Papers

We study unfoldings of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves. We prove theorem of existence for unfoldings of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves for one-parameter families. We prove theorem of completeness for unfoldings of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves. We also present unfoldings of foliated nonsingular varieties defined by locally free subsheaves of cotangent sheaves in the language of functors of Artin rings which is the algebraic version of unfoldings of foliated complex analytic structures. We identify first-order unfoldings and obstructions.

Let $E:y^2=x^3-x$. We prove that $\#E(\mathbf{F}_p)/8$ is squarefree and has at most two prime factors for $\gg X/(\log X)^2$ primes $p\le X$. The proof follows the Gaussian CM sieve framework of Iwaniec and Jiménez Urroz and incorporates the intersection terms identified by Xie. For a squarefree Gaussian ideal $\mathfrak{a}$, we separate its largest rational ideal factor and use the resulting divisibility of both integral coordinates to control the large-factor tail. This transfers an ideal Bombieri–Vinogradov estimate to rational norm divisibility at level $1/2$, with any fixed power of the divisor function as a weight. We prove the corresponding distribution estimate for the switched product sequence and apply the transfer to both sequences in Chen's weighted sieve. The primes counted may be restricted to $p\equiv5\pmod8$; their normalized orders are odd, squarefree, and supported on primes congruent to $1\pmod4$.

Scaling self-supervised contrastive reinforcement learning (CRL) to 1000-layer Euclidean neural networks has enabled state-of-the-art goal-reaching performance, yet wall-clock time scales linearly with depth. We find that changing the geometry achieves stronger performance with far shallower networks. This matters in complex navigation environments, where walls can make route distance deviate substantially from Euclidean distance. In a U-shaped maze, we show that the longer the route, the more it deviates from Euclidean distance. Hyperbolic CRL addresses this mismatch by replacing Euclidean distances in CRL's contrastive objective with geodesic distances in hyperbolic space. On hard navigation tasks, inductive bias beats scale: depth 64 hyperbolic agents outperform 1024-layer-deep Euclidean networks while training more than $12 \times$ faster and using about $13 \times$ fewer parameters. We observe that hyperbolic agents behave qualitatively better in hard humanoid mazes, where they follow corridors to reach goals, whereas Euclidean agents try straight-line shortcuts over walls. Our results show that matching the representation geometry to the task can substitute for substantial network depth. Code is at https://github.com/Probabilistic-and-Interactive-ML/HypGCRL .

For all sufficiently large $n$, we prove that every Latin square of order $n$ can be made to avoid any prescribed partial Latin square of order $n$ by permuting only its rows and columns. The proof combines Hall's theorem with elementary estimates for a random permutation.

We study the minimax estimation of the pseudo-spectral gap $\gamma_{ps}$ of a Markov chain from the observation of a single trajectory. Extending earlier works on finite-state space chains by Wolfer and Kontorovich (2024), we provide an estimator in the case of a compact, continuous state-space. Our estimator appears to be minimax-optimal on the class of chains with a transition operator with Lipschitz density, up to logarithmic terms. Our estimator, combined with Paulin's version of Bernstein inequality, allows to derive empirical concentration inequalities for Markov chains. This unlocks many applications, from confidence intervals, to active learning and generalization bounds. In particular, for rested bandits with Markov rewards, we provide a version of UCB with a regret bound similar to the one in the iid case, without assuming any knowledge on the pseudo-spectral gap of the chains.

We study the relation between the exact-WKB Delabaere–Dillinger–Pham Stokes jumps and the multi-instanton expansion of a one-dimensional potential with $N$ equal-depth minima. For a single barrier this relation is the Bogomolny–Zinn-Justin cancellation between the instanton–anti-instanton ambiguity and the Borel ambiguity of perturbation theory, and we extend it to every consecutive barrier interval and to the integration contours themselves. In the dilute large-$T$ regime at fixed nonperturbative grade, we derive the quasi-zero mode integral over the instanton separations from the Euclidean path integral and reduce it to Gamma and Hankel periods. With the local Weber level function of each well and one normalization per well fixed by the local Weber connection matrix, these periods reproduce the two lateral connected exact-WKB coefficients. The two lateral contours of each instanton–anti-instanton separation differ by exactly one Hankel cycle, and this linear Picard–Lefschetz relation reproduces the Stokes jump of the connected coefficient. A numerical solution of the Schrödinger equation for a symmetric triple well confirms the resulting two-barrier level splitting. At higher fixed grade, the map from ordered event sequences to normalized scalar coefficient products is not one-to-one. At leading dilute order the scalar weight of a closed event sequence depends only on the numbers of barrier crossings, $\mathcal I\bar{\mathcal I}$ pairs, and $\mathcal I\mathcal I$/$\bar{\mathcal I}\bar{\mathcal I}$ pairs. We construct two inequivalent grade-six triple-well event sequences with the same product of local Weber amplitudes, which embed in every potential with $N\ge3$ wells. Our results hold within the regular dilute, one-loop, local Weber approximation.

We characterize a source of photon pairs simultaneously entangled in polarization and in the transverse spatial degree of freedom, with the spatial measurements performed in intermediate planes of the fractional Fourier transform. The source consists of two crossed type-I crystals in a non-collinear geometry, such that the transverse variables carry information about the crystal of origin. Detection in the crystal plane reveals this information completely and destroys the polarization entanglement, so that a scheme using the near and far fields as conjugate spatial bases cannot support a parallel polarization channel. In intermediate planes both spatial bases combine position and momentum, such that the which-crystal information is partially erased, and the polarization entanglement survives in all four combinations of spatial bases. Applying Bell and Einstein-Podolsky-Rosen criteria to the same photon pairs certifies both degrees of freedom simultaneously. We evaluate quantum bit error rates and asymptotic secret key rates in each degree of freedom, and find that the source delivers two simultaneously usable channels per photon pair.

Transformers can perform in-context reinforcement learning without parameter updates. However, existing theoretical constructions often require context sizes that grow with the number of sampled transitions, leading to increasing memory costs in large-scale problems. This raises a fundamental scalability question: how much context is intrinsically necessary to carry out such learning? We answer this question for fixed-policy TD evaluation. We show that, with a shared linear decoder, the minimum context dimension required to exactly reproduce the first $L$ TD iterates is the Krylov dimension induced by the sampled transitions. When the available context dimension $m$ falls below this threshold, exact reproduction is no longer possible. In this regime, we characterize the optimal approximation error and show that it decays as $\Theta(\gamma^m\max\{1,1/(m(1-\gamma))\})$ where $\gamma$ is the discounting factor. Moving from representation complexity to actual memory cost, we characterize the optimal retained memory for exact continuous representations as $\Theta(nk_L)$ and show experimentally that on a sampled task whose exact rank is full but whose numerical rank is small, a context accurate to $10^{-2}$ is $88.7\%$ smaller than the sparse sufficient statistics of TD. Finally, we establish the learnability of the shared executor through pretraining by proving global convergence of an auxiliary objective over effective attention parameters under suitable task diversity. Experiments across diverse TD evaluation problems validate parameter recovery and the predicted context–accuracy tradeoff.

Human supervisors rely on coding agents' reports to assess progress, verify repairs, and decide when to intervene. But a supervisor cannot investigate a failure they were never told occurred. Failed attempts, workarounds, and unresolved risks may disappear from routine reports, leaving supervisors unaware of issues that warrant scrutiny. We conduct CTXGAP, a measurement study quantifying the gap in task-state recoverability between agents' accumulated user-facing reports and their full recorded execution contexts. Across 37 coding tasks and 4,799 valid probes, agents answer 93.6% of task-state questions on demand; full-context readers recover 92.7%, while report-only readers recover 52.0%, a 40.7-point gap. At the final checkpoints of 30 verified fixes, the gap remains 34.8 points. Human annotation identifies states mentioned without enough supporting detail as the largest category of unrecovered information, and model-assessed supervision-relevant states retain substantial gaps. Full terminal scrollback improves recovery to 73.6% but still trails full context. Under matched report lengths, context-informed rewrites improve recovery by about eight points, without a detectable targeted-over-natural advantage. Positive gaps persist across harnesses and independent DeepSeek execution and measurement. CTXGAP establishes report-based task-state recoverability as a complementary evaluation dimension.

Recent analyses combining DESI baryon acoustic oscillation (BAO) measurements with other cosmological observations have reported indications of departures from the $\Lambda$CDM model within dynamical dark energy scenarios. We investigate whether such deviations are reproduced consistently in $f(T)$ gravity and whether they persist when different cosmological observations are combined. Using the DESI DR2 BAO measurements, Cosmic Microwave Background (CMB), and Pantheon+ Type Ia Supernovae (SNe Ia) sample, we constrain three representative $f(T)$ models: the power-law, exponential, and hyperbolic-tangent models. We compare the constraints obtained from different combinations of data, distinguishing changes in marginalized parameter distributions from changes in the relative statistical performance of the models. For the power-law model, the mild deviation from the $\Lambda$CDM model obtained from BAO+CMB disappears after the SNe Ia data are included. In contrast, the exponential model retains a preference for a nonzero deformation parameter in the full analysis, while the allowed parameter space of the hyperbolic-tangent model changes substantially after the CMB distance priors are included. Importantly, the parameter-level deviation found for the exponential model does not lead to a clear statistical preference over the $\Lambda$CDM model according to the information criteria, whereas the hyperbolic-tangent model shows a poorer statistical performance for the combined data. Despite these differences in the model constraints, the inferred transition redshift $z_T$ remains relatively stable among the three models and data combinations. Our results show that the departures from the $\Lambda$CDM model indicated by DESI analyses are not a generic feature of $f(T)$ gravity, but depend strongly on the specific modified gravity parameterization and on the combination of cosmological observations.

Let $n\geq2$ and $1<p<n$. We study sharp Sobolev inequalities on the round sphere $\mathbb{S}^n$ for functions $u\in W^{1,p}(\mathbb{S}^n)$ satisfying $\int_{\mathbb{S}^n}x|u|^{np/(n-p)}\,d\sigma=0$. We prove that \[ \|u\|_{L^{np/(n-p)}(\mathbb{S}^n)}^p \leq 2^{-p/n}S_{n,p}^p\|\nabla u\|_{L^p(\mathbb{S}^n)}^p+B\|u\|_{L^p(\mathbb{S}^n)}^p \] holds for all such functions with some $B=B(n,p)<\infty$ if and only if $1<p\leq2$ or $2<p<n<3p-2$, where $S_{n,p}$ is the optimal Euclidean Sobolev constant. This determines the exact range in which Aubin's inequality holds with the sharp gradient coefficient. For every $2\leq p<n$, we also establish the corresponding inequality with squared norms and gradient coefficient $2^{-2/n}S_{n,p}^2$. The gradient coefficient in each inequality is optimal. The proof combines stereographic averaging with quantitative stability of the Euclidean Sobolev inequality to obtain local second moment estimates that allow absorption of the localization errors.

We study finite-maturity American puts in the Heston model without the Feller condition. At positive variance, the exercise boundary is smooth and strictly decreasing in remaining maturity and variance. At zero variance, the endpoint is locally Lipschitz in remaining maturity, the boundary moves with bounded speed in maturity uniformly down to zero variance, and it extends continuously to the axis. Each boundary slice at positive maturity has a regular $C^1$ closure tangent to the zero-variance axis, its stock threshold has variance derivative tending to minus infinity, and both partial derivatives of the inverse boundary graph vanish along the slice. The continuation premium at the axis contact stock is of order variance, and as a consequence smooth fit fails in the variance direction at the axis. At expiry the whole boundary, including its axis edge, converges to the strike. We give quantitative boundary gap bounds, an exact limiting weighted boundary Hessian, and vertical identities retaining the maturity term. Exact boundary gap equivalents remain open.

Legal-content reproduction requires matching the source exactly in section hierarchy, verbatim text, citations, and amendment spans; a dropped citation or mislabeled amendment carries into the products lawyers rely on. We report an industrial pipeline converting U.S. state legislation into strict, schema-constrained XML, and hypothesize that its accuracy comes from harness engineering, the deterministic tooling built around a model, more than the model itself. Our controlled ablation crosses three harnesses (zero-shot, a single tool-using agent, and a decomposed multi-agent pipeline) with three Claude models on 153 gold bills from two jurisdictions. The harness narrows the model gap: the weakest model in the full pipeline beats the strongest model alone (0.948 versus 0.806, paired bootstrap p less than 0.001). Because the two agentic harnesses share identical tools, their gap comes from orchestration: decomposition plus the persisted state and per-agent context it brings. Across four scored dimensions, the model spread at zero-shot (0.68 to 0.81) narrows to 0.95 at the full pipeline; the residual sits almost entirely in citation normalization, where the canonical identifier is retrieved from an external authority rather than generated from the weights, so a deterministic lookup tool supplies the exactness that a larger model does not.

Augmented Reality can reshape our visual perception (e.g., simulating color blindness) yet such effects are typically pre-programmed by experts, making visual augmentation static and inflexible. We introduce ShadAR, an AR system that allows users to describe expected visual augmentations (e.g., "Blur all people") for changing their vision in real time. ShadAR can graphically manipulate AR passthrough in real time via shader injection. When receiving a user's prompt, ShadAR uses LLMs (GPT5.2) to interpret the user's intent, analyzes the user's view with object detection (YOLOv26), generates a pixel shader, and applies it to the targeted region on AR passthrough. To understand usage patterns, we invited 21 HCI researchers to use ShadAR, identifying four application domains (design, health, productivity, and social interaction) and two prompt-framing patterns: procedural prompts specifying what to modify and how ("Highlight the book."), and end-goal prompts conveying desired mental or behavioral states ("Make me laugh."). Such end-goal prompts suggest a future in which users specify outcomes (e.g., "Make me not miss my flight") while the system determines how to realize them, though often with inherently indeterminate results. We discuss design opportunities for ShadAR and ethical concerns.

Efficient sequence modeling remains a fundamental challenge in machine learning. Despite the success of Transformers, recurrent neural networks (RNNs) have recently re-emerged as efficient alternatives. Particularly, Echo State Networks (ESNs) offer a compelling framework of randomized RNNs that have been extensively studied for time-series forecasting and dynamical systems. This paper presents an empirical analysis of distributional, architectural, and optimization variants of ESNs evaluated on few-shot sequence-to-sequence (seq2seq) pattern tasks. These synthetic tasks utilize discrete symbol sequences, characterized by delays, inactive phases, and triggered markers to isolate and evaluate memory and pattern recognition capabilities. The minimality of these tasks is crucial to understand the strengths and limitations of machine learning models. Furthermore, when the input signal utilizes a one-hot encoding, ESN dynamics are governed strictly by their weight distributions, offering an ideal lens through which to study reservoir computing. Our findings map the performance boundaries of ESNs relative to state-of-the-art (SOTA) models. We show that preserving a linear state regime is vital for memory tasks, such as pattern completion or simple copy. We propose a data-dependent modulation mechanism for nonlinear input injection into the reservoir (G-ESN). G-ESN functions as a statistical filter capable of solving tasks that conventional ESNs cannot (the adding problem and bracket matching). Finally, for Deep ESNs, we evaluate a shared readout mechanism across layers that outperforms conventional concatenated readouts on structural tasks, including selective copy, sorting problem, and bracket matching.

Large language models often produce fluent but factually incorrect output. Gray-box detectors identify such hallucinations from the token-probability statistics emitted during generation, avoiding the need for white-box access to hidden states that frontier models do not provide. These detectors, however, typically rely only on probability-derived representations and ignore the identities of the candidate tokens. They can therefore observe how uncertain the model is, but not what it is uncertain about. Consequently, they cannot distinguish between a model that is torn between "big" and "small" and one that is torn between "big" and "huge" when the two pairs receive the same probability distribution. We introduce GURI (Gray-box Uncertainty Representations with Identity), a lightweight gray-box hallucination detector that combines candidate-token probabilities with per-token vector identifiers. This exposes which alternatives the model considered while preserving the accessibility and transferability of the gray-box setting. Across three open-weight models and three question-answering benchmarks, GURI increases mean test ROC-AUC from 72.09% to 79.09% over a probability-only baseline. On the closed-source GPT-4.1 model, mean ROC-AUC increases from 67.39% to 71.84%. Among the three open-weight models, GURI outperforms transferred probability-only baseline in 17 of 18 directed cross-model transfers without fine-tuning. For the three open-weight models, cross-dataset transfer between HotpotQA and TriviaQA with the model held fixed improves mean ROC-AUC over transferred LOS-Net by 3.76 points.

We investigate the role of different correlation functions in reconstructing the phase diagram of the spin-1 bilinear-biquadratic chain, parametrized by the angle $\theta$ between the bilinear and biquadratic couplings, using our previously developed label-free machine-learning pipeline \cite{Ming21, Ming23}, in which an autoencoder compresses iDMRG correlation matrices into latent features and K-means clustering with silhouette analysis groups them into phases. Without any prior labels, the scheme reveals four natural phases, the Haldane, dimerized, critical trimerized, and ferromagnetic phases, and accurately locates the Takhtajan-Babujian ($\theta=-\pi/4$), $SU(3)$-symmetric Uimin-Lai-Sutherland ($\theta=\pi/4$), and Ferro-Dimer ($\theta=5\pi/4$) transition points; a shallow supervised refinement further sharpens the phase boundaries. We find that different correlation functions play fundamentally different roles in phase identification. The $SU(3)$-symmetric correlation alone selects a four-cluster partition that can be associated with the four phases, whereas the spin-spin and dimer-dimer correlations must be combined to recover the complete phase diagram. In contrast, the trimer-trimer correlation behaves as a marginal correlation: it is insufficient for phase identification by itself but improves the precision of the predicted phase boundaries when combined with the other correlations. Furthermore, the latent representations and silhouette values reveal signatures that we relate to the $\theta$ dependence of the correlations within each phase and, tentatively, to the underlying renormalization-group structure. These results demonstrate that machine learning can not only identify quantum phases but also uncover the physical information encoded in different correlation functions.

High-resolution precipitation datasets are essential for weather forecasting, hydrological modeling, and climate risk assessment, yet observation-based gridded products often lack the spatial detail required for regional applications. Here we present PrecipCorrDiff, a diffusion-based downscaling framework built by modifying the original corrector diffusion based archetecture that incorporates a customized loss function and a multi-scale residual generation strategy to improve the precipitation especially the representation of extremes. PrecipCorrDiff generates 0.0625{\deg} resolution India Meteorological Department (IMD) precipitation fields from 0.25{\deg} gridded data while preserving precipitation statistics and reconstructing fine-scale spatial variability. Validation against independent IMD station observations, comparing the proposed framework with the SRGAN-based model and the original CorrDiff model, indicates that the proposed framework substantially improves the representation of spatial precipitation patterns, rainfall intensity distributions, and extreme rainfall events, while eliminating the systematic negative bias present in the previously developed SRGAN-based model. Across four climatologically distinct homogeneous Indian monsoon regions, PrecipCorrDiff reproduces observed precipitation characteristics with near-zero spatial bias, demonstrating the potential of diffusion-based downscaling for generating physically consistent, high-resolution precipitation datasets for hydrological, meteorological, and climate applications.

Reconstruction of 3D structures using under-ice AUV video footage has been inaccurate, although it is not certain whether this is due to shortcomings in the methods or an inherent deficiency of the capture system. We test four different models for reconstruction on the same Antarctic under-ice AUV video (Icefin, Kamb Ice Stream): classical feature matching SfM, differentiable photometric splatting, volumetric depth fusion, and a pose-free feed-forward foundation model. All four fail, each in a distinct way: near-total registration yielding almost no structure, geometrically broken output despite healthy photometric metrics, a persistent coverage artifact, and unstable cross-chunk scale. We show that all four are consistent with the same capture regime, a small camera-baseline-to-scene-depth ratio. Following our analysis, we propose a detection system that flags this type of failure in minutes, rather than hours of wasted compute.

Law students are examined through open-text answers: case solutions, essays and memos that apply the law to a set of facts. Such answers are expensive to grade, human graders disagree widely on them, and large language models (LLMs) now offer to grade them at scale, both in teaching and in the benchmarks that test legal AI. This paper systematizes how open-text legal answers can be evaluated. We derive eight desiderata from educational measurement and from the law that governs exams, among them validity, reliability, reasons, credit for defensible alternative solutions, robustness to gaming and human oversight. We then compare eight evaluation paradigms against them, from unstructured human grading to LLM judges with instance-specific rubrics and pairwise comparison. Three findings stand out. A rubric written for the task is the strongest lever for agreement between an LLM judge and human graders, but the reported figures answer different questions and rarely come with the human agreement they must be read against. The central validation gap lies where examination law is most demanding: defensible alternative solutions, follow-on errors, and the examiner's correction of model judgments. And rubric-centred grading is the most auditable family, because it makes the standard, its application and the final decision checkable. From this we derive the trustworthy rubric judge: an instance rubric that credits arguments and alternative paths,evidence checked in code, probes, repeated runs, calibration against independent human grades,logging, and an examiner who reads every script. We state what each safeguard rests on, what remains to be tested, and the uses the evidence supports today.

As artificial intelligence (AI) increasingly supports human decision-making in high-stakes domains, designing effective human-AI collaboration becomes critical. Successful collaboration requires humans to reflect on decision-making processes and adjust their strategies accordingly. Current research typically investigates interventions, such as explainable AI, that target reflection during the decision-making process (reflection-in-action). Less attention has been given to retrospective examination of decisions (reflection-on-action). This work explores reflection-on-action by studying the impact of feedback on reliance behavior and human-AI team performance. Specifically, we investigate feedback interventions targeting overrides of AI advice through a behavioral study (N=244). We find that such feedback influences reflection-on-action, while unidirectional feedback risks reducing subsequent reflection-in-action. Furthermore, we identify opposing shifts in reliance behavior that leave overall human-AI team performance unchanged but show context-dependent effects. Balanced feedback did not show these risks, preserving both concurrent and retrospective reflection, offering insights for designing for appropriate reliance on AI.

We give an elementary proof that if $f$ is a locally constant function on a topologically mixing subshift of finite type, then the equilibrium measures for $tf$ converge as $t\to\infty$, a result due to Brémont. The proof shows that the limit measure assigns an algebraic number to every cylinder.

Classical novae eject material during thermonuclear eruptions on accreting white dwarfs, forming expanding shells whose structures and kinematics provide insights into the eruption mechanism and binary interaction. However, their intrinsic three-dimensional (3D) structures remain poorly constrained. We present the first 3D catalog of nova shells based on integral field spectroscopy, enabling us to investigate their structures beyond projection effects. We compiled observations of 19 nova shells obtained with MUSE at the VLT, complemented by new MEGARA observations at the GTC and WEAVE observations at the WHT. We constructed 3D visualizations of selected emission lines to explore the diversity of shell morphologies and kinematics. The shells reveal a wide variety of structures, including ellipsoidal or spherical shells, rings, bipolar features, and clumpy or irregular components, many of which are not apparent in 2D projections. Approximately 60% of the sample exhibits predominantly ellipsoidal or spherical shapes, while about 30% shows ring-like or bipolar structures. Several objects display strong line-dependent morphological differences, with [O III] or [N II] tracing distinct kinematic components from those seen in H${\alpha}$. Compared with imaging studies, integral field spectroscopy reveals a larger fraction of bipolar or toroidal shells. The differences between emission lines highlight the diverse excitation mechanisms and physical conditions within the ejecta. Finally, comparing expansion velocities in the plane of the sky and along the line of sight suggests low intrinsic aspect ratios, consistent with estimates from projected shell shapes.

The spherical code problem asks how to place $N$ points on a unit sphere in $D$ dimensions to maximize the smallest distance between two points. It is a non-smooth and non-convex max-min problem, where few problem sizes have known optimal solutions, some have constructions, but most still only have numerically obtained solutions. We use a simple solver, with seven parameters, that anneals a LogSumExp surrogate, minimizes with L-BFGS, applies a random perturbation between anneals, and repeats with no acceptance test until a fixed limit is reached. We search the solver parameters for every instance with $N \le 1024$, $3 \le D \le 32$, and $N > 2D$, a total of 29,670 instances, using 32 parameter trials. We then run the solver for 60 seconds with the best parameters for each instance, without utilizing any symmetries or existing solutions. After enforcing monotonicity of the results and numerically polishing the solutions, we improve the best known code in 4,577 instances. When a single set of solver parameters is used for all instances, with one 60-second episode, the number of improvements is still 3,649. From our records, we find 24 codes with an exact algebraic cosine. Among our ablations, we find that 1) the repeated anneal contributes most to improvements, 2) at our $32 \times 60$-second budget only fitting one parameter per-instance is better than fitting all seven, and 3) good parameters can be predicted from the instance size. We provide the solver code, the fitted parameters, and the codes.

In hybrid monostatic-bistatic orthogonal frequency-division multiplexing (OFDM) sensing, each sensing node receives a superposition of monostatic and bistatic target returns, whose phase-noise (PN) characteristics differ fundamentally due to oscillator sharing. While monostatic sensing benefits from partial PN cancellation through the common transmit-receive oscillator, bistatic sensing involves independent oscillators and is therefore more strongly affected by PN. In this paper, we characterize the joint residual differential PN statistics of these superimposed components in a centralized two-node sensing system and quantify their impact on target localization via theoretical lower bounds considering three different receivers: (i) PN-free ideal receiver, (ii) PN-aware receiver that exploits the joint PN statistics, and (iii) PN-ignorant receiver that neglects PN.

Here, we study an undamped Rayleigh beam subjected to a concentrated transverse impulse applied at a point far from the boundaries. For a simply supported finite beam struck at its midpoint, using the modal expansion of the impulse response, a short-time asymptotic series for the response of the struck point is derived. The infinite beam impulse response is obtained as a spectral integral in two ways. The spectral integral is evaluated exactly in terms of modified Bessel functions via Laplace transform calculations. The corresponding short- and long-time asymptotics of the exact integral are consistent with the power series developed earlier and the Euler-Bernoulli beam, respectively. Direct numerical evaluation of the integral with a pre-specified error bound and numerical contour integration is also presented for verification. An ad hoc approximation is given which matches the response well for all times. The relationship between finite and infinite beam responses is examined in some detail. There are three consequences of finite length. First, the coefficients in the power series expansion contain components that decay exponentially with increasing length of the beam. Second, a smooth deviation from the infinite beam response is observed, consistent with the greatest group velocity in the beam, wherein a small precursor is detected which is consistent with broader theory. Third, a slope discontinuity is observed, corresponding to the high-frequency limit of group velocity. The full time series plot from modal expansion satisfactorily shows both the arrival and passage of these high frequency components. This paper opens the door to impact studies that are envisaged in future work.

We investigate the statistical properties and ergodic behavior of the absolute area below the trajectory of a particle subject to stochastic resetting and moving according to a scaled Brownian motion with exponent $\beta$. We compare the contributions to the absolute area from the waiting times between resets and the backward recurrence time. Using the renewal formalism we derive the lower-order moments of the absolute area under both Poissonian and heavy-tailed power-law resetting waiting time distributions with exponent $\alpha$. For Poissonian resetting, stochastic interruption suppresses fluctuations, driving the area to be ergodic, regardless of the value of $\beta$. In contrast, power-law resetting yields a rich spectrum of ergodic regimes. For $0 < \alpha < 1$, the system displays weak ergodicity breaking and the maximum contribution dominates the area accumulated during the waiting times. For $1+\beta < \alpha < 2$, we unveil an ergodic transition exclusive to the subdiffusive regime at $\alpha=1+\beta$, where ergodicity is restored. These results clarify the subtle interplay between anomalous diffusion and power-law waiting times between resettings. Our results are tested by numerical simulations exhibiting an excellent agreement.

Inspired by a pioneering work by Caffarelli, Gidas, and Spruck, we study positive solutions to the Yamabe equation with nonremovable isolated singularities. It is known that there is a formal one-to-one correspondence between the collection of the solutions to the Yamabe equation with nonremovable isolated singularities and the collection of harmonic functions through formal infinite series determined by the harmonic functions. In this paper, we prove that these formal infinite series yield actual solutions to the Yamabe equation with a nonremovable isolated singularity. These solutions to the Yamabe equation have designated asymptotic behaviors determined by the harmonic functions. In a broader context, we provide a general method of constructing all solutions near their isolated singular points and all solutions near infinity.

We develop a boundary value and potential theory for elliptic operators \[ L=-\Div(A\nabla)+\B\cdot\nabla \] with form-bounded drift in one-sided chord-arc domains. Assuming that the positive part of $\Div\B$ has sufficiently small form bound, we establish local regularity estimates and solvability of the Dirichlet problem, and construct the associated harmonic measure. Under the additional assumption $\Div\B\le0$, we construct a Green's function and prove pointwise and gradient estimates. For the characteristic distributional identity with a Dirac mass at the pole to hold, we will additionally assume either $\Div\B=0$ or $\B\in L^{n,1}_{\textup{loc}}(\Omega)$. Moreover, in the divergence-free case, we obtain a local lower bound for the Green's function, CFMS estimates, doubling of harmonic measure, a comparison principle for nonnegative solutions with zero trace, and a change of pole formula. Finally, using AI, we construct, in the upper half-space, an operator of the form $-\Delta+\B\cdot\nabla$ with $\B$ form-bounded and divergence-free, whose harmonic measure exhibits dimension drop.

We study a spherical perfectly matched layer (PML) method for the time-harmonic Maxwell scattering problem with large wavenumber. Using vector spherical harmonics and a modal transformation for the coupled transverse magnetic components, we establish the sharp wavenumber-explicit stability estimate for the truncated PML problem and prove that its inf-sup constant is of order $O(\kappa^{-1})$. We then analyze the first-order Nédélec edge element method and its continuous interior penalty counterpart. Under the condition that $\kappa^3h^2$ is sufficiently small, the preasymptotic errors in the energy norm and the $\kappa$-scaled $\mathbf{L}^2$ norm are of orders $O(\kappa h+\kappa^3h^2)$ and $O(\kappa^2h^2+\kappa^3h^2)$, respectively. Numerical experiments confirm the theoretical results.