Papers

The classical semiparametric convolution theorem characterizes the limiting distributions of regular estimators of pathwise differentiable parameters. We extend this analysis to directionally pathwise differentiable functionals. Regularity along a subspace or convex cone forces the directional derivative to agree there with a bounded linear functional, and every such estimator has a Gaussian convolution factor determined by that functional. We characterize the relevant subspaces, compare their efficiency bounds, and identify the influence functions of efficient estimators. These comparisons show when a stronger regularity requirement increases the variance bound. Under an additivity condition, the Gaussian limit experiment separates into a linear estimation problem and a nonlinear one. Their minimax risks add under squared error; for more general losses, the Gaussian component enters through convolution of the loss. Applications to treatment values, instrumental-variable bounds, and calibration give explicit efficient influence functions, compare existing procedures, and establish feasible attainment in finite-stratum models.

We disprove Laugesen's conjecture that the ratio of the first two Robin eigenvalues decreases with the positive boundary parameter. On any prescribed compact interval of positive parameters, the ratio is strictly increasing on a suitable connected polyhedral domain in every dimension at least two; in the plane, the domain can be a simply connected polygon. In contrast, we prove strict decrease on every rectangular box. Both results follow from a comparison of the relative growth of interval eigenvalues.

A tree poset $T$ is a poset whose Hasse diagram is a tree. Bukh proved that if a family $\mathcal F\subseteq 2^{[n]}$ contains $(h(T)-1+\varepsilon)\binom{n}{\lfloor \frac{n}{2}\rfloor}$ sets, then $\mathcal F$ contains a weak copy of $T$, where $h(T)$ is the height of $T$, the number of elements in a longest chain of $T$. Several strengthenings and generalizations of this result have been obtained. We prove the following robust variant. For a tree poset $T$ and $x\in T$, the $b$-blow-up $T(x,b)$ rooted at $x$ is the tree poset that we obtain from $T$ by replacing every element $u$ by $b^d$ new elements, where $d$ is the distance $d(x,u)$ in the Hasse diagram of $T$ and an edge $uv$ with $v$ being closer to $x$ is replaced by edges such that every new copy of $v$ is joined to $b$ new copies of $u$ such that these new copies form pairwise disjoint sets for the copies of $v$. We prove that for any tree poset $T$, $x\in T$, and $\varepsilon>0$ there exists $\delta$ such that if $\mathcal F\subseteq 2^{[n]}$ contains $(h(T)-1+\varepsilon)\binom{n}{\lfloor \frac{n}{2}\rfloor}$ sets, then $\mathcal F$ contains a weak copy of $T(x,\lfloor \delta n\rfloor)$. This settles a conjecture of Treglown and the author. As applications, we derive the known asymptotic counting and random versions of Bukh's theorem from this stronger embedding result, and obtain new maximal anti-Ramsey results for tree posets.

In many soft-matter and biological systems, task completion relies on the cumulative arrival of multiple searchers rather than the speed of a single pioneer. The completion kinetics are therefore set not only by the first arrival, but by the full ordered sequence of first-passage times. Here, we determine how stochastic resetting optimizes these ordered arrivals for all arrival ranks. We construct an exact finite-$N$ reference for non-interacting Brownian searchers and obtain the mean ordered first-passage time $\langle T_{(k)} \rangle$ and its optimal resetting rate $r_k^*$. For searchers with identical initial conditions, $r_k^*$ increases monotonically with arrival rank and, with increasing population size, approaches the known large-$N$ quantile limit where a finite optimum appears only above a critical rank fraction $\phi_c \simeq 0.412$. Spatial heterogeneity qualitatively reorganizes this sequence, shifting its maximum from late to early ranks even without particle interactions. We then compare this baseline with Brownian colloid experiments, interacting active Brownian particles, and a collective autochemotactic search with persistent environmental memory. Across these systems, sensitivity to resetting increases strongly with arrival rank, while deviations from appropriate non-interacting references reveal the influence of direct interactions, finite return overhead, and environmental memory. Our results show that optimal resetting in multiparticle search is governed by the required completion rank and must be evaluated relative to protocol- and geometry-matched baselines.

Multimodal 3D object detection is fundamental to robust perception in autonomous driving because it integrates complementary information from LiDAR and camera sensors. However, existing methods often fail to maintain robustness under out-of-distribution (OOD) corruptions caused by sensor noise, adverse weather, and environmental changes. To address this problem, we propose RoboDistill, a robust and generalizable multimodal 3D object detection framework that leverages visual foundation models (VFMs), such as the Segment Anything Model (SAM). First, we introduce SAM-AD, a domain-specific pretraining strategy that fine-tunes SAM on autonomous-driving imagery to extract feature representations with rich semantic information. Second, we design the AD Feature Pyramid Network (AD-FPN) to refine and upsample SAM features at multiple scales for seamless fusion with LiDAR features. Third, we develop the Depth-Guided Wavelet Attention (DGWA) module, which suppresses high-frequency sensor noise while preserving critical contextual information. Finally, we introduce KD Fusion, in which the pretrained SAM-AD serves as a teacher that distills high-quality visual knowledge into a lightweight point-cloud network, thereby improving robustness under noisy conditions. Extensive experiments across 27 challenging OOD corruption settings show that RoboDistill generally delivers stronger or competitive detection performance and robustness relative to representative state-of-the-art methods. This work bridges the gap between VFMs and 3D object detection and advances robust multimodal perception for real-world autonomous-driving applications.

The Komlós problem asks for signs $\varepsilon\in\{-1,1\}^n$ with $\|A\varepsilon\|_\infty$ bounded by an absolute constant whenever the columns of $A\in\mathbb R^{m\times n}$ have Euclidean norm at most $1$. We give the first polynomial-time algorithm for this problem in the unit-cost real-RAM model. Our deterministic algorithm achieves discrepancy at most $8272$. As a consequence, we obtain a deterministic $O(\sqrt t)$-discrepancy algorithm for the Beck-Fiala problem on set systems in which each element belongs to at most $t$ sets. Our algorithm rounds a fractional coloring while keeping the row sums bounded. Its key idea is to keep the number of nearly tight row constraints small relative to the number of active coordinates. We achieve this through a spectral bound based on a weighted Gram matrix of squared row entries. We maintain the bound using a cubic spectral potential and negative-curvature updates that satisfy the required linear constraints.

Agents may need capabilities exposed through external agent endpoints or service APIs. When a requester is not already bound to a provider, it must discover advertised capabilities matching its task and interface requirements. Over exact-key distributed hash tables (DHTs), broad retrieval transfers large candidate lists, whereas selective retrieval may miss relevant providers or require more replication and lookups. Open publication also lets providers inflate their exposure unless publication bounds are enforceable. We present SemDHT, a certified semantic index for discovering agent-accessible capabilities over exact-key DHTs. A two-layer semantic sketch uses coarse cells to group nearby descriptors and residual codes to narrow candidate selection. Providers publish at a bounded set of derived keys, while requesters probe precision keys before broader recall keys within a lookup budget. Anchor committees certify each descriptor's publication-key set, enabling storage services and requesters to enforce descriptor-to-key consistency. On real API descriptors and task queries, SemDHT achieves recall@10 of 0.955 against exact embedding-space neighbors and 0.947 against ToolBench relevance labels. On a corpus with controlled density augmentation, it matches the candidate exposure of tuned locality-sensitive hashing (LSH) over a DHT at recall 0.95 with 7.7x fewer lookups and reduces publication fan-out from 16 to 10. A Go/libp2p prototype deployed on same-region and cross-region 200-peer cloud overlays replays 299 Internet queries. With parallel probes and cold certificate caches, SemDHT achieves mean completion-time speedups of 3.64x and 4.11x over LSH, respectively.

Let $\pi:\X\to\D$ be a proper holomorphic submersion with connected fibers, and suppose that $X_t$ is biholomorphic to a fixed smooth Moishezon manifold $S$ for every $t\ne0$. If $\kappa(S)\ge0$, we prove that $X_0\simeq S$.

Let $I_{\inf}(a):=\lim_{n\to\infty}I_{n}(a)=\inf_{n\ge1}I_{n}(a)$ be the asymptotic isoperimetric profile for product of a weighted Riemannian manifold satisfying $\mathrm{CD}(0,\infty)$ and, more broadly, for product of a nonsmooth subspace with density that can be approximated by a sequence of densities satisfying $\mathrm{CD}(0,\infty)$. We establish the following identity: \[ \lim_{a\downarrow0}\frac{I_{\inf}(a)}{a\sqrt{2\ln(1/a)}}=\sqrt{K_{\mathrm{LS}}}, \] where $K_{\mathrm{LS}}$ is the optimal log-Sobolev constant. The same argument also yields the large-deviation and moderate-deviation asymptotics for isoperimetry.

CUDA Graphs eliminate launch overheads by replaying tensor operations over static virtual addresses. However, FP8 pipeline training continuously alters the scaling states, microbatches, and deferred backward tasks that those fixed addresses represent. Split-backward schedules (e.g., 1F1B, Zero-Bubble) decouple input-gradient ($dI$) and weight-gradient ($dW$) computations to minimize bubbles, breaking traditional LIFO lifecycles. Standard dataflow graphs cannot inform the runtime of hidden numerical updates, non-LIFO work ownership, or cache validity, leading to silent cross-stream data corruption. We present QEffect, an explicit state and resource contract runtime for low-precision pipeline training. QEffect formalizes four foundational invariants: (1) temporal serialization of hidden scaling updates, (2) generational ownership of retained backward resources, (3) validity versioning for cached weights across optimizer boundaries, and (4) bidirectional caller-graph stream completion synchronization. These invariants uniformly govern eager and captured execution, allowing ephemeral graph resources to be cleanly rebuilt across process restarts. Leveraging work-ownership semantics, we also introduce an affine direct-gradient placement mechanism that eliminates redundant memory copies. Integrated with TorchTitan and NVIDIA Transformer Engine, QEffect maintains strict bitwise parity with native baselines across delayed-scaling rollovers, deterministically traps cross-stream ordering violations, and enables flawless cold-start resumption. On NVIDIA H800 GPUs, captured Transformer layers achieve 1.82–2.79x speedup over eager execution, while direct gradient placement delivers an additional 1.132x gain by eliminating 96 matrix copies per rank-step.

Causal discovery from observational data is a fundamental yet challenging task in scientific research. While existing approaches are primarily based on conditional independence tests, structure scores, or restrictive functional assumptions, we propose Decoupled Causal Discovery (DCD), a novel decoupling-based perspective that does not rely on these methodologies. DCD directly identifies the Markov boundary (MB) by decoupling non-target variables via weighting functions, such that only variables within the MB preserve dependence with the target under the decoupled distribution. Building on this, DCD iteratively constructs the Completed Partially Directed Acyclic Graph (CPDAG) by exploiting structural asymmetries within the MBs. We establish the theoretical identifiability, soundness, and completeness of DCD. Empirical evaluations demonstrate that DCD achieves strong performance, particularly excelling in challenging noise regimes.

Language-based 3D localization retrieves the point-cloud submap containing a target position from descriptions of nearby objects and their spatial relations. Existing methods typically compress queries and submaps into global descriptors, potentially obscuring object-level semantics and cross-description spatial coherence. We propose Position-Conditioned Evidence Localization (PosEviLoc), a query-position-aware framework for coarse text-to-point-cloud localization. Instead of relying on global matching, PosEviLoc evaluates each candidate submap using explicit semantic and spatial evidence. It models direction as a relation jointly determined by an object position and a hypothetical query position. The resulting Query-Position Spatial Evidence Field (QSEF) measures the fraction of query descriptions supported at each hypothetical position, explicitly capturing their agreement without using the ground-truth query pose to construct the evidence field. A Multi-Level Evidence Readout (MER) summarizes this evidence in a compact representation, which a lightweight MLP converts into a retrieval score. Across five benchmarks, PosEviLoc outperforms MNCL by an average of 17 percentage points in Recall@1. When used as a plug-and-play reranker, it improves MNCL by an average of 16 percentage points. Moreover, PosEviLoc introduces substantially fewer parameters and achieves faster inference speed than existing methods.

Multimodal classifiers can converge to modality-dominant solutions in which one modality dominates the joint prediction, suppressing the learning of others. Existing balancing methods mainly adjust losses, gradients, or modality contributions, largely treating modality imbalance as an optimization problem while implicitly treating the weak modality as under-optimized but representationally intact. In this work, we find that this assumption does not always hold, as persistent modality dominance can induce a representation-level collapse of the weak modality, which we term manifold modality collapse (MMC). MMC manifests as a coupled geometric degradation in which weak-modality representations collapse onto fewer directions within each class and become less separable across classes. Motivated by this observation, we propose GeoBalance, a geometry-aware framework that monitors these two geometric properties and reconstructs the weak modality representation only when it exhibits signs of MMC. Once triggered, GeoBalance uses a fixed Simplex-ETF class scaffold and spectral regularization to restore class separation while preventing collapse onto a few feature directions. To preserve reconstruction during joint training, asymmetric gradient projection removes the joint-gradient component conflicting with reconstruction, leaving non-conflicting optimization unchanged. Extensive experiments across six multimodal benchmarks demonstrate great improvements over competitive balancing methods, validating its effectiveness.

Let $\mathbf{G}=\{G_1,\dots,G_{n}\}$ be a collection of not necessarily distinct $n$-vertex graphs with a common vertex set $V$. A cycle $C$ with $V(C)\subseteq V$ and $|E(C)|\leq n$ is called rainbow in $\mathbf{G}$, if there exists an injection $\phi\colon E(C)\to [n]$ such that $e\in E(G_{\phi(e)})$ for each $e\in E(C)$. The graph collection $\mathbf{G}$ is said to be rainbow pancyclic if it contains a rainbow cycle of every length from 3 to $n$. In this paper, we show that if $e(G_i)\ge \binom{n-1}{2}+1$ for each $i\in[n]$ with $n\ge 3$, then $\mathbf{G}$ is rainbow pancyclic, apart from three explicitly described exceptional graph collections. This answers Problem $1$ of [Discrete Math., 348(2025), 114600] and strengthens the result from rainbow Hamiltonicity to rainbow pancyclicity. As a consequence, we obtain that if $\rho(G_i)>n-2$ for each $i\in[n]$, then $\mathbf{G}$ is rainbow pancyclic unless $G_1=G_2=\dots=G_n\cong K_1\vee(K_{n-2}\cup K_1)$, which improves Theorem $5$ of [Discrete Math., 348(2025), 114600]. We also characterize all graph collections that are not rainbow pancyclic under the condition $\rho(G_i)\ge n-2$ for each $i\in[n]$.

We study the following curvature equation with two singular sources on a flat torus $$ \Delta u+e^u=8\pi(\delta_p+\delta_{-p})\quad\text{on } E_\tau:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau), $$ where $\delta_p$ denotes the Dirac measure at $p$. The case $\wp'(2p)=0$ was solved by Kuo (J. Differential Geom. 2026). In this paper, we study the general case $\wp'(2p)\neq 0$. A novel phenomenon for $\wp'(2p)\neq 0$ is that the Hill discriminant of the associated generalized Lamé equation has an essential singularity. We develop new ideas to determine the intersection of two conditional stability sets completely, which is quite surprising since it seems impossible to determine the exact structure of these two sets due to the essential singularity. This leads to a sharp criteria for the existence and non-existence of solutions when $E_\tau$ is a rectangular torus and $\wp(p)\in \mathbb{R}$.

We study weighted pluricomplex extremal functions with a fixed weight and a variable logarithmic growth parameter. We show that the regular parameters at a point form the empty set, the smallest contact parameter alone, or the whole contact ray. For continuous weights, we represent the right parameter derivative by probability measures satisfying quasi-everywhere inequalities. This gives a criterion for equality with the unweighted Green function. We construct two counterexamples to preservation of weighted regularity, using smooth plurisubharmonic weights and polynomially convex compact sets that are non-pluripolar and non-plurithin at every point. For the product example, we compute the derivative on the entire exceptional polydisc through planar Green functions and determine exactly where equality holds. The second example is globally pluriregular but has a positive right derivative. We also characterize the points where local pluriregularity fails without plurithinness, using negligible subsets and the plurifine topology, and prove preservation of regularity under local strict plurisubharmonicity of the extending weight.

Neural operators have emerged as powerful surrogates for solving partial differential equations (PDEs), yet their reliability under distribution shift remains a critical barrier to deployment. Existing approaches to out-of-distribution (OOD) generalization in operator learning are largely empirical and black-box: they report aggregate error metrics without explaining why errors arise or when they will grow. We propose a structure-preserving framework that makes OOD generalization predictable and auditable. Our key idea is to parameterize the learned solution operator as a spectral filter $h_\theta(\lambda)$ acting on the eigenvalues of the underlying elliptic operator, implemented via Chebyshev polynomial expansions and trained with a weak-form objective. This parameterization admits an exact decomposition of the energy-norm error into two observable components: a model-dependent spectral approximation term and a distribution-dependent spectral weighting term induced by the input. From this decomposition we derive three diagnostics: a conservative in-band supremum $\vareps_{\mathrm{sup}}$, a global RMS proxy $\vareps_{\mathrm{rms}}$, and a sample-dependent effective metric $\vareps_{\mathrm{eff}}(f)$. These diagnostics can be computed without access to ground-truth solutions. Through four controlled experiments, we show that $\vareps_{\mathrm{eff}}(f)\|f\|$ consistently predicts energy error under in-distribution, in-band spectral shift, out-of-band tail, and compound shifts, whereas global metrics can be systematically misleading. Our framework shifts OOD assessment of neural operators from black-box benchmarking to operator-structure diagnostics, providing a practical route to auditable scientific machine learning.

In this paper, we study the homological critical points for persistence hypergraphs and persistence simplicial complexes. With the help of the relative homology groups, we define the homological critical points for persistence morphisms between persistence hypergraphs and persistence simplicial maps between persistence simplicial complexes. We prove some commutative diagrams of subset relations for the homological critical points of persistence hypergraphs as well as persistence morphisms between them. We also prove some commutative diagrams of subset relations for the homological critical points of persistence simplicial complexes as well as persistence simplicial maps between them. As examples, we use the parametric configuration spaces to construct the parametric independence complexes as the space of parametric packings and construct the parametric dominating hypergraphs as the space of parametric coverings.

We construct a transferred $L_\infty$-structure on the shifted Chouhy–Solotar cochain complex of a finite-dimensional skew-gentle algebra and give explicit basis-level formulas for its binary operation $l_2$ in all shifted degrees. We further prove that the Hochschild dg Lie algebra of an $A_n$-type skew-gentle algebra is homotopy abelian, and hence $L_\infty$-formal, and contrast this with Wong's non-formality result for the dual numbers $\mathbb{K}[\theta]/\langle \theta^2 \rangle$, leading to a general formality classification problem.

We provide direct constructions of nontrivial root elations of thick generalized quadrangles embedded in thick polar spaces of higher finite rank and show that all of these can be expressed as self-projectivities of length $4$. This reveals a connection between the little projective groups and the special projectivity groups of thick polar spaces of finite rank explicitly. We also prove that the constructed root elations extend to the ambient polar space. Furthermore, we introduce point elations and line elations and use them to give an equivalent definition of the Moufang property for thick polar spaces, without using the notion of apartments. This, in turn, allows us to define the Moufang property for thick polar spaces of infinite rank. Consequently, we show that all thick polar spaces of finite and infinite rank are Moufang.

Discrete speech tokens provide a compact interface between speech encoders and large language models for automatic speech recognition, but single-tokenization systems remain sensitive to the chosen encoder. We propose multi-view discrete token augmentation, a simple strategy that augments each training utterance by generating alternative token sequences from fixed SSL encoders, such as HuBERT, WavLM, and MMS-300M. These tokenizations are treated as complementary training views for a shared LLM decoder, exposing it to more diverse discrete speech representations without requiring multi-encoder inference. At test time, the model can operate with a single encoder. On LibriSpeech, the approach consistently improves all encoders over independently trained baselines, with WavLM reaching 3.30% WER on test-clean and 8.13% on test-other. Budget-matched controls show that the gains come from encoder diversity rather than data volume. ROVER over multi-view hypotheses further improves WER to 3.03% and 7.38%.

In clinical trials, randomization of treatment and control groups is typically used to ensure the groups have similar covariate distributions on average, resulting in unbiased causal effect estimation. Such balanced covariate distributions are hard to achieve in practice, however, due to recruitment costs, patient dropouts, and more. One possible solution to this problem is to include control patients from external, real world databases. In this paper, we propose DBML, a method that selects control units from a real world database by matching the distribution between the treatment group and the potential control group, instead of matching units between two groups. DBML has three steps. First, we detect anomalous database units (with respect to the treatment distribution) using a variational autoencoder. Second, we re-weight the remaining database units to match the distribution of the treatments. Finally, we use these weights to sample units for our control group. The proposed method is compared to alternative matching-based algorithms and weighted algorithms, achieving superior performance in covariate balance.

This series of papers aims to establish a universal generalized Brillouin zone (GBZ) theory for higher-dimensional non-Hermitian systems. As the starting point of this series, we emphasize a fundamental question: while the conventional one-dimensional (1D) GBZ condition, $|\beta_p| = |\beta_{p+1}|$, is well known to fail in two dimensions, how does this breakdown actually occur as a system gradually crosses over from 1D to 2D? Investigating this question reveals that the existing 1D GBZ theory itself remains incomplete. In this first paper, we therefore systematically review the 1D spectral approach and clarify where the underlying difficulties lie. Our work identifies the key challenges that motivate the wavefunction approach developed in Paper~II.

In this paper, we construct nonunital universal enveloping Hom-associative algebras for multiplicative Hom-Lie algebras equipped with $\alpha$-derivations. We prove that the prescribed derivation extends to the enveloping algebra and that the resulting universal property yields a left adjoint to the commutator functor between the corresponding categories. When the twisting map is bijective, we establish a canonical isomorphism between this enveloping algebra and the twist of the nonunital universal enveloping algebra of the associated untwisted Lie algebra. This isomorphism preserves the twisting maps and the derivations. Consequently, the classical Poincaré-Birkhoff-Witt theorem identifies the associated graded algebra for the untwisted product with the positive-degree symmetric algebra. When the Hom-Lie algebra is finite-dimensional, the resulting PBW basis consists of ordered monomials of positive length. The twisting map, the extended $\alpha$-derivation, and its untwisted counterpart preserve the PBW filtration.

Residential traffic classification supports service management, but learning across homes must account for heterogeneous traffic and privacy constraints. Privacy-aware training may impose uneven costs across traffic categories. We study this tradeoff in simulated two-client federated learning using 1.62 million preprocessed gateway-collected flows across six categories. We compare a full-feature baseline, feature suppression (FS), and differentially private stochastic gradient descent (DP-SGD) under one fixed record-level privacy setting. FS-mild excludes four timing features from 16 model inputs; it provides no formal privacy guarantee. With size-proportional aggregation, FS-mild achieves higher combined macro-F1 and worst-group F1 (the minimum per-class F1 across homes) than DP-SGD in all five seeds at both model capacities under stratified and temporal splits. The tested DP-SGD configuration incurs pronounced minority-category losses, especially in the smaller home, but FS-mild does not uniformly improve on the full-feature baseline. On stratified-split models, loss-based and shadow-model membership probes show near-chance aggregate discrimination without a consistent ranking across probes; this does not establish equivalent privacy. These findings support FS as an input-minimization baseline, not a substitute for formal privacy.

Let $m,n\ge 2$. For $m\times n$ complex matrices $A$, $C$ and an $n\times m$ matrix $B$, Nobori (Linear Algebra Appl. 725 (2025) 135–144) proposed the following conjecture on the generalized Böttcher-Wenzel inequality: $$ \|ABC-CBA\|_F^2 \le 2\|B\|_2^2\|A\|_{(2),2}^2\|C\|_F^2, $$ where $\|\cdot\|_F$ is the Frobenius norm, $\|\cdot\|_2$ is the spectral norm, and $\|\cdot\|_{(2),2}$ is the $(2,2)$-norm defined by $\|X\|_{(2),2}=\sqrt{\sigma_1(X)^2+\sigma_2(X)^2}$, in which $\sigma_1(X)$ and $\sigma_2(X)$ are the largest and second largest singular values of $X$, respectively. This paper proves the conjecture and, as a consequence, establishes the Kronecker product inequality that Nobori derived from it. In addition, we show that for each fixed $m,~n\ge 2$, the constant $2$ is sharp. We also give necessary and sufficient conditions for equality, namely, the equality condition in the Böttcher–Wenzel inequality and some matrix identities derived from a fixed singular value decomposition.

The proximal point algorithm for finding a zero of a maximal monotone mapping is the engine that drives augmented Lagrangian methods and various splitting schemes in convex optimization, where proximal terms that are merely positive semidefinite can be helpful in preconditioning. Here a semidefinite extension of that algorithm is developed with new features which guarantee convergence, typically even linear convergence, in broader territory than previously foreseen. Moreover, inexact minimization is allowed in the subproblems in a variable-metric implementation which might open up quasi-Newton-like approaches. The key to the extension is applying an advanced form of the positive definite proximal point algorithm to a projected monotone mapping derived from the given mapping, and combining that with a completion step. For this to work, the maximality of the projected mapping is essential, but maximality is not automatic. The circumstances that guarantee maximality are definitively analyzed and convenient criteria for maximality are thereby identified. Consequences of the extended algorithm are worked out for application to convex inf-projection, augmented Lagrangian steps, and bracketed progressive decoupling based on Spingarn's partial inverse as a scheme for problem decomposition.

We prove weak existence and uniqueness in law of $[0,1]$-valued solutions for a class of scalar stochastic heat equations with measurable drifts $b : [0,1] \to \mathbb{R}$, driven by Wright–Fisher noise. The only assumptions we require on the drift are Borel-measurability, and the one-way inequalities $-Cu\leq b(u)\leq C(1-u)$ for some $C<\infty$, compatible with the necessary condition $b(1)\leq 0\leq b(0)$. These are vastly more general conditions than were previously available. Our proof relies on a stochastic duality between the solution of the stochastic heat equation, and a voting scheme running along the graph of a branching-coalescing particle system, generalising the class of drifts for which a dual process is available. Our results and their proof put the regularisation-by-noise result of Barnes, Mytnik and Sun, and its explanation, on a much more general footing.

Detecting RTL design bugs in open-source RISC-V CPU implementations is critical for ensuring system reliability. Traditional detection approaches inherently rely on predefined artifacts. In this paper, we leverage the official,natural-language RISC-V specifications as an effective information source for bug detection. We present VSpector, a specification-driven bug detection pipeline that directly checks whether CPU register-transfer level (RTL) implementations adhere to official specification rules, without requiring specialized construction of reference models, formal properties, or custom bug patterns. To resolve the key technical trade-off between broad context scope and model reasoning accuracy when using Large Language Models (LLMs), VSpector employs a stepwise context refinement scheme across a four-stage pipeline: rule extraction, implementation localization, candidate identification, and sequential violation auditing. We evaluate VSpector on two industrial-strength RISC-V CPUs, CVA6 and XiangShan. Out of 217 reported candidates, manual inspection confirmed 148 true violations, representing a 68.2% precision. These violations correspond to 73 distinct bugs, including 42 previously unknown bugs. In our comparative experiments, DiveFuzz, a state-of-the-art CPU fuzzer, detected none of these new bugs during 24-hour runs per CPU. All 42 new bugs have been reported upstream, with developers already fixing 19 and confirming an additional 11 (30 in total), demonstrating that specification-driven auditing is a practical and complementary strategy for CPU bug detection.

Structural causal models for time series recover contemporaneous and lagged effects, but most methods require complete observation windows and become misspecified when samples are missing. We introduce ITSY, the first continuous-optimization method for causal discovery from irregular time series under a linear model. ITSY reformulates the structural equation so that prediction uses the nearest available history rather than the possibly missing current slice, and jointly imputes missing values while learning both graphs. A weighted reconstruction objective corrects the noise transformation induced by this reformulation. Across synthetic regimes varying missingness, scale, graph density, and noise, and on a real world benchmark, ITSY consistently improves graph recovery over representative SCM-based baselines, demonstrating the effectiveness of the proposed method. The results establish a focused solution for irregular linear first-order dynamics and clarify the assumptions required for nonlinear or higher-order extensions.