Papers

We empirically investigate Question Under Discussion based modelling in naturalistic dialogue by studying whether the salience of generated potential questions predicts their subsequent resolution. Building on Wu et al. (2024), we construct a dataset of 7,124 questions automatically generated from utterances and preceding context from the British National Corpus, and annotated for salience and answerability. We find a robust but low positive correlation between salience and answerability in dialogue, indicating that more salient questions are more likely to be addressed. However, this effect is markedly weaker than in monologic text, suggesting that conversational structure is less predictable. We further observe that structured interactions exhibit stronger alignment between annotators than less organised dialogues.

The sharing and contagion of emotions on social media influence online interactions and users' psychological states. This study analyzes emotional dynamics on Vent, an emotion-sharing social media platform where users explicitly assign emotional labels to their own posts. Using a large-scale dataset, we examine how emotions in users' pre-posting timelines are associated with their subsequent emotional labels and how such associations vary across users. Our analysis reveals three key findings. First, users' subsequent emotional labels are associated with the emotional composition of their pre-posting timelines, with same-category emotions being overrepresented before posts in all analyzed categories. Second, several cross-category associations are observed; for example, Surprise was overrepresented before Fear posts; Affection and Happiness were overrepresented before Anger posts; and Affection was overrepresented before Sadness posts. Third, users differ in their degree of alignment with timeline emotional fluctuations, and highly aligned users tend to be located close to one another in the network. These findings provide large-scale observational evidence that emotional expression on Vent is associated with both short-term timeline context and network structure.

Frequency-based adversarial attacks have recently grown popular by exploiting spectral sensitivities shared across neural architectures. Unlike spatial perturbations, frequency-based attacks expose deeper vulnerabilities, making them especially valuable for robust evaluation of safety-critical and security-sensitive applications. Yet, existing approaches are typically not derived as solutions to an optimization problem that explicitly captures transform-domain structure. In this paper, we propose a methodology for crafting principled frequency-based adversarial attacks, via a dedicated optimization framework. A cornerstone of our method hinges on the introduction of a perturbation constraint set, tied to highly structured non-orthogonal transforms, well-known for their flexible, non-predefined frequency handling. We prove that the attacks emerge as weighted $\ell_2$-projections onto this set, yielding a general and controlled attack generation mechanism. By this, we provide a clear geometric attack characterization, ensuring alignment between the optimization objective and the perturbation constraint. We assess our framework on standardized datasets, for pretrained and adversarially robust models. Results highlight that our attacks, being solutions to an optimization problem, over a structured perturbation set, are highly effective, even across different, unseen architectures. Our methodology could serve as a theoretical baseline for designing and analyzing transformed-based attacks, targeting fundamental model vulnerabilities, instead of mere architecture-specific artifacts typically studied in the robustness literature.

To design systems that are protected against events much rarer than the observational record, extreme-value methods are needed to extrapolate distribution tails. Tail-index estimators such as the Hill estimator are central to this extrapolation, but overestimating the tail exponent can lead to substantial underestimation of rare-event probabilities. Motivated by this, we derive large- and moderate-deviation asymptotics for the Hill estimator and use them to construct estimators whose probability of exceeding the true tail index decays at a controlled exponential rate (the decay rate). In the large-deviations regime, we show that a simple rescaled version of the Hill estimator achieves an optimal balance between bias and decay rate among scale-invariant estimators based on the same top $k$ order statistics. Under a second-order condition, we quantify the effect of the Hill bias, analyze a bias-corrected estimator, and identify sufficient conditions for moderate deviations in the boundary case where the second-order parameter $\rho$ equals zero.

We present a systematic approach to characterize Devaney chaos in the solution of homogeneous one dimensional constant-coefficients PDEs on Herzog spaces. We show that for equations where the highest-order time derivative lacks mixed spatial derivatives, there always exists a weighted space in which the solution is chaotic. This analysis is extended to include mixed partial derivatives in the highest-order temporal term. Moreover, we link the chaotic behavior in Herzog spaces to the semidiscretization of the equations by identifying isometric isomorphisms between the two contexts, providing also a characterization for chaos on these semidiscretizations. The work concludes with two applications that provide new insights on the dynamics of the fourth-order Moore-Gibson-Thompson equation and the viscous van Wijngaarden-Eringen equation.

Stochastic processes underpin dynamics across biology, physics, epidemiology, and finance, yet accurately simulating them remains a major challenge. Classical approaches such as the Gillespie algorithm are exact for Markovian, time-independent systems, where propensities depend only on the current state and agents of a given type are statistically identical. While efficient, this framework misses a defining feature of many real systems: heterogeneity and memory at the level of individual agents. Cells may divide or differentiate on distinct intrinsic timescales, individuals may preferentially interact with specific partners, and inter-event-time distributions can deviate strongly from the exponential. We introduce MOSAIC (Modeling of Stochastic Agents with Individual Complexity), a general and scalable framework that embeds agent-specific properties directly into the dynamics. MOSAIC unifies heterogeneous rates, dynamic interaction preferences, and both Markovian and non-Markovian waiting-time distributions within a single stochastic formalism, while retaining Gillespie-like computational cost.

By distilling and combining earlier arguments, we give a self-contained and entirely elementary proof of the strong and real linear polarization inequalities established by Martínez and Ortega-Moreno. Specifically, for any unit vectors $u_1,\ldots,u_n\in\mathbb{R}^d$, there exists a unit vector $v\in\mathbb{R}^d$ such that $$\sum_{i=1}^n \frac{1}{\langle u_i,v\rangle^2}\le n^2. $$ As an immediate consequence, we recover the real linear polarization inequality $$\prod_{i=1}^n |\langle u_i,v\rangle|\ge n^{-n/2}. $$

Deriving emergent cosmology from the discrete geometry of Loop Quantum Gravity remains an open challenge. We solve the two-vertex model exactly, including general non-symmetric configurations, by introducing macroscopic global variables that form a closed Poisson subalgebra. This yields an intrinsic minimal area gap from the Casimir invariants of the subalgebra. In the semiclassical limit, a generalized Friedmann equation emerges incorporating graph anisotropies and quantum geometry corrections, which lead to a Big Bounce. This global formulation opens a coarse-graining route toward statistical descriptions of quantum spacetime.

Activation steering is a powerful training-free paradigm for controlling large language models at inference time. However, standard approaches estimate a per-layer steering direction from contrastive data and apply it on the layer's entire representation space, which may couple the intervention to off-target properties present in the contrastive data and degrade unrelated capabilities. To mitigate this issue, we introduce LocUS (Localized Unembedding Steering), a method which grounds activation steering to the model's own output vocabulary subspace. By identifying a property-specific linear subspace within the unembedding matrix, LocUS enforces a geometric constraint that restricts the steering transformation to a specific subspace and at the same time localizes its application to a sparse subset of attention heads. Extensive evaluations across three model families on toxicity mitigation, sentiment redirection and sycophancy suppression show that LocUS matches or outperforms state-of-the-art baselines while intervening on under 6% of parameters and better preserving general capability.

Chain-of-thought (CoT) monitoring is a promising safety technique for reasoning models, enabling detection of problematic reasoning before models act. A key concern is encoded reasoning, where models hide their true reasoning in ways that monitors and humans cannot interpret. Optimization pressure from CoT monitors during reinforcement learning is considered a likely driver of such behavior. We investigate this by training reasoning models to perform a main task and a side task, while penalizing them when a monitor detects reasoning about the side task. Surprisingly, models learn to evade monitors without encoding their reasoning. Instead, they learn to phrase and format their chains of thought such that monitors fail to flag side task reasoning, while the reasoning remains completely transparent to human readers. We call this phenomenon monitor jailbreaking. We find that monitor jailbreaking arises across different model sizes, monitors, and tasks. Jailbreaks generalize to monitors not seen during training, including both less and more capable monitors, and transfer across different monitor prompts. While jailbreaking strategies appear simple, manually replicating them does not reliably fool monitors. Finally, we show that paraphrasing is an effective defense: paraphrasing a jailbroken CoT allows the same monitor to correctly flag it, while still allowing the model to perform both tasks.

The production of a Higgs boson with a jet is one of the most important processes studied at the LHC. It serves as a sensitive probe of high-energy dynamics and provides a powerful testing ground for the Standard Model, while also offering sensitivity to possible new physics effects. In this work, we perform a phenomenological study of Higgs + jet production in proton-proton collisions at 13 TeV center of mass energy, using the MadGraph5_aMC@NLO framework. We compare predictions obtained from the full loop induced Standard Model with those derived within the HEFT and SMEFT frameworks. Taking the Higgs transverse momentum as a probe, we study normalized distributions and ratio spectra to test the effective field theories. Our results show that effective approaches successfully reproduce the Standard Model behavior in the low transverse-momentum region. However, sizable deviations emerge at high pT, signaling the breakdown of the effective approximation. These findings highlight the importance of carefully accounting for kinematic effects when applying EFT-based methods to precision Higgs studies and provide a quantitative estimate of their range of validity.

Resource disaggregation separates memory and accelerators from compute nodes and makes them remotely accessible. This improves resource sharing, but also removes the local kernel from the resource-access path. Under an untrusted host, compromised host software may use stale authority, exceed delegated authority, or reuse authority provisioned for another process. Prior work identifies capability-based access control as well suited to these architectures. Our systematization of twenty-two prior capability systems finds that none combines host-independent validation of process authority, authoritative enforcement at the resource, and revocation that remains effective while remote authorization state is stale. We present SADRA, a distributed capability-based access-control system for resource-disaggregated architectures. SmartNIC hardware isolated from host software independently checks every inter-node request at two points, first at the compute node against the requesting process's authority and again at the resource against the current authoritative access state. Linked process, compute, and resource capabilities allow these checks to use local state without coordination on the access path. When distributed authority is revoked, SADRA denies subsequent dependent accesses at the resource without waiting for remote nodes to update, while stale capability state is reclaimed separately. We prove capability safety, authority safety, revocation soundness, and strong isolation for a formal architectural model, and model-check the formalization with SPIN. Our FPGA SmartNIC prototype sustains 89.5 Gbit/s aggregate throughput, within run-to-run variation of a non-enforcing baseline that peaks at 90–91 Gbit/s. Cleanup of a 128-capability subtree completes in 528 ns, while access denial does not depend on completion of that cleanup.

Accurate and computationally efficient waveform models of compact binary coalescences are a fundamental requirement for gravitational wave data analysis. This work presents differentiable and hardware-accelerated implementations of the machine-learning surrogate models mlgw and mlgw-bns within the JAX ecosystem. The proposed framework enables JIT compilation, vector paralellization, native GPU acceleration, and automatic differentiation, while preserving the original surrogate training infrastructure and its applicability to arbitrary non-precessing waveform approximants. Benchmarks are performed in the binary black hole case with a newly trained surrogate of the SEOBNRv5HM approximant; on CPU, they show speed-ups in waveform evaluation time with respect to the original mlgw implementation that exceed one order of magnitude; further, when exploiting GPU acceleration and large-batch vectorization these reach approximately two orders of magnitude. The benchmarks performed in the case of binary neutron stars with a TEOBResumSPA surrogate model achieve similar gains on GPU. Our JAX-based surrogate modeling framework can be integrated into Bayesian parameter estimation pipelines through GWgpu-jax, an open-source package that connects JAX-compatible gravitational waveform generators to JAX-native sampling algorithms. We demonstrate this capability with a nested-sampling pipeline built upon BlackJax-NS, performing parameter estimation analyses of the GW150914 and GW170817 events with mlgw and mlgw-bns. The analyses require approximately twelve and seventeen minutes, respectively, on a single GPU and yield posterior distributions consistent with those reported by the LIGO-Virgo-KAGRA Collaboration. Beyond nested sampling, JAX automatic differentiation provides efficient gradient and Hessian evaluations of the waveform models, enabling integration with gradient-based Bayesian inference methods.

Service robots remain difficult to deploy in domestic environments, partly because fully autonomous operation is not yet reliable in unpredictable surroundings, and partly because conventional control methods remain inaccessible to novice users. Extended Reality (XR) enables operators to visualize robot information overlaid onto the real world and to interact with augmented elements. Yet, common XR control methods, such as motion controllers and hand gestures, are still perceived as unintuitive. This paper presents a control interface that uses a commercial XR pen to command a semi-autonomous mobile robot in Augmented Reality (AR): the operator points at a position in the room, selects it, and drags an augmented arrow to set the desired orientation of the robot at this destination. Two additional interfaces, based on the XR motion controllers and hand gestures, were developed within the same framework. To assess the performance and users' perception of these interfaces, and of the XR pen in particular, a study with 10 participants compared four control methods, i.e., the XR pen, the XR motion controllers, hand gestures, and a computer-based baseline RViz, in navigation tasks performed in a home-like environment. Results show that the XR pen significantly outperforms the other methods in task selection time with the most consistent selections, and that the XR motion controllers obtain the best perceived workload and usability scores, ahead of the computer-based baseline, supporting XR-based control as an intuitive alternative for novice users. However, technical limitations in the integration of the recently released XR pen currently hold back its user experience.

We study nonparametric multiple change-point detection for high-dimensional sequences, aiming to identify time points at which the underlying distribution changes. While many existing methods perform well when change-points are well-separated, their performance can deteriorate when structural breaks are densely clustered. To address this challenge, we propose gBottomup, a graph-based bottom-up framework for multiple change-point detection in high-dimensional settings. gBottomup constructs a hierarchical segmentation by proposing merges of adjacent segments and verifying them through an unmerge rule that combines absolute significance with relative local heterogeneity, thereby adaptively refining partitions and estimating the number of change-points. Simulation results demonstrate that gBottomup performs reliably across a range of structural configurations and is particularly effective in frequent change-point settings, where existing top-down procedures may lose sensitivity. We illustrate the proposed method through an analysis of high-dimensional sensor data from an experimental hydraulic system.

Light-sheet microscopy is ideal for the 3D imaging of large biological specimens, yet systematic documentation on adapting microscope designs to diverse experimental requirements remains limited. We present the iterative process used to design and build a modular light-sheet microscope and adapt it to different sample types and experimental conditions, including live and fixed zebrafish, Drosophila, and mouse gastruloids, as well as cellulose nanocrystals. Starting from a stripped-down OpenSPIM platform, we gradually made our microscope more versatile by adding hardware such as additional laser channels and optical filters, and new protocols for sample handling. In its final iteration, FlowSPIM is a new flow-based incubation technique for 3D tissue culture.

Neural algorithmic reasoning aims to train neural networks to follow known algorithms and generalize beyond the input sizes seen during training. However, correct final outputs and intermediate supervision do not necessarily show that a model follows the intended execution. We study this problem using insertion sort. Our analysis of the CLRS30 baseline NAR shows that the hint objective is weakly optimized and that hint accuracy remains low. Moreover, many intermediate representations can already be decoded into sorted sequences before the reference insertion-sort execution terminates, suggesting that the model learns a shortcut to the final output. Motivated by these findings, we introduce Discrete Neural Insertion Sort. Our model represents the sequence as a chain, separates scalar exchanges from control-state transitions, and projects node representations back to discrete states after every processor step. When trained only on sequences of length 16, the model achieves $100\%$ sorted-sequence accuracy on sequences of length 64 and 128. However, an ablation shows that discretization and graph structure alone are insufficient: without additional supervision of the global inner-loop state, the model fails even at the training length. Our results show that discrete execution can support strong length generalization, while also highlighting the problem-specific inductive bias required to learn a faithful algorithmic execution.

We constrain a scalar-vector-tensor (SVT) dark-energy model in which the vector sector drives the dark-energy equation of state below $-1$ at earlier times, while a canonical scalar field subsequently drives it across $-1$ toward a present-day value above $-1$. We confront the model with DES Year 5 supernovae, DESI DR2 baryon acoustic oscillations, and compressed cosmic microwave background information, and then include Gold-2017 redshift-space-distortion (RSD) measurements to probe structure growth. The background posterior assigns substantial weight to crossing histories and gives a better best fit than the flat $\Lambda$-cold-dark-matter ($\Lambda$CDM) model. The RSD analysis preserves this qualitative background evolution, constrains the clustering amplitude, and increases the posterior support for the directional crossing criterion. Since the growth analysis also imposes additional perturbative support conditions, this increase cannot be attributed to the RSD likelihood alone. Matched PolyChord calculations give Bayes factors larger than unity for SVT relative to $\Lambda$CDM under both prior-volume conventions considered here. The quantitative Bayes factor depends on the prior normalization assigned to the regular radiation-era branch and is larger in the RSD analysis under either convention. Overall, current expansion and structure-growth data are consistent with a stable dynamical crossing of the phantom divide in the SVT model.

Vision-language models (VLMs) enable open-vocabulary reasoning for robot manipulation, but their high inference latency limits responsiveness in dynamic scenes. Many scene changes, however, alter object geometry without invalidating task intent. We present DualManip, a dual-path framework that decouples infrequent semantic reasoning from responsive geometric adaptation. The semantic path decomposes the task and grounds task-relevant interactions, followed by a constraint-solving module for pose optimization. During execution, the geometric path continuously updates template-to-observation correspondences from live RGB-D observations via a shape-adaptive network. These correspondences transfer task-relevant grasp contacts across observations, enabling online grasp reconstruction under object motion and non-rigid deformation. The Information Interaction Module bridges the two paths by initializing task-relevant grasps from semantic grounding, validating geometric updates, and triggering semantic replanning upon update failures. Real-world evaluation spans six manipulation tasks covering non-rigid deformation, articulated reconfiguration, rigid motion, and high-precision assembly across three settings: static, single-change, and continuous dynamic. DualManip demonstrates superior manipulation robustness, particularly under continuous scene changes, while achieving geometric adaptation approximately 46$\times$ faster than agentic verification and semantic replanning. Our project page: https://lichengxi1.github.io/Dualmanip.

We explore the energy extraction from a Kerr-like regular black hole coupled with a cloud of strings (CoS) via the Comisso-Asenjo mechanism. We study the effects of the black hole spin (a), black hole charge (g) and CoS parameter (b) on the ergo-region of the black hole and the magnetic reconnection process, which is governed by the Comisso-Asenjo mechanism. We also analyze the phase space involving negative energy density of the relativistic magnetized plasma at infinity. The rate of energy extraction in the magnetic reconnection process is also computed and compared with that of the Blandford-Znajek mechanism.

Large language models are increasingly used as high-level planners for mobile robots, robot manipulators, and autonomous vehicles. Recent studies show that these systems can be influenced through malicious text, speech, visual instructions, retrieved documents, and poisoned sensory context. Most defenses ask whether a proposed action is physically safe. This paper studies a different problem: an action may be physically safe and still violate the mission authorized by the user. An attacker may redirect a delivery robot, replace an approved object, extend a robot's operating region, activate an unnecessary sensor, or delay a mission without creating an immediate physical hazard. We call this attack safety-compliant mission hijacking. We propose MissionPAIR, an adaptive attack framework that searches for executable plans that pass a safety gate while violating an authenticated mission. We also propose AuthGuard-R, a deterministic authorization layer that binds every executable action to a signed mission, robot identity, object and region scope, current state, time, and input provenance. AuthGuard-R operates with an independent safety gate, giving a dual-gate architecture. We formalize mission policies over robot traces, define security games, and prove authorization soundness, mission non-escalation, replay resistance, robot binding, provenance separation, threshold-approval security, audit-log tamper evidence, and trace-level composition. We report a preliminary cross-model evaluation with Claude Haiku~4.5 and the open-source Qwen2.5~7B planner. Across 240 live attack trials, the planners followed an injected mission deviation in 109 trials; AuthGuard-R rejected all 109 resulting unauthorized actions. A separate hand-constructed suite of eleven protocol- and policy-level attacks was also blocked completely.

In modern large-scale surveys, young stellar objects (YSOs) are mostly identified by the infrared excess in their spectral energy distributions (SEDs). To this end, YSO identification and classification were mostly based on certain colours, resulting in wavelength-specific methods. Using all available data allows us to construct a uniformly classified YSO catalogue by applying the same identification method to heterogeneous samples, reducing systematic differences and enabling more consistent studies of YSO populations and star formation. Our goal is to develop a method that takes advantage of the heterogeneous data available in the VizieR database and to provide a tool that can be used to identify YSOs with high accuracy. We also aim to create the largest homogeneous catalogue of YSOs. We use the CDS VizieR and NASA/IPAC IRSA to collect data for a large number of objects of various types. The data collected included SEDs, light curves, and direct imaging and were used to build different image representations. These were used to teach deep learning algorithms to differentiate between YSOs and all other types of sources. Architectures from the PyTorch library and custom-built convolutional neural networks were applied to create an ensemble of binary classifiers specialised at identifying YSOs. The resulting method is able to automatically identify YSO candidates with an accuracy higher than 90% while contamination is identified with an accuracy higher than 99%. We applied our classifier to numerous YSO catalogues to confirm the youth of the stars and to create the largest catalogue of homogeneously identified potential YSOs ever – the NEMESIS General YSO catalogue contains 274\,408 unique, high reliability YSOs. Their positions on the colour-magnitude diagrams, and their distribution relative to the local Galactic structures, were also used to test their young nature.

High dynamic range (HDR) video reconstruction from al\-ter\-na\-ting-exposure sequences remains challenging, especially in regions with extreme luminance variation. We propose a novel HDR reconstruction framework based on dual-stream registration and accurate pyramid fusion. Given three consecutive frames, our method computes optical flow directly with the central frame, while introducing a complementary midpoint displacement strategy to handle cases with severe overexposition. A pyramid fusion stage then merges the resulting radiance and LDR images into a final HDR output. Experimental results demonstrate that our approach consistently outperforms state-of-the-art methods.

We study Bayesian optimization (BO) through the lens of information geometry. Pulling back the Fisher information metric through the surrogate posterior map yields a local sensitivity tensor on the input space, which leads to an upper bound on the gradient of reparameterizable acquisition functions (AFs). This view explains vanishing-gradient behavior in high-dimensional BO and provides a common interpretation of heuristics such as RAASP and dimension-scaled lengthscales. The upper bound can also serve as a generic recipe to understand the behavior of new acquisition functions or surrogates in high input dimensions as well. Using the bounds, we also identify necessary conditions for an input location to be an optimizer of the AF for a univariate Gaussian predictive posterior. Building on this analysis, we propose FITR, a trust-region-based BO method that replaces lengthscale-based scaling with local pullback-Fisher weights. FITR is not restricted to GP kernels with explicit lengthscales and easily generalizes to non-isotropic surrogates. FITR shows competitive performance on several benchmarks.

We consider the formation of multiphase gas in cosmic web objects (CWOs) through cooling and fragmentation within the shock-heated medium of cosmic sheets and filaments, and the circumgalactic medium (CGM) of haloes. This leads to dense, cool ($\sim10^{4}\,\mathrm{K}$), small-scale cloudlets in pressure equilibrium with their surroundings. Such ionized overdensities can affect the propagation of radio waves emitted by fast radio bursts (FRBs), allowing us to probe the small-scale structure in CWOs by examining their impact on the scintillation pattern of the observed FRB flux. While we find that the probability of encountering a high-temperature filament or sheet with cold cloudlets is rather small and usually confined to high redshifts, the expected high rates of FRBs may still permit detection through scintillation along rare sightlines. We suggest scintillation from the Milky Way (MW) interstellar medium as a promising probe: CWO plasma screens can suppress it, placing constraints on their cold-gas properties. Filaments and sheets can suppress MW scintillation from sources at $z_\mathrm{S}\gtrsim2$ and $3$, respectively, while cold and turbulent cloudlets in the CGM can be probed by sources at redshifts as low as $z_\mathrm{S}\sim0.5$, constraining remarkably low volume-filling fractions, $f_\mathrm{v}$ (e.g. $f_\mathrm{v}\gtrsim10^{-5}$ for $z_\mathrm{S}\gtrsim2$). To provide testable predictions, we estimate the fraction of FRBs expected to encounter CWOs that may suppress MW scintillation below a given frequency, and express this as a function of the average extragalactic dispersion measure. Future FRB samples could map these constraints onto bounds on the existence and properties of cloudlets in CWOs, such as their density, size, $f_\mathrm{v}$, and turbulence.

Let $C_n^m$ be the $m$-th power of the cycle on $n$ vertices, and let $\Gnm$ be its complement, where $m\ge1$ and $n\ge3m+1$. We prove that every induced subcomplex of the clique complex of $C_n^m$ is homotopy equivalent either to a disjoint union of contractible complexes or to a circle, and we establish this dichotomy for a larger class of circular interval graphs. As a consequence, we obtain a closed formula for the linear strand of the Betti table of $R/I(\Gnm)$. We also prove that $R/I(\Gnm)$ is a Buchsbaum ring of depth $2$ and that the independence complex of $\Gnm$ is a triangulated $m$-manifold with boundary for $m\ge2$, namely a Möbius band or an annulus when $m=2$. For the complement $\overline H$ of every graph $H$ in the above class, we show that $\reg(I(\overline H)^k)=2k+\indm(\overline H)-1$ for all $k\ge2$, where $\indm$ denotes the induced matching number. In particular, some power of $I(\Gnm)$ has a linear resolution if and only if $n\ge4m+1$. In this range, we further prove that the colon ideals $(I^{k+1}:M)$, where $I=I(\Gnm)$ and $M$ is a minimal monomial generator of $I^k$, have regularity $2$.

Andrews and Newman \cite{AN19, AN20} defined the minimal excludant or mex of a partition. The mex of a partition is the smallest positive integer which is absent from the partition. Baruah, Bhoria, Eyyunni and Maji \cite{BBEM} studied the sum of mex split according to parity, together with its $\ell$-th moments. Ballantine and Merca \cite{BM20} generalized mex to the least $s$-gap, which is the smallest natural number that does not occur minimum $s$ times in the partition. Baruah and Talukdar \cite{BT} conjectured a corresponding asymptotic equivalence between the sums of odd and even least $s$-gaps for every $s>1$, generalizing a theorem of Barman and Singh \cite{BS} for the classical mex. In this article, we derive exact formulas for the $\ell$-th moments of $\smex(\pi)$ for every fixed $\ell\ge 1$, in terms of partition functions. We give a complete proof of the conjecture of Baruah and Talukdar \cite{BT} for every natural number $s>1$. We also generalize an identity of Hopkins, Sellers and Stanton \cite{HSS} to the least $s$-gap setting.

Spatial reasoning with metric constraints requires linking objects to geometric measurements and preserving their numerical content during language reasoning. We present DepthEvidence, a 4B model that uses its own dense metric predictions as object-grounded evidence for language generation. A camera-conditioned decoder predicts full-resolution metric depth using multi-scale visual features and high-resolution RGB refinement. A dense-to-language interface converts predicted depths and decoder features into object-aligned continuous geometry tokens anchored to object identifiers. Geometric supervision encourages metric information to remain recoverable before and after language-context interaction, while instruction tuning supports object measurement and compositional reasoning. We introduce a Depth-VQA benchmark evaluating object-depth queries, relative comparisons, and decisions combining spatial and numerical constraints. Across nine datasets, DepthEvidence achieves the highest average dense $\delta_1$ among evaluated methods, competitive with specialized estimators. It also leads the evaluated methods in instance-level metric depth estimation and overall accuracy on both relative and metric reasoning tracks, while broadly preserving general VQA performance and improving spatial understanding relative to the base model.

We study a class of non-negative spatio-temporal random fields that exhibit self-exciting clustering, which we refer to as affine pure-jump Volterra fields. They are defined via stochastic integration of a Volterra kernel against a thinned Poisson random measure and the thinning is such that the field admits a representation as a stochastic integral of the same kernel, but against a random measure whose compensator has a density that is pointwise affine in the field itself. This representation leads to affine transform formulas characterizing the Laplace transform of functionals of the field up to the solution of a deterministic non-linear Volterra integral equation. Affine pure-jump Volterra random fields extend a non-negative finite-first-moment subclass of pure-jump affine Volterra processes to the random-field setting. The framework includes non-negative finite-variation ambit fields, marked Hawkes systems, and branching-type models as special cases, and we make these connections explicit through examples.

The flatness of the loss landscape at a minimizer is a widely used heuristic for reasoning about neural-network generalization, yet evidence for this relation is mostly empirical and controversial. We study this relation in a teacher-student tree committee machine, where both the ERM estimator and the Hessian spectrum are analytically tractable in the proportional high-dimensional limit. First, we use a zero-temperature Gibbs formulation to obtain predictions for the observables of the typical minimizers of the empirical loss. Secondly, we use Edwards-Jones formalism to derive the limiting Hessian resolvent around these typical minimizers. All predictions agree with finite-size gradient-descent simulations. Finally, we study three measures of flatness, namely the left and right edges and the spectral mean, and check if a decrease in generalization error as the dataset size is increased corresponds to an increase in flatness. We find that the answer strongly depends on the learning task and on the ratio of the number of parameters to the number of data points. In regression, the spectral mean and right edge correlate with the generalization error, while the left edge does so only in the overparametrized regime. In classification this correlation reliably holds only in the highly overparametrized phase, while for underparametrized networks it can even reverse.