Papers

Conditional diffusion models can generate high-quality images that align closely with a given prompt. However, samples generated from different random seeds may remain visually similar, limiting creative exploration. Existing approaches can increase variation, but often require additional optimization or sampling overhead, while stronger perturbations may introduce distortions that degrade image quality or prompt alignment. We propose a training-free method that leverages the local pullback geometry induced by the mapping from the conditioning-embedding space to the image space to identify directions of maximal output sensitivity. By steering different sampling trajectories along these directions, we encourage distinct generations while maintaining consistency with the conditioning signal. Experiments on MS-COCO and BrushNet show the efficacy of the method in improving diversity on prompt-conditioned image generation and inpainting.

Video diffusion models increasingly serve as priors for 4D generation, but the generated videos often lack geometric consistency. Recent approaches address this through post-training by designing geometric rewards, yet this can degrade texture and fine detail, requiring auxiliary appearance rewards and careful reward balancing. To address this challenge, we propose GeoFix, a unified framework that improves geometric consistency and visual quality with a single geometric reward, applicable to both training and inference-time reward alignment. Our reward measures cross-view depth consistency in static scene regions, targeting structural inconsistencies without directly constraining appearance or penalizing valid object motion. Crucially, depth agreement increases with camera field-of-view overlap even in real videos and 4D datasets, allowing a smaller camera movement to be mistaken for better geometry. We account for this effect by calibrating the reward against the overlap-dependent trend. GeoFix achieves a 39% improvement in WorldScore 3D consistency and a 4% increase in VBench Quality Score.

For a graph $G$, we investigate sufficient degree conditions under which either $G$ or its complement $\overline{G}$ is $k$-connected. Motivated by a theorem of Hellwig and Volkmann, we introduce the notion of complement connectivity admissible pairs and define the parameter $\mathcal{C}(k)$ as the minimum degree threshold guaranteeing that either $G$ or $\overline{G}$ is $k$-connected whenever $\delta(G)\ge\mathcal{C}(k)$ and $\Delta(G)\le |V(G)|-1-\mathcal{C}(k)$. We prove that $\mathcal{C}(k)=2k-2$ for every $k\ge2$, thereby determining the exact degree threshold. As an application, we derive a Nordhaus–Gaddum type product condition, showing that for $n\ge4k-5$, the condition $\delta(G)\delta(\overline{G}) \ge (2k-3)(n-2k+2)$ guarantees that either $G$ or $\overline{G}$ is $k$-connected. Finally, we show that no analogous product condition depending only on $k$ can exist.

We classify graphs of complex polynomials in one variable up to isometry, both for the inner (length) distance and for the outer (Euclidean) distance of $\mathbb{C}^2\cong\mathbb{R}^4$. For polynomials $p,q$ with $\deg p\ge 2$ we show that the graphs of $p$ and $q$ are inner isometric if and only if they are outer isometric, if and only if $p(z)=\alpha\,q(\beta z+h)+k$ or $p(z)=\alpha\,\bar q(\beta z+h)+k$ for some $\alpha,\beta,h,k\in\mathbb{C}$ with $|\alpha|=|\beta|=1$, where $\bar q$ denotes the polynomial with conjugate coefficients. Moreover, every inner isometry between such graphs is the restriction of an isometry of $\mathbb{C}^2$; in this sense graphs of complex polynomials are isometrically rigid. We contrast this with the real case, where the two distances behave very differently: the graphs of any two real polynomials are inner isometric, whereas, for degree at least $2$, outer isometries are generated by translations, reflections in horizontal and vertical lines and half-turns.

The inner workings of deep neural networks are still not well understood. The information bottleneck framework, which characterizes deep networks by the mutual information between their hidden representations and the network's input, has been used to identify compression of input information with respect to an input image and output labels. We instead study how much of a layer's own input is recoverable from the layer's output in convolutional neural networks (CNNs). A convolutional layer applies its filters to one input patch at a time, so we evaluate recoverability on sampled input patches with a reconstruction-based probe and not on the entire feature map. The probe's size thus depends on channels and kernel size and not on feature map size, which lets our method scale to deep CNNs with wide feature maps. For each layer, we compare pretrained, random, and reconstruction-optimized encoders, using linear and linear-plus-residual-MLP decoders. We measure recoverability as $R^2$ between input patches and their reconstructions on a held-out set. Across nine ImageNet CNNs, recoverability generally decreases with depth but is more consistently predicted by the compression of the layer. Learning-attributed recoverability, the difference between pretrained and random encoders, is not uniform across architectures. We introduce single-property random controls, which transfer one pretrained property onto a random encoder, and find that each control changes recoverability with a consistent sign across models: copying pretrained singular vectors tends to raise recoverability, while copying the pretrained singular-value spectrum or activation sparsity lowers it.

LLM watermarking has emerged as a promising approach for detecting downstream model misuse in centralized settings, but extending it to federated learning (FL) presents unique challenges. In particular, FL systems often rely on strong robust aggregators to defend against poisoning attacks. We empirically show that a malicious server can exploit these same aggregators as effective watermark-removal mechanisms during training, causing representative existing LLM watermarking methods to fail detection in the final model. To address this limitation, we formalize the problem as a constrained optimization task and propose CamoMark, an aggregator-agnostic watermarking scheme for embedding robust watermarks in FL. CamoMark optimizes watermarked updates to become inliers, maintaining bounded deviation from the distribution of clean updates, while preserving sufficient watermark signal for reliable detection. Evaluated on Qwen fine-tuned on the Alpaca dataset, CamoMark achieves a detection confidence of $p$-value $= 1.82\times10^{-2}$ using only 3.3% watermarked data. The watermark also remains robust to a range of post-training removal attacks, including continued fine-tuning on clean data, pruning, and model distillation. At higher watermark ratios, detection confidence improves substantially, reaching $p$-value $= 5.11\times10^{-11}$ at a 13.3% watermark ratio.

The cvxoptdes R-package provides a unified framework to perform convex optimal experiment design and online response optimization for linear, generalized linear and additive, nonlinear, and random intercept mixed models, using exclusively open-source solvers. It supports optimization and augmentation of weighted, penalized and convex constrained continuous designs across a discrete grid of candidate points and rounding to integer-valued designs using point-exchange algorithms or mixed-integer programming for orthogonally constrained designs, including the class of orthogonal minimally aliased response surface (OMARS) designs. Models and design constraints are defined and updated in an object-oriented context with the aim to facilitate the process of sequential (penalized) design augmentation analogous to online design procedures known from active learning and Bayesian optimization.

We study minimal positive cubature formulas on the two-dimensional sphere of degrees four and six, with ten and eighteen nodes, respectively. We prove the regularity of Popov's symmetric formulas and establish that their local families of deformations are two-dimensional, up to rotations. We obtain algebraic relations between the weights and pairwise inner products of the nodes. For degree four, we construct a one-parameter family with half-turn symmetry connecting the formulas of Popov and Heo–Xu. We also give a self-contained proof that eighteen nodes are minimal for degree six.

Super-resolution techniques in microscopy are opening new frontiers in biology, and super-resolution for telescope imaging may provide similar benefits for astronomy and space domain awareness. Telescope super-resolution techniques such as multi-plane light conversion and photonic lanterns have been inspired by the ability of mode sorting measurements to extract classical Fisher information closer to the quantum bound than standard telescope imaging. Fisher information analyses for super-resolution imaging systems commonly consider the case of imaging point sources, but often only consider estimating a small subset of the parameters. Furthermore, there has been no demonstration of an estimation algorithm with performance that follows the limit set by the Fisher information when all point source parameters are considered. We provide a framework for defining and analyzing super-resolution using the classical and quantum Fisher information for the multi-parameter estimation problem and demonstrate an algorithm that can achieve the classical Fisher information bound for all the parameters simultaneously.

High-frequency switching and chattering are well-known problems in sliding-mode control. Although the super-twisting algorithm produces a continuous control input, the sliding variable may still cross zero repeatedly near the sliding manifold. This paper studies how fast these crossings can occur and whether they can accumulate in finite time for the scalar homogeneous super-twisting algorithm. Two lower bounds on the time required to return to zero are derived, together with a bound on the time needed to complete a full excursion. These results give minimum separation times and upper bounds on the number of crossings associated with excursions above a prescribed amplitude. Although zero crossings may accumulate in finite time, the amplitudes of the complete excursions considered here must vanish and satisfy a summability condition. Thus, crossings associated with excursions above any prescribed nonzero amplitude cannot accumulate in finite time. When the perturbation-rate bound is zero, the estimates recover the scaling imposed by homogeneity. For the classical super-twisting algorithm, the square-root scaling also holds for nonzero perturbation-rate bounds. Numerical simulations illustrate the results.

Here we perform a Kaluza-Klein dimensional reduction from $D+1$ to $D$ dimensions of a partially broken Fronsdal model for spin-3 and spin-4. The reduced models describe massive spin-3 and spin-4 particles without extra propagating modes. They differ from Singh-Hagen models by quadratic terms which vanish on shell, similarly to their higher dimensional massless parents. A covariant description of the reduced models in terms of gauge invariant field combinations naturally leads to higher derivative models, contrary to lower spin cases ($s\le 2$). For both $s=3$ and $s=4$ we have a rather peculiar massless limit. Although their particle content corresponds to the $\pm j$ helicities with $j=0,1, \cdots s$, as expected, they appear intertwined in a non trivial way preventing diagonalization through any local field redefinition.

TESS light curves encode a star's pulsation, rotation, binarity and oscillations, among other physical properties, but the survey produces far more light curves than can be labelled by hand. Astronomers have engineered a few dozen physically motivated features from them, which together remain a strong baseline. Previous work has trained encoders on light curves and judged them by their own downstream scores, but has not asked what they add to the engineered features. We pre-train a convolutional encoder on unlabelled TESS light curves to predict each window's latent representation from those of its neighbours. We then freeze it, add its features to the 25 engineered features, and read the combination out with one fixed linear probe on ten tasks: five variability classes, four asteroseismic targets and rotation period. Adding the learned features improves the readout on 7 of 10 tasks over the engineered features alone, where an untrained encoder does not. The combination also outperforms two networks trained end-to-end on the same labelled stars on most tasks. An encoder trained without labels therefore gives astronomers new features that complement the ones they already use rather than replace them.

We show that the maximal number of nonsimultaneous elastic collisions among $n$ identical hard disks moving freely on the plane is at least exponential in $n$.

Diffusion models have achieved remarkable success in image super-resolution (SR) by leveraging strong generative priors to enhance perceptual quality. However, for textual content, these priors often produce text hallucinations: non-existent or distorted text, which alter scene semantics and compromise model reliability. While recent methods focus on enhancing text generation, they do so indiscriminately, leaving hallucinations fundamentally unresolved. We argue the core problem is a lack of selective control. Therefore, we reformulate text SR as a selective generation problem: text is resolved only when reliable generation is expected, while other scene content is fully super-resolved. We introduce TGSR (Text-Grounded Super-Resolution), a test-time framework that progressively identifies and suppresses likely hallucinated text regions during generation. To enable this, we train dedicated detectors on a large-scale dataset curated from diverse diffusion-based SR outputs. TGSR consistently improves text fidelity across multiple backbones, increasing recognition precision by up to 17.68 points without degrading perceptual quality, underscoring the value of selective control in generative SR.

Emerging dataflow accelerators offer exceptional energy efficiency and throughput for machine learning workloads, yet remain largely opaque to developers for performance analysis. Unlike CPUs and GPUs, where instrumentation frameworks enable fine-grained introspection, dataflow hardware exposes little information. This lack of transparency limits the ability of compiler engineers, system designers, and application developers to analyze, tune, or debug accelerator behavior. In this paper, we introduce DFIT (DataFlow Accelerator Instrumentation Tool), an instrumentation framework designed specifically for dataflow architectures. DFIT directly instruments operations, regions, and compiler transformations at the graph IR and backend-lowering stages using semantic program representation. DFIT captures high-level, semantically meaningful data that reflect the true dataflow of computation. These data can be exported for visualization and analysis. Our approach demonstrates how instrumentation can be generalized beyond traditional hardware to support heterogeneous architectures, providing portability without modifying tool logic. By opening the black box of dataflow execution, DFIT establishes a foundation for new classes of performance analysis tools for AI accelerators. It provides a practical path toward reproducible, architecture-aware introspection across heterogeneous systems, bridging the long-standing visibility gap between compiler design, runtime scheduling, and hardware realization. Evaluation across Cerebras, SambaNova, and TPU shows practical overhead of 3.53%-13.62% across instrumentation modes with robust cross-platform portability.

Learning long-range dependencies with recurrent neural networks (RNNs) is difficult because of the vanishing gradient problem. Gated recurrent neural networks like GRUs and LSTMs address this problem and are commonly explained by their ability to create additive copy paths through time. This explanation is correct, but incomplete: useful RNN gate values must themselves be learned, which is typically done with another RNN. We study this gate-learning problem through carefully controlled sequence classification tasks. Our analysis shows that gated architectures struggle when all training samples require long-range memory and thus equally suffer from the vanishing gradient problem. We intervene by adding short-dependency samples which let gated models learn solutions that capture long-dependency samples, but also generalize across different class labels. When properly learned, gated RNNs match or even exceed the test accuracy of modern architectures like Mamba and xLSTM. Overall, our results highlight gate learning as a central part of learning long-range memory and show how training data can support this process by exposing the same task over shorter dependencies.

In this paper, we introduce and study the (involutive) representation homology of involutive algebras, following the development of representation homology of associative algebra in our earlier papers. Our main motivation is to understand the hyperoctahedral homology of involutive algebras. We show that over a field of characteristic $0$, the hyperoctahedral homology of an involutive algebra is isomorphic (as a graded commutative algebra) to its one dimensional involutive representation homology. This result enables the explicit computation of the hyperoctahedral homology of the universal enveloping algebras of involutive (DG) Lie algebras. For group algebras $k[\Gamma]$ equipped with the involution $g \mapsto g^{-1}$, we relate the involutive representation homology of $k[\Gamma]$ to the representation homology of the group $\Gamma$ with coefficients in $\mu_2$, and we show that the latter is the homology of the loop space of the mod $2$ Dold-Thom space of $B\Gamma$; more generally, we compute the representation homology of $\Gamma$ with coefficients in an arbitrary diagonalizable group scheme. As a consequence, the hyperoctahedral homology of $k[\Gamma]$ is concentrated in degree $0$ in characteristic $0$, for every $\Gamma$.

Tensor parallelism ends every layer with an all-reduce that sits between two dependent matrix multiplications, so its cost cannot be hidden and grows as compute outpaces interconnect bandwidth. Sending 4-bit instead of 16-bit values would cut this cost by nearly 4x, but the all-reduce must add the values it carries, and rounding the sum injects an error at every call. Model quality depends on how these errors accumulate over layers, yet existing methods treat each call on its own. We present Echo, a 4-bit all-reduce for training and inference that controls the accumulated error. Echo moves data in 4 bits and sums in high precision, so each value is quantized at most twice per call for any number of GPUs. The summation reveals each injected error, and Echo feeds it back into later calls, so old errors cancel instead of accumulating. We prove that the accumulated error stays bounded independently of depth. On A100 GPUs, Echo is 2.09x faster than the 16-bit NCCL all-reduce on 1 GB messages. It speeds up the time to first token by up to 1.16x and the training step by up to 1.09x on an idle interconnect, and by up to 1.44x and 1.38x as background traffic grows. It keeps the average accuracy over seven benchmarks within 0.4 points of 16-bit inference on Qwen3-8B. In 10B-token pretraining, its final loss is less than 0.9% above that of 16-bit training. Echo needs no FP8 or FP4 hardware.

Let $E$ be a vector bundle on a normal complex projective variety. We prove that slope stability of $\mathop{Sym}^2E$ and Zariski density of the reductions of $E$ modulo $p$ that are not strongly geometrically stable imply that the bundle $E$ is projectively unitary flat. This generalizes Shepherd–Barron's result from rank $2$ to arbitrary rank. We also show that if $E$ has rank $\le 3$ we can replace stability of $\mathop{Sym}^2E$ by stability of $E$ but we cannot do so if $E$ has rank $4$. We also generalize these results to reflexive sheaves on some mildly singular normal complex projective varieties.

Let $\Phi$ be an irreducible root system normalized so that long roots have length $\sqrt{2}$, and let $\mu\leq\lambda$ be dominant weights. We associate to $(\Phi,\mu,\lambda)$ the convex polytope $$P^{\Phi}_{\mu,\lambda} = (\mu+C)\cap(\lambda-C)\cap(\overline{D}-\rho),$$ where $C=\mathbb{R}_{\geq 0}\Phi^+$ and $\overline{D}$ is the closed dominant Weyl chamber. We prove that if $P^{\Phi}_{\mu,\lambda}$ and $P^{\Phi'}_{\mu',\lambda'}$ are isometric, then $$K^{\Phi}_{\lambda\mu}(q)=K^{\Phi'}_{\lambda'\mu'}(q),$$ where $K^{\Phi}_{\lambda\mu}(q)$ is Lusztig's $q$-analogue of the $\mu$-weight multiplicity in the irreducible highest-weight representation $V(\lambda)$, often called the generalized Kostka-Foulkes polynomial. Moreover, we construct the polynomial directly from the metric shape of the polytope, and we classify when two such polytopes are isometric. Such coincidences are abundant and can occur in different Dynkin types, and in different Dynkin families. By the Lusztig-Kato theorem, the main theorem gives a metric analogue, in the spherical setting, of the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.

Image-based crack analysis is essential for automated infrastructure inspection to reduce the cost, subjectivity, and safety risks of manual surveys. Despite advances in visual crack detection, methods that translate predictions into physically meaningful and engineering-interpretable crack condition information remain limited. To bridge this gap, this study proposes Inspect Any Crack (IAC), an innovative end-to-end crack recognition and condition assessment framework that automatically derives physical geometric measurements and continuous severity scores from concrete-crack imagery. At its core, IAC introduces SACv2-Tiny, an efficient crack segmentation model adapted from a vision foundation model. Trained on the challenging OmniCrack30K dataset, SACv2-Tiny achieved a $62.49\%$ F1-score and $45.45\%$ IoU on the test set and a mean zero-shot F1-score of $69.89\%$ across three unseen datasets, demonstrating robust segmentation that enables downstream analysis with limited post-processing. IAC further processes the predicted masks to quantify physical crack width and affected area using ground-sampling-distance calibration. Validation at 41 caliper-referenced locations yielded a mean relative width error of $7.33\%$. Beyond geometric quantification, IAC introduces a novel continuous crack-condition metric, the Crack Severity Index (CSI), bounded on $[0,100]$, which combines maximum crack width and affected-area ratio into a unified, standards-informed severity measure. Through these developments, IAC advances crack inspection beyond pixel-level recognition toward interpretable and automated infrastructure condition assessment.

We construct an explicit realization of finite-dimensional irreducible representations $V_{\lambda}$ of $\mathfrak{gl}_{n}$ using the representation theory of $\mathrm{Y}(\mathfrak{gl}_{2})$. We consider $V_{\lambda}$ as the evaluation representation of $\mathrm{Y}(\mathfrak{gl}_{n})$ and restrict it to the subalgebras $\mathrm{Y}(\mathfrak{gl}_{2})^{(i)}$. Then $V_{\lambda}$ is completely reducible as a representation of $\mathrm{Y}(\mathfrak{gl}_{2})^{(i)}$. It allows us to use the explicit formulas for irreducible representations of $\mathrm{Y}(\mathfrak{gl}_{2})$.

In electronic devices, the quality of the dielectric material is determined by figures of merit such as the band gap, leakage onset field, breakdown field and the number of built-in defects, which can all limit device performance. An additional and important metric to consider is the strength of the dielectric constant. Hexagonal boron nitride (hBN) is the benchmark gate dielectric for two-dimensional van der Waals heterostructures, owing to its wide bandgap, and atomically flat interface. At the same time, its dielectric constant (2–4) highlights the opportunity to explore alternative higher-permittivity van der Waals dielectrics that provide stronger electrostatic control. Previous work established lanthanum oxybromide (LaOBr), a layered dielectric with a predicted dielectric constant of 12.5, as a potential dielectric for van der Waals heterostructures. Using graphene field-effect devices with LaOBr, we show that the leakage-onset and breakdown fields decrease systematically with thickness. Dual-gate capacitance measurements combined with a dead-layer model reveal a bulk dielectric constant of $\varepsilon_{\mathrm{bulk}} = 12.2$, in agreement with theory, while interfacial dead-layers cause the effective dielectric constant to vary from 6 to 11 across multiple thicknesses. Benchmarked against established gate dielectrics, LaOBr combines breakdown fields comparable to hBN with dielectric constants 2 to 3 times larger. By establishing the operating regime of devices with various thicknesses, we further confirm the advantage of LaOBr over hBN for high-capacitance gate-dielectric operation. These results validate LaOBr as a promising dielectric for integration into two-dimensional heterostructures, and establish thickness-dependent benchmarking as essential for evaluating emerging layered dielectric materials.

We extend the study of Hamilton-Jacobi equations on networks from a single equation to a weakly coupled system of two equations on a star graph, and we develop a numerical scheme for it. Instead of prescribing the boundary values and imposing a junction condition, we compute the solution on each branch from its outer vertex toward the junction and determine the boundary values for which these branchwise solutions agree at the junction. On each branch, we replace the right-hand sides by their cell averages and solve the resulting problems exactly on every cell. This gives an iteration whose limits we compute in closed form. From these limits, the values of the discrete solution at the junction are affine functions of the boundary values on the branch, with a $2 \times 2$ matrix whose rows sum to one and whose determinant factors into explicit contributions of the cells. We use these formulas to characterize the boundary values that give solutions continuous at the junction and monotone on every branch: when the matrices are nonsingular, the admissible boundary values of one branch form a convex polygon, bounded by conditions on the difference of the two components and by positivity conditions, and the boundary values on the other branches follow from them. We also treat the singular case, and we illustrate the results with numerical experiments.

Hybrid reasoning models such as the Qwen3 series switch between a THINK mode and a NO-THINK mode through a few chat-template tokens. A popular recipe trains such models with reinforcement learning in NO-THINK mode and reports improved reasoning performance. We provide evidence that much of this gain can be explained by thinking leakage: the gain comes largely from partially reactivating the reasoning abilities of the original model's THINK mode instead of learning enhanced reasoning behavior. We train Qwen3-1.7B, 4B, and 8B with GRPO in NO-THINK mode. As accuracy improves, responses become longer and more reflective, are increasingly judged by the original model to resemble its own thinking, and the model's internal activations shift toward its THINK mode. We then investigate at the level of weights, activations, and attention heads: scaling the RL update further elicits THINK-mode behavior and approaches THINK-mode accuracy but does not exceed it; in Qwen3-8B, replacing a low-dimensional THINK-related activation subspace of the RL-trained model with its original-model value removes most of the RL gain, and the reverse exchange transfers a considerable fraction of the gain to the original model; and the RL-induced shift passes through the attention heads that read the empty think block. We further identify a possible mechanism in the GRPO training signal: responses favored by the advantage-weighted update are more THINK-like, and a gradient step computed from this signal moves the model toward its THINK state. Thus, gains in NO-THINK mode can largely reflect reactivation of reasoning abilities already present in the original model.

Humanoid face robots can now move their lips in sync with speech and show detailed facial expressions. Give such a robot the face of a real person and drive it with generated speech, and you get a new kind of presentation attack: a face that talks and reacts in front of the camera, recorded as a real object with no edited pixel. We call this a RoboFake and ask whether current defenses are ready for it. We test three public face presentation attack detection (PAD) models and six public deepfake detectors on video of a physical lip-syncing face robot, on a synthesized video of the same robot, and on rendered robot faces. Thresholds are fixed on human data (BPCER 5% on FaceForensics++), and every result is compared with real human faces processed the same way as the robot videos and turned to the same head angle. A small mobile PAD model detects every physical robot frame (APCER 0%) and is not affected by compression, but its BPCER on processed human video rises to 14%; the two PAD models trained on CelebA-Spoof miss the robot in the direct capture, and one of them also misses most panel frames. Among the deepfake detectors, three flag almost every robot frame and two fewer than 7%; masking shows that the response comes from the face but does not track synthesis, and JPEG compression pushes most robot frames below the threshold for some detectors. Our results suggest that current models catch today's robots because their faces do not look human, not because they recognize the attack. We state three testable hypotheses about realistic, interactive RoboFakes.

The scalability and performance of edge federated learning are fundamentally constrained by synchronization bottlenecks. Clients operate under heterogeneous compute, time-varying communication conditions, and, in some settings, privacy constraints, yet synchronization is often still handled with fixed schedules. However, synchronization decisions are often made outside the client, even though the client directly observes local progress, resource state, and communication conditions. We present a client-side reinforcement learning synchronization framework in which each client decides during training whether to upload, download, synchronize bidirectionally, or defer communication. The framework places a lightweight scheduler alongside the local model and formulates synchronization as a resource-aware control problem driven by local learning and system state. We study this framework on two representative edge workloads, spatiotemporal traffic forecasting and image classification. Across both settings, adaptive client-side synchronization improves the accuracy-cost trade-off, reducing system cost in traffic forecasting while maintaining competitive cost and improving final accuracy in image classification. We also consider privacy-constrained settings, where synchronization must account for not only latency and energy but also privacy expenditure. Overall, our results show that synchronization in edge federated learning is better treated as an adaptive client-side control problem than only as a server-side coordination mechanism.

We establish characterizations of planar quasiconformal mappings in weighted Bergman spaces $A_\alpha^p(\mathbb D)$. For $p>0$ and $\alpha>-1$, our characterization involves the convergence of certain integrals involving superlevel set areas, the Hausdorff measure of boundary superlevel sets, and the averaged derivative. These include quasiconformal analogues of characterizations established by Karafyllia and Karamanlis for conformal mappings. Moreover, for $-1<\alpha\leq 0$, we obtain an additional equivalent characterization in terms of the Hausdorff measure of level sets. The power $\alpha$ is sharp: for every $\alpha>0$, we construct a bounded $K_\alpha$-quasiconformal mapping for which the weighted level-set length integral with exponent $\alpha+2$ diverges.

Motivated by the role of piecewise linear functions in ReLU neural networks, we study systems of dilated functions in the Lebesgue spaces $L_q([0,1]^d)$, $q\in(1,\infty)$. For a sequence $\tau\in\ell_1$ we consider the operator $T_\tau g(x)=\sum_{k=1}^\infty \tau_k g(kx)$ and the function system $\mathcal Z^\tau$ obtained by applying it to the trigonometric system. We show that $\tau\mapsto T_\tau$ is an isomorphism of algebras from $\ell_1$ with the Dirichlet convolution as multiplication onto an algebra of operators and connect the inverse of $T_\tau$ to the inverse of $\tau$ with respect to the Dirichlet convolution. We give explicit conditions on $\tau$, under which the resulting system $\mathcal Z^\tau$ is a Schauder basis of $L_q([0,1]^d)$ and, for $q=2$, a Riesz basis for which we estimate the Riesz constants in terms of $\tau$. We characterise the Riesz bases of this type in terms of a sequence space of Dirichlet convolution multipliers, and we treat mixed systems of cosine and sine functions with different sequences. Finally, we apply these results to the system generated by two piecewise linear functions inspired by ReLU networks and improve the known lower Riesz constant to $2/3$, independently of the dimension.

Recent advances in multi-modal large language models have shifted toward the "thinking with images" paradigm to improve reasoning faithfulness, yet existing methods rely heavily on explicit coordinate- based grounding. We identify two fundamental limitations in this approach, i.e., a semantic gap where sparse regressions miss high-dimensional latent structures, and a resolution gap due to the granularity mismatch between patch-level tokens and pixel-level coordinates. To address these, we propose TiCO (Thinking with images without COordinates). It reformulates visual reasoning as a semantic evidence discovery process within the latent space, and evidence is injected as token-level regions without pixel coordinates. Specifically, TiCO utilizes a dedicated probe token to estimate similarity distributions and performs importance sampling over spatially connected components to generate diverse, semantically dominant evidence rollouts. Extensive experiments across multiple high-resolution and general visual reasoning benchmarks demonstrate that our TiCO consistently outperforms state-of-the-art coordinate-based methods with significantly improved inference efficiency.