Papers

In this paper, we establish global-in-time Strichartz estimates for Klein-Gordon equations on non-trapping asymptotically hyperbolic manifolds. Our result is a positive-mass counterpart of the global Strichartz estimates of Sire, Sogge, Wang, and Zhang for shifted wave equations, and recovers the same range of admissible pairs. The main new feature is the low-frequency analysis, where the positive mass changes the spectral phase and leads to a different dispersive mechanism. Combining the resulting low-frequency estimates with microlocalized high-frequency estimates yields the full global Strichartz estimates. As applications, we obtain small-data global existence and scattering for power-type and certain derivative semilinear equations, as well as small-data global existence for a class of smooth semilinear equations without imposing a null condition.

Object hallucination remains a major obstacle for large vision-language models (LVLMs) to generate reliable content. An intuitive mitigation strategy is to suppress hallucination-related components in hidden representations. However, these components may also contain useful information, and suppressing them can weaken the model's multimodal capabilities. In this paper, we propose ResOT, a training-free method that repairs representations at inference time through localized distribution alignment. Specifically, ResOT projects dominant hallucinated directions away from the faithful subspace, forming a low-dimensional residual subspace for intervention. Within this subspace, ResOT uses Gaussian optimal transport (OT) to align the hallucinated distribution with the faithful one. The resulting map defines repair targets with minimal changes to the original representations. At inference, ResOT adaptively controls how far each token state moves toward its OT target. Experiments on three representative LVLMs show that ResOT substantially reduces object hallucination while improving image caption quality and multimodal performance across multiple benchmarks. Code will be released.

Detecting intra-operative speech impairment during awake craniotomy is essential for preserving language function. However, automated detection remains challenging because operating-room recordings contain substantial acoustic interference, clinically relevant speech events are rare, and available cohorts are small and heterogeneous across speakers. This study presents a systematic component-wise evaluation of a pipeline for distinguishing dysarthric from no-trouble speech in the DATABRASE corpus of awake-craniotomy recordings. The pipeline incorporates speaker diarization to isolate patient speech, a multi-view representation combining handcrafted acoustic descriptors with multilayer wav2vec 2.0 embeddings, speaker-conditional normalization and transferability-based feature selection to improve cross-speaker robustness, and a cascaded classifier comprising a gradient-boosted first stage and a neural second stage. Evaluation was conducted under strict speaker-independent conditions using leave-one-speaker-out cross-validation. The results show that cross-speaker performance is influenced more strongly by the speech representation than by classifier choice. The AUCs of three classifiers differed by no more than 4.7%, whereas replacing conventional acoustic descriptors with the multilayer self-supervised representation produced AUC improvements of 18.2%-26.1%. Diarization-conditioned feature extraction and the proposed classifier cascade provided additional consistent gains. These findings indicate that reliable patient-specific speech isolation and strong pretrained representations are more important than increased classifier complexity in low-resource intra-operative settings. They also quantify the potential performance gains that may be achieved through patient-specific preoperative calibration.

Urban cycling is a sustainable mode of transportation, but cyclists remain highly vulnerable in dense traffic because of limited situational awareness and the lack of active safety assistance on conventional bicycles. This paper presents SmartBike, a low power Internet-of-Things (IoT) platform for cyclist safety, combining blind spot monitoring, rider usage detection, localization, and wireless edge-to-gateway communication. The proposed system consists of a bike-mounted IoT node and a gateway assisted monitoring interface. The node integrates three time-of-flight sensors for blind spot detection, an IMU module for motion sensing and on-board speed estimation, a GNSS module for localization, and a saddle mounted piezo-electric transducer for event-driven wake up. To satisfy the energy constraints of the IoT node, the design employs piezo-triggered activation, application specific sensing rates, and compact connectionless Bluetooth Low Energy (BLE) advertisements for low overhead transmission. The firmware is implemented in Zephyr RTOS using a lightweight state machine architecture, while the gateway performs passive BLE scanning, payload decoding, and real-time dashboard visualization. The BLE payload is reduced to 11 bytes, and power analysis shows that sensing accounts for more than 90% of the total energy budget, while wireless communication represents less overall consumption. With a 12600 mWh battery, the estimated lifetime reaches approximately 21 months under typical commuting conditions (Average 40 min per day).

The rank of the tensor of $3\times 3$ matrix multiplication over the field with two elements is at most $23$ by Laderman's algorithm, and Rudich and Rousseau recently proved that it is at least $22$. We prove that it equals $23$. Hence Laderman's algorithm uses the fewest multiplications among all bilinear algorithms over $\mathbb{F}_2$ and among all bilinear algorithms with integer coefficients. The proof uses the substitution method in the form developed by D'Ambrosio, Wang and Yang et al., which bounds the number of first factors of a decomposition that lie in a given subspace of the space of first factors. We sharpen these bounds for $111$ of Wang's $496$ classes of subspaces, constrain the ranks of the first factors through a $27\times 27$ flattening of the tensor, and list, up to symmetry, all sets of $22$ first factors that satisfy these constraints. An exact completion search shows that none of them is the set of first factors of a decomposition. The computation emits certificates, which are checked in the Lean 4 proof assistant. In one formal proof the largest checks are evaluated as compiled code, and in a second one the kernel evaluates every check, so that its final theorem depends only on the three standard axioms of Lean. The Lean proofs build on the formalization of Rudich and Rousseau and import their table of bounds and their reduction for block algorithms.

We represent (weighted-)selection-elliptical distributions as an affine combination of the $q$ selection variables plus an elliptical term whose direction alone is independent. This form suffices for $q+2$ fund separation via first-order stochastic dominance, inter alia relaxing Simaan's (1993) three-fund assumptions.

We study constrained nonsmooth nonconvex stochastic optimization in federated settings, where clients access only stochastic function evaluations. Existing federated zeroth-order methods primarily combine local zeroth-order updates with model averaging. However, they struggle with client drift induced by local projected updates and sampling variance in stochastic zeroth-order estimates. In this paper, we propose a federated zeroth-order framework based on direction aggregation, equipped with two local sampling schemes. Specifically, FedZOO constructs a local direction from a minibatch shared across multiple spherical queries, thereby controlling the stochastic errors arising from data sampling and gradient approximation. In contrast, FedVRZO forms its local estimator from independent sample–direction pairs, so that its sampling error is controlled directly by the number of sample–direction pairs. In both algorithms, clients compute the mean of the directions evaluated along their projected local trajectories, and the server performs a single projected update using the weighted aggregate of the client directions. Furthermore, we establish convergence guarantees for FedZOO and FedVRZO, respectively. Experiments on black-box adversarial attacks demonstrate the effectiveness of the proposed methods.

We study the approximation by a finite volume scheme of the stationary Gross-Pitaevskii equation with angular momentum rotation. We prove that, under standard mesh regularity assumptions, the discrete global minimizers converge to the continuous one as the mesh diameter tends to zero. Moreover, under a spectral gap assumption on the Hessian of the discrete energy, we prove that solutions to a normalized continuous gradient flow converges exponentially fast to such discrete minimizers. Additionally, we show that this previous discrete spectral gap assumption can be derived from similar hypotheses in the continuous setting. We support our theoretical results with numerical simulations, which broaden the scope of the analysis.

In this paper, we study a family of weighted Gagliardo–Nirenberg interpolation inequalities associated with Grushin–type operators on $\mathbb{R}^n \times \mathbb{R}^m$. Specifically, we examine how to control the weighted norm $\||x|^\beta u \|_{L^B}$ by balancing fractional derivatives $D_x^s u$ with weighted derivatives $|x|^b D_y^r u$. Using two–parameter scaling, spatial translations, and Littlewood–Paley decompositions, we map out exact parameter conditions and demonstrate that the scaling exponents are both necessary and optimal. For multi-parameter weighted extensions, we leverage Khintchine-type random sums to show why $Q \ge 2$ is strictly required. When $x$ and $y$ are radially symmetric, two key advantages emerge: the allowable parameter range for the weights expands, and compact embeddings of the energy space into $L^p$, $2 < p < B$, are completely restored. We establish this compactness through two complementary tools, geometric covering arguments and a generalized three–norm version of Strauss's lemma, that yields explicit pointwise weighted decay estimates. Finally, in the scale–invariant case $\beta=0$, a concentration–compactness analysis proves that the inequality's sharp constant is achieved by an optimal function, with minimizing sequences remaining precompact up to the natural three-parameter group of symmetries.

While massive, closed-source Vision-Language Models (VLMs) set strong benchmarks for document understanding, their dependence on commercial APIs limits adoption in institutional archives due to data autonomy concerns, recurring costs, and the environmental footprint of hyperscale computing. This is especially acute in heritage digitization, where documents include historical handwriting, domain-specific terminology (e.g., jewelry, prehistory, architecture), and non-standard layouts requiring high-dimensional structured extraction. We present a comparative study of eight open-source lightweight VLMs (up to 7B parameters) for Optical Character Recognition (OCR)-to-structure across three university heritage collections. Given a document image, models must extract text and generate schema-compliant JSON, enabling automatic validation and downstream use. We evaluate models under a constraint-aware protocol across zero-shot, few-shot, and fine-tuning settings, measuring extraction fidelity and structured-output quality using Character Error Rate (CER), Approximate Normalized Levenshtein Similarity (ANLS*), and mean Average Precision F1 (mAP-F1). Against a fine-tuning baseline, we further test the independent impact of (i) hyperparameter optimization, (ii) classical image preprocessing (illumination flattening, denoising, and CLAHE), and (iii) multi-stage training. Finally, we analyze the trade-off between dataset-specific fine-tuning and a single multi-dataset checkpoint, where joint training enables one model to operate across collections but can shift performance between datasets. Overall, we show that carefully adapted VLMs with up to 7B parameters can provide a sustainable, private, high-performing alternative to manual transcription or commercial black-box systems, and we offer actionable guidance for heritage institutions seeking institution-controlled OCR-to-JSON extraction.

Let $f:\mathbb{P}^N\to\mathbb{P}^N$ be a postcritically finite endomorphism of degree at least two. Combining Yamanoi's hyperbolicity results with the dynamical property of $f$, we prove that the complement of its postcritical divisor is taut unless $f$ is a monomial power map or admits a nontrivial equivariant rational fibration with a polarized divisorially postcritically finite base map. We establish analogous alternatives for polarized PCF endomorphisms of rationally connected projective manifolds and, for Kobayashi hyperbolicity, on normal rationally connected projective varieties. In dimension two, the non-taut case can be described, up to iteration and rational semiconjugacy, by monomial power maps and skew products with monomial fiber maps.

Dense retrievers have made significant progress on text and image corpora, but whether these capabilities extend reliably to mixed corpora containing text, image, and fused text-image documents remains unclear. In this paper, we systematically examine retrievers across architectures and find that their performance is highly sensitive to modality composition. As image documents are progressively replaced with semantically corresponding text representations, retrieval performance follows a pronounced V-shaped curve, remaining strong on single-modality corpora but degrading substantially when modalities coexist. In particular, irrelevant text causes more severe degradation than an equal number of irrelevant images, a phenomenon we term Chaos in the Text. Further analysis reveals modality preference, whereby text representations receive systematically higher similarity scores, allowing irrelevant text to outrank relevant images. To mitigate this bias, we introduce Trident, which constructs text, image, and fused text-image views of each document as co-equal positives and jointly optimizes relevance discrimination and positive-view balance through Multi-Positive View InfoNCE. Experiments across visual document and natural image benchmarks show that trident improves mixed-modality retrieval on both CLIP-based and VLM-based architectures, reduces sensitivity to modality composition and text distractors, and increases average single-modality retrieval performance.

Tracking a novel object's 6D pose over long horizons currently requires either expensive onboarding or a reconstruction maintained throughout the sequence. This makes current trackers impractical for robotic manipulation and augmented reality, which need trackers that are ready to use and run in real time. We show that a lightweight tracking module can be applied on top of a wide range of correspondence estimation methods to keep drifts bounded while maintaining fast runtime. Our key idea is to avoid point-based optimization in the pose graph and operate only on relative pose constraints, which we weight by a derived uncertainty from the geometric alignment. This makes optimization independent of the number of correspondences while avoiding the direct inclusion of noisy point measurements, leading to fast and robust long-term tracking. Across four real-world benchmarks, our approach achieves tracking accuracy comparable to reconstruction-based trackers with a fraction of the optimization cost. Overall, these results suggest that a compact and reliable pose graph optimization can provide long-horizon consistency at substantially lower computational cost.

We study the relation between equal-weight circle quadrature, Gauss–Chebyshev quadrature, and planar Minkowski reconstruction. The regular $N$-point grid is an $\mathbb S^1$-design of strength $N-1$; a half-step rotation of the $2M$-point grid projects to the classical $M$-point Gauss–Chebyshev rule. A common family of quadrature-error functionals gives exact identities for integration error, mixed-area error, and support-function error. For $N\ge3$, the centered Minkowski polygon $P_N$ with perimeter $2\pi$ has the exact Hausdorff distance from the unit disk $B^2$ \[ d_H(P_N,B^2)=1-\frac{\pi}{N}\cot\frac{\pi}{N} =\frac{\pi^2}{3N^2}+O(N^{-4}). \] With Steiner centering, we obtain $d_H(K_\mu,K_\nu)\le\pi W_1(\mu,\nu)$ for balanced, nondegenerate probability measures, where $W_1$ uses geodesic distance. Its exponent is sharp even for smooth positive densities near the uniform measure. The identity $W_1(\nu_N,\sigma)=\pi/(2N)$ yields a general $O(N^{-1})$ bound for the regular grid. The sharper $O(N^{-2})$ rate follows from the sparse discrepancy spectrum and the inverse multiplier $(1-k^2)^{-1}$. Finally, a perturbation estimate gives sufficient conditions for preserving the quadratic rate and its leading constant.

We present an X-ray analysis of the supernova remnant (SNR) MSH 15-52 associated with the young rotation-powered pulsar PSR B1509-58, using Chandra, XMM-Newton, and eROSITA, complemented by MeerKAT radio data. We report the discovery of the missing southern counterpart to the bright northern RCW 89 region and identify a foreground X-ray source that may be a previously unidentified SNR candidate. Imaging and spectroscopy with XMM-Newton and eROSITA reveal the shell properties of MSH 15-52 and suggest that both the southern and northern shell-like structures are consistent with shocked interstellar medium associated with the forward shock, while their low ionization timescales and possible hard power-law components indicate possible influence from particle injection or additional heating by the pulsar wind nebula (PWN) jets. New Chandra spectral maps of the southern jet reveal a region of hard non-thermal emission, with a photon index of approximately 1.6, suggesting interaction between the PWN and the reverse shock and possible re-acceleration of the pulsar outflow. Spatially resolved spectroscopy of the RCW 89 northern clumps reveals high-velocity, metal-rich ejecta with enhanced Ne and Mg abundances. Comparison with core-collapse nucleosynthesis models favors a massive progenitor with an initial mass of approximately 29 solar masses in one-dimensional yield models, while asymmetric explosion models show that lower-mass progenitors can also reproduce the observed abundance ratios. Our study supports a picture in which RCW 89 is part of a larger asymmetric SNR shell interacting with energetic pulsar jets, making the system a valuable laboratory for probing pulsar-PWN-SNR interaction.

We study deterministic relative approximation of the total variation distance between two Gibbs distributions induced by spin systems on the same bounded-degree graph. For the hard-core model, we give a deterministic polynomial-time $\varepsilon$-relative-error approximation when both external-field vectors lie in $[b,(1-\eta)\lambda_{\mathrm c}(\Delta)]^V$, where $b>0$ and $\eta\in(0,1)$ are fixed and $\lambda_{\mathrm c}(\Delta)$ is the hard-core uniqueness threshold. For the Ising model, we obtain deterministic polynomial-time algorithms in two settings: the ferromagnetic Lee–Yang regime and the antiferromagnetic correlation-decay regime. We develop a new deterministic framework that reduces this task to estimating suitably chosen partition functions.

A lazy cellular automaton $\tau:A^G\to A^G$ is determined by a finite neighborhood $S\subseteq G$ containing the group identity $e$, a pattern $p\in A^S$, and a writing symbol $a\in A\setminus\{p(e)\}$: its local rule is equal to the projection to $e$ except when it reads the pattern $p$, in which case it writes $a$. For a nonempty subset $T$ of the fiber $S_a:= p^{-1}(a)$, we introduce compatibility conditions on $p$ along the right translates $St$, $t\in T$. We show that, over every group, the existence of a compatible set is necessary for $\tau$ to be non-idempotent, and that it is also sufficient whenever $st = ts$ for all $t \in S_a$, $s \in S_{p(e)}$. In particular, compatibility characterizes non-idempotence over abelian groups, which reduces deciding idempotence to a search over subsets of $S_a$. Finally, for every dihedral group $D_n$ with $n\geq 4$, we construct an idempotent lazy cellular automaton that admits a compatible set, showing that the commutation hypothesis cannot simply be dropped.

Existing approaches to multi-view spatial reasoning operate largely on sparse input views. Vision-language models (VLMs) are thus restricted to understand a scene and infer spatial relations within these fixed views, leading to fragile cross-view alignment and geometry-to-language bottleneck. To address these issues, we formulate a novel Seek-and-View reasoning approach to find implicit cross-view spatial evidence by locating a question-relevant view to support the spatial reasoning. To realize this approach, we propose Vantage, a training-free model-agnostic reasoning framework that pairs a VLM with a 3D foundation model: a viewpoint-grounded reasoning stage for question analysis and view planning, followed by a geometry-grounded evidence augmentation stage to effectively synthesize and incorporate visual evidence into the final reasoning. Comprehensive experiments on six VLMs demonstrate consistent improvements on five benchmarks without fine-tuning. Overall, by revealing spatial evidence through view-grounded reasoning, Vantage can largely reduce reliance on language-based cross-view alignment and improve multi-view spatial understanding. Our code is available at https://github.com/q1xiangchen/Vantage.

Robust navigation in cluttered environments remains a fundamental challenge for quadrotors, particularly when strong wind disturbances arise, which perturb vehicle dynamics, limit control authority, and substantially increase collision risk. Existing learning-based navigation policies typically rely on obstacle perception and proprioceptive observations, requiring the policy to infer time-varying disturbance effects implicitly and thereby limiting robustness under partial observability. This paper proposes WAND (Wind-Aware Navigation with Disturbance Estimation), a reinforcement learning framework for navigation under time-varying wind disturbances in dense obstacle fields. Specifically, WAND estimates wind-induced disturbance acceleration from historical proprioceptive states using a Temporal Convolutional Network (TCN). This estimation is integrated into the policy via a zero-initialized residual module, WindAdapter, while simultaneously providing feedforward compensation for low-level control. The dual use of the estimate couples disturbance-conditioned navigation with feedforward disturbance rejection. Across 12 wind-disturbed simulation settings, WAND improved the observed success rate by 8.3 percentage points on average relative to feedforward compensation alone. Controlled opposite-crosswind experiments further showed wind-direction-dependent trajectory adaptation. In indoor fan-induced flight tests, WAND succeeded in 18 of 20 trials, demonstrating the feasibility of real-time onboard navigation.

We consider long-range percolation (LRP) on $\mathbb{Z}^d, d \ge 1$ with connection probabilities $\mathbf{p}(x, y) \approx \frac{\beta}{\|x-y\|^s}$ for $\beta>0$ and $s > \min(2d, d+2)$. Our main result is that the mixing time of the random walk on the giant component of supercritical LRP inside a box of side length $n$ is of order $n^2$. To establish the upper bound, we show that a $d$-dimensional isoperimetric inequality holds almost surely simultaneously for all connected sets inside the giant component that are large enough.

Altermagnetism, characterized by compensated collinear magnetic moments and even-parity momentum-dependent spin splitting, has emerged as a distinct form of magnetic order. However, realizing odd- and mixed-parity magnetism in fully compensated collinear magnets remains challenging. Here, we demonstrate light-induced odd- and mixed-parity magnetism in conventional collinear antiferromagnets. Combining Floquet theory with first-principles calculations, we show that circularly polarized light (CPL) induces odd-parity $f$-wave spin splitting in monolayer CoS and CoSe, whereas elliptically polarized light (EPL) induces $p$-wave spin splitting. Uniaxial strain combined with CPL offers an alternative route to the $p$-wave state. Furthermore, combining an electric field with CPL or EPL generates mixed-parity spin splitting. Light-induced breaking of $\mathcal{PT}$ symmetry, together with spin-orbit coupling, gives rise to a finite anomalous Hall response. Our findings establish a versatile approach to engineering spin splitting and controlling electronic and magnetic properties in compensated antiferromagnets.

We introduce a new and tractable class of non-Gaussian isotropic random fields on the sphere, which we call, for brevity, Grey Spherical Random Fields (GSRFs). The proposed framework provides a flexible extension of the classical Gaussian theory, while preserving strong isotropy and embedding Gaussian random fields as a limiting special case. The construction is based on random spherical harmonic coefficients obtained as Gaussian mixtures driven by multivariate subordinators evaluated at independent random times (with Mittag-Leffler or more general distribution). This representation allows us to derive explicit characteristic functions, mixed moments and cumulants of arbitrary order. We also investigate asymptotic high-frequency behaviour and identify general conditions under which suitably normalized spherical harmonic coefficients converge to Gaussian scale mixtures, recovering several important models, including stable, Gamma and inverse Gaussian cases, as particular examples. Proving that GSRFs emerge as scaling limits of multivariate continuous-time random walks, with heavy-tailed waiting times, provides a microscopic foundation for the proposed models. We further obtain quantitative bounds measuring the deviation from Gaussianity of the spherical harmonic coefficients. The resulting framework substantially broadens the class of analytically tractable isotropic random fields on the sphere, while preserving explicit probabilistic representations and flexible dependence structures.

This work studies the multidimensional asymptotic agreement problem in dynamic networks. We assume that nodes have $d$-dimensional inputs and need to converge arbitrarily close to each other's inputs while their vectors stay inside the convex hull of all inputs. We present a new upper bound of $\sqrt{2d/(3d+1)}$ on the contraction rate of asymptotic agreement in the non-split network model, where $d$ is the dimension of the input and $\sqrt{2d/(3d+1)}\rightarrow \sqrt{2/3}\approx 0.816$ for $d\rightarrow\infty$. To this end, we adapt the BallMidpoint algorithm (Melnyk, 2026) recently introduced for the all-to-all communication model. This algorithm lets the nodes choose the midpoint of the smallest enclosing ball of the received vectors. This bound is strictly worse than the $\sqrt{1/2}\approx 0.707$ contraction rate for fault-tolerant approximate agreement in all-to-all communication networks because the convex hulls of the nodes do not have a common intersection, and only intersect pairwise. Our bound improves over the previously best-known contraction rate of $\sqrt{7/8}\approx 0.935$ for dynamic networks via the MidExtremes algorithm (Függer and Nowak, 2018). We present the first multi-dimensional lower bound of $2/3$ for the contraction of coordinate-free memoryless anonymous deterministic algorithms in non-split networks, for $d\ge 8$. This result shows that the previously best-known lower bound of $1/2$ on the contraction rate for $n\ge 3$ and $d=1$ is not tight in higher dimensions. If $n$ is infinite, we extend this lower bound to $\sqrt{1/2\cdot d/(d+1)}$, which converges to $\sqrt{1/2}$ for $d\rightarrow \infty$. We further show that for a single round, the one-round contraction is at least $\sqrt{\frac{2d-2}{3d+7}}$, which converges to $\sqrt{2/3}$ for $d\rightarrow\infty$, showing that the BallMidpoint strategy is asymptotically optimal for one round.

In this paper, for a finite group $G$ and a positive integer $k$, we introduce the function $$\psi''_k(G)=\dfrac{\psi_k(G)}{|G|^{k+1}}, \quad \text{where} \quad \psi_k(G)=\sum_{g \in G}o(g)^k,$$ as the $k$-th generalization of the function $\psi''$, introduced by M. Tărnăuceanu. Different thresholds for cyclic, abelian, nilpotent and (super)solvable groups in terms of $k$ are obtained. It is shown that these thresholds often outperform similar thresholds derived from other parameters like $\psi'$ and $\psi''$. Moreover, considering $\psi''_k$ as a $(0,1]$-valued function, we study its injectivity, surjectivity and density.

Self-improving learning systems adapt data selection, optimization, and auxiliary symbolic components, inducing nonstationary objectives outside standard learning assumptions. We introduce PRAXIS, a co-evolutionary framework that models generators, learners, and symbolic archives as interacting dynamical processes. We prove that KL-constrained generator updates and controlled archive-weight movement bound one-step objective drift, that archive updates suppress a program relative to any fixed comparator with a persistent cumulative utility advantage under sub-Gaussian noise, and that stochastic gradient descent achieves an average-stationarity guarantee whose degradation is governed by cumulative objective drift. Experiments across visual robustness, relational graph reasoning, and algorithmic graph reasoning exhibit generator stabilization, decreasing learner loss, and archive concentration consistent with these theoretical mechanisms.

We show that integers act invertibly on a derived scheme $S$ if and only if they act invertibly on the reduced motivic suspension spectrum of the affine line in $\mathrm{MS}_S$. As a consequence, over regular locally noetherian schemes and derived schemes on which some nonzero integer acts as zero, the motivic sphere spectrum is $\mathbf{A}^1$-invariant after inverting primes that are not invertible in the base scheme. In particular, over regular locally noetherian $\mathbf{Q}$-schemes, the motivic sphere spectrum is $\mathbf{A}^1$-invariant.

In real-world software development, the primary challenge in vulnerability detection is often not finding suspicious code, but identifying which alerts among the large number of candidate alerts produced by static analysis truly warrant attention. Existing learning-based methods mainly identify suspicious patterns at the function or line level, making it difficult to extract complete program evidence centered on an individual alert. Although large language models can infer risk sources, dangerous operations, protection conditions, and state preconditions from local program facts, such semantic information cannot be reliably aligned with specific program nodes, dependency relations, and propagation paths, and is therefore insufficient to verify whether the corresponding safety obligations truly affect the current alert. To address this problem, we propose ObliVul, an alert-conditioned safety obligation modeling and bidirectional counterfactual validation framework for vulnerability detection. For each candidate alert, ObliVul first extracts a Local Evidence Pack (LEP) from the Code Property Graph (CPG) and uses a large language model to recover candidate safety obligations. It then aligns the safety obligations with program nodes, dependency edges, and path scopes to construct a Local Safety Obligation Graph (LSOG). Finally, VAFI aggregates complementary verified alert evidence while suppressing redundant or weaker evidence to produce a function-level vulnerability prediction. Experimental results show that ObliVul effectively distinguishes vulnerable versions from fixed versions and reduces persistent false positives on fixed code. Ablation studies further confirm the necessity of each component for safety obligation recovery, risk response validation, and function-level vulnerability inference.

For a complex simple Lie algebra $\mathfrak{g}$ with compact conjugation $\tau$, Killing form $B$, and Hermitian form $H(x, y) = -B(x, \tau y)$, it is shown that $H([x,y],[x,y]) \leq (h^{\vee})^{-1}H(x,x)H(y,y)$, where $h^{\vee}$ is the dual Coxeter number, with equality exactly for $H$-orthogonal pairs in $\tau$-stable long-root subalgebras isomorphic to $\mathfrak{sl}(2, \mathbb{C})$. Equivalently, the comass of the Cartan $3$-form with respect to $H$ equals its comass on the compact real form: the spectral norm of the Cartan $3$-form does not increase under complexification. For the Frobenius norm on the image of a representation of Dynkin index $\ell$ the optimal constant is $2/\ell$; this contains the Böttcher–Wenzel and Bloch–Iserles inequalities, extends the latter to complex skew-symmetric matrices, and improves them substantially for the exceptional Lie algebras of types $F_{4}$, $E_{6}$, $E_{7}$, and $E_{8}$. The proof uses the Nahm algebra of $\mathfrak{g}$ with the conjugation induced by $\tau$: critical points of the relevant function correspond to $\tau$-twisted idempotents, and its Hessian is governed by $\tau$-twisted multiplication operators, whose anticommutation with multiplication by $i$ turns the one-sided second order condition at a maximum into a two-sided spectral bound. Subalgebras isomorphic to $\mathfrak{sl}(2, \mathbb{C})$ of Dynkin index $j$ give critical points with critical value $1/(jh^{\vee})$, whose Morse indices are computed, and the set of maximum points is a nondegenerate critical manifold. Geometrically, the maximal complex sectional curvature of a compact simple Lie group equals its maximal sectional curvature.

Flare activity in red giants is quite common when continuously monitored with high-precision space photometry. In fact, the most powerful flares, also known as "superflares" have been observed on red giant stars. KIC 2852961 is a red giant star producing the highest-energy flares in the Kepler field, and is also known for its large-amplitude starspot-induced light variations. Our goal is to find out what causes the extreme flare activity and flare energies in this star. For our investigations, we use ground-based spectroscopic measurements and space photometric data, which we process using standard tools. We also collect relevant data from databases of large surveys such as Gaia. Our new radial velocity data confirm that the red giant star is part of a close binary system, in which tidal forces play a crucial role in controlling the evolution of the binary system. Our results show that tidal evolution has driven the system close to orbital circularization and rotational synchronization, but it did not completely constrain the differential rotation of the red giant, in which the magnetic dynamo can operate extremely efficiently. We find that high-energy (super)flares are primarily controlled by the internal magnetic activity of the active giant component, rather than phase-dependent interactions with the secondary star. The large spot-induced rotational light modulations indicate extensive starspots that can store enormous amounts of magnetic energy. This provides the conditions for the formation of extremely large flares, confirming the close relationship between large spot coverage and high flare energies and/or frequency. Behind all this, differential rotation may play a dual role in the flare activity of KIC 2852961: it contributes both to the formation of large starspots and to the shearing and twisting of coronal magnetic fields that can power energetic superflares.

Softmax attention, at the heart of Transformers, has demonstrated remarkable capabilities. Yet its underlying mechanisms remain only partially understood. Recent theoretical work studies Gaussian prompts, where the infinite-prompt limit reduces softmax attention to a linear map, but also removes the query-dependent selection that distinguishes it from linear attention. This work studies the infinite-prompt limit of softmax attention on Gaussian mixtures, which retain the tractability of Gaussian data while introducing latent structure, multimodality, and nonlinear dependencies. We show that softmax attention can represent and learn, via gradient-based methods, optimal solutions to a range of statistical tasks, including supervised classification and denoising. Our results highlight two complementary capabilities of softmax attention: it can recover linear tasks as effectively as its simpler linear counterpart, while also exploiting query-dependent context selection to solve nonlinear tasks beyond the reach of linear attention.