For fully connected populations of diffusively coupled bistable elements, we identified three qualitatively distinct mechanisms of noise-induced escape as coupling strength varies [H. Ishii and H. Kori, arXiv:2512.01388 (2025)]. Here we generalize these results to a class of networked systems and demonstrate that degree heterogeneity (i.e., variability in node degree) shapes escape mechanisms alongside coupling strength. In applied contexts, networks of noisy bistable elements provide a minimal conceptual framework for understanding abrupt state transitions in complex systems. Theoretically, a quantitative approach to escape is challenging because nonlinearity, network interactions, and dynamical noise jointly shape the collective dynamics. We extend the analytical framework developed for the fully connected model to a class of networked systems based on the annealed network approximation. We derive three effective one-dimensional descriptions of collective escape dynamics. We validate our theoretical predictions for mean escape times by direct numerical simulations. Our analysis reveals that the validity and quantitative behavior of the reduced descriptions depend on degree heterogeneity in addition to coupling strength. This work extends the classification of escape mechanisms to networked bistable elements. Furthermore, our analytical framework provides tools for understanding synergistic phenomena arising from the interplay of nonlinearity, diffusive coupling, and dynamical noise.

Multiple long-term conditions (MLTC) are increasingly observed in clinical practice globally. Clustering methods to group diseases into commonly co-occurring clusters have been of interest for further understanding of how MLTC group together and their associated impact on patient outcomes. However, such approaches require large, often population-scale datasets. Bayesian Profile Regression (BPR) is a statistical model that combines a Dirichlet Process Mixture model with a hierarchical regression model, in order to form clusters of items conditional on covariates and an outcome of interest. We developed a BPR model using full-rank Stochastic Variational Inference (SVI) for application in large-scale data. We assessed it's performance using simulation studies comparing fits using the No-U-turn (NUTS) sampler and full-rank SVI. We then fit a BPR model to find clusters of MLTC in a population-scale data held in the Secure Anonymised Information Linkage (SAIL) databank. We found results from full-rank SVI compared well with results from NUTS in a simulation study, and the improved fitting performance allowed for fitting models in population-scale datasets. There were 1,296,463 individuals in our electronic health record (EHR) cohort. The clustering model was conditioned on age at cohort entry, socioeconomic deprivation and sex with mortality as the outcome. We used the Elixhauser comorbidity index disease definitions, and found there were 33 disease clusters. We found that clusters featuring metastatic cancer and cardiovascular diseases, such as congestive heart failure, were most strongly associated with the probability of mortality. Our findings show that SVI can be a useful and accurate method for fitting Bayesian models, especially when the dataset size would make Monte Carlo methods prohibitively time consuming or impossible.

Interest-based learning (IBL) is a paradigm of instruction in which educational content is contextualized using learners' interests to enhance content relevance. IBL has been shown to result in improved learning outcomes. Unfortunately, high effort is needed for instructors to design and deliver IBL content for individual students. LLMs in the form of AI tutors may allow for IBL to scale across many students. Designing an AI tutor for IBL, however, first requires an understanding of how IBL is implemented in teaching scenarios. This paper presents a study that seeks to derive this understanding from an analysis of how human instructors design and deliver IBL content. We studied 14 one-to-one online tutoring sessions (28 participants) in which tutors designed and delivered a lesson tailored to a student's self-identified interest. Using lesson artifacts, tutoring transcripts, interviews, and questionnaires, findings include themes on how tutors integrate interests during instruction and why. Finally, actionable design implications are presented for LLM-powered AI tutors that aim to deliver IBL at scale.

This paper is the second part of a two-part article where we generalize Sarnak's program to sets where we remove congruence classes modulo some infinite set $\mathcal{B}$ of ideals of an étale $\mathbb{Q}-$algebra $K$, which we denote by Erdős sieves. Given a sieve $R$ we define the set $\mathcal{F}_R$ of algebraic integers in $K$ not contained in any of the congruence classes of $R$. We associate to each sieve two measure-theoretical dynamical systems $X_R$ (the orbit closure of $\mathcal{F}_R$) and $Ω_R$ (the set of $R-$admissible sets) and show how they are related. We show that the system associated to $Ω_R$ is isomorphic to an ergodic rotation of a compact abelian group, and compute its spectrum. As applications we show results about infinite sumsets in the integers, investigate the case where $\mathcal{F}_R$ is the squarefree values of some polynomial, and show a prime number theorem for $R-$free numbers.

Within the family of altermagnets, CrSb is a metallic, collinearly ordered material that exhibits particularly strong symmetry-induced spin splitting in its band structure. In this study, we combine electrical magnetotransport measurements up to 68 T on microfabricated single-crystalline CrSb with first-principles calculations to investigate its Fermi surface. Notably, we study the temperature and field-orientation dependence of magnetic quantum oscillations observed in the magnetoresistance. The observed frequency spectrum agrees well with results from density-functional-theory calculations. Our results confirm the predicted electronic band structure of altermagnetic CrSb and highlight the importance of high magnetic fields for accurately mapping the Fermi surfaces of unconventional emergent materials.

In this work, we study a system of degenerate parabolic equations modeling the dynamics of multiple cell populations in intestinal crypts. The model describes cell division, differentiation, and migration through a strongly coupled system of reaction-cross-diffusion equations with degenerate diffusion. By working with initial data in BV, we first consider a regularized form of the system and establish uniform BV estimates. Using these bounds, we then pass to the limit to obtain the existence of weak solutions.

This paper is the first part of a two-part article where we generalize Sarnak's program to sets where we remove congruence classes modulo some infinite set $\mathcal{B}$ of ideals of an étale $\mathbb{Q}-$algebra $K$, which we denote by Erdős sieves. We define some light tail conditions on a sieve $R$, and show how these are related to the genericity under the Mirsky measure of the set of $R-$free numbers, which are the algebraic integers of $K$ not contained in any of the congruence classes in $R$. We also show that Erdős $\mathcal{B}-$free systems in any étale $\mathbb{Q}-$algebra satisfy these light tail conditions, so our results generalize Sarnak's program to Erdős $\mathcal{B}-$free systems over any étale $\mathbb{Q}-$algebra.

As terahertz (THz) frequencies emerge as promising candidates for next-generation wireless networks, accurate characterization of propagation mechanisms in indoor/outdoor environments becomes essential for system design and performance optimization. This article presents an experimental and theoretical investigation of structure-induced indoor surface scattering on THz channels, examining how material properties and structural configurations jointly govern channel power and angular distribution. Six representative indoor surfaces are characterized, revealing that intrinsic structural inhomogeneity -- particularly the quasi-periodic earlywood-latewood arrangement in pine wood -- induces measurable angular scattering whose dominant lobes and angular shifts are reproduced by a beam-propagation modeling (BPM) framework. Material-covered surface configurations are further investigated, demonstrating that thin dielectric covering layers can substantially modify reflection characteristics through thickness- and frequency- dependent thin-film interference effects. Wide-angle bistatic measurements conducted in a conference-room environment reveal that structured indoor elements, such as folded curtains, can enhance angular scattering and extend spatial coverage. These findings establish that structure-induced surface scattering mechanisms offer potential for constructing non-line-of-sight THz links in indoor environments.

Cubic A15-type superconducting alloys continue to fascinate the academic and industrial fields because they mainly support the largest market for low-temperature superconducting applications and show exotic physical properties. Medium-/high-entropy alloys (MEAs-HEAs) can be employed stably under extreme conditions due to their high mechanical hardness and excellent irradiation tolerance. Combining with the features of the A15-type superconductor and MEAs-HEAs, we design a series of previously unreported A15-type V3(Os1-2xSixGex) (x = 0.333, 0.375, 0.425) MEA superconductors, which can be obtained by an arc melting method. Resistivity, magnetic susceptibility, and specific heat measurements indicate that all of them are type-II bulk superconductors. The superconducting transition temperature (Tc) exhibits an upward trend with the systematic reduction of Os concentration. Additionally, the upper critical field of the V3(Os0.333Si0.333Ge0.333) sample is larger than the Pauli limit, suggesting it may be robust against magnetic fields due to spin-orbit coupling induced by the heavy Os atoms. These findings not only advance our understanding of emergent phenomena in entropy-stabilized A15-type alloys but also expand the members of new superconductors.

We study dynamical constraints arising from Embedded Contact Homology (ECH) in the spatial isosceles three-body problem. For energies below the critical level, the dynamics on the energy surface is identified with a Reeb flow on the tight three-sphere. We obtain quantitative estimates for the Euler orbit, including monotonicity of its transverse rotation number and a strict inequality comparing its action with the contact volume. Combined with the ECH classification of Reeb flows on the tight three-sphere with two simple periodic orbits, these estimates rule out the two-orbit scenario, thus forcing every compact energy surface below the critical level to have infinitely many periodic orbits. The result admits a dynamical interpretation via disk-like global surfaces of section bounded by the Euler orbit. In this setting, the rotation number and the contact volume define a non-trivial twist interval which encodes the relative winding of periodic orbits. For energies above the critical level, where the energy surface is non-compact, we prove the existence of infinitely many periodic orbits and infinitely many parabolic trajectories via twist estimates near infinity.

Independent media are central to democratic decision-making, yet recent technological developments, such as social media, pseudonymous identities, and generative AI, have made them more vulnerable to coordinated influence campaigns--usually referred to as Coordinated Inauthentic Behavior. By automatically generating large numbers of similar messages and news reports, such campaigns create an illusion of widespread support, and exploit the tendency of human observers and aggregation mechanisms alike to treat frequency as evidence of credibility or consensus. Clone-robust weighting functions offer a solution to this problem by assigning influence in a way that is insensitive to arbitrary duplication or near-duplication, as measured by a metric. This axiomatic framework rests on three principles: symmetry (equivalent elements are treated equally), continuity (weights vary smoothly under perturbations), and clone-robustness (adding duplicates or near-duplicates does not distort the overall distribution). We provide a general construction of clone-robust weighting functions that applies to arbitrary metric spaces, is entirely independent of the underlying topology, and admits efficient computation. Our approach identifies radius graphs as a natural invariant under cloning, and builds on graph weighting functions that satisfy a basic locality condition. We explore the resulting design space, starting with a simple family that satisfies the core axioms, and then identify explainability as a guiding criterion for navigating this design space. To this end, we introduce sharing coefficients that enable meaningful comparison and interpretation of different constructions, but require additional axiomatic principles. We then consider alternative constructions based on clique-covers, and unveil approaches using clique-partitions that are grounded in information-theoretic principles.

Recent studies have shown that novel collective behaviors emerge in complex systems due to higher-order interactions. However, the way in which the structural correlations of these interactions shape such behaviors remains a significant gap in current research. To address this, we use signatures of higher-order behaviors (HOBs) to identify the underlying dynamical rules, or higher-order mechanisms (HOMs). In this work, we compare several HOB measures derived from information theory. Utilizing a simplicial SIS contagion model, we demonstrate that simpler, computationally efficient measures can serve as robust indicators of HOMs. We uncover the novel phenomenon of cross-order induced behaviors, where behavioral signatures emerge at interaction orders where no direct mechanism is present. Crucially, these cross-order HOBs are not simply induced by structural correlations -- such as nestedness and hyperedge overlap -- but they appear in the neighborhood of any HOM. Among the information-theoretic measures we tested, synergy is the most reliable indicator of the true order where the underlying mechanism is at play. These findings offer new insights into the relationship between the network structure and observed dynamics of higher-order systems.

Speckled illumination enhances widefield fluorescence microscopy by enabling optical sectioning and super resolution. In random illumination microscopy, sequences of speckled illumination patterns are used to excite fluorescent samples and images are reconstructed based on a statistical analysis of the intensity fluctuations. Although random illumination microscopy has been shown to give excellent performance, its widespread implementation is hindered by the high cost and complexity of the generation of suitable speckled illumination patterns, which is achieved using digital micro-mirror devices or spatial light modulators. Here, we present a zwitterion-doped liquid crystal (LC) device capable of generating independent, high-contrast speckle patterns with a tunable decorrelation time in the 0.1 s to 0.1 ms range under visible laser illumination. This LC-based dynamic speckle generator is applied to widefield random illumination fluorescence microscopy of tissue and cell samples, where it enables optical sectioning with a 2 micron axial resolution, and a 1.5-fold improvement in lateral spatial resolution. Owing to its low cost and simplicity, this LC speckle generator offers an attractive alternative to digital micro-mirror and spatial light modulator devices for implementing widefield random illumination microscopy.

Using the modern perspective of noncommutative algebraic geometry we survey some recent progress in the theory of stability conditions and moduli spaces with applications in hyperkähler geometry and classical algebraic geometry.

In this article we explicitly determine the Weierstrass semigroup at any place of some $\mathbb{F}_{q^2}$-maximal Fermat function fields $\mathcal{F}_m$, namely for $m=(q+1)/2$ and $m=(q+1)/3$. These famous function fields arise as Galois subfields of the Hermitian function field, and even though they have been intensively studied in the literature, the Weierstrass semigroup at every place is still not fully known. Surprisingly enough this problem is in fact quite involved and $\mathcal{F}_m$ has many different types of Weierstrass semigroups. Moreover, its set of Weierstrass places is much richer than its set of rational places.

Property checking of RTL designs is a central task in formal verification. Among available engines, IC3/PDR is a widely used backbone whose performance critically depends on inductive generalization, the step that generalizes a concrete counterexample-to-induction (CTI) cube into a lemma. Prior work has explored machine learning to guide this step and achieved encouraging results, yet most methods adopt a per-clause graph analysis paradigm: for each clause they repeatedly build and analyze graphs, incurring heavy overhead and creating a scalability bottleneck. We introduce LeGend, which replaces this paradigm with one-time global representation learning. LeGend pre-trains a domain-adapted self-supervised model to produce latch embeddings that capture global circuit properties. These precomputed embeddings allow a lightweight model to predict high-quality lemmas with negligible overhead, effectively decoupling expensive learning from fast inference. Experiments show LeGend accelerates two state-of-the-art IC3/PDR engines across a diverse set of benchmarks, presenting a promising path to scale up formal verification.

Parton Distribution Functions (PDFs) are a key ingredient in theoretical predictions for Large Hadron Collider (LHC) observables and play a central role in the extraction of precision Standard Model (SM) and Beyond the SM (BSM) parameters from LHC data. Recent analyses demonstrate that the determination of fundamental SM parameters such as $α_s(m_Z)$, $m_W$, $m_t$, and $\sin^2θ_W$ is strongly influenced by the choice of input PDFs. In this contribution, we present the status and challenges of PDF determination from the NNPDF perspective, both in stand-alone fits and in joint extractions with (B)SM parameters. We place particular emphasis on results for $α_s(m_Z)$, $m_t$, and Wilson coefficients in the SM Effective Field Theory (SMEFT) framework.

For three Winter Olympics in a row, tiny nation Norway has out-medalled everyone else, in 2026 winning 18 golds, 12 silvers, 11 bronzes, i.e.~41 medals, compared to e.g.~12 + 12 + 9 = 33 for the USA, 10 + 6 + 14 = 30 for home team Italy, 8 + 10 + 7 = 26 for powerhouse Germany, etc. Never before have we [pluralis proudiensis] or anyone else won as many as 41 medals at a Winter Olympics. But how impressive is this, really, when we factor in that the number of events has increased so drastically?

Microwave filtering for superconducting qubits is a key element of quantum computing technology, enabling high coherence and fast state detection. This work presents the design and implementation of novel microwave Purcell filters for superconducting quantum circuits, integrated within a multilayer printed circuit board (PCB). The off-chip design removes all filter components from the qubit substrate, reducing device complexity, improving layout footprint and allowing better scalability to large qubit counts. Each embedded filter can couple up to nine readout resonators, enabling efficient multiplexed readout. Electromagnetic simulations of the filter predict a thousand-fold improvement in qubit isolation from the readout port. The design was experimentally validated under cryogenic conditions in conjunction with a 35-qubit device, demonstrating compatibility of the PCB-based filter with high-coherence superconducting qubits. The comparison of the measured qubit median T1 of 84 $μ$s with the expected radiative limit from electromagnetic simulations validated the presence of Purcell filtering in the system.

We present the first lattice QCD calculation of electromagnetic form factors of a tetraquark, focusing on the $T_{bb} = bb\bar u \bar d$ with quantum numbers $I(J^P) = 0(1^+)$. The electromagnetic current probes the charge monopole, magnetic dipole and the electric quadrupole distributions within the tetraquark. From it, we find evidence that its structure consists of a compact heavy diquark $[bb]$ in spin one, color-antitriplet configuration, and a light antidiquark $[\bar u \bar d]$ in spin zero, color-triplet configuration. The computations were performed on a single CLS ensemble with $N_f = 2+1$ dynamical quarks at a lattice spacing $a\approx 0.064$ fm and with a pion mass $m_π\approx 290$ MeV.

Approximate nearest neighbor search (ANNS) on GPUs is gaining increasing popularity for modern retrieval and recommendation workloads that operate over massive high-dimensional vectors. Graph-based indexes deliver high recall and throughput but incur heavy build-time and storage costs. In contrast, cluster-based methods build and scale efficiently yet often need many probes for high recall, straining memory bandwidth and compute. Aiming to simultaneously achieve fast index build, high-throughput search, high recall, and low storage requirement for GPUs, we present IVF-RaBitQ (GPU), a GPU-native ANNS solution that integrates the cluster-based method IVF with RaBitQ quantization into an efficient GPU index build/search pipeline. Specifically, for index build, we develop a scalable GPU-native RaBitQ quantization method that enables fast and accurate low-bit encoding at scale. For search, we develop GPU-native distance computation schemes for RaBitQ codes and a fused search kernel to achieve high throughput with high recall. With IVF-RaBitQ implemented and integrated into the NVIDIA cuVS Library, experiments on cuVS Bench across multiple datasets show that IVF-RaBitQ offers a strong performance frontier in recall, throughput, index build time, and storage footprint. For Recall approximately equal to 0.95, IVF-RaBitQ achieves 2.2x higher QPS than the state-of-the-art graph-based method CAGRA, while also constructing indices 7.7x faster on average. Compared to the cluster-based method IVF-PQ, IVF-RaBitQ delivers on average over 2.7x higher throughput while avoiding accessing the raw vectors for reranking.

We discuss invariants in equivariant birational geometry.

This proceedings contribution summarises projected constraints from the QUEST-DMC concept, a surface-based direct-detection experiment using superfluid $^3$He operated below the millikelvin regime and instrumented with nanomechanical resonators read out by SQUIDs. The low recoil-energy threshold (down to sub-eV for the SQUID configuration) enables sensitivity to sub-GeV dark matter across a wide set of interaction structures beyond the canonical spin-independent and spin-dependent limits. We present projections in the non-relativistic Effective Field Theory (EFT) framework, scanning the standard set of fourteen Galilean-invariant operators and expressing reach in terms of effective dark matter-nucleon (or dark matter-neutron) cross sections. Because QUEST-DMC operates at the surface, we also account for suppression of the incident flux due to scattering in the atmosphere and Earth, which produces an interaction-dependent sensitivity ceiling at large couplings. Finally, we outline how the non-relativistic results map onto representative relativistic EFT dark matter-nucleon bilinears, enabling a compact interpretation of the projected reach in terms of UV-motivated coupling structures.

We consider a dynamical elasto-plasticity system with Kelvin--Voigt viscosity and linear kinematic hardening of Melan--Prager type. The model is formulated in a variational framework in which a constraint set for the stress evolves in time and is translated by an internal (backstress) variable. As a consequence, the flow rule is coupled with an equation of motion through a quasi-variational structure, since the constraint set depends on the unknown internal variable. To construct solutions, we employ Rothe's method and introduce a projection-based time discretization. Each time step consists of solving a linear viscous-elastic subproblem to obtain a trial stress, followed by a projection onto the translated constraint set. We establish stability of the resulting discrete solutions under suitable norms. By compactness and passage to the limit as the time step tends to zero, we prove existence of a weak solution in the variational sense, and uniqueness follows from an energy argument. The results cover time-dependent yield bounds without assuming spatial continuity or a strictly positive lower bound, and the discretization provides a constructive basis for numerical approximation.

Cooperative sensing with uncrewed aerial vehicles (UAVs) is a key enabler for low-altitude wireless networks (LAWNs), where sensing accuracy critically depends on the spatial configuration of the UAV formation. In this paper, we study formation design and control for Cramer-Rao lower bound (CRLB)-optimal cooperative target sensing. We first establish a sensing performance model based on range measurements and derive the Fisher information matrix (FIM) of the target location. By adopting the A-optimality criterion, we analytically characterize the formation geometry that minimizes the CRLB of the estimation error. The optimal formation is shown to exhibit isotropic Fisher information in the horizontal plane, leading to a regular polygon geometry with an elevation angle determined by the tradeoff between path loss and geometric diversity. Building on this result, we further develop a distributed formation control strategy that steers UAVs from arbitrary initial deployments toward the sensing-optimal configuration while maintaining formation motion and obstacle avoidance. Numerical results demonstrate that the proposed scheme consistently outperforms benchmark formations in terms of CRLB and achieves reliable convergence under practical constraints.

Intelligent behavior in life-like systems often arises from the ability to gather, process, and act on information. While active matter provides a framework for studying life-like dynamics, it typically omits internal information-processing and decision-making. Here we introduce an adaptive active particle model that uses minimal information processing capabilities in order to navigate towards a distant target. By combining renewal-based intermittent motion with the Cramér-Rao inequality, we derive a bound on the navigation speed valid for a wide range of information processing strategies. The framework captures hallmark features of cognitive systems, including optimal sensing durations and a speed-accuracy trade-off that balances noise and reliability. Allowing stored information to degrade before action reveals that although deterioration slows navigation, the trade-off remains governed primarily by external orientational noise and is remarkably insensitive to memory decay.

Datasets that exhibit non-Gaussian characteristics are common in many fields, while the current modeling framework and available software for non-Gaussian models is limited. We introduce Linear Latent Non-Gaussian Models (LLnGMs), a unified and computationally efficient statistical modeling framework that extends a class of latent Gaussian models to allow for latent non-Gaussian processes. The framework unifies several popular models, from simple temporal models to complex spatial-temporal and multivariate models, facilitating natural non-Gaussian extensions. Computationally efficient Bayesian inference, with theoretical guarantees, is developed based on stochastic gradient descent estimation. The R package \texttt{ngme2}, which implements the framework, is presented and demonstrated through a wide range of applications including novel non-Gaussian spatial and spatio-temporal models.

Timely availability of high-fidelity entanglement is essential for emerging quantum networks. This paper introduces the Age of Entanglement (AoE) as a novel performance metric that captures the freshness of bipartite entanglement under continuous distribution in quantum repeater chains. AoE extends classical Age of Information (AoI)-based metrics to quantum networking by capturing storage, decoherence, and probabilistic entanglement generation and swapping. We study a satellite-assisted quantum repeater network in which entangled pairs are generated probabilistically, stored in quantum memories that suffer from decoherence, and combined to form end-to-end entangled links. Satellite-ground connectivity is intermittent and modeled as a two-state Markov chain. The resulting AoE minimization problem is formulated as an infinite-horizon Markov decision process (MDP), where control actions determine when to generate, store, or swap entangled pairs under stochastic link availability and memory degradation. Using relative value iteration, we characterize AoE-optimal policies and evaluate their performance numerically. Our results highlight the impact of decoherence, imperfect operations, and visibility dynamics, and show that the proposed dynamic policies significantly outperform swap-as-soon-as-possible and greedy entanglement generation strategies. Our results provide practical design and control guidelines for satellite-enabled quantum repeater chains supporting continuous entanglement distribution.

The ionization mechanisms of low-ionization nuclear emission-line regions (LINERs), which are common in the local Universe, have been debated for decades. Our nearest large neighbor, M31, is classified as a LINER based on its optical emission line properties within the central kpc. In this work, we present a detailed photoionization modeling of the circumnuclear ionized gas in M31, explicitly tailored to its well-constrained physical conditions, including the absence of ongoing star formation and a currently inactive active galactic nucleus (AGN). Using spatially resolved CFHT/SITELLE observations, we find that photoionization by hot, evolved low-mass stars distributed throughout the bulge can roughly reproduce the observed radial intensity profiles of Hα, H\b{eta}, and [NII]. However, these models fail to match the observed [OIII] emission, producing radial profiles and [O III]/H\b{eta} ratios that are significantly steeper than observed. This discrepancy indicates a deficit of high-energy ionizing photons in standard stellar photoionization models, even with extended ionizing sources. We explore whether this tension can be alleviated by invoking either a bulge-filling, low-density ionized medium surrounding a denser Hα-emitting disk, or enhanced AGN activity in the recent past. While both scenarios can partially increase the [O III] emission, neither provides a fully satisfactory explanation under physically plausible conditions. Together with our earlier results for M81, these findings underscore persistent challenges in explaining LINER-like emission solely through conventional photoionization mechanisms.

We propose a method for predicting the nonlinear saturation level of convective instabilities in neutral and magnetized fluids. The method combines Gardner's restacking algorithm, which computes the available energy and ground states of collisionless plasmas in phase space, and Lagrangian relaxation, where fluid elements find lower-energy equilibria while preserving local invariants. For the incompressible Rayleigh-Taylor instability, the problem is formally equivalent to Gardner's and the restacking algorithm directly applies in configuration space. To treat compressibility, we follow restacking with Lagrangian relaxation to obtain the ground state, and the results show excellent agreement with direct numerical simulations. Successful extension to the $m=0$ interchange instability in a Z-pinch demonstrates the method's potential as a general framework for estimating the nonlinear extent of convective instabilities, which can facilitate the design and operation of fusion reactors.