Recent advances in Retrieval-Augmented Generation (RAG) have revolutionized knowledge-intensive tasks, yet traditional RAG methods struggle when the search space is unknown or when documents are semi-structured or structured. We introduce a novel end-to-end Graph RAG framework that leverages both Labeled Property Graph (LPG) and Resource Description Framework (RDF) architectures to overcome these limitations. Our approach enables dynamic document retrieval without the need to pre-specify the number of documents and eliminates inefficient reranking. We propose an innovative method for converting documents into RDF triplets using JSON key-value pairs, facilitating seamless integration of semi-structured data. Additionally, we present a text to Cypher framework for LPG, achieving over 90% accuracy in real-time translation of text queries to Cypher, enabling fast and reliable query generation suitable for online applications. Our empirical evaluation demonstrates that Graph RAG significantly outperforms traditional embedding-based RAG in accuracy, response quality, and reasoning, especially for complex, semi-structured tasks. These findings establish Graph RAG as a transformative solution for next-generation retrieval-augmented systems.

Accurate prediction of RNA secondary structure underpins transcriptome annotation, mechanistic analysis of non-coding RNAs, and RNA therapeutic design. Recent gains from deep learning and RNA foundation models are difficult to interpret because current benchmarks may overestimate generalization across RNA families. We present the Comprehensive Hierarchical Annotation of Non-coding RNA Groups (CHANRG), a benchmark of 170{,}083 structurally non-redundant RNAs curated from more than 10 million sequences in Rfam~15.0 using structure-aware deduplication, genome-aware split design and multiscale structural evaluation. Across 29 predictors, foundation-model methods achieved the highest held-out accuracy but lost most of that advantage out of distribution, whereas structured decoders and direct neural predictors remained markedly more robust. This gap persisted after controlling for sequence length and reflected both loss of structural coverage and incorrect higher-order wiring. Together, CHANRG and a padding-free, symmetry-aware evaluation stack provide a stricter and batch-invariant framework for developing RNA structure predictors with demonstrable out-of-distribution robustness.

Fluorescence-based Ca$^{2+}$-imaging is a powerful tool for studying localized neuronal activity, including miniature Synaptic Calcium Transients, providing real-time insights into synaptic activity. These transients induce only subtle changes in the fluorescence signal, often barely above baseline, which poses a significant challenge for automated synaptic transient detection and segmentation. Detecting astronomical transients similarly requires efficient algorithms that will remain robust over a large field of view with varying noise properties. We leverage techniques used in astronomical transient detection for miniature Synaptic Calcium Transient detection in fluorescence microscopy. We present Astro-BEATS, an automatic miniature Synaptic Calcium Transient segmentation algorithm that incorporates image estimation and source-finding techniques used in astronomy and designed for Ca$^{2+}$-imaging videos. Astro-BEATS outperforms current threshold-based approaches for synaptic Ca$^{2+}$ transient detection and segmentation. The produced segmentation masks can be used to train a supervised deep learning algorithm for improved synaptic Ca$^{2+}$ transient detection in Ca$^{2+}$-imaging data. The speed of Astro-BEATS and its applicability to previously unseen datasets without re-optimization makes it particularly useful for generating training datasets for deep learning-based approaches.

Polymers are attractive in applications like flexible electronics and thermal interface materials due to their mechanical compliance and processability. However, conventional polymers have low thermal conductivity (TC), limiting their heat dissipation performance. Identifying polymers that simultaneously achieve high intrinsic TC and mechanical flexibility (i.e., low modulus) remains a challenge. Here, we develop an active learning (AL) framework based on multi-objective Bayesian optimization (MOBO) to discover polymers exhibiting both high TC and low bulk modulus. Initially, a high-throughput molecular dynamics (MD) pipeline generated an initial dataset, and independent Deep Kernel Learning (DKL) surrogate models were trained for TC and bulk modulus to predict properties and uncertainties. Using the parallel noisy expected hypervolume improvement (qNEHVI) acquisition function, the framework iteratively screens a larger unlabeled polymer database, systematically recommends new polymer candidates for MD evaluation, and updates the DKL models with newly acquired data. Ultimately, six candidates were identified on the Pareto front, representing optimal trade-offs between TC and modulus. Interpretability analysis further revealed molecular features associated with these trade-offs, and synthesizability assessment supported the practical relevance of the selected candidates. By combining MD simulations with AL-enabled MOBO, our workflow mitigates data scarcity, reduces development time, and provides actionable guidance for designing multifunctional polymers tailored for different applications.

We study a periodically driven spin-$1/2$ Ising chain with a nearest-neighbour coupling and longitudinal field while a weak transverse field induces single-spin flips. Through Floquet perturbation theory (FPT), we obtain signatures of Hilbert space fragmentation (HSF) and an unconventional form of dynamical localisation which we call the Floquet freezing. Our analysis suggests that these observations emerge due to a single Floquet selection rule that dictates the prethermal dynamics. For a special value of the field-to-interaction strength ratio together with commensurate drive periods, this rule permits only a constrained subset of bulk spin flips, leading to prethermal HSF in the full spin-$1/2$ Hilbert space. Under open boundary conditions, the same rule suppresses boundary spin flips up to higher order in perturbation and produces long-lived prethermal edge memory, which is neither topological in origin nor is a strong zero mode. Furthermore, under periodic boundary conditions, the largest surviving fragment is exactly the PXP sector at leading order and therefore exhibits Floquet-inherited scar phenomenology in the prethermal window. At higher commensurate ratios of field strength to interaction strength, all first-order single-spin-flip channels are suppressed and the system enters a regime of Floquet freezing. Hence, our study leverages the selection rules obtained through Floquet perturbation theory to obtain exotic prethermal phenomena at different parameter regimes.

A $t$-spanner of a point set $X$ in a metric space $(\mathcal{X}, δ)$ is a graph $G$ with vertex set $P$ such that, for any pair of points $u,v \in X$, the distance between $u$ and $v$ in $G$ is at most $t$ times $δ(u,v)$. We study the problem of maintaining a spanner for a dynamic point set $X$ -- that is, when $X$ undergoes a sequence of insertions and deletions -- in a metric space of constant doubling dimension. For any constant $\varepsilon>0$, we maintain a $(1+\varepsilon)$-spanner of $P$ whose total weight remains within a constant factor of the weight of the minimum spanning tree of $X$. Each update (insertion or deletion) can be performed in $\operatorname{poly}(\log Φ)$ time, where $Φ$ denotes the aspect ratio of $X$. Prior to our work, no efficient dynamic algorithm for maintaining a light-weight spanner was known even for point sets in low-dimensional Euclidean space.

Future quantum computing architectures require electro-optic materials that maintain a strong, stable performance at cryogenic temperatures. In conventional electro-optic materials, large electro-optic coefficients are often confined to narrow temperature windows near structural phase transitions, where small changes in temperature lead to large changes in the electro-optic response. Using thermodynamic analysis, phase-field simulations, experimental growth and cryogenic optical measurements we show that quantum fluctuations can be harnessed to overcome this trade-off. By tuning the ferroelectric phase boundaries down to 0 K, quantum fluctuations induce a saturation regime in which a large electro-optic response becomes nearly temperature-independent below 25 K. We demonstrate that the phase boundaries can be tuned through either strain in BaTiO3 or through chemical composition in Ba1-xCaxTiO3, leading to a large, temperature insensitive, cryogenic electro-optic effect comparable to bulk BaTiO3 at room temperature; the performance exceeds BaTiO3-on-Si by over an order of magnitude. These findings establish a general design principle for engineering high-performance electro-optic materials for cryogenic applications.

We show that to every small and decaying solution of the linearized constraint equations about Minkowski spacetime, one can add a quadratically small correction to obtain a solution of the full constraint equations. Near spacelike infinity, the correction is given by Kerr black hole initial data, up to a term that decays faster than the linearized solution, and that has Schwartz decay if the linearized solution has Schwartz decay. Using a recent result, we obtain that the solutions of the Einstein equations with these initial data admit a regular conformal compactification along null and timelike infinity. The construction is based on a right inverse (up to necessary integrability conditions) for the linearized constraint operator about Minkowski initial data obtained previously, that has optimal mapping properties relative to weighted b-Sobolev spaces, where the weights measure decay towards infinity. On an algebraic level, we show that the constraint equations can be derived using the homotopy transfer theorem, rather than using the geometric Gauss and Codazzi equations.

Stablecoins serve as the fundamental infrastructure for Decentralised Finance (DeFi), acting as the primary bridge between fiat currencies and the digital asset ecosystem. While peg stability is well-documented, the structural role stablecoins play in transmitting systemic risk to the broader market remains under-explored. This study uses copula-based approaches to quantify the transmission of volatility and activity from stablecoin to cryptocurrency markets. We demonstrate in-sample causality across daily, weekly, and monthly horizons. Furthermore, we show that incorporating stablecoin factors significantly reduces Mean Squared Error in cryptocurrency forecasting. Specifically, we link stablecoin volume and upside volatility to broader market volatility, indicating its role as dry powder. Finally, we establish economic value by demonstrating reduced risk in a cryptocurrency volatility targeting model when stablecoin factors are employed.

Intercavity polaritons, hybrid quasiparticles with spatially separated photonic and excitonic components, provide a platform to engineer structured light-matter states. We show that resonant driving of the middle polariton branch leads to a qualitatively distinct dynamical regime in which coherent Rabi oscillations are suppressed, and the system evolves monotonically toward its steady state. Including interactions, we demonstrate that this regime supports Bogoliubov excitations with a phonon-like dispersion at low momenta. These collective modes inherit interactions from the excitonic fraction, while preserving the intrinsically intercavity nature of the quasiparticles.

Understanding how electric fields destabilize biological membranes is important for electroporation-based technologies and bioelectronic interfaces. However, theoretical descriptions of this phenomenon remain fragmented. Existing theories treat either electrostatics in membranes of finite thickness or electrohydrodynamic flows at idealized zero-thickness interfaces, leaving unresolved a unified description that simultaneously incorporates finite membrane thickness, surface charge, and bulk electrohydrodynamics. Here, we apply a recently-developed, dimension-reduction framework that captures the coupled electrohydrodynamic and mechanical effects governing height fluctuations of a charged lipid bilayer of thickness $δ$ in an electrolyte characterized by Debye screening length $λ$. We derive voltage- and charge-dependent renormalizations of the effective surface tension and bending rigidity, along with a dispersion relation governing undulatory instabilities. A wide range of prior theoretical results arise as limiting cases of our more general theory when finite-thickness effects are neglected or screening is asymptotically strong. The key new contribution arises from traction moments generated across the finite membrane thickness, which are absent in zero-thickness descriptions. Under physiological screening ($δ/λ\sim 4$), these contributions account for more than $>70\%$ of the total electrostatic correction to both surface tension and bending rigidity. The theory further reveals that surface charges can stabilize the membrane at physiological ionic strengths, increasing the effective tension and shifting electroporation thresholds in a manner that depends on charge asymmetry between the leaflets.

We consider a quantum switch with a finite number of quantum memory registers that aims to serve multipartite entanglement requests among $N$ users. We propose scheduling policies that aim to optimize the average number of requests served per unit time by efficiently utilizing the switch's available memory. To measure the performance of the scheduling policies, we employ the newly introduced metric of age of entanglement establishment (AoEE). We formulate the scheduling problem in a restless multi-armed bandit (RMAB) framework. We show that the scheduling of entanglement requests is indexable. Subsequently, we find a closed-form expression of the Whittle index for all possible request-age pairs. By modeling the Whittle index of each request as its reward and its cardinality as its cost, we formulate the memory-constrained scheduling problem as a $0$-$1$ knapsack problem and solve it via dynamic programming. Furthermore, we consider two low-complexity sequential greedy policies that leverage two different modified Whittle indices.

Acoustic holography provides a practical means of flexibly controlling acoustic wavefronts. However, high-fidelity shaping of acoustic fields remains constrained by the numerical-physical gap inherent in conventional phase-only designs. These approaches realize a two-dimensional phase-delay profile as a three-dimensional thickness-varying lens, while neglecting wave-matter interactions arising from the lens structure. Here, we introduce an end-to-end, physics-aware differentiable structural optimization framework that directly incorporates three-dimensional lens geometries into the acoustic simulation and optimization loop. Using a novel differentiable relaxation, termed Differentiable Hologram Lens Approximation (DHLA), the lens geometry is treated as a differentiable design variable, ensuring intrinsic consistency between numerical design and physical realization. The resulting Thickness-Only Acoustic Holograms (TOAHs) significantly outperform state-of-the-art phase-only acoustic holograms (POAHs) in field reconstruction fidelity and precision under complex conditions. We further demonstrate the application of the framework to spatially selective neuromodulation in a neuropathic pain mouse model, highlighting its potential for non-invasive transcranial neuromodulation. In summary, by reconciling numerical design with physical realization, this work establishes a robust strategy for high-fidelity acoustic wavefront shaping in complex environments.

Search engines (SEs) and large language models (LLMs) are central to political information access, yet their algorithmic decisions and potential underlying biases remain underexplored. We developed a standardized, privacy-preserving, bot-and-proxy methodology to audit four SEs and two LLMs before the 2024 European Parliament and US presidential elections. We collected answers to approximately 4,360 queries related to elections in five EU countries and 15 US counties, identified political entities and topics in those answers, and mapped them to ideological positions (EU) or issue associations (US). In Europe, SE results disproportionately mentioned far-right entities beyond levels expected from polls, past elections, or media salience. In the US, Google strongly favored topics more important to Republican voters, while other search engines favored issues more relevant to Democrats. LLMs responses were more balanced, although there is evidence of overrepresentation of far-right (and Green) entities. These results show evidence of bias and open important discussions on how even small skews in widely used platforms may influence democratic processes, calling for systematic audits of their outputs.

We propose the Locally Pumped Dark Energy (LPDE) mechanism in which cosmic acceleration is triggered by the emergence of non-linear dark matter structure. In an effective-field-theory description, coarse-graining over the density contrast profile, whose short-wavelength modes grow during halo formation, induces a shift in the local equilibrium point of a second, sufficiently heavy scalar field $χ$. At early times, the pump mechanism is negligible and $χ$ remains fixed at the origin, contributing no DE. As structures form, the equilibrium value of $χ$ is locally displaced within halos, generating a vacuum energy whose global contribution, in a mean-field picture, is controlled by the halo volume filling factor. If the $χ$ field is sufficiently heavy, with a Compton wavelength limited by halo scales, its response is localised, and spatial gradients are exponentially suppressed on large scales. After volume-averaging over the halo population, the resulting contribution on large scales behaves as a homogeneous DE component. Using the halo mass function of a fiducial $Λ$CDM cosmology, we show that vacuum-energy domination generically emerges at $z\sim\mathcal{O}(1)$, naturally correlating cosmic acceleration with structure formation. For reference, we present an explicit realisation of such a mechanism and show that, by naturally featuring a transient acceleration epoch, it can be in excellent agreement with the most recent cosmological data, including the Dark Energy Spectroscopic Instrument (DESI).

AI agents -- systems that can independently take actions to pursue complex goals with only limited human oversight -- have entered the mainstream. These systems are now being widely used to produce software, conduct business activities, and automate everyday personal tasks. While AI agents implicate many areas of law, ranging from agency law and contracts to tort liability and labor law, they present particularly pressing questions for the most globally consequential AI regulation: the European Union's AI Act. Promulgated prior to the development and widespread use of AI agents, the EU AI Act faces significant obstacles in confronting the governance challenges arising from this transformative technology, such as performance failures in autonomous task execution, the risk of misuse of agents by malicious actors, and unequal access to the economic opportunities afforded by AI agents. We systematically analyze the EU AI Act's response to these challenges, focusing on both the substantive provisions of the regulation and, crucially, the institutional frameworks that aim to support its implementation. Our analysis of the Act's allocation of monitoring and enforcement responsibilities, reliance on industry self-regulation, and level of government resourcing illustrates how a regulatory framework designed for conventional AI systems can be ill-suited to AI agents. Taken together, our findings suggest that policymakers in the EU and beyond will need to change course, and soon, if they are to effectively govern the next generation of AI technology.

Large language models (LLMs) have shown promising results for software engineering applications, but still struggle with code reasoning tasks such as vulnerability detection (VD). We introduce ConceptCoder, a fine-tuning method that simulates human code inspection: models are trained to first recognize code concepts and then perform reasoning on top of these concepts. In prior work, concepts are extracted by multimodal models or LLMs to explain vision and natural language models. Our work is the first to formulate concepts for code. We define code concepts as human-understandable semantic properties of code and train models to learn such concepts. Our evaluation shows that this approach significantly improves VD accuracy, from 66.32 to 72.15 F1 on average over 9 open-source LLMs. ConceptCoder achieves the best VD performance compared to state-of-the-art (SOTA) baselines, including fine-tuned SOTA open-source LLMs and prompted proprietary models such as GPT-5.2 and Claude-Opus-4.5. Our approach also scales: concepts defined from four types of vulnerabilities benefit general vulnerability datasets with 134 CWEs. We further demonstrate that concept-based fine-tuning generalizes beyond VD and improves branch prediction. We release our code and datasets at https://figshare.com/s/1decab8232c653b44f71.

We ask under what conditions a finite brickwork circuit of random gates retains local information about the initial state. To answer this question we measure the averaged Frobenius distance between the reduced states obtained by evolving two arbitrary initial states and tracing out a portion of the system. By characterising this distance exactly at all times we find that the information is retained if the environment -- the subsystem traced out -- is smaller than half of the system and washed away otherwise. We also find that, while the dynamics of the Frobenius distance depends on the specific initial states chosen, this dependence becomes increasingly weak for large scales and eventually the Frobenius distance attains a universal form as a function of time. Finally, we show that by introducing weak enough boundary dissipation, one can observe a phase transition between a memory preserving phase and one where the information is completely lost.

We establish an information-theoretic scaling law for generic autoregressive neural quantum states, determined by the middle-cut mutual information of the wavefunction amplitude. By formalizing the virtual bond as an effective information channel across a sequence bipartition, we rigorously prove that exact autoregressive representation of a quantum state requires the virtual-bond dimension to scale with the amplitude mutual information. For stabilizer-state families, we show that this law yields an explicit, analytical rank formula. Applying this framework across quantum-state tomography, ground-state and finite-temperature learning, our numerical experiments expose precise exponent matching, architecture-dependent scaling differences between recurrent and Transformer neural quantum state, and the critical role of autoregressive basis ordering. These results establish a rigorous physical link between the intrinsic structure of a quantum many-body state and the corresponding neural-network capacity required for its faithful representation.

We survey recent advances in non-abelian Hodge theory in the "mixed" setting of non-proper algebraic varieties. We then describe how these tools are used to construct algebraic Shafarevich morphisms and prove a version of the linear Shafarevich conjecture for any algebraic variety.

Density functional theory (DFT) offers an exceptional balance between accuracy and efficiency, but practical density functional approximations face an unavoidable trade-off among simplicity, accuracy, and transferability. A systematic protocol is therefore needed to develop functionals that are reliably most accurate within a chosen application domain. Here we present such a protocol by combining constraint enforcement, flexible functional forms, and modern optimization. Applying this strategy to the range-separated hybrid (RSH) meta-GGA framework, we obtain the carefully optimized and appropriately constrained hybrid (COACH) functional. Across broad molecular benchmarks, COACH improves both accuracy and transferability relative to leading RSH meta-GGAs, including \omegaB97M-V, while retaining the computational practicality of its rung. Finally, our analysis of the remaining trade-offs and saturation behavior suggests that further systematic progress will likely require the incorporation of genuinely nonlocal information.

Compounding error, where small prediction mistakes accumulate over time, presents a major challenge in learning-based control. A common remedy is to train multi-step predictors directly instead of rolling out single-step models. However, it is unclear when the benefits of multi-step predictors outweigh the difficulty of learning a more complex model. We provide the first quantitative analysis of this trade-off for linear dynamical systems. We study three predictor classes: (i) single step models, (ii) multi-step models, and (iii) single step models trained with multi-step losses. We show that when the model class is well-specified and accurately captures the system dynamics, single-step models achieve the lowest asymptotic prediction error. On the other hand, when the model class is misspecified due to partial observability, direct multi-step predictors can significantly reduce bias and improve accuracy. We provide theoretical and empirical evidence that these trade-offs persist when predictors are used in closed-loop control.

Funayama proved that a lattice embeds into a complete Boolean algebra in such a way that all existing joins and meets are preserved if and only if the lattice satisfies the join-infinite and meet-infinite distributive laws. There are several proofs of this classic result in the literature. In this note, we provide a new and purely order-theoretic proof of Funayama's theorem, as well as of generalizations of the theorem.

The TRAP protocol solves rational agreement by combining accountable consensus with a one-shot BFTCR finalization phase. We present SNARE (Scalable Nash Agreement via Reward and Exclusion), the adaptation of TRAP to $n=5f{+}1$, and prove $ε$-$(k,t)$-robustness for rational agreement tolerating coalitions up to ${\approx}73\%$ with deposits under $0.5\%$ of the gain. A central finding is that appending a single all-to-all broadcast round with the $4f{+}1$ threshold after predecisions yields $ε$-$(k,t)$-robustness for coalitions up to $3f$ (${\approx}60\%$) without any deposit: we need not model or know the utility function of deviating players, only that they participate in the protocol. These players can be \emph{deceitful} (arbitrary unknown utility), not just rational, and the finalization structure prevents disagreement regardless of their motivation. This observation is protocol-agnostic, applies to any $5f{+}1$ protocol at the cost of one message delay that runs concurrently with the next view, and does not require commit-reveal mechanisms. Above $60\%$, the full baiting mechanism with deposits under $0.5\%$ extends tolerance to ${\approx}73\%$. A second finding is that valid-candidacy, the property preventing reward front-running, holds unconditionally regardless of the quorum threshold, removing both the $n>2(k{+}t)$ and $n>\frac{3}{2}k{+}3t$ constraints from the original TRAP. This retroactively extends the $3f{+}1$ bound from $C<n/2$ to $C<5n/9$. The binding constraint in both models is the winner consensus operating on $2f$ residual players after excluding $3f{+}1$ detected equivocators. We explore avenues for relaxing this limit.

We prove that every Mahler series, over a field of characteristic $0$, with multiplicative coefficients is regular in the sense of Allouche and Shallit. We also obtain an explicit characterization of such series. This yields a joint extension of the characterization of rational series with multiplicative coefficients (by Bézivin and Bell--Bruin--Coons) and of multiplicative automatic sequences (by Konieczny--Lemańczyk--Müllner). Both of these results are used in our characterization, so we do not obtain new proofs of these special cases.

A search for new particles decaying to top quark-antiquark pairs is performed using proton-proton collision data at a centre-of-mass energy of 13 TeV. The data set recorded with the CMS detector between 2016 and 2018 is used, corresponding to an integrated luminosity of 138 fb$^{-1}$. Final states with 0, 1, and 2 leptons are analyzed, covering all decay modes of the top quark-antiquark pairs. Heavy Z' bosons with relative widths of 1, 10, and 30% are excluded for masses in the ranges 0.4$-$4.8, 0.4$-$6.2, and 0.4$-$7.4 TeV, respectively. A Kaluza$-$Klein gluon in the Randall$-$Sundrum model and a dark-matter mediator are excluded for masses between 0.5$-$5.5 and 1.0$-$4.2 TeV, respectively. These results set the most stringent limits to date for the considered models in the $\mathrm{t\bar{t}}$ final state. In addition, in the two-Higgs-doublet models, upper limits are set on the coupling strength modifier for scalar and pseudoscalar Higgs bosons with relative widths of 2.5, 10, and 25% in the mass range of 0.5$-$1.0 TeV.

We study perturbations of relative cubic Dirac operators for basic classical Lie superalgebras within the uniform formalism of the colour quantum Weil algebra. This perspective leads to three complementary classes of perturbations and resulting invariants. First, we define semisimple perturbations that assign to each finite-dimensional simple supermodule a finite collection of semisimple orbits, together with canonically defined vector spaces measuring the degree of atypicality. Second, we introduce nilpotent perturbations parametrized by the self-commuting variety of a quadratic Lie subsuperalgebra; the resulting family of cohomology theories combines Dirac cohomology and Duflo--Serganova cohomology. Third, we deform the cubic Dirac operator by a Weil-covariant differential built from the universal $1$-form in the colour quantum Weil algebra and the Weil differential, producing a Chern-type invariant that assigns to each finite-dimensional module a natural class in the cohomology of the Weil complex.

For a general $k$-gonal curve $C$ with a morphism $f: C \rightarrow \mathbb{P}^1$ of degree $k$, we consider the refinement of the Brill-Noether schemes $W^r_d(C)$ by means of the Brill-Noether degeneracy schemes $\overlineΣ_{\overrightarrow {e}}(C,f)$. The schemes $\overlineΣ_{\overrightarrow {e}}(C,f)$ as sets are closures of subsets $Σ_{\overrightarrow {e}}(C,f)$ of $\Pic (C)$ and as a scheme $Σ_{\overrightarrow {e}}(C,f)$ is a smooth open subscheme of $\overlineΣ_{\overrightarrow {e}}(C,f)$. In this paper we describe naturally defined open subsets of $\overlineΣ_{\overrightarrow {e}}(C,f)$ in general strictly containing $Σ_{\overrightarrow {e}}(C,f)$ such that $\overlineΣ_{\overrightarrow {e}}(C,f)$ is smooth along them. As an application we describe all invertible sheaves $L$ on $C$ having an injective Petri map. Some of those sets $\overlineΣ_{\overrightarrow {e}}(C,f)$ are the irreducible components of $W^r_d(C)$. In those cases we prove $W^r_d(C)$ is smooth at a point $L$ of those larger open subsets of $\overlineΣ_{\overrightarrow {e}}(C,f)$ unless $L$ belongs to at least two irreducible components of $W^r_d(C)$ (such points exist). On the other hand in general the singular locus of the schemes $W^r_d(C)$ is not equal to the complement of the union of $W^{r+1}_d(C)$ and the intersections of two different components of $W^r_d(C)$.

This work studies the synthesis of active perception policies for predictive safety monitoring in partially observable stochastic systems. Operating under strict sensing and communication budgets, the proposed monitor dynamically schedules sensor queries to maximize information gain about the safety of future states. The underlying stochastic dynamics are captured by a labeled hidden Markov model (HMM), with safety requirements defined by a deterministic finite automaton (DFA). To enable active information acquisition, we introduce minimizing k-step Shannon conditional entropy of the safety of future states as a planning objective, under the constraint of a limited sensor query budget. Using observable operators, we derive an efficient algorithm to compute the k-step conditional entropy and analyze key properties of the conditional entropy gradient with respect to policy parameters. We validate the effectiveness of the method for predictive safety monitoring through a dynamic congestion game example.

Missing values are ubiquitous in (data) science, with potential detrimental consequences for any statistical analysis. As a consequence, a wealth of methods and theoretical results have been developed in recent years. Still, many questions remain open, in particular in the case of general non-monotone missing at random (MAR). In this work, we extend nonparametric Bayesian theory to this MAR setting. We introduce a general theorem of posterior contraction under MAR and an additional mild positivity condition. Using this result, we are able to show that, despite the missing values, the density of the uncontaminated data can be estimated with the minimax posterior contraction rate up to log factors. To the best of our knowledge, this is the first nonparametric result showing that the uncontaminated distribution can be consistently estimated under Rubin's MAR definition. As a consequence, we obtain an algorithm that takes data contaminated with missing values and returns a sample from a provably consistent estimate of the uncontaminated distribution.