Papers

We give two-sided moment estimates for decoupled chaoses of any order generated by independent symmetric random variables with regular moment growth: $\|X\|_{2p}\le K\|X\|_p$ for every $p\ge1$. The estimates involve only the coefficient tensor and the one-dimensional tails. They use the partition norms of Adamczak and Latała, with constants depending only on the order and on $K$. The proof starts with log-concave tails. Gaussian smoothing gives the estimate for the expected norm needed for induction, and a product comparison reduces the remaining distributions to this case. The extra tensor indices introduced by the comparison are removed using Rademacher moments. A comparison with products of log-concave-tail variables then extends the formula to regular moment growth. We also give tail bounds and explicit Rademacher and Weibull formulas for $r\ge1$, including fourth-order examples. This article was developed with the assistance of GPT Astra-6.

Large language models (LLMs) offer new opportunities for public opinion research by enabling early prediction of survey responses, potentially reducing the cost and time of traditional surveys. However, many existing steering approaches rely on target-domain human data for fine-tuning or prompting that is costly to collect and raises privacy concerns. In this paper, we study demographic group-level survey simulation, where personas induced from heterogeneous, anonymized public behavioral data condition agents that simulate responses of individuals from specific demographic groups. We examine whether representative personas can be induced from diverse sources and analyze how the domain, scale, and granularity of the source data affect survey simulation alignment. We find that personas induced from out-of-domain sources rarely outperform simulations conditioned only on basic demographic information, largely due to population mismatch. However, when personas are accurately assigned to the target demographic groups, alignment improves substantially. Finally, personas induced from target-domain survey data generalize better as more survey question history becomes available, suggesting that richer behavioral evidence enables more stable persona trait inference that transfers to better unseen questions simulation alignment.

Estimating causal effects in climate science, such as the effect of anthropogenic warming on crop loss, is challenging because of complex spatio-temporal dependence and the high-dimensional nature of the treatment. To address this dependence and the resulting poor overlap between observed and counterfactual scenarios, we develop a spatio-temporal stochastic-intervention framework for estimating causal effects from climate observations. We introduce a regularized estimator of the stochastic-intervention treatment effect that trades a controlled bias for a reduction in the weight variance caused by poor overlap. Simulation studies show that this estimator attains lower mean squared error than alternative weighting estimators and removes the confounding bias of an unadjusted estimator. We apply the framework to estimate the effect of historical warming on vapor-pressure deficit, a driver of crop stress, adjusting for precipitation, which confounds the effect by affecting both temperature and humidity. In GISS-E2-1-G climate-model simulations, the global effect is distinguishable from zero in every year from 1995 onward, and omitting the precipitation adjustment inflates the global estimate by 47%. Adjustment reverses the sign of the estimate over 8% of global cropland (125 million hectares), where an unadjusted analysis could misdirect adaptation between heat-focused and moisture-focused measures.

Continuous ingestion can expose new retrieval-augmented generation (RAG) content to retrieval before vetting completes, creating a temporal attack surface that conventional admission decisions do not capture. We present a fail-closed provisional-visibility protocol that admits new content under a deadline, hides items whose verification has not committed in time, and supports lineage-scoped containment. The protocol bounds unvetted exposure by the configured visibility budget plus query-path enforcement delay; poisoning exposure remains conditional on verifier correctness. We evaluate the design on five workloads of encoded natural-language documents using an exact backend and Milvus. Under verifier backlog, the deadline reduced median poisoned retrievals from 34 (29–34) to 7 (6–7) relative to asynchronous admission without a deadline, with the same reduction in displaced clean results. Verify-before-visible avoided provisional exposure but delayed clean first visibility to 6.3 s, whereas provisional admission made content visible within 3 ms; expired clean items could nevertheless experience temporary availability gaps. Replaying decisions from a recipe-specific detector illustrated the protocol boundary: false promotions left poison visible, while false refusals excluded clean content from retrieval. Standalone Milvus tests characterized enforcement delay and concurrent HNSW retrieval on one node. These results clarify which guarantees come from the protocol and which outcomes depend on detector quality, while quantifying the freshness and availability trade-offs.

Classification problem in the context of highly imbalanced data is a major challenge in many real-world applications (e.g., FinTech, healthcare, etc.). In these cases, the vast majority of instances belong to a single class and a small fraction represent the minority class (often the most critical class). Recently, diffusion models have emerged as powerful approaches to reduce the degree of “imbalanced-ness” in the dataset; they work by generating synthetic data by capturing complex data distributions using iterative transformations. However, standard diffusion models are not inherently suited to highly skewed or heavy-tailed data, due to inbuilt quadratic error loss, which lacks the structural sensitivity to capture rare, extreme values, and minority-class nuances. We propose a novel approach, namely, Quantile-TabDDPM, based on a quantile-regularized denoising objective that combines the standard quadratic error loss with a quantile loss term to explicitly capture rare events while preserving the theoretical grounding of the original denoising objective. We extensively evaluated our approach on a real-world credit card transaction dataset characterized by extreme class imbalance. The results demonstrate that the integration of diffusion-based synthetic data generation with a quantile-regularized denoising objective provides a robust and effective framework for fraud detection in highly imbalanced datasets.

We extend the Wentzell operator to differential forms on compact manifolds with boundary, investigate its spectrum, and establish a variational characterization of the first eigenvalue. We explicitly compute the eigenvalues on the unit ball in Euclidean space. We also derive several eigenvalue estimates for the first eigenvalue, some of which are shown to be sharp. Finally, we obtain a lower bound for the gap between the lowest eigenvalues in consecutive degrees.

Personal AI assistants are beginning to transact for consumers, and early adopters capture real savings. Whether those savings survive, and what happens to consumers who have no assistant, depends on how sellers respond – a question single-user evidence cannot answer. We build an agent-based rental market in which language models play consumers, assistants, and six adaptive sellers guided by an algorithmic pricing tool. Half the population receives an assistant under an advisory or an executing mandate; the contract pairs a fee only the renter's physical action avoids with a pre-selected add-on an authorised assistant can cancel online. An analytical benchmark and a behaviourally calibrated rule market supply ex-ante predictions, and paired branches with frozen versus adaptive sellers separate adoption effects from market feedback. Across thirty simulated markets, executing assistants cut adopters' spending by 13.7 USD per renter-day when sellers are frozen; adaptation claws back about a third, leaving 8.7, with the gains arriving both as lower bills and as rentals completed at all. Sellers raise headline rates while cutting fees, and the calibrated forecast of the burden on unassisted consumers (+3.6) does not transfer: their mean spending change is +0.4, confidence interval -0.6 to +1.3. Seller-model swaps and a within-market transfer of fee-setting to the pricing tool show that fee conduct, and with it the division of the gains, is decided on the seller side. Assistants, we conclude, should be evaluated at market level – completion, total spending, and non-users included – and the comparison layers locate exactly where a calibrated behavioural forecast fails in a language-model market.

Romantic relationships shape physical, mental, and social well-being, and relationship quality is built through everyday interactions. Addressing minor issues early and reinforcing positive interactions help maintain relationship quality, and ubiquitous computing creates new opportunities to sense and support couples' relationships in daily life. However, existing systems remain largely developed with limited involvement from couples, raising questions about which types of sensing and support couples may accept in their private lives. To address this gap, we conducted exploratory needs-assessment interviews and co-design sessions with five co-located couples (10 participants) to identify their needs and preferences for relationship-attunement-enhancing technologies, including the use of LLMs. Couples' own designs converged on systems that detect meaningful moments—such as conflict onset—and offer subtle mediation rather than AI-generated advice, highlighting dyadic concerns around partners' agency, authenticity, and control over shared interaction data. We offer preliminary design implications for ubiquitous sensing and intervention systems that center the couple as a unit rather than the individual.

We investigate global consistency conditions for five-dimensional warped braneworlds in Palatini and metric-affine gravity. Starting from the generalized Gibbons-Kallosh-Linde framework, we derive integrated sum rules for cosmological warped geometries and examine how independent connection degrees of freedom modify the standard Einstein-gravity obstruction to smooth compact braneworlds. The analysis separates geometric, matter, and non-Riemannian contributions and clarifies the conditions under which the usual no-go result can be modified. We show that the generalized consistency relations provide necessary global constraints on admissible configurations and identify the terms responsible for possible departures from the Riemannian case. The results establish a systematic framework for testing the global consistency of smooth five-dimensional braneworlds in gravitational theories with independent affine structure.

We develop structure-preserving methods for the compressible Euler equations with gravity in vector-invariant form and potential temperature as a prognostic variable within the flux-differencing discontinuous Galerkin spectral element method (DGSEM) framework. By discretizing the nonconservative terms as symmetric and antisymmetric products, we derive two-point numerical fluxes that conserve both the thermodynamic entropy and the total energy. Moreover, we design an entropy-stable numerical flux that is well-balanced for both isothermal and isentropic background states. All properties are shown to carry over to the high-order DGSEM on general curvilinear meshes. Several numerical examples confirm the theoretical findings and show the robustness and accuracy of the scheme for use in modern dynamical cores for atmospheric flows.

Transporting unsecured payloads with legged robots over uneven terrain requires balancing locomotion performance and payload stability, since aggressive motion can destabilize the payload even when the robot remains stable. We study quadrupedal transportation of unsecured stacked boxes on an edgeless torso-mounted board without dedicated payload sensors or active carrier mechanisms. To address this trade-off, we propose Payload-Adaptive Multi-Objective Reinforcement learning for Transportation (PAMORT). PAMORT trains a multi-objective base policy conditioned on a preference vector that weights locomotion and payload-stability reward groups, then trains a weight adjuster on the frozen policy to adapt this preference online from proprioception. In simulation, PAMORT achieves comparable or better overall transportation success than a corresponding single-objective baseline across different payload configurations, including an unseen three-box stack, despite training only with two boxes. Real-world experiments on a Unitree Go2 demonstrate zero-shot transfer to slopes and steps at or beyond the training difficulty, with mean success rates of 0.850 for PAMORT and 0.675 for the baseline across eight tasks. These results demonstrate robust unsecured-payload transportation with online adaptation of the locomotion–payload trade-off from proprioceptive information.

Vision-language-action models (VLAs) act on instructions without being able to refuse, so screening harmful requests falls to monitors. We test whether these monitors catch ordinary robot tasks requested for harmful reasons, holding the task fixed while varying only how explicitly the intent is stated. $\pi_{0.5}$ completes the task at every level of explicitness, as often as for harmless controls. Text guards flag nearly every blunt request but few implied ones: up to 95% of implied-harm runs end with the task done and no flag raised, and most still leak after recalibrating on robot instructions at low false-positive rates. Monitoring the model's activations does not close this gap. Linear probes separate harmful from harmless instructions almost perfectly in the base language and vision-language models, but this separation weakens after robot training in two model families, most for implied harm.

Text revision has become an integral component of large language models. This paper formulates revision such that it admits a Nash equilibrium: Token positions are players, vocabulary items are actions, and each player's utility is the language model's log conditional probability. We motivate the revision by showing that Nash equilibria can have exponentially higher likelihood than autoregressive outputs as the sequence length grows. We further propose Nash decoding, an algorithm that reaches an $\varepsilon$-Nash equilibrium in $O(1/\varepsilon)$ time given access to the joint probability of tokens conditioned on a prompt. In practice, we run Nash decoding using conditional probability estimates from large language models and evaluate the resulting equilibria on question-answering benchmarks. On CLAPNQ, PubMedQA, and CoQA, Nash equilibria obtained from masked language models achieve higher F1 and ROUGE scores than autoregressive models up to $18\times$ larger, without any fine-tuning or retraining, at the cost of additional test-time computation.

Image and video diffusion models allocate equal computation to every region, even when the intended scene calls for varying levels of detail. The spatial distribution of detail can often be anticipated before generation, indicating where computation can be reduced. We introduce Level-of-Token (LoT) Diffusion, a framework that turns this knowledge into an explicit multiresolution token layout (Level-of-Token layout) for adaptive and efficient generation. Tokens represent rectangular patches of varying sizes and shapes, allocating finer tokens where detail is needed and coarser tokens elsewhere. We adapt pretrained diffusion transformers to LoT layouts through a patch-wise asymmetric flow parametrization and embeddings for multiresolution tokens, preserving full-resolution flow prediction at every denoising step while processing only a reduced token sequence. LoT Diffusion enables layout-adaptive generation while preserving pretrained generative priors. We demonstrate LoT with layouts derived from semantic masks, bounding boxes, texture variance, and depth-of-field cues, as well as agentic plans. Across image and video generation, LoT offers favorable quality-efficiency tradeoffs, with significant speedups determined by the layout's token budget. Our project website is at https://georgenakayama.github.io/lotdiffusion/.

Continuous language flows generate text by denoising all positions of a target canvas together. The natural way to add reasoning to such a model is to write a trace ahead of the answer, but the trace length changes from question to question. The answer start is therefore unknown during denoising, and the model has to decide the trace length, the place of every trace token, and the answer at the same time. We propose Plan Canvas to fix the boundary between the trace and the answer. A plan region of fixed capacity holds a compact trace, supervised padding fills its unused positions, and the answer starts at a fixed position. The fixed regions also allow separate denoising clocks for the plan and for the answer. With the trace text, backbone, and canvas length of the free-trace baseline held fixed, Plan Canvas improves accuracy on ProsQA and on Deep ProsQA, a graph benchmark with longer proofs. On Deep ProsQA, accuracy rises from 73.0% to 87.0%, the share of questions answered with a valid path rises from 30.8% to 59.1%, and the gain is largest on the longest proofs.

We prove a weighted Wasserstein spectral gap for the three-dimensional primitive equations on $\mathscr E=\{u\in H^1\cap L^8: \partial_zu\in L^4\}$ under finite-rank additive Brownian forcing that is phase-complete finite Fourier and satisfies a quantitative quadratic saturation condition. In particular, four real forcing directions suffice and this number is optimal within the phase-complete class. Consequently, the system is uniquely ergodic and exponentially mixing in the $H^1$ topology. The main difficulties stem from the lack of sufficiently strong moment bounds and the highly degenerate nature of the noise, placing the problem outside the direct reach of standard methods. We introduce a new quantitative stable–compact asymptotic coupling criterion that captures the delicate balance between the available control of the growth of initial perturbations and the accuracy–cost of compensating them by perturbing the noise. For the primitive equations, this leads to a quantitative infinite-dimensional Hörmander analysis requiring tracking the degree and coefficient cost of Lie-bracket polynomials with the Fourier frequency.

Given a finite Cartesian set $S=X \times Y$ and a decreasing set of monomials $\mathcal M$, we call extremal polynomials for $\mathcal M$ over $S$ those that have the maximum number of zeros in $S$ and whose support belongs to $\mathcal M$. Coordinate factorizations give a family of extremal polynomials; we call them canonical. If $\max(\mathcal M)$, taken with respect to divisibility, is a single monomial, all extremal polynomials are canonical. If $|\max(\mathcal M)|=2$, either all extremal polynomials are canonical, or the problem reduces to the case where $\max(\mathcal M)=\{x^{d_x},y^{d_y}\}$. In the latter case, we prove that the existence of noncanonical extremal polynomials depends on finding families of subsets whose elementary symmetric functions agree. This condition is more restrictive than the classical Prouhet–Tarry–Escott problem, which asks for two sets whose elementary symmetric functions agree. We determine the number of triples $(X, Y, h)$, where $h$ is a quadratic noncanonical extremal polynomial. We apply extremal polynomials to coding theory via minimum-weight codewords.

Process control of advanced semiconductor nodes is not only pushing the limits of metrology equipment requirements in terms of resolution and throughput but also in terms of the richness of data to be extracted to enable engineers to finetune the process steps for increased yield. The move towards 3D structures requires extraction of critical dimension parameters from structures which can vary largely from layer to layer. For in-line process control, the necessary automation forces the development of layer and equipment-specific dedicated image processing algorithms. Similarly, with the increase in stochastic defects in the EUV era, detection of defects at the nm scale requires the identification of features captured in low resolution to meet the throughput requirements of HVM fabs, which can again lead to custom algorithm development. With the emergence of ML-based image processing methods, this process of algorithm development for both cases can be accelerated. In this work, we provide the general framework under which the images obtained from high-speed scanning probe microscopy-based systems can be used to train a network for either feature detection for parameter extraction or defect identification.

Battery systems and algorithms are developed and evaluated primarily on laboratory cycling data, but fielded battery systems are subjected to duty cycles, temperatures, and measurement conditions that cannot be reproduced in a lab. Field data from batteries in service has begun to appear in public releases attached to individual papers, but each release has its own format, units, labels, and other gaps that make it difficult to compare against others. We present a curated collection of 35 public field-data releases from 27 research groups, covering applications that span passenger EVs, buses, light electric mobility, grid-scale storage, distributed storage, consumer electronics, industrial robotics and satellites. Where the releases are complete enough to measure, this heterogeneous collection contains 256 GB of data from at least 149,188 lithium-ion cells, 3,187 batteries, 115.6 MWh of energy storage capacity, and 3,040 unit-years of service. Critically, it also contains 6,465 labeled or reported faults, which provide greater visibility into what real battery performance and safety issues look like outside the lab. We release a public repository of metadata and code for each of the field datasets. For each dataset it provides a registry entry with sources, license and access links, checksums to confirm a download is complete, a metadata table, and code for how to load it. We also report where released datasets differ from their papers. We hope the collection can help serve researchers (human or otherwise) as a useful platform for improving battery field diagnostics, fault detection and health estimation. Better field evidence can also feed back to improve manufacturing quality, inform second-life screening and residual-value estimates, and broadly accelerate electrification.

Machine learning (ML)-powered poisoning attack vectors are adversarial maneuvers whereby an attacker intentionally inserts, corrupts, or alters training data to distort an ML model's learning process. The objective is to diminish model efficacy, instill biases, induce misclassifications, or include concealed backdoors that may be attacked during implementation. In streaming contexts, poisoning attacks pose significant risks since models perpetually update based on incoming streams of data. An assailant may incrementally introduce harmful samples into this data stream, leading the model to assimilate erroneous features over time without timely identification. Therefore, this study is aimed at evaluating the efficacy of the adversarial training (AT) defense approach against poisoning attacks (label flip and noise injection) using an online AutoML pipeline for Internet of Things (IoT) networks. Specifically, poisoning attacks (label flip and noise injection) were applied to streaming-capable AutoML learners (Hoeffding Tree (HT), Leveraging Bagging (LB), Adaptive Random Forest (ARF), Hoeffding Adaptive Tree (HAT), and Streaming Random Patches (SRP)). Under the strongest poisoning rate (PR = 1.0), AT-SRP achieved the highest F1-score against label flip poisoning (0.904), while AT-LB achieved the highest F1-score against noise-injection poisoning (0.933). Finally, several drift detection methods were used for rolling accuracy and prequential evaluation.

Advanced semiconductor nodes are pushing the limits of feature sizes and require metrology with sub-nm resolution without compromising on the throughput as needed for in-line process control. Recently, high-throughput scanning probe microscopy (SPM) based metrology and inspection tools capable of meeting these needs have been introduced to the market and qualified for use in HVM. While innovative measurement methods and tool architecture have allowed for a leap of improvement in throughput, the next step in further reducing imaging time can be obtained through the application of machine learning for enhancing the resolution of measured images for extraction of relevant parameters. In this work, we provide the general framework under which a neural network-based resolution enhancer is designed and used for SPM images. We showcase the effectiveness of this framework using measurements performed on Line/Space structures with a pitch of 200 nm. For the reusability of a pre-developed pre-trained model, we additionally leverage transfer learning and show that a new model for slightly differing structures can be re-trained and calibrated with a smaller data set of measurements performed on Line/Space structures with a pitch of 100 nm.

Scientific design often requires jointly satisfying multiple objectives and constraints. Pretrained masked diffusion models provide a generative foundation for this task, but fine-tuning them to meet these objectives and constraints incurs additional training costs, motivating inference-time guidance with frozen models. However, such guidance faces two challenges: pass-or-fail constraints and black-box reward models may provide no useful gradients, while jointly satisfying multiple requirements can leave a small feasible region, making feasible designs difficult to find within a limited inference budget. To address these challenges, we introduce DiMOS, a training-free framework for multi-objective scientific design. Using joint rewards from candidate completions, DiMOS performs approximate Doob-guided local resampling without requiring reward gradients. To allocate computation efficiently, it uses budget-efficient trajectory search to focus computation on promising continuations. Across six DNA, protein, and RNA tasks, DiMOS attains the highest joint success rate at comparable generation times, up to $1.98\times$ the strongest baseline on DNA and protein, while maintaining high sequence uniqueness and naturalness.

Protocol choices can change the conclusions drawn from log anomaly detection benchmarks even when detector settings are fixed. We present a joint empirical study of split construction, representation visibility, and component costs using six fixed count, sequence, and semantic configurations on Hadoop Distributed File System (HDFS) and Blue Gene/L (BGL) logs. Random splits place several configurations near the average-precision ceiling, whereas group-disjoint HDFS and chronological BGL evaluation produce lower scores and different observed orderings. At a fixed BGL cutoff, parser choice spans 0.124 in semantic XGBoost mean average precision while preserving its lead over count XGBoost; the earliest rolling period reverses that ordering. A two-factor cross-system ablation contrasts source-only representations with offline transductive access to unlabeled target templates through the representation corpus and inverse document frequency: HDFS-to-BGL mean average precision moves from 0.191 with source-only access to 0.325 with union-corpus, target-IDF access, and the intermediate conditions reveal direction-dependent interactions in average precision and retrieval at fixed review budgets. Component-level profiling separates parsing and representation costs from classifier training, prediction, and storage. Together, these findings connect detector comparisons to the test population, preprocessing state, visible information, and measured pipeline stages, and identify the protocol fields needed alongside a score to support interpretable comparisons of log anomaly detection accuracy and resource use.

Higher-order networks (HONs) represent interactions among groups of objects, supporting the study of collective behaviour that pairwise models can miss. In simplicial models of these networks, Hodge Laplacian linear systems provide a common mathematical framework for analysis, for example, for solving statistical ranking problems and investigating long-term stability in coupled oscillator systems. The large number of variables associated with edges, triangles, and higher-dimensional simplices can make these systems costly to solve. Recent quantum algorithms address this bottleneck. However, comparisons with particular classical algorithms do not decision output whether solving these linear system problems establishes a quantum complexity foundation for provable quantum advantage. We prove that preparing a quantum state encoding the normalized minimum-norm solution of Hodge Laplacian linear systems is $\mathsf{BQP}$-hard under the specified sparse oracle model and parameter promises. Our sequence of reductions maps an arbitrary polynomial-time bounded-error quantum computation to such linear systems. Each reduction allows efficient recovery of the normalized minimum-norm solution of the preceding linear system. Together with an efficient quantum algorithm for the associated decision problem from the approximate solution state, we show that this problem is $\mathsf{BQP}$-complete. These results provide a worst-case complexity foundation for evaluating quantum advantage in HON analysis and show that the $\mathsf{BQP}$-hardness of the quantum linear system problem (QLSP) persists even when the matrices are restricted to Hodge Laplacians.

We introduce temporal metric spaces, a framework for formalizing metric and topological structures on continuously evolving domains. Rather than defining a single metric on a fixed space, we construct a family of interval-dependent metrics on an associated trajectory space, enabling distances to be measured between continuous trajectories over finite time intervals. We establish the fundamental topology of the trajectory space, prove that it inherits completeness from the underlying metric space, and develop the corresponding temporal hyperspace together with its completeness. Finally, we develop a theory of iterated function systems acting on these evolving spaces, proving that their attractors evolve coherently with the underlying dynamics and remain stable even when the evolving geometry causes the generating mappings to lose their classical contraction property.

Approximate solutions are found for Feller-like neutral multi-type branching diffusions $\big{(}\mathbf{X}(t)\big{)}_{t \in \mathbb{R}_{\ge 0}}$ in the limit of small mutation rates. The method employed involves solving approximations to the Laplace transformed forward Kolmogorov equation by integrating along characteristics. To leading order in the scale $\theta$ of the overall mutation rate the super-critical diffusion is found to collapse onto a line density aligned with the stationary left eigenvector of the rate matrix following a rapid change of behaviour at a critical time $t_{\rm c}$, which has a weak logarithmic dependence on $\theta$. First order in $\theta$ approximations to the density and moments of $\mathbf{X}(t)$ are also determined. The first-order approximation corresponds to allowing at most one mutation in the coalescent tree of a sample and is valid for $t < t_{\rm c}$ in the supercritical case, and more generally for the critical and sub-critical cases.

Let $R$ be a Noetherian standard $\mathbb{N}$-graded ring, and let $I_1,\ldots,I_r$ be homogeneous ideals of $R$. Let $L,M$, and $N$ be finitely generated $\mathbb{Z}$-graded $R$-modules with $N\subseteq M$. For $\underline{n} :=(n_1,\dots,n_r)\in\mathbb N^r$, set ${\bf I}^{\underline{n}}:=I_1^{n_1}\cdots I_r^{n_r}$. We study the asymptotic behaviour of the (Castelnuovo-Mumford) regularity of two families of Ext and Tor modules: those obtained from the quotients ${\bf I}^{\underline{n}}M/{\bf I}^{\underline{n}}N$, and those obtained from the quotients $M/{\bf I}^{\underline{n}}N$, with $L$ as the first argument in both the families. For each fixed homological degree, we prove that the regularity of the former family is eventually given by the supremum of finitely many linear functions of $\underline{n}$, where the coefficient of $n_i$ belongs to the set of degrees of generators of $I_i$. For the second family, when at least one of the ideals $I_i$ is generated by homogeneous elements of positive degree, we establish, under suitable conditions, a dichotomy: the regularity is either eventually equal to the regularity of the Ext or Tor module of $(L,M)$, or exhibits the same asymptotic behaviour as in the first family. These results follow from a theorem describing the asymptotic behaviour of the regularity of $(U+{\bf I}^{\underline{n}}V)/{\bf I}^{\underline{n}}W$, under suitable conditions, where $U$, $V$, and $W$ are graded submodules of a finitely generated graded $R$-module with $W\subseteq V$.

We consider the focusing nonlinear fourth-order Schrödinger equation \begin{equation*} i \partial_t u + \mu \Delta u - \Delta^2 u = - |u|^{p-1} u, \ \ \ (t,x) \in \mathbb{R} \times \mathbb{R}^d, \end{equation*} where $\mu \ge 0$. The scattering of radial solutions below the ground state is shown by Guo (2016) and Dinh (2021) in the case where $d \ge 2$ and $1 + 8/d < p < 1 + 8/(d-4)_{+}$. In the present paper, we prove the scattering of non-radial solutions below the ground state in the case where $d \ge 2$ and $1 + 8/(d-1) \le p < 1 + 8/(d-4)_{+}$. Our proof is based on the concentration compactness argument by Kenig–Merle (2006). To overcome difficulties in the non-radial case, we use the virial identity in the direction orthogonal to the momentum vector. The condition $p \ge 1 + 8/(d-1)$ is required to show the positivity of the functional in that virial identity.

We propose a Gauss-map variation (GMV) model for image denoising that measures the spatial variation of the tangent-plane projectors of the scaled image graph. We establish an equivalent representation of the regularizer in terms of the corresponding Gauss map and, using differential geometric tools including tubular coordinates and the Frenet frame, analyze its behavior across general $C^2$ and piecewise $C^2$ boundaries. The resulting estimates provide edge- and corner-contrast preservation properties. To solve the proposed model, we introduce a bilinear decomposition involving a unit normal field and a scalar magnitude field and develop an Anderson-accelerated majorization–minimization algorithm. The normal field subproblem admits an explicit pointwise majorization–minimization update, which is combined with an Anderson acceleration. For both $L^1$ and $L^2$ data fidelity terms, we establish sufficient decrease and boundedness of the iterates and prove that the generated sequence converges to a critical point of the penalized model. Numerical experiments on synthetic and natural images demonstrate the boundary preserving capability of the proposed model and its competitive performance in removing Gaussian and impulsive noise.

Low-Rank Adaptation (LoRA) achieves parameter-efficient fine-tuning by constraining model updates to a low-rank subspace and has been widely used in practice. However, LoRA typically employs a shared low-rank update across tokens, which limits its ability to fully exploit the adaptation subspace for tokens from different sequences. To address this issue, we propose an adaptive utilization of Low-Rank Adaptation (U-LoRA), which employs conditioned gating to explicitly learn effective token-level utilization of the limited low-rank adaptation subspace. Specifically, U-LoRA generates utilization coefficients along low-rank directions for each token and jointly coordinates and constrains them using sequence-level contextual information, thereby inducing more consistent adaptive patterns within a sentence. To further enhance training stability, we introduce a bias-corrected exponential moving average (EMA) historical prior that calibrates utilization signals across optimization steps, suppressing noise caused by batch-to-batch fluctuations. The effectiveness of our method arises from a better utilization of the existing low-rank subspace via input-conditioned strategies, rather than from expanding the subspace. Experiments on mathematical reasoning and natural language understanding benchmarks demonstrate that U-LoRA achieves competitive performance under comparable parameter budgets when with strong LoRA baselines and recent variants.