Papers

This research describes the adaptation of Large Language Models (LLMs) for economic monitoring in the public sector to automatically determine whether an article discusses Economic Policy Uncertanity (EPU) and to identify its specific type. Previous studies either rely on keywords, which often result in a high count of false positives, or use machine learning approaches that require a large number of quality human labeled data that is costly and time consuming to acquire. In this study, we propose approaches based on weak supervision techniques, using generative LLMs to create synthetic labels through prompting, making the approach both cost-effective and scalable. Additionally, we propose methods for for multi-label and hierarchical classification of articles related to EPU.

Optimizing deployed AI systems increasingly amounts to editing prompts, skills, harnesses, and code rather than model weights. Existing approaches commonly optimize these artifacts through propose-evaluate-select procedures, where candidate configurations are evaluated and only those meeting an acceptance criterion are selected. Yet our analysis shows that discarded candidates often contain information critical for subsequent optimization. Discarding them causes later proposals to revisit the same failure modes. We introduce Mara Chain, a refinement procedure that turns rejected candidates into stepping stones. Rather than discarding a rejected candidate, Mara Chain retains and iteratively refines it using evidence accumulated across preceding attempts. The procedure limits each refinement chain to a fixed depth and applies Pareto-filtered Top-N selection to bound the candidate pool. Across AppWorld skill optimization, TerminalBench 2.1 harness optimization, and MuSiQue retrieval-pipeline optimization, Mara Chain delivers greater task-performance gains with fewer rollouts. It outperforms GEPA, ACE, and SkillOpt-Lite by up to 20.5% in relative performance on AppWorld, reaching the target score with 65.5% fewer rollouts than GEPA. It improves the pass rate by 20.2 and 22.5 percentage points over AHE and Meta-Harness on TerminalBench 2.1, respectively, and improves MuSiQue test nDCG@10 and Recall@10 by 0.104 and 0.131 over a hand-written retrieval pipeline.

In the field of Machine Learning, many papers contain empirical results supporting claimed statements or illustrating the performance of a proposed method. However, most practitioners know that (1) results are generally hard to reproduce, and increasingly so, (2) code is not often available to do so, and (3) it hinders the development of research. In this position paper, we analyze and quantify these issues, and make concrete proposals to improve result checkability, if not reproducibility. Code and supporting materials are available at https://github.com/giddyyupp/position-enforce-verifiability.

We show that, in $\MG^{-}$ — $\MG$, the Hilbert plane, deprived of the Side-Angle-Side axiom — a hypotenuse-angle criterion for right triangles, together with the existence of a perpendicular from an external point to a line and a ray correspondence principle governing the ordering of angles, suffices to reconstruct SAS in full — and hence, as standard consequences, SSS and SAA as well. This offers a more elementary alternative to Hilbert's own choice of SAS as an axiom, confined throughout to right triangles and provable without any continuity assumption. Several recent papers have proposed related alternatives on comparable grounds; we compare our approach to theirs in the concluding remarks. The result builds on, and is best read alongside, two earlier papers of ours on related reconstructions of SAS \cite{Volpe2026I,Volpe2026II}.

Pooled statistics of Transformer weights obscure how magnitude is distributed across functional channels, while individual weights are too numerous to compare directly. We study the mesoscopic level between them: row and column scale fields, the median-centred log-RMS profiles of a weight matrix over its channels, which together with a global scale and a full balanced core represent the matrix exactly. Across public Pythia checkpoints at four sizes and controlled runs from three initialization families, balancing reveals similar measured core magnitude profiles. A mixture bridge, with its form fixed before the analysis and its coefficients fitted, predicts the pooled-shape departure from field width on held-out runs and data arms of the controlled grid. The indexed fields retain further structure: they align across projections that share a functional channel, and query/key profiles follow reassigned RoPE frequencies rather than fixed matrix coordinates. Training trajectories show early field formation followed by component-dependent broadening or recession. Extending the channel-based analysis to AdamW's second moment reveals related functional organization in its log-space row and column factors. Finally, edits of a frozen checkpoint separate reciprocal scale balance, which preserves the forward computation, from relative channel gain: flattening the gain increases in-distribution loss while preserving matrix norms and the balanced core. Row and column scale fields thus connect pooled magnitude statistics to channel organization and provide coordinates for tracking and testing trained weight structure.

We show that there exists a subadditive bijection $f:(0,\infty)\to(0,\infty)$ such that $\liminf_{x\to 0}f(x)=0$ and $\limsup_{x\to 0}f(x)=1$. This answers a question by Matkowski and Swiatkowski in [Proc. Amer. Math. Soc. 119 (1993), 187–197].

Ionizing radiation is a major source of correlated errors in superconducting quantum processors, yet the mechanisms responsible for the qubit response to particle interactions remain only partially understood. In this work, we operate superconducting qubits as particle detectors while simultaneously monitoring the deposited energy with an independent semiconductor cryogenic sensor. This complementary measurement provides an absolute determination of the particle interaction rate in the chip, enabling the first direct measurement of the detection efficiency of superconducting qubits exposed to environmental radiation. We measure individual qubit detection efficiencies of about 30-40$\%$, increasing to approximately 50$\%$ when combining the response of two qubits. Under the conservative assumption that the efficiency loss is entirely determined by a finite detection threshold, we derive an upper limit on the effective energy threshold of the detector of $70\pm20~\mathrm{(stat)}\pm30~\mathrm{(syst)}~\mathrm{keV}$. Measurements with radioactive sources producing different deposited-energy spectra further show that the detection efficiency decreases for lower-energy interactions. We further investigate the origin of the qubit response by injecting controlled thermal pulses with a resistive heater. Although these pulses deposit energies comparable to those released by particle interactions, they do not reproduce the radiation-induced signatures, demonstrating that the qubit response to radiation cannot be explained by a transient increase in substrate temperature alone. Our results establish an experimental framework for quantitatively connecting particle energy deposition to radiation-induced responses in superconducting quantum circuits.

Guo and Igusa introduced the derived and sub-derived delooping levels as refinements of the delooping level and asked how these two invariants compare, whether the derived delooping level always coincides with the opposite big finitistic dimension, and, if not, how large their difference can be. We answer these questions. First, we prove that every Artin algebra $A$ satisfies $\operatorname{ddell} A\leq \operatorname{sub-ddell} A$. This gives a positive answer to the ordinary-degree comparison problem, whereas a single explicit eight-dimensional algebra $C$ satisfies $k\text{-}\operatorname{sub-ddell} C=0<1=k\text{-}\operatorname{ddell} C$ for $k\geq 2$. We then compute, for every $k\geq 1$, all $k$-derived and $k$-sub-derived delooping levels of the monomial family of Barrios–Lanzilotta–Mata. In particular, that family satisfies $k\text{-}\operatorname{ddell} A_{n,s}=\operatorname{Findim}(A_{n,s}^{\mathrm{op}})$ for $k\geq 1$. In ordinary degree, this recovers an equality previously announced by Lam. We also construct a two-vertex, eleven-dimensional monomial algebra $\Lambda$, over an arbitrary field, such that $\operatorname{Findim}(\Lambda^{\mathrm{op}})=1<2=\operatorname{ddell}\Lambda=\operatorname{dell}\Lambda$. The same strict inequality is obtained by Gao, Liu and Xu using different algebras. A ten-dimensional non-monomial variant with a commutative corner algebra is also given. Finally, we construct a family of finite-dimensional monomial algebras $B_r$ satisfying $\operatorname{Findim}(B_r^{\mathrm{op}})=1$ and $\operatorname{ddell} B_r=2r+1$. Thus the difference $\operatorname{ddell} A-\operatorname{Findim}(A^{\mathrm{op}})$ is unbounded even among monomial algebras with fixed opposite big finitistic dimension.

On-board compression of synthetic aperture radar (SAR) phase history is bandwidth-critical, and block-adaptive quantization (BAQ) remains the operational standard. We test whether a small convolutional autoencoder, with its encoder on the sensor, can compete with BAQ on complex phase-history patches from the AFRL GOTCHA collection. Every method is charged for all transmitted bits, rates are reported in bits per complex sample (b/cs), and detection is scored by one-to-one matching of CA-CFAR detections. The autoencoder (28,656 encoder parameters) loses at every rate. At 16 b/cs it reaches -2.87 dB NMSE, against -35.5 dB for 8-bit BAQ with $\pm 3\sigma$ clipping and -41.0 dB with a tuned clipping range. It also loses to a $16 \times 16$ block Karhunen-Loève transform (KLT), a local linear coder with a tenth of its encoder cost (-5.39 dB). Running the network in a companded Fourier domain helps, but its detection F1 remains bounded at 33%. The evidence points to this model, its normalization, and its objective, not to a fundamental limit of learned coding. Per patch, the data have modest lag-1 coherence ($|\rho| \approx 0.3$) and patch-specific spectral concentration. Two findings concern evaluation itself. First, 97% of CFAR crossings on raw $64 \times 64$ patches are border artifacts of the zero-padded detector. Second, on interior cells BAQ's clipping range decides detection: 8-bit BAQ keeps 69% F1 with tuned clipping but 17% at $\pm 3\sigma$, and at 8 b/cs or less adaptive FFT thresholding preserves more detections than BAQ. We close with an evaluation protocol for learned radar compression.

We present a constructive atlas of finite simple groups with proofs of their orders, simplicity, and structural properties. The eight completed families are cyclic groups of prime order, alternating groups, the classical series $A_r(q)$, $B_r(q)$, $C_r(q)$, $D_r(q)$, and the exceptional series $G_2(q)$ and the small Ree groups $ {}^2G_2(3^{2m+1})$, $m\geq1$. The fifteen sporadic entries are $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$, $M_{24}$, $\mathrm{Co}_1$, $\mathrm{Co}_2$, $\mathrm{Co}_3$, $\mathrm{McL}$, $\mathrm{HS}$, $\mathrm{Suz}$, $J_2$, and $\mathrm{Fi}_{22}$, $\mathrm{Fi}_{23}$, $\mathrm{Fi}_{24}'$. The simple parameter ranges and exceptional cases are stated explicitly. The models arise from codes, lattices, forms, algebras, and finite geometries, including the split octonions and the Conway–Parker algebra. They retain natural actions, stabilizers, central quotients, and comparison maps for subsequent group theory. Several classical comparison isomorphisms relate the models, while involution-class counts distinguish the equal-order orthogonal and symplectic families in odd characteristic and rank at least three. Drawing on classical sources and companion mathematical work, the project develops the construction side of finite simple group theory, not the exhaustiveness proof of the classification. The completed models and stated proofs are formalized in Lean for verification and reuse.

We classify the simultaneous half-bandwidths $p,q$ of a symmetric oscillatory matrix $A$ and its inverse. At order $n\ge2$, the possible pairs are $(1,n-1)$, $(n-1,1)$, and all pairs with $2\le p,q\le n-1$. Let $e_k(A)$ be the least positive integer $m$ for which all minors of order $k$ of $A^m$ are positive, and let $e(A)=\max_{1\le k<n}e_k(A)$. Put $a=\min\{p,q\}$ and $b=\max\{p,q\}$. At every feasible pair of actual half-bandwidths, \[ \max_A e(A)= \max\left\{\left\lfloor\frac n2\right\rfloor, n-a-\left\lfloor\frac{\max\{n-p-q,0\}}{b}\right\rfloor\right\}. \] We also determine the maximum of $\sum_{k=1}^{n-1}e_k(A)$ for every pair. When $p+q\ge n$, one realization maximizes all $e_k$ simultaneously. Write $A=U^T\Delta U$, with $U$ unit upper triangular and totally nonnegative, and $\Delta$ positive diagonal. Let $\ell(U)$ be the least number of elementary upper bidiagonal factors with positive parameters whose product is $U$. At fixed $(p,q)$, its minimum is $\max\{n-1,p+q-1\}$. One symmetric integer matrix of determinant one simultaneously attains this minimum, the maximum exponent, and the maximum exponent sum. We prove $\ell(U^{r+1})-\ell(U^r)=\#\{k:1\le k<n,\ e_k(A)>r\}$ for every integer $r\ge\lfloor(n-1)/2\rfloor$, with the starting point optimal. With $\delta=\ell(U)-n+1$, the exponent sum satisfies $\delta\le n(n-1)/2-\sum_{k=1}^{n-1}e_k(A) \le\lfloor(n-1)/2\rfloor\delta$, with both constants sharp. The lower deficit bound is attained at every admissible factor count. Within the symmetric class, a total sum of $n(n-1)/2$ characterizes basic matrices. In the basic oscillatory class, without a symmetry assumption, a positive integer vector $(f_1,\ldots,f_{n-1})$ is realizable if and only if $f_k\le\max\{k,n-k\}$ and $f_k+f_{n-k}\ge n$ for $1\le k<n$, and $\lvert f_{k+1}-f_k\rvert\le1$ for $1\le k<n-1$. Explicit factor orders realize every such vector.

Macroeconomic productivity metrics, such as Total Factor Productivity, register technological breakthroughs with multi-year reporting lags due to administrative survey intervals and national accounting conventions. This paper introduces Hyperspherical Semantic Trajectory Analysis (HSTA), an unsupervised quantitative methodology that tracks technology diffusion directly from unstructured scientific and commercial text streams. We analyze 30,000 filtered document records spanning academic preprints from arXiv and patent application records from the USPTO. By projecting high-dimensional Transformer sentence embeddings onto unit hyperspheres using Spherical K-Means clustering across eight primary sub-topics and UMAP manifold reductions, HSTA formalizes two quantitative metrics: (1) Semantic Centroid Vector Drift, which tracks vocabulary shifts between temporal sub-corpora to identify structural paradigm transformations; and (2) Commercialization Offset, which evaluates cross-corpus peak density alignments between scientific discovery and intellectual property filings. Linking quarterly topic volume velocity with physical hardware metrics from the Epoch AI database, Vector Autoregressive F-tests demonstrate that quarterly paper volume velocity alone does not Granger-cause frontier compute allocation surges at conventional statistical significance levels, highlighting the necessity of conditioning textual signals on physical capital constraints. Empirical results reveal that sub-topics covering Large Language Models (with a drift metric of 0.332) and Artificial Intelligence Systems (with a drift metric of 0.234) undergo the highest rate of semantic evolution, offering an objective, real-time mechanism to complement traditional economic statistics.

Pair arithmetic omits the renormalization stage performed at the end of each operation of multi-component multiple-precision arithmetic, reducing the operation count and, by eliminating conditional branches, easing SIMD vectorization. Renormalization cannot be omitted entirely, however, because iterative solvers may then fail to converge. We propose RenormBF-pair, a branch-free fixed-trip-count renormalizer parameterized by the word count \(K\) and the input length \(n\): a VecSum sweep, a tail fold, and \(r\) rounds of a FastTwoSum chain, costing \(6(n-1)+(n-K)+3r(K-1)\) flops. Over \(200{,}000\) trials per condition, the smallest round count producing no non-overlap violation was \(r=1\) for \(K \leq 3\) and \(r=2\) for \(K=4\), whereas the same cost spent on a VecSum\(K\)-style TwoSum chain fails for \(K=4\). We delimit with explicit counterexamples both the input condition under which the sum relation holds and the fact that the scheme does not, in general, guarantee a correctly rounded lowest component. Integrating it into CG and BiCGStab over pair arithmetic in binary64 and binary32, we sweep \(240\) paired runs including real matrices screened from SuiteSparse. The proposed scheme and VecSum\(K\) returned bitwise identical words wherever both succeeded, so they are indistinguishable for CG in binary64; differences appear only where VecSum\(K\) breaks, namely binary32 and BiCGStab. There the \(22\)–\(16\) win–loss record is not significant (sign test, \(p=0.42\)); the clear bias is structural, non-overlap violations dropping from \(66\) rows to \(37\). The cost is \(+3.4\%\) for the whole CG solver.

Temporal dynamics in learning behavior can be revealed through period segmentation in Transition Network Analysis (TNA). Cristea et al. demonstrated that segmenting courses into halves and quarters reveals how learning strategies evolve and relate to academic performance. Building on this approach, we investigate whether Topological Data Analysis (TDA), specifically connected components ($\beta_0$) change points from Zigzag Persistent Homology, can provide data-driven period boundaries that identify intervention windows. Analyzing 22 courses from the Open University Learning Analytics Dataset, we find that $\beta_0$-based segmentation captures greater between-period variation than time-based segmentation (median variance ratio (VR) = 4.10$\times$; 21/22 courses show VR $>$ 1.1). Permutation tests confirmed significance ($p < 0.05$) in 18% of individual courses, with 73% showing positive improvement over random breakpoints. Only one course showed better performance with time-based segmentation. However, analysis of academic outcomes reveals a key insight: final-period behavior shows lower correlation with outcomes in $\beta_0$-based segmentation than in time-based segmentation, reflecting a marked behavioral collapse where engaged learners rapidly disengage. We interpret this as evidence that final-period behavior reflects rather than causes outcomes: students who will pass maintain engagement, while those who will fail disengage. This reframes the value of $\beta_0$: rather than improving prediction, change points identify intervention windows. These are periods where behavioral structure shifts and targeted support may be most effective.

Powered by expert guidance, agents can operate in interactive environments; however, it is unclear whether they can learn autonomously from their own experience. To evaluate such self-improvement methods, we introduce GameBoyWorlds, a testbed for agentic self-improvement in video games. GameBoyWorlds-Execution evaluates task execution on a collection of 5 distinct game series. Agents are allowed access to dedicated training games but are provided no demonstrations, documentation, or rewards. Agents must ground themselves in the environment through self-directed exploration and by inferring actionable knowledge from their own experience. At test time, agents must complete short-horizon tasks that evaluate their ability to navigate, interact, and engage with game-specific mechanics in unseen games. Out-of-the-box frontier models complete fewer than 50% of the 500 tasks due to failures in multimodal grounding, establishing that self-improvement methods have room to push performance. We demonstrate that contemporary approaches to self-improvement are lacking, with world modelling and autonomous skill discovery failing, and a novel strategy that uses curiosity-based exploration to write guides achieving only partial success. GameBoyWorlds-Playthrough tests end-to-end game completion in two fan-made Pokémon games. We show that while frontier models have been pre-exposed to official releases such as Pokémon Red, they lack essential information on the games in our testbed. Instead of relying on their parametric knowledge to succeed, agents must learn from their own experience and autonomously improve over the course of the playthrough. We show that a sophisticated agentic pipeline with multimodal memory and hierarchical subgoals fails to reach even the first major milestone in both games, establishing GameBoyWorlds as an ambitious target for self-improving agents.

We present a framework for evaluating and improving a large-scale, multi-agent shopping assistant in production, and report lessons from its use. Offline evaluation of such a system faces three obstacles. (i) A logged conversation cannot be replayed against a modified system, because a different response changes every turn that follows. (ii) The unchanged system itself varies from run to run. Its LLM components are stochastic, and in product search the available products, their prices, and the customer's personalization signals change. (iii) Aggregate quality scores combine distinct behaviors, so they show that quality has changed but not which behavior caused the change. Our framework addresses each obstacle in turn. For a reported behavior, an Evaluation Harness generates targeted assertions and a fixed cohort of customer scenarios. It then reproduces the behavior in a local instance of the assistant through grounded user simulation. Instead of replaying the log, the simulator writes new customer turns conditioned on the recorded messages and context. Repeated runs of the unchanged system form a stored baseline. An Improvement Orchestrator turns the assertion results into hypotheses, implements each as an isolated modification, and compares it with the baseline using paired percentile bootstrap intervals over scenario-level differences. When an investigation ends, the harness may propose revisions to future evaluations, subject to human approval and without altering past decisions. We report production investigations with this framework. Assertion profiles showed which positions of a product carousel a failure affected, and repeated runs distinguished a real improvement from run-to-run fluctuation. Audits of the evaluation itself found a judge that lacked the evidence it needed and a model setting that was configured but not applied.

AI agents are stateless across sessions by default and therefore operationally amnesic: each session begins with little durable knowledge of prior failures, repairs, preferences, or successful strategies. As a result, agents repeat the same mistakes and discard hard-won experience. The dominant fix is bespoke memory—retrieval, persistence, and learning logic hand-wired into one agent and bound to one storage engine. This creates a fragmented landscape where memory cannot be swapped, shared, isolated, or reasoned about independently of the agent that owns it. We argue that this is a middleware problem: agent memory deserves a first-class, pluggable layer, just as data access, messaging, and persistence each became middleware concerns. We develop this vision through six systems challenges: two-sided pluggability, host-native interposition, multi-tenant isolation, write-path consistency, federated sharing with provenance, and lifecycle governance. We present ALTK-Evolve, a reference implementation of memory middleware for self-improving agents, and use it to motivate a broader research agenda for future memory middleware.

We prove sharp lower bounds for the Schrödinger maximal function on the torus T^d for all dimensions d at least 2, and as a corollary obtain sharp regularity conditions for pointwise convergence of the periodic Schrödinger equation. Combined with sufficiency results established by Compaan-Lucá-Staffilani, this yields a full resolution of Carleson's problem up to endpoint for the periodic Schrödinger equation in all dimensions at least 2, and disproves the conjectured regularity condition of Miao-Yuan-Zhao, in contrast to the corresponding question in R^d. We also prove sharp estimates on the dimensions of divergence sets for the equation for high dimensional tori. Our approach uses complex multiplication on abelian varieties.

Multiple internal reflections enhance laser absorption in vaporization-induced cavities, but connecting ray histories, cavity geometry, and absorptance typically requires full ray tracing. We develop a reduced-order framework based on Beer-Lambert-type attenuation governed by an effective per-reflection optical depth (EPROD). A power-weighted formulation defines an absorptance-equivalent interaction count under constant effective single-interaction absorptance, while a cumulant expansion separates the EPROD-independent mean interaction count from ray-path heterogeneity corrections. Across nine simulation groups spanning three alloys and three spot diameters, the mean interaction count follows a unit-intercept linear depth relation with R-squared values of 0.7824-0.9843. Its slope varies by only 3.4%-6.3% across alloys at fixed spot diameter and decreases with increasing spot diameter. Fixing this slope from ray statistics yields an absorptance-depth model with one fitted parameter and a median prediction error (MAPE) of 2.61%. An independent synchronized Ti-64 experiment yields model parameters within approximately 2.4% of corresponding simulation values. This framework separates geometric interaction accumulation from optical attenuation, enabling efficient energy-coupling prediction and absorptance-based keyhole characterization.

Every solution for the classical four body problem in space which ends in total collision has a limiting shape. We prove that when that shape is planar the entire solution lies in a fixed plane and when that shape is collinear the entire solution lies in a fixed line.

Let $g=w|dz|^2$ be an affine special Kähler metric on a punctured disc and let $\Xi=q(z)\,dz^3$ be its associated holomorphic cubic differential. We prove that if the $g$-area of concentric punctured neighbourhoods grows at most polynomially, then $q$ is meromorphic at the puncture. More precisely, an area bound of order $\eps^{-N}$ forces every pole of $q$ to have order strictly less than $N+3$. Hence an essential singularity of the cubic differential rules out every polynomial area upper bound. The proof associates to $(g,\Xi)$ a curvature $-1$ conformal pseudometric and applies the Ahlfors–Schwarz lemma, followed by the submean inequality for holomorphic functions. We also discuss the identically zero cubic differential and the borderline logarithmic growth responsible for the strict pole-order bound.

The clustering of dark matter halos depends not only on halo mass but also on secondary halo properties, a phenomenon known as halo assembly bias. Coupled with variations in halo occupation, this dependence impacts galaxy clustering, an effect termed galaxy assembly bias. While previous studies have found only weak cosmological dependence of assembly bias within the $\Lambda$CDM framework, its sensitivity to the underlying dark matter physics remains largely unexplored. We use the matched dark-matter-only and hydrodynamical simulations from the AIDA-TNG suite to measure assembly bias across four dark matter models: cold dark matter, warm dark matter, and self-interacting dark matter with constant cross-sections and velocity-dependent cross-sections. We find that at z = 0, halo assembly bias is nearly identical across the four models for concentration-selected halos, as is the concentration dependence of the halo occupation distribution. The resulting galaxy assembly bias signals likewise show only weak model dependence. For stellar-mass-selected galaxy samples with $n = 0.0586 h^{3} Mpc^{-3}$ at z = 0, the velocity-dependent self-interacting dark matter scenario exhibits the largest deviation from the fiducial cold dark matter model: its assembly bias contribution to galaxy clustering is enhanced by up to $\sim 6\%$, corresponding to a $\sim 3\%$ difference in the assembly bias amplitude. Although statistically distinguishable, this difference is small and unlikely to be detected in current analyses. The weak model dependence persists across different galaxy number densities and redshifts. Overall, within the TNG galaxy-formation framework, these results indicate that assembly bias is largely insensitive to the dark matter models explored here, providing a robust baseline for future studies of the galaxy-halo connection in alternative dark matter scenarios.

The analysis of experimental or observational data often requires controlling for the effects of variables of secondary interest. Two common approaches to account for such parameters are grouping experimental or observational units into homogeneous groups (blocking), and observing quantitative variables on each one of them (covariance analysis). Often, both can be used and; therefore, comparing their performance poses a research question. Specifically, under randomized complete block designs where the blocks are defined by quantitative variables, there is interest in comparing the two-way Anova (TWA) with analysis of covariance (Ancova) models. Motivated by real-life problems coming from statistical consulting, we compared the performance of linear models considering block effects and those considering covariate effects from analytical and empirical perspectives. Models assumed Gaussian errors and included the effects of T treatments. The formal comparison relied on the theory of orthogonal projections and vector spaces under containment relationships of column spaces of the design matrices induced by conditions on the experimental design or data structure. We found asymptotic and small-sample conditions for the TWA to be at least as precise as the Ancova, as well as for these models to be equivalent. As to the empirical perspective, models were compared through a simulation study motivated by agronomical experiments. The results suggested that when the covariates show complex heterogeneity patterns, the Ancova outperforms the TWA with blocks defined by traditional approaches in terms of precision of location parameters, but not in hypothesis testing accuracy.

A square is a word $xx$, where $x$ is nonempty. We show that a circular word of length $n$ contains at most $\lfloor 3n/2 \rfloor$ distinct squares of length at most $n$. The proof combines known results relating squares to circuits in Rauzy graphs. The coefficient $3/2$ agrees with the known lower bound.

Deploying AI systems requires deciding when to delegate tasks and when humans should intervene to monitor and mitigate risk induced by AI operations. These decisions are challenging when failures cluster: a hallucination or harmful output can trigger further errors, creating periods of elevated risk. We introduce a continuous-time framework for learning adaptive human oversight under such turbulent AI risk. Existing oversight and delegation formulations condition on history but do not model incident clustering, or its suppression by supervision effort, jointly with the delegation decision and this study fixes this gap. The self-exciting dynamics capture how risk events increase the likelihood of subsequent events, making their timing and history central to decision-making. We formulate a stochastic control problem combining human actions, monitoring effort, and switching between human-AI-assisted operation and full AI delegation, balancing operational rewards against oversight costs, and cascading AI-failures and induced uncertainty. Human participation is an endogenous component of risk management: the policy determines both when oversight is needed and how much effort to allocate. We study a relaxed switching formulation and propose Hawkes-PPO, a policy-gradient method that uses a bank of exponential filters of observed incident times. In a synthetic environment it attains a higher risk-adjusted objective than either fixed regime and approaches an approximate full-information oracle. We illustrate our results with numerical simulations by examining how cascade risks influence intervention and delegation, connecting reinforcement learning with adaptive human oversight of AI systems. In particular, we illustrate the benefit of our switching strategy and Hawkes-PPO algorithm to monitor the project efficiently along time, reducing turbulent risks occurrences and costs.

Voice agents depend on accurate automatic speech recognition (ASR) to act on what callers say, yet ASR is evaluated on read, English-centric speech with word error rate (WER), which penalizes surface rather than semantic differences. We introduce mu-bench, a dataset of 4,270 caller utterances from 250 phone calls to an AI banking agent in English, Spanish, Turkish, Vietnamese, and Mandarin, centered on form-field inputs such as names, email addresses, and confirmation codes. We release Utterance Error Rate (UER), an LLM judge of whether a transcript preserves meaning, calibrated against human raters, together with an LLM normalizer that makes WER comparable across providers' output formats. On 1,847 human-rated transcripts, UER agrees with annotators at $\kappa$ = 0.78, versus 0.53 for exact-match WER on normalized text. We rank six commercial providers on a public leaderboard; the best reaches 11.9% UER, and Mandarin is hardest for all six.

Empowered by advances in Language Model agents, systems have made substantial strides in code generation and understanding. However, these approaches often rely on read access to the relevant code, an assumption which does not hold when dealing with external APIs. In this work, we introduce the PAU (Python API Understanding) benchmark, where we provide models with black-box, API-level access to code snippets. Models must query the API with exploratory inputs and draw insights from the resulting outputs, with the goal of describing the snippet's true functionality. By treating the code snippets as external tools that must be understood via interaction alone, PAU studies the more general problem of unsupervised tool understanding, specifically for tools implemented as Python methods. Despite recent progress in coding agents, even frontier models struggle to achieve high performance on PAU, with the best model (Claude-4-Opus) failing to understand over 45% of the PAU test set. An investigation into the common error modes reveals that models are overconfident; they often overrate the quality of their current hypothesis, leading to insufficient exploration and premature termination. Finally, we take inspiration from the Asymmetric Actor Critic (AAC) paradigm, frequently used in robot learning, to post-train models for interactive code understanding. Models trained with AAC conduct more active exploration of the APIs, with an AAC-tuned Qwen3-8B model matching the performance of GPT-5-mini.

Stochastic density functional theory (SDFT) provides a powerful framework for studying the dynamics of many-body systems both in and out of equilibrium. In its standard form, it can be viewed as a density-based formulation of coupled Langevin equations with white noise. Here, we extend SDFT to a class of systems, ranging from magnetized electrolytes to chiral active matter, that possess a chiral overdamped limit in which a relaxation timescale and an oscillation period are simultaneously taken to zero. We derive this formulation, which we dub “odd SDFT", by taking the overdamped limit of a fluctuating hydrodynamic description of underdamped particles subject to a Lorentz force. We establish the validity and utility of the resulting framework by calculating the collective mobility tensor of interacting particles using both direct linear response and a Green-Kubo relation. We then apply the theory to the Hall conductivity of electrolytes in magnetic fields. Going beyond mean-field theory and using two distinct Green-Kubo relations, we derive the full conductivity tensor, extending the relaxation part of the Debye-Hückel-Onsager theory to the case of finite magnetic fields.

This work establishes that determinant-based manipulability is an exact objective for fixed-task redundancy optimization, despite its dependence on task coordinates and the choice of task-space metric used for volume measurement. On every regular task fiber, the determinant proxy, its representation in any task chart, and every metric-completed manipulability differ only by positive constants. They consequently induce the same complete ordering, constrained extrema, gradient directions, critical points, and local optimality classifications. For comparisons and trajectory optimization across task fibers, this work introduces fiber-normalized manipulability: the capability attained at an internal state divided by the best capability available on the same fiber. The resulting dimensionless scalar is invariant under coordinate changes on the internal-state and task manifolds and independent of the task-space metric. Its associated loss provides an intrinsic objective for physically admissible cross-fiber trajectories, including problems with prescribed or free task evolution and temporal coupling. Planar-manipulator and redundant aerodynamic-allocation examples demonstrate fixed-fiber equivalence, task-dependent cross-fiber differences, and invariant fiber-normalized trajectory optimization.

We study filter-Laver forcing on omegaxomega with successor sets containing final quadrants. The observation retaining the minimum coordinate and orientation is a complete projection. Its finite nontrivial hidden-label quotients are Cohen, whereas the full offset quotient has Boolean density equal to the ground-model dominating number dV . For the associated grid-meager ideal I_box, we prove cov(I_box) = add(M) for the usual meager ideal M, and obtain further cardinal bounds. We also analyze threshold games, oracle degrees and the distinction between positive and filter-large certificate sets: recurrent filter-large acceptance admits refinements whose every branch succeeds. As an arithmetic application, we prove Wilfs inequality c \leq 13e for numerical semigroups with thirteen elements below the conductor c, where eis the embedding dimension. A complete arithmetic proof, including a bounded exact computation, is given in the appendix. Its terminating certificate procedures supply a computable quadrant tree and uniform refinements for every forcing condition. Finite support kernels describe persistence of the registered proofs; arithmetic absoluteness explains precisely why forcing organizes these certificates without replacing their arithmetic justification.