This paper provides a theoretical account of memorization in stochastic interpolation models. By leveraging closed-form expressions for the optimal velocity field and the associated score function, we show that, in the continuous-time oracle setting, both deterministic and stochastic generation processes recover training samples. Under Euler discretization, generated samples remain centered around training samples, with deviations controlled by the step size. We further analyze generation in the presence of estimation errors and show that accumulated estimation errors control the endpoint deviation from the training set. These results imply that the generated sample admits a representation as a training sample perturbed by three controlled terms: a discretization-induced bound, an estimation-error-induced bound, and stochastic Gaussian noise. Based on this characterization, we provide theoretical definitions of overfitting and underfitting in generative models. Synthetic simulations support our theoretical findings.

Relational deep learning (RDL) converts relational databases (RDBs) into heterogeneous graphs, but graphs derived directly from database schemas are often not well suited for how graph neural networks (GNNs) perform relational reasoning. We study what makes a relational graph suitable for deep learning and show that schema-derived graphs suffer from two systematic failures: information overload and semantic fragmentation. Our empirical analysis reveals that the desired graph is not the raw schema, but a result of controlled structural adaptation. Performance depends on balancing two operations: mitigating information overload via filtering, and repairing semantic fragmentation via injection. Specifically, filtering serves as a bias-variance knob with non-monotonic effects, while injection improves performance only when it explicitly restores the relational dependencies missing from the original schema. Based on these findings, we develop an end-to-end structural optimizer that applies both operations to adapt relational graphs automatically. Across 26 tasks spanning classification, regression, and recommendation, the optimized graphs consistently improve accuracy while often reducing inference cost.

Diffusion and continuous flow-based language models have emerged as the leading non-autoregressive alternatives to language modeling. Progress in both paradigms is overwhelmingly tracked by generative perplexity (gen-PPL): the per-token negative log-likelihood of samples under a frozen autoregressive (AR) scorer such as gpt2-large, typically paired with an empirical-entropy guardrail to rule out low-entropy collapse. We argue that this metric is unsound. By construction, gen-PPL measures only predictability under the scoring AR, not grammaticality or semantic coherence -- and the set of predictable but still low-quality sequences is combinatorially large. To make this concrete, we construct a suite of zero-parameter, deliberately naive samplers that achieve state-of-the-art gen-PPL on LM1B and OpenWebText at non-degenerate entropy, surpassing recently published diffusion and continuous-flow models while producing text that is incoherent by construction. We recommend evaluation suites that directly quantify the distributional divergence between generated and reference text, and use such a suite to re-benchmark recent non-autoregressive models, recovering a more faithful picture of the current state of the art.

Network dynamics - including spreading, influence maximisation, and epidemic modelling - remain largely confined to the transductive paradigm, where models are trained on a single network and cannot be reused on unseen graphs without retraining. We argue that inductive cross-network generalisation is a necessary prerequisite for Graph Foundation Models (GFMs) in this domain and propose four design properties towards this goal. As a proof of concept, ts-net (TopSpreadersNetwork), trained solely on synthetic multilayer networks (MLNs), demonstrates zero-shot generalisation to real-world MLNs of varying size and layer count, outperforming classical heuristics and transductive baselines on three of four metrics. Based on ts-net's performance, we further outline five open challenges towards building GFMs for network dynamics: scale, many-layer generalisation, self-supervised pretraining, cross-task transfer, and node-attribute integration.

Many real-world queries over personal data span multiple applications and require structured planning, as individual tools expose only partial information. While LLMs show strong reasoning and tool use, reliably executing multi-step, cross-tool queries remains challenging. We introduce a system that converts natural language queries into structured graphs and executes them via a deterministic planner. Our approach uses depth-first search to resolve dependencies and combine results across tools, improving reliability and enabling queries beyond traditional keyword-based search. We demonstrate high accuracy even with smaller or locally hosted LLMs.

Large language models (LLMs) and agentic systems are increasingly proposed for financial trading, yet their reported performance remains difficult to compare because studies vary in data provenance, temporal split discipline, execution timing, turnover treatment, and transaction-cost modeling. This article presents a targeted topical review and reproducibility audit of execution realism in LLM-based trading research. A coded evidence matrix covering 30 trade-relevant primary studies is used to assess point-in-time controls, split transparency, held-out evaluation, cost and turnover treatment, execution semantics, universe definition, and artifact release. Across the audited sample, architecture reporting is generally clearer than the evaluation assumptions needed to judge whether a trading result is economically interpretable or reproducible. A 10-equity worked example is included only as a methodological scaffold to illustrate how explicit friction and timing choices can materially compress active-strategy results. The main conclusion is that the next useful step for LLM trading research is not only better agent design, but also clearer reporting standards for execution realism, reproducibility, and evaluation comparability.

Large language models (LLMs) can generate factually inconsistent claims, motivating accurate and scalable hallucination detectors. Prior work largely enlarges training sets via synthesis or new annotations, introducing increasing cost and potential bias while underusing the consistency implied by semantically equivalent paraphrases. We propose Consistency-Constrained Hallucination Detector (CCHD), which formulates training as a constrained optimization problem. The standard cross-entropy on original document-claim pairs is complemented by (i) paraphrase-consistency constraints bounding divergence across paraphrased views, and (ii) label-preservation constraints tying paraphrases to ground truth. We solve the problem by gradient descent-ascent over model parameters and per-view Lagrange multipliers, adding only a few scalar dual variables and no inference-time overhead. With DeBERTa and Flan-T5 backbones, CCHD consistently outperforms strong baselines (FactCG, MiniCheck, and AlignScore) on standard factuality benchmarks, demonstrating its superiority on hallucination detection.

Large language models (LLMs) frequently generate hallucinations, which are unsupported by a source document. To avoid costly LLM-as-evaluator pipelines and the heavy annotation demands of existing classifiers, we propose CPIL (Cross Paraphrastic Invariance Learning), a two-stage Siamese framework that maximizes the utility of existing labeled data. Concretely, CPIL constructs informative training pairs by: (i) generating paraphrastic views of each document-claim example as positives, and explicitly aligning their representations to enforce invariance to surface form; and (ii) mining same-document, opposite-label pairs as hard negatives to sharpen document-sensitive decision boundaries. Then CPIL conduct a two-stage model training: Stage 1 performs contrastive pretraining to learn a paraphrase-invariant, grounding-aware embedding space; and Stage 2 attaches a lightweight classifier for binary groundedness. On the LLM-AggreFact benchmark (11 tasks), CPIL surpasses strong baselines concerning F1 scores with only ~1% labeled data, showing its prediction superiority and label efficiency.

Tool-augmented large language model agents increasingly rely on external APIs, but standard tool schemas describe how to call a tool, not when the tool is causally appropriate or what task state it produces. Causal tool filtering addresses this gap by using lightweight contracts that specify each tool's preconditions, effects, risk level, and cost. However, manually writing and maintaining such contracts does not scale to large or changing tool ecosystems. We introduce Contract2Tool, a framework for inferring tool contracts from metadata, schemas, documentation, and execution traces. Contract2Tool converts observable tool evidence into normalized symbolic contracts that can be evaluated intrinsically and deployed inside downstream causal tool filtering. We evaluate learned contracts against gold preconditions, effects, and risk labels, and measure their downstream utility on multi-step agent tasks. Our results show that hybrid documentation-and-trace evidence produces contracts accurate enough to preserve most of the reliability and efficiency benefits of gold contracts. Learned-contract CMTF achieves 0.980 downstream success, close to 0.990 for gold-contract CMTF, while reducing visible tools from 100 to 1 and reducing average token usage from 26,172 to 2,528 relative to all-tools exposure. These results suggest that learned contracts can provide a scalable contract layer between tool schemas and reliable agent execution.

Current approaches to aligning large language models (LLMs) aggregate diverse human preferences into a single reward signal, effectively optimizing for a hypothetical ``average user'' who represents no real person particularly well. This position paper argues that LLMs should learn personalized, individual preferences rather than aggregated ones. We show that aggregation masks critical information about preference diversity, individual values, and contextual dependencies, which is a limitation both theoretically grounded in social choice theory and empirically evident across demographic groups. We analyze the rich structure that human preferences encode, survey technical approaches to personalization, and systematically address counterarguments on scalability, shared standards, and manipulation risk. While personalization introduces genuine safety challenges including filter bubbles, value lock-in, and psychological manipulation, we argue these are manageable through bounded personalization frameworks that preserve universal safety constraints while accommodating legitimate individual variation. We conclude with a concrete research and policy agenda for developing preference-aware models that respect both individual autonomy and collective safety.

Scaling laws provide a fundamental framework for understanding the performance of Language Models (LMs), yet deriving them requires prohibitively expensive evaluations across thousands of checkpoints or millions of inference samples. To address this, we introduce Item Response Scaling Laws (IRSL), a unified framework that integrates Item Response Theory (IRT) within the scaling law framework. Unlike traditional approaches that treat each model-benchmark pair in isolation, IRSL disentangles latent model ability from question characteristics, factorizing the scaling law estimation for $M$ models and $N$ questions to significantly reduce parameter complexity from $O(M \times N)$ to $O(M + N)$. We instantiate IRSL with Beta-IRT, which leverages the empirical probability responses of LMs -- such as token probabilities in pre-training and pass rates in test-time sampling -- to capture richer signals than binary responses. We validate our approach across two prevalent scaling paradigms: (1) pre-training downstream scaling, using 6,612 LM checkpoints and 37,682 questions from 10 benchmarks; and (2) test-time scaling, using 12 LMs and 120 questions from 4 benchmarks with up to 2,500 samples per question. Given a one-time calibration on existing model responses, IRSL yields more reliable scaling estimates using only 50 questions per benchmark (a 99.9\% reduction), achieving comparable or superior decision accuracy to traditional approaches. Furthermore, we show that the estimated latent model abilities are generalizable, enabling accurate performance forecasting across benchmarks that share the same measurement objective.

In this digital world, data is everything and significantly impacts our everyday lives. Interestingly, in this small world, everything is part of an ecosystem, where everything is connected, directly or indirectly. The same thing happens to data as well. In most cases, it may seem like a particular topic does not have any connection with another one, but in reality, they are connected through a mutually related topic. Therefore, in this research, we will discuss an adaptive model modified from the ABC model by Don R. Swanson, a Literature-Based Discovery (LBD) Model, to find the hidden connections between Concepts of Interest. The model demonstrates that two topics, A and C are different and have no relationship. But they have a common topic, B that can be used to connect topics A and C This famous model will be used in this discussion to connect Medical Concepts.

Multimodal Large Language Model (MLLM)-driven image restoration agent demonstrates effectiveness in degradation coupling scenarios by flexibly selecting tools and determining removal orders. However, their zero-shot planning often fails without experience, necessitating severe trial-and-error overhead to achieve satisfactory outcomes. Currently, two paradigms are employed to address this issue, yet a dilemma persists: Training-based methods embed intrinsic experience into parameters, achieving high inference efficiency but lacking compatibility with new tools or degradation. In contrast, training-free methods utilize explicit experience storage for compatibility but still incur trial-and-error overhead due to naive experience. To resolve the dilemma, we propose EvoIR-Agent, which first systematically formulates the experience components of a training-free image restoration agent. Subsequently, a hierarchical experience pool is constructed, which enables coarse-to-fine guidance for diverse tools and removal orders. Furthermore, a self-evolving mechanism is introduced to update the pool from scratch using accumulated records, thereby greatly improving performance and efficiency. Extensive experiments reveal that EvoIR-Agent achieves a significant lead in the full reference metrics and yields a remarkable Pareto-optimal balance between performance and efficiency compared to the state-of-the-art methods.

Robots must learn from both what people do and what they say, but either modality alone is often incomplete: physical corrections are grounded but ambiguous in intent, while language expresses high-level goals but lacks physical grounding. We introduce QuickLAP: Quick Language-Action Preference learning, a Bayesian framework that fuses physical and language feedback to infer reward functions in real time. Our key insight is to treat language as a probabilistic observation over the user's latent preferences, clarifying which reward features matter and how physical corrections should be interpreted. QuickLAP uses Large Language Models (LLMs) to extract reward feature attention masks and preference shifts from free-form utterances, which it integrates with physical feedback in a closed-form update rule. This enables fast, real-time, and robust reward learning that handles ambiguous feedback. In a semi-autonomous driving simulator, QuickLAP reduces reward learning error by over 70% compared to physical-only and heuristic multimodal baselines. A 15-participant user study further validates our approach: participants found QuickLAP significantly more understandable and collaborative, and preferred its learned behavior over baselines. Code is available at https://github.com/MIT-CLEAR-Lab/QuickLAP.

Behavioral interview evaluation using large language models presents unique challenges that require structured assessment, realistic interviewer behavior simulation, and pedagogical value for candidate training. We investigate chain of thought prompting for interview answer evaluation and improvement through two controlled experiments with 50 behavioral interview question and answer pairs. Our contributions are threefold. First, we provide a quantitative comparison between human in the loop and automated chain of thought improvement. Using a within subject paired design with n equals 50, both approaches show positive rating improvements. The human in the loop approach provides significant training benefits. Confidence improves from 3.16 to 4.16 (p less than 0.001) and authenticity improves from 2.94 to 4.53 (p less than 0.001, Cohen's d is 3.21). The human in the loop method also requires five times fewer iterations (1.0 versus 5.0, p less than 0.001) and achieves full personal detail integration. Second, we analyze convergence behavior. Both methods converge rapidly with mean iterations below one, with the human in the loop approach achieving a 100 percent success rate compared to 84 percent for automated approaches among initially weak answers (Cohen's h is 0.82, large effect). Additional iterations provide diminishing returns, indicating that the primary limitation is context availability rather than computational resources. Third, we propose an adversarial challenging mechanism based on a negativity bias model, named bar raiser, to simulate realistic interviewer behavior, although quantitative validation remains future work. Our findings demonstrate that while chain of thought prompting provides a useful foundation for interview evaluation, domain specific enhancements and context aware approach selection are essential for realistic and pedagogically valuable results.

Are two distributions close to each other with statistical significance? Distribution closeness testing (DCT) formalizes this question by testing whether the distance between a distribution pair is at least epsilon-far. Existing DCT methods mainly measure discrepancies between distribution pairs defined on discrete spaces, for example using total variation, which limits their application to complex data such as images. To extend DCT to more types of data, a natural idea is to introduce maximum mean discrepancy (MMD), a powerful measure of distributional discrepancy between complex distributions, into DCT scenarios. However, empirical results indicate that many distribution pairs can have the same MMD value despite having different norms in the same reproducing kernel Hilbert space (RKHS). These pairs may exhibit different finite-sample distinguishability and reflect different practical closeness levels, making MMD less informative for DCT. To mitigate this issue, we design a new measure of distributional discrepancy, norm-adaptive MMD (NAMMD), which scales the MMD value using the RKHS norms of distributions. Based on the asymptotic distribution of NAMMD, we propose NAMMD-based DCT to assess the closeness level of a distribution pair. Theoretically, we prove that NAMMD-based DCT has higher test power than MMD-based DCT while maintaining bounded type-I error. This is further validated by extensive experiments on multiple types of data, including synthetic noise and real images. Our code is available at https://github.com/zhijianzhouml/NAMMD.

Using speaker embeddings as conditioning can strengthen speech enhancement, but most methods either require clean enrollment audio or rely on embeddings extracted from noisy speech, which are fragile under noise and domain shift. We propose G-MaP-SE, a guided enhancement framework that builds a clean-speech embedding prior with a Gaussian Mixture Model (GMM) and refines a noisy conditioning embedding by matching it to this prior. The matched prior embedding is then injected into a time-frequency enhancement backbone via a lightweight gated fusion module. Experiments on VoiceBank+DEMAND and DNS Challenge 2020 datasets show that the proposed prior matching consistently outperforms noisy conditioning and substantially narrows the gap to an oracle clean-conditioning upper bound, while requiring no enrollment audio at inference time. The code, audio samples, and checkpoint are available.

LLM judges increasingly turn verdicts into system commitments. Under mixed evidence (claims with both supporting and refuting sources) this is unsafe: when the schema exposes CONFLICTING as the authorized non-directional verdict, returning SUPPORTS/REFUTES is an unauthorized directional commitment, a failure we name Cherry-pick Override (CCO). We define CCO under an explicit task contract and report it with a same-denominator diagnostic protocol paired with matched-coverage bootstrap and an apples-to-apples random-veto null. On AVeriTeC's Conflicting subset (N_C = 150), three-option judges return a directional verdict on more than 84% of mixed-evidence claims; under the typed schema, three-judge majority voting amplifies direction-on-conflict on AVeriTeC (0.887 vs. 0.840; 95% CI [+0.013, +0.080]) but does not replicate on VitaminC-Mixed. Walking an intervention ladder of common single-channel fixes (typed vocabulary, panel aggregation, confidence thresholding, validator-only filtering), each leaves a distinct residual failure: panel aggregation suppresses single-judge CONFLICTING dissent in 48% of CCO cases; the panel is well-calibrated for direction (ECE = 0.07 on pure-S/R) so confidence cannot operationally separate CCO from correct directional commits; validator-as-classifier nearly halves pure-evidence accuracy. A minimal two-channel reference probe reaches operating points neither single channel reaches; under the random-veto null its promotion to CONFLICTING is structurally targeted on AVeriTeC (empirical p < 1/2001) and weaker but in the same direction on VitaminC-Mixed, a selectivity result rather than a magnitude one. We argue for an external commitment-control layer that separates verdict generation from commitment authorization, using structural evidence and confidence as orthogonal channels and NO-COMMIT as a routed controller state.

Transfer learning addresses the challenge of transfering knowledge from one domain to another. Traditional transfer learning focuses on adapting models trained on a source domain (with a lot of observations) to improve performance on a target domain (with few observations). In this work we consider the case of a model shift and we focus on the transfer learning applied to a causal forest namely HTERF. This causal forest aims to estimate the Conditional Average Treatment Effect (CATE). The approach considered is the offset method presented by Wang (2016) adapted to a causal context. This method relies on the use of intermediate models in order to estimate the offset between source and target distributions. Our main result is a bound on the CATE error of HTERF on target depending on the error of the intermediate models. Simulation studies show the good performances of this approach in different settings on simulations and on a real-world dataset.

Regulatory stress testing frameworks, including the Comprehensive Capital Analysis and Review (CCAR) and the Internal Capital Adequacy Assessment Process (ICAAP), require robust Stressed Value-at-Risk (SVaR) estimation under forward-looking macroeconomic scenarios. Traditional parametric approaches often exhibit numerical instability under extreme shocks, reducing the reliability of capital projections. This paper extends the Hybrid Gaussian Process Regression Historical Simulation (GPR-HS) framework of Vadrevu (2026) to forward-looking stress scenarios, demonstrating stability across three regimes: West Asia War, Climate Risk, and AI Bubble/Regulation. A key contribution is the Scenario-Averaged Covariance Stabilization (SACS) framework, which constructs stress covariance as a weighted aggregation of historical crisis regimes, providing stable and interpretable dependence structures. Stressed return paths are generated over a 252-day horizon using deterministic drift and stochastic residuals, while volatility is modeled via Gaussian Process Regression with Aggressive Noise Initialization (ANI). The framework exhibits consistent convergence across all assets and scenarios. SVaR ranges from -2.1020% to -2.2231%, with the coherence property |SES| > |SVaR| preserved. The results support GPR-HS with SACS as a stable and regulator-aligned approach for forward-looking SVaR and SES estimation in CCAR and ICAAP applications.

Frontier artificial intelligence is reshaping all aspects of society, from economic output or military capability to democratic institutions. The EU is entering this transformation from a position of structural dependence: frontier models originate almost exclusively from the United States or China, the US holds approximately sixteen times the EU's AI supercomputing capacity, and only 15% of global hyperscale data centre capacity resides within EU borders. Although the European Commission has accelerated its policy response, existing initiatives remain fragmented and lack a cohesive vision for securing strategic autonomy across the full frontier AI value chain. Here we propose a unified framework connecting five sovereignty pillars (economic competitiveness, resilience, security and defence, European values, and foreign relations) to a decomposition of the frontier AI stack comprising five layers, 26 components, and 29 sub-components. This framework allows the identification of critical gaps, redundancies, and inter-pillar trade-offs that current EU policy leaves implicit. Our analysis of the AI Gigafactory Initiative illustrates how a sovereignty-centred lens reveals conflicts that narrowly economic framings obscure. Moreover, this framework offers policymakers a structured basis for designing, evaluating, and prioritising frontier AI interventions across multiple dimensions of European strategic autonomy across the 92 initiatives from four major Commission communications we. identify, and beyond.

Estimating derivatives from noisy sampled data is fundamental to control, human--computer interaction, and biomedical engineering. Causal FIR derivative filters offer a natural approach for this challenge, yet their performance depend on their length. While short filters amplify noise, long filters introduce smoothing bias. We present SURDE (SURE Derivative Estimator), which addresses this tradeoff at each time step by evaluating a data-driven cost derived from Stein's Unbiased Risk Estimator (SURE) across a bank of candidate lengths and soft-combining their outputs via exponential weighting. We prove a minimax-optimal oracle inequality for the soft-combined estimator and use it to derive the optimal weighting temperature in closed form. Thus, the only tuning parameter for SURDE is the noise variance. Via numerical simulations we show that SURDE consistently outperforms alternative adaptive methods (the Intersection of Confidence Intervals (ICI) rule and the Adaptive Windowing Velocity Estimator (AWVE)) for first-derivative estimation. We further show that \surede{} is robust to noise-variance misspecification (9\% degradation over a $4\times$ range), and that it is superior to ICI and AWVE also over real data scenarios (the EuRoC MAV dataset). SURDE is causal, computationally light, and requires only a rough estimate of the noise variance.

We investigate the triplon excitations of the pinwheel valence-bond-solid phase on the deformed kagome lattice compound Rb2Cu3SnF12. Using bond-operator mean-field theory, we compute the triplon band structure, dynamical structure factor, Berry curvatures and the associated thermal Hall response. We show that the presence of Dzyaloshinskii-Moriya interactions and an external magnetic field are key for endowing triplon bands with nontrivial Chern numbers. We find good qualitative agreement of the low-energy dynamical structure factor with neutron-scattering experiments. An applied magnetic field can isolate the lowest triplon Chern band leading to a tunable thermal Hall conductivity for accessible temperature and field regimes. Our results establish the deformed kagome pinwheel valence-bond solid as a promising platform for topological triplon physics and for observing the associated thermal Hall effect.

We analyze a class of linear Ricci--trace deformations of Einstein's field equations in which the relative weight between the Ricci tensor and the scalar-curvature trace sector is modified while the metric remains the only gravitational field. The purpose of the analysis is structural rather than phenomenological: we classify the corresponding field-equation class, fix the parameter dictionaries commonly used in the Rastall-gravity literature, and identify which equivalence statements survive after Newtonian calibration. We show that two frequently used parametrizations, \[ (1-ε)R_{μν}-\frac12 g_{μν}R=κ_εT_{μν}, \qquad R_{μν}-\frac{1-λ}{2}g_{μν}R=κ_λT_{μν}, \] are algebraically isomorphic only if both the deformation parameter and the bare gravitational coupling are transformed simultaneously. This algebraic equivalence, however, is not automatically an operational equivalence. Once the same laboratory stress tensor and the same measured Newton constant are fixed, the parameter map is a passive reparametrization only at the Einstein point. We further identify the \(λ\)-representative with the standard Rastall equation, clarify the role of the conserved effective source, derive the corresponding FLRW perfect-fluid sector, and discuss degenerate cases including vacuum, trace-free matter, radiation, dust, and the singular traceless point. Finally, we distinguish the Ricci--trace class from Unimodular Gravity (UG): although both involve the trace sector, UG follows from a restricted variational principle and produces the cosmological constant as an integration constant, rather than from an algebraic Ricci--trace deformation. The result is a compact operational classification of Rastall-type Ricci--trace models.

In this note, we use a simple argument to show the existence of large values of conjecturally sharp size for Dirichlet $L$-functions attached to primitive characters of fixed order at $σ\in (1/2, 1]$. More precisely, for every fixed integer $g\geq 2$ we prove the existence of a primitive character $χ$ of order $g$ and conductor $Q\asymp x$ such that $|L(1,χ)| \geq e^γ\left(\log\log x+\log\log\log x-\log(2\log g)+o(1)\right). $ We also show that for every fixed $1/2<σ<1$ there exists a primitive character $χ$ of order $g$ and conductor $Q\asymp x$ such that $\log |L(σ,χ)| \geq \left(C_g(σ)+o(1)\right) (\log x)^{1-σ}(\log\log x)^{-σ}, $ for some explicit positive constant $C_g(σ).$ Previously, such bounds were known only conditionally on the Generalized Riemann Hypothesis, and even then only in the special cases $g=2$ and $g=3$.

The framework of $α$-potential games has recently been introduced as a tool to analyze finite-player dynamic games, reducing the challenging task of finding approximate Nash equilibria to a control problem of minimizing a single function called $α$-potential function. In this work, we investigate the limiting behavior of $α$-potential games as the number of players $N$ tends to infinity. We show that potential mean field games (MFGs) arise naturally as this limit. Specifically, both the optimal values and the minimizers of normalized $N$-player $α_N$-potential functions converge to those of a mean field control (MFC) problem with measure-valued controls. We establish the equivalence of $\lim_{N\to\infty}α_N= 0$ with the existing conditions for potential MFGs, and provide an unified approach to construct the potential function for MFGs using the techniques from differential geometry in Wasserstein space. We further demonstrate that the objective of the limiting MFC problem serves as a potential function for the corresponding MFGs, an extension of the analogous finite-player setting. This connection yields new constructions of potential MFGs from a finite-player game, through the asymptotic condition $\lim_{N\to \infty}α_N= 0$. As a by-product, we establish propagation of chaos for $N$-player games converging to MFGs for general controlled diffusions with common noise and non-separable control interactions.

Recent materials advances have made it possible to fabricate twisted transition metal dichalcogenide homobilayers. These systems have been shown to host integer and fractional Chern insulating states. Because of spontaneous time reversal symmetry breaking, their ground state harbors intriguing spin-polarized currents with whirlpools on the moiré length scale that can be measured by scanning probe methods. We first provide a quantitative analysis of these persistent currents and then show that the maximum of the amplitude of the current density in the bulk of the sample is an accurate tracker of topological order. We conclude by calculating how the quantization of the Hall conductance is affected by finite-size effects.

The second data release of the European Pulsar Timing Array (EPTA) collaboration provides evidence for the presence of a gravitational-wave (GW) background. In this work, we explore a potential cosmological interpretation of this signal in terms of inflationary scenarios. We parametrize the tensor power spectrum in terms of the tensor-to-scalar ratio $r$, the tensor spectral index $n_t$, the reheating temperature $T_{\text{rh}}$, and the cut-off frequency $f_{\text{end}}$. We incorporate all relevant observational constraints, including those from the Cosmic Microwave Background, Big Bang Nucleosynthesis, and LIGO-Virgo-KAGRA observations. We demonstrate that imposing these constraints consistently reduces the region of parameter space that provides a viable interpretation of the EPTA signal, to $-11.66 \lesssim \log_{10}r \lesssim -1.45$, $1.32 \lesssim n_t \lesssim 2.47$, $1.78\text{ MeV} \lesssim T_{\text{rh}} \lesssim 28.2\text{ GeV}$, and $75.86\text{ nHz} \lesssim f_{\text{end}} \lesssim 14.45\text{ Hz}$ at the 95% confidence level. This favours the scenario in which the GW spectrum in the EPTA frequency band originates from tensor modes that re-entered the Hubble radius during the radiation-dominated era, allowing for a higher $r$ and a flatter spectrum. However, $T_{\text{rh}}$ must take very low values, which are challenging to explain theoretically.

We investigate the high Mach number limit for the scaled compressible Navier--Stokes system in the critical Besov framework. In the scaled momentum equation, the pressure force is represented by the term \(\varepsilon^2\nabla a^\varepsilon\), where $\varepsilon$ is the inverse Mach number; as \(\varepsilon\to0\), the formal limiting system is the compressible pressureless Navier--Stokes system. The analysis is complicated by the absence of density dissipation in the limiting model and by the highest-order coupling created by the viscous terms. For \(d\geq2\), we prove the global well-posedness of the scaled system for small initial data and obtain estimates that are uniform with respect to $\varepsilon$. A crucial ingredient is a parameter-dependent lower-order estimate for \(\varepsilon a^\varepsilon\), which compensates for the purely transport nature of the density equation and allows the uniform bounds to be closed. Based on these estimates, we justify the high Mach number limit and recover a global strong solution to the pressureless Navier--Stokes system. For \(d\geq3\), we further derive quantitative error estimates between the scaled solutions and the pressureless limiting solution. More precisely, on each fixed finite time interval, if the initial discrepancy is of order \(\mathcal{O}(\varepsilon)\), then the corresponding lower-order critical Besov error satisfies the same rate, which yields a quantitative justification of the pressureless limit.

The spaces in the title are holomorphic function spaces on the unit disk containing a function with the bizarre property that, although it can be approximated in norm by polynomials, it cannot be approximated by the partial sums of its own Taylor series, nor by any linear combinations thereof. The search for concrete examples of such spaces leads to an elementary problem about convergent series that is still unsolved.