measurement directions other than z-z, there may be many possible orderings of the post-measurement probabilities that yield an active state. However, as we illustrate, for specific cases (e.g. x-x), a definite ordering may be obtained with the projective measurements. Thus, we find that the five-stroke engine exploiting ergotropy outperforms both its four-stroke as well as three-stroke counterparts.

Due to the limited qubit number of quantum devices, distributed quantum computing is considered a promising pathway to overcome this constraint. In this paradigm, multiple quantum processors are interconnected to form a cohesive computational network, and the most natural set of free operations is local operations and classical communication (LOCC). However, designing a practical LOCC protocol for a particular task has been a tough problem. In this work, we propose a general and flexible framework called dynamic LOCCNet (DLOCCNet) to simulate and design LOCC protocols. We demonstrate its effectiveness in two key applications: entanglement distillation and distributed state discrimination. The protocols designed by DLOCCNet, in contrast to conventional ones, can solve larger-sized problems with reduced training time, making the framework a practical and scalable tool for current quantum devices. This work advances our understanding of the capabilities and limitations of LOCC while providing a powerful methodology for protocol design.

Using the multisymplectic Hamiltonian formalism, we propose a Poisson bracket for the electromagnetic field that, in addition to satisfying the restricted principle of relativity, reproduces well-established results from the standard formulations of field theories. To this end, it is necessary to translate the multisymplectic description of the system to the field's momentum representation, where a bilinear and Lorentz-covariant form emerges in a straightforward manner. Consequently, we establish a description for a system consisting of an electromagnetic field and a point particle interacting with each other that is based on a single Lorentz-covariant Poisson bracket with respect to all the canonical variables of the system. The results obtained from this proposal offer a solid foundation for the development of a Lorentz-covariant quantization process.

The ringdown phase of the binary black hole (BBH) merger provides a clean and direct probe of strong-field gravity and tests of the nature of black holes. The quasinormal mode (QNM) frequencies in modified gravity theories, as well as their amplitudes and phases, might deviate from the Kerr ones in general relativity. Charged black holes (BHs) in Einstein-Maxwell theory provide an excellent example of a beyond-Kerr solution with direct astrophysical and fundamental physics applications. In this work, we extract the ringdown mode amplitudes and phases for charged BBH mergers based on fully general relativistic simulations with charge-to-mass ratio up to 0.3. Our results suggest that even though the inspiral phase of charged BBHs can be significantly accelerated or decelerated, the ringdown mode excitation only changes mildly. We further explore the charge detectability with the ringdown-only signal for the Einstein Telescope and Cosmic Explorer. We find that previous studies may have overestimated the charge detectability and including higher modes in charged waveforms is necessary for future ringdown analysis. This constitutes the first such analysis based on waveforms generated by numerical relativity simulations of charged BHs in full Einstein-Maxwell theory.

We show that Jacobson's thermodynamic derivation of Einstein's equations remains valid when local Rindler horizons are treated as finite heat-capacity systems, resolving the unphysical infinite-bath assumption of standard Unruh thermodynamics. The resulting entropy takes the form of Rényi entropy with nonextensivity parameter $λ\sim C^{-1}$, or equivalently, a new "Einstein entropy" that exactly preserves the Einstein equations for all heat capacities. In both cases, the Unruh temperature is modified as \begin{equation*} T_\text{mod}=\frac{\hbarκ}{2π}\left(1+\frac{S}{C}\right), \end{equation*} establishing a universal link between finite-capacity thermodynamics and nonextensive entropy. We further obtain a corrected scalar Einstein equation with an upper bound on horizon energy flux, pointing to testable signatures in heavy-ion collisions, accelerator spin polarization, and analog gravity experiments. These results reinforce the robustness of the emergent-gravity paradigm and connect spacetime dynamics to generalized entropies of quantum information theory.

It is well-known that the trace-free Einstein tensor of a pseudo-Riemannian metric cannot arise by variation of a local diffeomorphism-invariant action functional with the (inverse) metric as field variable. We show that this statement remains true even for general local actions, without the assumption of diffeomorphism invariance.

We present realistic eccentricity distributions for extreme mass ratio inspirals (EMRIs) forming via the two-body relaxation channel in nuclear star clusters, tracking their evolution up to the final plunge onto the central Schwarzschild massive black hole (MBH). We find that EMRIs can retain significant eccentricities at plunge, with a distribution peaking at $e_\mathrm{pl} \approx0.2$, and a considerable fraction reaching much higher values. In particular, up to $20\%$ of the forming EMRIs feature $e_\mathrm{pl} > 0.5$ for central MBH masses $M_\bullet$ in the range $10^5 \mathrm{M_\odot} \leq M_\bullet \leq 10^6 \mathrm{M_\odot}$, partially due to EMRIs forming at large semi-major axes and "cliffhanger EMRI", usually neglected in literature. This highlights the importance of accounting for eccentricity in waveform modeling and detection strategies for future space-based gravitational wave observatories such as the upcoming Laser Interferometer Space Antenna (LISA). Furthermore, we find that the numerical fluxes in energy and angular momentum currently implemented in the FastEMRIWaveforms (FEW) package may not adequately sample the full parameter space relevant to low-mass MBHs ($M_\bullet < 10^6 \mathrm{M_\odot}$), potentially limiting its predictive power in that regime. Specifically, for $M_\bullet=10^5 \mathrm{M_\odot}$ we find that about $75\%$ ($50 \%$) of EMRIs at 2 years (6 months) from plunge fall outside the currently available flux parameter space. Our findings motivate the development of extended flux grids and improved interpolation schemes to enable accurate modeling of EMRIs across a broader range of system parameters.

Living systems often function with regulatory interactions, but the question of how activity, stochasticity and regulations work together for achieving different goals still remains puzzling. We propose a stochastic model of an active tracking particle with information processing, where the entropy production and information flow are discussed, with the generalised fluctuation theorem serving as a benchmark for verifying the probability setup. Based on the model, the system performance, in terms of the first passage steps and the total energy consumption, are analysed in the variable space of (measurement error, control field), leading to discussions on optimal controls of the system. Not only elucidating the basic concepts involved in a stochastic active system with information processing, this prototypical model could also inspire more elaborated modelings of natural smart organisms and industrial designs of controllable active systems with desired physical performances in the future.

We experimentally demonstrate that pairs of time-delayed ultrabright and ultrashort X-ray pulses of two different colors, delivered by modern X-ray Free Electron Lasers, can provide two time-delayed snapshots of a sample. We introduce Dichography, a method that algorithmically separates the diffraction signals overlapping on the detector and independently retrieves the two images of the specimen. We employ Dichography to reconstruct two views of individual xenon-doped helium nanodroplets with 20 nm spatial resolution. The consistency of structures observed in both images at delays up to 750 fs provides evidence that, under these illumination conditions, significant structural damage only occurs at longer timescales. We further validate the method by imaging pairs of silver nanoparticles intercepted by the same light pulse. Dichography enables a new class of experiments across physics, chemistry, and materials science, making a significant step toward the original promise of X-ray free-electron lasers to capture ultrafast movies of nanomatter.

Single-shot coherent diffractive imaging (CDI) with intense short-wavelength light pulses enables the structural characterization of individual nanoparticles in free flight with high spatial and temporal resolution. Conventional CDI assumes that the target object exhibits a linear scattering response and static electronic properties. Here, we extend this approach to investigate transient laser-driven modifications of the electronic structure in individual nanoparticles, imprinted in their time-resolved diffraction patterns. In the presence of a near-infrared laser pulse, we observe a pronounced reduction in the diffraction signal from helium nanodroplets when probed with ultrashort extreme ultraviolet (XUV) pulses. This effect is attributed to a light-field-induced modification of the electronic structure of the droplets, which substantially increases their XUV absorption. Our results demonstrate that single-particle diffraction can capture ultrafast light-driven electron dynamics in nanoscale systems. This paves the way for the spatiotemporal tracking of reversible changes in the electronic properties of nanoscale structures with potential applications in ultrafast X-ray optics, materials science, and all-optical signal processing.

Ambient air pollution measurements from regulatory monitoring networks are routinely used to support epidemiologic studies and environmental policy decision making. However, regulatory monitors are spatially sparse and preferentially located in areas with large populations. Numerical air pollution model output can be leveraged into the inference and prediction of air pollution data combining with measurements from monitors. Nonstationary covariance functions allow the model to adapt to spatial surfaces whose variability changes with location like air pollution data. In the paper, we employ localized covariance parameters learned from the numerical output model to knit together into a global nonstationary covariance, to incorporate in a fully Bayesian model. We model the nonstationary structure in a computationally efficient way to make the Bayesian model scalable.

We consider the impact on cosmological first-order phase transitions (FOPTs) of low-temperature thermal corrections to the effective potential. These are corrections from degrees of freedom whose field-dependent masses in the true vacuum are much larger than the nucleation temperature, though in the false vacuum the field-dependent masses may be much smaller than the nucleation temperature. Although the general form of these corrections to the thermal effective potential can be quite complicated, we argue that the net effect of all such corrections can be well-modeled with a single new parameter. We determine the shift in the parameters of the FOPT in terms of this new parameter, and the impact on gravitational wave signals and cosmological observables.

The use of atomic systems as accurate magnetic field probes requires precise characterization of the particle's magnetic properties. Insufficient knowledge of the spatial and temporal characteristics of the external magnetic field often limits the determination of the corresponding atomic parameters. Here, we present a quantum logic scheme mitigating systematic effects caused by temporal magnetic field fluctuations through simultaneous co-magnetometry. This allows measurement of the ground state $g$ factors of a single $^{48}$Ti$^+$ ion with uncertainties at the $10^{-6}$ level. We compare experimentally determined $g$ factors with new theoretical predictions using a combination of configuration interaction (CI) and second-order many-body perturbation theory (MBPT). Theory and experiment agree within the expected level of accuracy. The scheme can be applied to many atomic species, including those that cannot be directly laser cooled.

We study mean field games with unbounded coefficients. The existence of a solution is proved. We propose a new approach based on Fokker-Planck-Kolmogorov equations, the Ambrosio-Figalli-Trevisan superposition principle, the method of doubling variables and a~priory estimates with Lyapunov functions.

Asymptotic uniform confidence bands are constructed for a multivariate nonparametric regression model with heteroscedastic noise, employing histogram estimators under flexible partition conditions. The construction is especially applicable to unsmooth regression functions of Hölder regularity less than one. While the radius of the confidence bands could be approximated via the Gumbel distribution, our construction does not depend on an extreme value distribution, but instead can be explicitly calculated for the chosen partition.

Structured adaptive mesh refinement (AMR), commonly implemented via quadtrees and octrees, underpins a wide range of applications including databases, computer graphics, physics simulations, and machine learning. However, octrees enforce isotropic refinement in regions of interest, which can be especially inefficient for problems that are intrinsically anisotropic--much resolution is spent where little information is gained. This paper presents omnitrees as an anisotropic generalization of octrees and related data structures. Omnitrees allow to refine only the locally most important dimensions, providing tree structures that are less deep than bintrees and less wide than octrees. As a result, the convergence of the AMR schemes can be increased by up to a factor of the dimensionality d for very anisotropic problems, quickly offsetting their modest increase in storage overhead. We validate this finding on the problem of binary shape representation across 4,166 three-dimensional objects: Omnitrees increase the mean convergence rate by 1.5x, require less storage to achieve equivalent error bounds, and maximize the information density of the stored function faster than octrees. These advantages are projected to be even stronger for higher-dimensional problems. We provide a first validation by introducing a time-dependent rotation to create four-dimensional representations, and discuss the properties of their 4-d octree and omnitree approximations. Overall, omnitree discretizations can make existing AMR approaches more efficient, and open up new possibilities for high-dimensional applications.

We theoretically investigate the mechanism of spin current generation induced by elastic waves in nonrelativistic magnets referred to as altermagnets. By analyzing an $f$-wave altermagnet formed by a three-sublattice noncollinear antiferromagnetic structure breaking the spatial inversion symmetry on a two-dimensional triangular lattice within the linear response theory, we show that the nonrelativistic antisymmetric spin-split band structure can give rise to spin current generation when either longitudinal or transverse elastic wave is applied. We find that the momentum dependence of the antisymmetric spin splitting leads to a characteristic direction-dependent spin current response. We also compare the present nonrelativistic magnetic-order-driven mechanism with the relativistic one in a nonmagnetic Rashba system. These findings highlight the potential of invesion-symmetry-breaking altermagnets as a spin current generator driven by elasticity without relying on the relativistic spin-orbit coupling.

Recent advances have showcased the extraordinary capabilities of Large Language Model (LLM) agents in tackling web-based information-seeking tasks. However, existing efforts mainly focus on single-fact retrieval and rely on outcome-only verification, thereby limiting their scalability in realistic knowledge-intensive scenarios that involve long-horizon web tasks requiring large-scale retrieval and synthesis of information from diverse sources. In this work, we introduce VeriWeb, a novel verifiable long-chain web benchmark designed to facilitate the evaluation and development of web agents within realistic web environments. Our benchmark emphasizes two critical dimensions: (1) long-chain complexity, encompassing both breadth- and depth-oriented search tasks to assess how effectively web agents ensure comprehensive information coverage and consistent context tracking in multi-hop reasoning; and (2) subtask-level verifiability, where tasks are decomposed into a sequence of interdependent verifiable subtasks. This structure enables diverse exploration strategies within each subtask, while ensuring that each subtask-level answer remains unchanged and verifiable. The benchmark consists of 302 tasks across five real-world domains, each with a complete trajectory demonstration, annotated by human experts. Extensive experiments on VeriWeb using various agents powered by different foundation models reveal significant performance gaps in handling long-horizon web tasks, highlighting the need for more powerful agentic information-seeking capabilities.

Let $\{Y_i\}_{i=1}^{\infty}$ be a stationary reversible Markov chain with state space $[N]$, let $(X, \| \cdot \|)$ be a real-valued Banach space and let $f_1, \ldots, f_n: [N] \rightarrow X$ be functions with mean $0$ such that $\|f_i(v)\| \leq 1$ for all $i$ and $v$. We prove bounds on the expected value of and deviation bounds for the random variable $\|f_1(Y_1)+\cdots+f_n(Y_n)\|$. For large enough $n$ that depends on the Banach space (and not $N$), these bounds behave similarly as known bounds for independent random variables. When the Banach space in question is the set of matrices equipped with the $\ell_2 \rightarrow \ell_2$ operator norm, for large enough $n$, our bounds on the expected value improve upon known bounds and match what is known for independent random variables up to a factor in the spectral gap.

We here present printable and castable magnetic nanocomposites containing superparamagnetic 11$\pm$3 nm $γ$-Fe$_2$O$_3$ particles in an insulating poly-vinyl alcohol polymer matrix. The nanocomposites feature well-dispersed particles with volume fractions between 10 and 45 \%, as confirmed by small-angle neutron scattering. The magnetic volume susceptibility is as high as 17, together with negligible hysteresis at low frequency, and constant AC-response up to the high-kHz range. Measured hysteresis curves at 100-900 kHz with up to 110 mT induced $B$-fields in the nanocomposite show that power losses depend on $B$-field squared, and frequency to the power of 1-1.3. The only loss mechanism in the nanocomposite is hysteresis losses at $>$100 kHz frequencies, where the largest particles in the 11$\pm$3 nm distribution transition from the superparamagnetic to blocked regime. To mitigate the resulting hysteresis losses (up 10$^2$-10$^5$ kW/m$^3$) a more narrow particle size distribution could be used for future materials. The presented material is eddy current-free and easily integrated into micro-fabrication protocols, as we demonstrate by fabrication of 3-turn print circuit board based inductors with cast/manual printed nanocomposite inductor cores, on which induction has been measured up to 100 MHz.

We investigate a two-component reaction-diffusion system with a slow-fast structure and spatially varying coefficients $f_1$ and $f_2$ appearing in the slow equation. Under mild boundedness and regularity conditions on $f_1$ and $f_2$ the system is shown to exhibit bi-stability in the form of two stable stationary heterogeneous background states. These background states can be connected by stationary and travelling front solutions. Travelling fronts feature an interface that moves with a non-uniform speed through the motionless spatially varying background states it connects. We construct both the background states and stationary fronts using an extension of Fenichel theory to the non-compact case. Additionally, we establish the existence of travelling front solutions and derive a leading-order expression for the dynamic position of the moving interface through a time-dependent spatial dynamics approach. This expression takes the form of a delay-differential equation, and its accuracy is validated through numerical simulations. A key contribution of our work lies in the general treatment of $f_1$ and $f_2$, which are neither (necessarily) asymptotically small nor restricted to specific forms such as periodic or localized structures. Furthermore, our derivation of the front position formula circumvents the traditional reliance on spectral analysis, enabling us to describe front dynamics beyond bifurcations from stationary fronts. This approach has the potential to be extended to other settings in which spectral properties at onset preclude conventional reduction techniques.

Turbulent flows, despite their apparent randomness, exhibit coherent structures that underpin their dynamics. Proper orthogonal decomposition (POD) has been widely used to extract these structures from experimental data. While periodic features like vortex shedding can be identified using POD mode pairs when periodicity dominates the flow, detecting such structures in complex flows is more challenging. The Hilbert proper orthogonal decomposition (HPOD) addresses this by applying POD to the analytic signal of the turbulent fluctuations, yielding complex modes with a $90^\circ$ phase shift between the real and imaginary components. These modes capture propagating structures effectively but introduce filtering artefacts from the Hilbert transform that is used to derive the analytic signal. The current work investigates the relationship between the modes of the POD and those of the HPOD on the velocity fluctuations in the wake of a sphere. By comparing their outputs, POD mode pairs that correspond to the same propagating structures revealed by HPOD are identified. Furthermore, this study explored whether computing the analytic signal of the POD modes can replicate the HPOD modes, offering a more computationally efficient method for determining the pairs of POD modes that represent propagating structures. The results show that the pairs of POD modes identified by the HPOD can be more efficiently determined using the Hilbert transform directly on the POD modes. This method enhances the interpretive power of POD, enabling more detailed analysis of turbulent dynamics without introducing the filtering from the Hilbert transform.