Modern computing systems process jobs with resource requirements such as CPU and memory, which are described by multiresource jobs (MRJ) queueing models. In practice, job resource requirements are spread out over so many values, that it is rare to see the same value twice. This pattern is best modeled by a continuous distribution of requirement values. However, the existing theoretical work on stability or throughput-optimality focuses on queueing models with class-based resource requirements. In class-based models, the number of distinct resource requirements must be small to demonstrate strong empirical performance, making them a poor match for these practical systems. We introduce the first throughput-optimal family of scheduling policies for the continuous MRJ model, with both preemptive and nonpreemptive variants. We further introduce several efficient policy families, which remain throughput-optimal while considerably improving computational efficiency, under some distributional assumptions. We use a discretization approach, where we choose the discretization granularity based on the system load and the distribution of resource requirements. We validate the real-world applicability of our policies by comparing them against existing index-based policies on parametrized distributions and on datacenter trace data from the Google Borg scheduler, demonstrating state-of-the-art performance.

We study the relation between primariness of Banach spaces and the stronger operator-theoretic notions of the primary factorisation property (PFP) and the uniform primary factorisation property (UPFP). We revisit several classical primariness arguments and isolate the additional information needed to factor the identity through arbitrary operators. In the separable setting, this recovers quantitative factorisation versions of the Casazza--Kottman--Lin method for spaces with symmetric bases and treats the exceptional cases of $\ell_1$ and $\ell_\infty$. We then develop support-reduction and free-selection tools for uncountable direct sums, allowing one to transfer primariness and the PFP/UPFP from countable building blocks to non-separable $\ell_p$-, $c_0$- and more general symmetric sums. As applications, we obtain, among others, the primariness of $C[0,1]^*$ under the negation of the Continuum Hypothesis and UPFP results for uncountable sums of ordinal $C(α)$-spaces. Finally, using the finite-block representation of $\mathcal B(\ell_p)$, we prove a uniform primary factorisation theorem for the Banach space $\mathcal B(\ell_p)$, $1<p<\infty$, and end with open problems concerning the gap between primariness and factorisation.

This paper investigates the control synthesis for continuous-time uncertain systems under nested Signal Temporal Logic (STL) specifications containing nested temporal operators. Control Barrier Functions (CBFs) are utilized herein to encode STL formulas into system constraints. However, traditional CBF designs fail to encode nested STL formulas, whereas recent reachability analysis-based methods capable of handling such formulas are inapplicable to uncertain systems and suffer from a severe computational burden. To overcome these challenges, a novel recursive CBF design procedure based on a modified STL tree (sTLT) is proposed to yield explicit parameterized CBFs. Within this framework, sliding window variables are introduced to capture complex temporal relationships. Crucially, satisfying the resulting CBF constraints is proven to guarantee the fulfillment of the STL specifications. To render the proposed recursive CBF design applicable to systems subject to uncertain disturbance, a novel controller based on reconstructed CBF using quadratic programming (QP) is proposed, ensuring strict CBF constraint satisfaction under disturbances. In contrast to existing methods, the proposed reconstructed CBF approach requires no prior knowledge of the disturbances while relaxing initial safety assumptions. Simulation results validate the efficacy of the proposed approach.

We prove a sharp regularity threshold for uniqueness in two anisotropic Calderón-type inverse problems in dimension $n\ge 3$. The main setting is the Riemannian Schrödinger problem with fixed scalar potential: for a prescribed nonconstant analytic function $V$, we study whether the Dirichlet-to-Neumann map of $-Δ_g+V$ on a domain $Ω\subset\mathbb{R}^n$ determines the unknown metric $g$. The natural gauge is the group of boundary-fixing diffeomorphisms preserving $V$. We show that, while analytic metrics are uniquely determined modulo this gauge by a minor adaptation of the Lassas--Uhlmann reconstruction theorem, uniqueness fails densely in every non-analytic Gevrey class $G^σ$, $σ>1$. In fact, our counterexamples are not isometric in the sense that they are not connected by the pushforward of any diffeomorphism of $\overlineΩ$. We also prove the analogous sharp threshold for the anisotropic Calderón problem at fixed nonzero frequency, thereby upgrading the previously known finite-regularity counterexamples to Gevrey and $C^\infty$ regularity. The two constructions use different scalar mechanisms: for fixed potentials, the nonconstant potential itself provides a local coordinate, while at nonzero frequency one uses a compactly supported prescribed-Jacobian lemma in Gevrey spaces. Thus analyticity is the exact threshold for uniqueness in both problems.

A well-known theorem by Milnor-Orlik provides a formula for the Milnor number of a weighted-homogeneous polynomial having an isolated singularity that depends only on the weights. In this paper we present a proof of that result using techniques from commutative algebra. Our approach is to obtain a free resolution of the Milnor algebra through the Koszul complex. The desired formula is then obtained from a Hilbert series calculation.

Neural control offers strong potential for handling highly nonlinear dynamics in shipboard microgrids (SMGs), yet its black-box nature can trigger abrupt control spikes and actuator saturation during initial transient shocks. This letter devises a formal verification method for SMG neural controller to assess its shock responses. Our contributions include: 1) a set-based SMG differential-algebraic equation(DAE) model compatible with set propagation; 2) a DAE-embedded bound propagation approach to compute tight envelopes of all possible neural control output. Extensive case studies demonstrate the effectiveness of the devised method in formally certifying SMG control performance under uncertain disturbances.

Superconducting two-dimensional materials, and in particular few-layer graphene, offer an exciting platform for low-power electronics, yet the origin of their unconventional superconductivity remains an open question. Prevailing theories, primarily rooted in the Bardeen-Cooper-Schrieffer (BCS) framework that assumes electron-phonon interactions are the main mechanism of superconductivity, struggle to account quantitatively for the observed phenomena. Recent studies point to a plasmonic pairing mechanism in graphene systems; however, disentangling the relative contributions of phonon- and plasmon-mediated pairing remains challenging due to the lack of a satisfactory first-principles framework capable of accurately capturing dynamical screening effects in the electronic channel. Here, we present a new theoretical framework that extends the quasiparticle self-consistent GW method to the superconducting phase by coupling it with the Eliashberg treatment of both phonon- and plasmon-mediated interactions. Our approach, termed "s-qpGW", is on par with the state-of-the-art Eliashberg theory of superconductivity when applied to bulk metals, and correctly predicts the absence of superconductivity in doped monolayer graphene. To differentiate s-qpGW from conventional Eliashberg approaches, we study a simple model system, graphene with an artificially enhanced density of states, and demonstrate that s-qpGW captures dynamical Coulomb screening effects in ways that standard BCS theory cannot.

State-space models (SSMs) are a broad class of probabilistic models for dynamical systems with many applications in engineering and science. Bayesian filtering is analytically tractable only in the linear-Gaussian setting, where the Kalman filter yields exact posterior distributions. For nonlinear or non-Gaussian SSMs, approximations are required. Two prominent families of approximate methods are Gaussian sum filters (GSFs), which rely on local Gaussian approximations and numerical integration schemes, and particle filters (PFs), which use sequential Monte Carlo sampling. Despite their success, GSFs can suffer from numerical instabilities and severe failures in strongly nonlinear regimes, while PFs are flexible and robust but often demand substantial computational resources to achieve accurate estimates. In this work, we propose the Augmented Gaussian Sum Filter (AGSF), a novel filtering framework that unifies GSFs and PFs through an augmented Gaussian approximation parameterized by latent variables and tunable covariance parameters. By adjusting these covariances, the AGSF interpolates continuously between GSF-like and PF-like behavior, recovering both as special cases. Building on this view, we develop an adaptive AGSF that automatically shifts its behavior according to the local nature of the nonlinearities, acting more like a GSF when Gaussian approximations are reliable and more like a PF when they are not. In a target-tracking application, we demonstrate that AGSF is efficient and robust to common failure modes of both GSFs and PFs. We empirically validate the switching behavior of the adaptive mechanism in a toy example.

This research paper investigates new and first insights into the magnetic and magnetocaloric properties of one-dimensional (1D) cobalt ferrite CoFe2O4 (CFO) nanofibers elaborated by sol gel based electrospinning technique, particularly focusing on their behavior at low temperatures for specific applications. The calcined CFO nanofibers microstructural, structural, magnetic, and magnetocaloric properties were explored. The nanofibers (NFs) microstructure, with an average diameter of 210 nm, was examined by scanning and transmission electron microscopies (SEM, TEM). The X-ray diffraction (XRD) of the CFO nanofibers showed a pure cubic close-packed (c.c.p) spinel crystalline structure with the F d 3 -m space group. The Raman spectroscopic studies further confirm the cubic inverse spinel phase. The Magnetic properties were explored as a function of temperature, ranging from 10 to 300 K, a ferromagnetic behaviour was observed with the highest saturation magnetization of 75.87 emu g(-1) and a coercivity of 723 Oe at room temperature. The variation of the magnetic entropy was measured indirectly using the Maxwell approach with an increasing magnetic field. A maximum of Delta(S)=1.71 J K-1 was reached around 32 K. At 180 K, the associated adiabatic temperature change, Delta (Tmax), was 0.93 K, with a large RCP value of 7.58 J kg-1 was measured, which is reasonably high for the corresponding nanoparticles (NPs). This work may suggest that 1D CFO nanofibers offer a promising route for the production of nanostructured magnetic materials, potentially impacting various electronic and electromagnetic device applications at low temperatures.

This paper studies empirical deep hedging for S&P 500 index options under a local downside-shortfall reward. It moves beyond performance comparison by asking what the learned hedge does, when it fails, and whether it can be made auditable. TD3 agents are compared with a daily-updated Black-Scholes delta hedge on the same option episodes. In walk-forward tests from 2015 to 2023, the agents usually learn a systematic delta haircut relative to Black-Scholes. The correction is explained by spot-implied-volatility co-movement and often improves accumulated reward and terminal downside variance, but it is regime-fragile: 2022 exposes losses in adverse daily states, while 2023 shows that underhedging can raise ordinary variance when option P&L is spot-dominated and the volatility channel is unusually weak. Symbolic regression distills the neural policies into compact formulas that can be traded out of sample; these formulas preserve much of the reward, downside-variance, and CVaR advantage over Black-Scholes, and sometimes sharpen it, but inherit the same fragility in difficult regimes.

This paper presents a passivity-based control framework for AC-DC converters supplying non-passive Information Technology rack loads in DC data centers. Unlike conventional cascaded proportional-integral controllers that ensure stability only near nominal operating points, the proposed method is derived from the system total energy balance using the Port-Hamiltonian formulation. By shaping the stored energy and injecting virtual damping through a lossless interconnection with a PH controller, the converter behaves as a passive system even when interfaced with non-passive loads or under grid disturbances. The closed-loop system guarantees asymptotic voltage regulation and strict energy dissipation without assuming constant grid voltage or frequency. Simulation studies under realistic load and fault scenarios validate that the proposed controller achieves smaller voltage deviations, faster recovery, and superior robustness, demonstrating its suitability for future high-efficiency DC data-center architectures.

Agile software development has been shaped by the interplay between academic research and industrial practice for over two decades, yet notable gaps persist between both domains. This paper focuses on three research-practice gaps: the theory gap, the time gap, and the transfer gap. To address these, the 2nd Agile Practice & Research Workshop was held at the International Conference on Agile Software Development (XP) 2026 in São Paulo, Brazil, bringing researchers and practitioners together to identify root causes and develop joint solutions. Building on two preceding sessions in which contributions of participants had been presented, participants engaged in a structured collaborative session, working in small groups on one of the three gaps and reflecting on possible causes and remedies. The organizers synthesized the results into four propositions for improving the research-practice intersection: (1) improving scientific communication, (2) aligning research more closely with emerging industrial needs, (3) creating stronger incentives for sustained collaboration, and (4) integrating educational approaches into research practice. From these, three calls for research were formulated: (a) broader adoption of open science practices for transparency, reproducibility, and cumulative evidence; (b) higher empirical quality standards through stronger theoretical grounding and rigorous design; and (c) more explicit, value-oriented contributions that clearly articulate their practical and scientific relevance. The paper offers both a summary of the workshop and a call to strengthen research-practice collaboration.

In this work, we investigate higher-derivative quantum field theories and the problem of Ostrogradsky instability within an open-system Keldysh-Lindblad framework. Coupling the ghost sector to dissipative baths generates non-perturbative effective masses and dissipative widths through self-consistent gap equations. Above a critical coupling, the nonequilibrium dynamics develops bifurcated dissipative branches, signaling the emergence of a dissipative phase transition and a nontrivial critical structure in parameter space. We find that the resulting dissipative dynamics can suppress ghost excitations through two distinct mechanisms: in one branch, a large dynamically generated effective mass preserves a quasiparticle-like excitation, while in the second branch, strong dissipative broadening destroys the quasiparticle character through overdamped dynamics. Our results suggest that dissipative effects may provide a nonequilibrium mechanism for the spectral suppression of Ostrogradsky ghosts. The comparison with the healthy sector indicates that the stabilization mechanism is intrinsically tied to the ghost-like spectral structure.

A method for measuring the altitude and particle size of noctilucent clouds, based on positioning and photometry from wide-angle three-color cameras, has been developed to determine the three velocity components, particle radius, and its derivative with respect to time for different cloud fragments. The updated method is applied to observational data of bright clouds during the summers of 2023-2025. Meridional motion of the cloud is found to be the principal factor driving the change in particle size. The effect of particle size evolution in the presence of a strong latitudinal temperature gradient is also studied.

This paper introduces a Doppler domain localized (DDL) implementation of the adaptive matched filter (AMF) for radar target detection in severely heterogeneous clutter environments with limited training data. The proposed detector uses the concept of a region of possible target detection (RPTD), a small set of Doppler bins that capture most of the target signal power. This RPTD-based DDL-AMF detector outperforms an earlier suggested DDL implementation of the generalized likelihood ratio (GLR) test, which employs the region of detection improvement (RODI) concept. Unlike the RODI-based DDL-GLR detector, the proposed DDL-AMF detector requires no information on clutter spectrum parameters and no measurements to determine the number and locations of RODIs. Moreover, the performance of the RODI-based DDL-GLR detector falls far below the optimum when the target Doppler frequency is unknown. In contrast, the RPTD-based DDL-AMF detector ensures rapid adaptive detection with near-optimum performance under unknown target Doppler frequency and multimodal clutter spectra.

We present a uniform, epoch-resolved analysis of soft X-ray observations of eight comets obtained with NICER, using Bayesian statistics to identify charge-exchange line components, measure relative ion fluxes, and infer nominal solar-wind freeze-in temperatures. The sample exhibits recurring spectral morphologies that fall into distinct families: carbon-dominated, intermediate, and nitrogen-/oxygen-dominated. Epoch-resolved flux ratios yield a robust separation between diagnostics: carbon-derived freeze-in temperatures cluster near T_freeze(C) about 1.4-1.7 MK, while nitrogen- and oxygen-derived diagnostics are systematically higher, typically T_freeze(N,O) about 2.0-2.3 MK. Short-timescale variability in inferred freeze-in conditions is common, indicating that instantaneous solar-wind charge-state fluctuations, rather than large changes in coma composition, dominate the spectral differences. We discuss instrumental and modeling limitations, demonstrate how our Bayesian fitting method mitigates degeneracies via physically motivated priors and Bayesian model selection, and recommend laboratory measurements and coordinated high-resolution X-ray observations to refine charge-exchange diagnostics and validate low-resolution inferences.

We present a new absolute age determination for NGC 6791, one of the Milky Way's oldest and most metal-rich open clusters. Its unusual properties make it an important probe of inner-disk evolution and asteroseismic calibration, but its age has remained difficult to determine because of coupled uncertainties in reddening, distance, photometry, and stellar-model physics. Gaia DR3 photometry together with detached eclipsing binaries (DEBs) in NGC 6791 are combined with 10,000 Monte Carlo isochrone sets (marginalizing over uncertainties in composition, convective mixing processes, opacities, diffusion, nuclear reaction-rates, distance modulus, and reddening) to determine the age of NGC 6791. For each isochrone we build a synthetic color-magnitude diagram (CMD) that matches the observed star count in the MSTO and subgiant-branch window and injects empirical photometric scatter perpendicular to the ridgeline, enabling CMD comparisons without artificial-star tests. We assess CMD morphology using a bootstrap-calibrated two-dimensional Kolmogorov-Smirnov statistic, and add an external check based on the nearest-point metric: a coeval DEB statistic in $(M,L)$ space. These statistics are mapped to probability-density weights via bootstrap-resampling and combined into a single isochrone weight. NGC 6791 is determined to have an age of $8.46\pm0.66$ Gyr, $[\mathrm{Fe/H}]=+0.280\pm0.079$, $Y=0.2968\pm0.0158$, $(m{-}M)_V=13.333\pm0.058$, and $E(B{-}V)=0.183\pm0.024$. Our error budget shows no single dominant contributor, and highlights differences between open-cluster and globular-cluster age errors. Combined with its super-solar metallicity, our age estimate favors an inner-Galaxy origin for NGC 6791 and subsequent outward migration, provides a benchmark for asteroseismic calibration at high metallicity, and extends the absolute cluster age--metallicity relation to an old, metal-rich open cluster.

An infinite structure has the finite length property (over a given field) if, for each of its finite powers, chains of equivariant subspaces in the corresponding free vector space are bounded in length. Prior work showed that the countable pure set and the countable dense linear order without endpoints have this property. We generalise these results to (a) any structure approximated by finite substructures with few orbits, provided the field is of characteristic zero, and (b) any Fraïssé limit with free amalgamation in a finite vocabulary consisting of unary and binary relations, possibly expanded with a generic total order. As a special case, we deduce the finite length property of the Rado graph using both methods. We also describe some connections with function spaces, weighted register automata, and orbit-finite systems of linear equations.

Component-wise accurate algorithms for computing the principal square root of an M-matrix are designed in terms of triplet representations. A triplet representation of an M-matrix $A$ is the triple $(P, {\bf u},{\bf v})$, where the matrix $P$ is such that $p_{ij}=-a_{ij}$ for $i\ne j$, $p_{ii}=0$, and ${\bf u}>0$, ${\bf v}\ge 0$ are two vectors such that $A{\bf u}={\bf v}$. It is shown that if $A$ is an M-matrix representable by a triplet, then its principal square root exists and is an M-matrix represented by a triplet as well. New versions of the Cyclic Reduction and the Incremental Newton iterations are provided in terms of triplets, to compute the principal matrix square root of $A$. It is shown that these algorithms are component-wise numerically stable independently of the singularity of $A$ and of its condition number. Numerical experiments are shown to confirm the component-wise stability.

Flaky tests pass and fail non-deterministically when run on the same version of code. Although many techniques have been proposed to detect, debug, and repair flaky tests, reproducing their failures remains a major challenge due to their inherent nondeterminism. Many flaky test datasets exist to help researchers study them, but these datasets are often composed of disjoint sets of flaky tests, where each dataset provides unique information, such as flaky tests of different categories, failure logs of flaky tests, or flaky tests reported by developers vs. flaky tests found by automated tools. In this work, we aim to create a reproducible dataset of flaky tests, curated from both developer issue reports and a popular dataset of flaky tests. Compared to prior flaky test datasets, our dataset is the first to provide (1) a reproducible environment to compile flaky tests, (2) scripts to reproduce failures, (3) scripts to automatically apply flaky test fixes and ensure that the tests are no longer flaky, and (4) execution logs of flaky test passing and failing. We present ReproFlake, a dataset of 1115 reproducible flaky tests across four flaky test categories. We create guidelines to help others contribute to this reproducible dataset, and demonstrate how to use our dataset to understand challenges in reproducing flaky test failures (e.g., challenges researchers may face when using prior flaky test datasets), the characteristics (e.g., location of the fix and its correlation with the flaky test category), and difficulties researchers may face in using our dataset to collect additional information (e.g., code coverage) about flaky tests. Our findings show that error information helps identify flaky test categories and guide repairs, that unresolved compilation failures highlight challenges in building legacy projects, and knowing typical fix locations can help prioritize repair efforts.

Stochastic cyber-physical systems (CPS) permeate critical infrastructure, from autonomous vehicles to medical devices. Yet, tools for runtime verification of such systems capturing the probabilistic dynamics in stochastic systems remain generally absent despite theoretical foundations established nearly a decade ago. In this paper, we present SENTIL, a novel runtime verification tool with provable statistical guarantees for the runtime monitoring of requirements expressed as Probabilistic Signal Temporal Logic (PrSTL). SENTIL combines an efficient Rust core with universal ecosystem integration, delivering performance exceeding existing deterministic monitors while providing rigorous probabilistic guarantees through statistical model checking, sequential probability ratio testing, and adaptive rare event estimation. SENTIL employs streaming algorithms for incremental robustness computation, parallel Monte Carlo sampling, and a language-agnostic C-ABI enabling seamless deployment across ROS, Apollo, MATLAB Simulink, and AUTOSAR platforms, and direct integration in C, C++, Python, and Java. To validate the effectiveness of the proposed tool, we validate SENTIL across various scenarios spanning autonomous vehicle monitoring, medical device validation, and biological networks, demonstrating 10-1,000$\times$ performance improvements over existing tools while maintaining provable confidence intervals. SENTIL is open source (\href{https://github.com/sedislab/SENTIL}{\texttt{sedislab/SENTIL}}) and it positions probabilistic runtime verification as a deployable infrastructure for all real-world safety-critical stochastic systems.

Large Language Models (LLMs) are widely deployed in diverse real-world settings, yet remain vulnerable to jailbreaking, where prompt-based attacks bypass safety filters. We present THREAT (Targeted Harmful generation via Reframing and Exploitation of Adversarial Tactics), a reasoning-driven framework that coordinates multiple LLMs in an iterative search loop to find textual jailbreak prompts. We formulate prompt discovery as a nonconvex optimization problem and provide an efficient solution that lowers runtime and improves attack effectiveness. Across diverse datasets and model architectures, THREAT delivers higher attack success rates with lower computational cost than prior methods. The crafted prompts were flagged as harmful in fewer than 1% of cases, compared with about 50% refusals for the corresponding unmodified prompts. These findings reveal previously undetected vulnerabilities in aligned LLMs and position THREAT as a practical tool for proactively strengthening the safety of foundation models.

This translational article documents the European Centre for Medium-Range Weather Forecasts (ECMWF) transition from a restricted data licensing model to open access under CC BY 4.0, completed in October 2025. The policy context included EU open data requirements and alignment with international data exchange frameworks. The transition was implemented through a tiered service model that kept core forecast data open while offering operationally supported delivery as a cost-recovered service. Between 2020 and 2025, ECMWF executed an iterative planning cycle: setting an annual target for revenue reduction, specifying additions to the open tier under that target, provisioning infrastructure, and assessing outcomes to update assumptions. Drawing on internal administrative records (2014 - 2025), we describe design choices, operational constraints, and early outcomes. In the six months following the end of the transition, more than 93% of previously paying organisations retained a Service Agreement, while open endpoint download volumes increased substantially. We discuss trade-offs in defining the open tier (resolution, parameters, schedule), the reduction of compliance overheads formerly associated with redistribution restrictions, and the scalability implications of global distribution. We note an emerging sustainability question as AI-based forecast products become freely available. The early evidence is consistent with the view that a tiered service model can be designed to reconcile open-access obligations with operational sustainability, subject to monitoring over longer contract renewal cycles (typically annual).

We introduce generalized Fofana spaces and we give some of their basic properties. These spaces are a kind of generalization of generalized Morrey spaces. As application, we establish the boundedness of the Hardy-Littlewood maximal operator on these new spaces.

We study several distinct but related Fourier analytic variants of the well-known Kakeya and Furstenberg set problems in the plane. For example, given $0<s,t<1$, we call a set $K \subseteq \mathbb{R}^2$ an $(s,t)$-Kakeya set if there exists a set of directions $E \subseteq S^1$ with Hausdorff dimension at least $t$ such that, for each $e \in E$, the set $K$ contains a subset of a unit line segment in direction $e$ whose Fourier dimension, viewed as a subset of $\mathbb{R}$, is at least $s$. For $Δ(s,t)$ defined to be the infimum of the Fourier dimension among all $(s,t)$-Kakeya sets in $\mathbb{R}^2$, we prove that \[ \frac{2st}{s+2t} \leq Δ(s,t) \leq \min\{s,2t\}. \] These bounds, though distinct, are asymptotically equivalent as either $s$ or $t$ tends to zero. We also obtain upper and lower bounds in the Furstenberg set version of the problem and in the case where the Hausdorff dimension of the collection of lines is replaced by the Fourier dimension.

We develop a relational duality for semilattices with adjunctions (SLatas) based on binary meet-relations. First, we introduce the category of MoS-spaces and establish a dual equivalence with modal semilattices. Then, by means of A-relations, we define the category RelSP and prove a dual equivalence between SLata and RelSP. To compare this framework with the multirelational semantics previously developed for SLatas, we introduce the notion of normal mS-space and show that, under this condition, the multirelational structure can be canonically recovered from a meet-relation, and conversely. As a consequence, we prove that the categories RelSP and SLataSp are isomorphic.

The study of arboreal Galois representations (that is, Galois groups arising from iteration of polynomial and rational functions) originated with work of Odoni in the 1980s. Beginning in the early 2000s it underwent a period of renewed interest, which continues to this day. Written in 2007, this survey article gives a sense of the subject from the early days of this renewal. It is presented here as a document of historical interest -- precisely as originally written -- and because some recent work has referenced specific pieces of it. It was written as an informal document, and not intended to be published. Much, though not all, of the content overlaps with the 2013 survey article ``Galois representations from pre-image trees: an arboreal survey" of the author.

In this paper, we study the long-time behavior of solutions to the two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane with Navier-slip boundary conditions. We prove that the total vorticity will approach \begin{align*} -1+\frac{M_2(ω_{0})}{ν^{3/2}(1+t)^{5/2}} \barΩ\left( \frac{x}{\sqrt{ν(1+t)^3}}, \frac{y}{\sqrt{ν(1+t)}} \right), \end{align*} where $-1$ is the vorticity of the Couette flow and $\barΩ$ is the kernel of a Fokker-Planck type operator $\mathcal{L}=\partial_Y^2+\frac32 X\partial_X+\frac12 Y\partial_Y+\frac52-Y\partial_X$. In the proof, we introduce a new idea of studying the spectrum of such type operators with boundary.

Frequency crowding is a fundamental limitation in superconducting quantum architectures, particularly in tunable-coupler systems. We present a framework that explicitly models both coherent spectator-induced errors and incoherent lifetime effects through an error budgeting approach. Using this model, we analyze how frequency crowding impacts gate fidelity as module size and connectivity scale, and formulate a constrained optimization problem to assign qubit and coupler frequencies under realistic separation and hardware constraints. We demonstrate scalable frequency allocation strategies that minimize spectator-induced errors. We further show that increasing qubit count and coupling density within a module leads to a fidelity-connectivity tradeoff. To explore the benefits at the system scale, we have developed a noise-aware transpilation approach called FINESSE, which minimizes error by selecting high-fidelity paths that satisfy connectivity via SWAP insertion while jointly optimizing downstream gate execution. We demonstrate this physics-informed architecture-transpilation co-design approach for a SNAIL-based third-order coupler that natively realizes the $\sqrt{iSWAP}$ basis with frequency aware gate fidelities. On SNAIL architectures, FINESSE achieves an average 8.9% reduction in log-infidelity cost and 6.8% reduction in circuit depth vs. SABRE. We also compare results on IBM Brisbane's architecture.

We provide some guidance and examples to clear up common misconceptions about special relativity. These misconceptions often come from trying to express the truths of special relativity in Newtonian terms rather than in terms more natural to special relativity itself. This conceptual stance can also help in attaining a better understanding of general relativity.