Papers

In the history of physics, the name scaffolding has been given to whatever served to raise a theory and is afterwards removed because, it is said, it is no longer needed. This article documents, from the originals, what it cost Maxwell's theory when that method was applied to it: what was removed from it, who did it, with what argument, and how over the years it had to be recovered because it was understood that it was needed after all. It also records what Maxwell himself missed in Ampère, what he calls the traces of the scaffolding: the template with which Ampère himself had given shape to his work.

We prove that the function $$ \log \frac{\Gamma(x+a+b)\Gamma(x)}{\Gamma(x+a)\Gamma(x+b)}-b\log\left(1+\frac{a}{x}\right) $$ is a generalized Stieltjes function of order $3$ for $a>0$ and $0<b<1$. A similar result is obtained when $a>0$ and $b>1$.

Modern conformal forecasting systems often maintain several adaptive pipelines that differ in base forecasters, conformity scores, calibration windows, and update rules. Comparing them is difficult because coverage is a hard constraint, whereas efficiency should be optimized only among feasible pipelines. We formulate this problem as sequential inference for a stochastic constrained argmin. At each time, the target is the set of minimum-cost pipelines satisfying multiple prefix-average conditional miscoverage constraints. We introduce Coverage-Constrained Sequential Model Confidence Sets (CC-SMCS), which separate certifiably feasible, possibly feasible, and possibly constrained-optimal pipelines. Using simultaneous martingale confidence sequences, CC-SMCS projects a rectangular confidence region onto the constrained argmin and admits an exact closed-form rule. With probability at least $1-\delta$, it contains every constrained-optimal pipeline simultaneously over all times. This finite-sample guarantee requires no stationarity or mixing assumptions and remains valid under data-dependent stopping. We also establish an impossibility result for safe certification at the coverage boundary and extend the construction to delayed multi-horizon feedback and outcome-dependent efficiency objectives.

We characterize boundedness of the Bergman projection on operator-weighted $L^p$ spaces over a separable Hilbert space, for $1<p<\infty$. Under the directional integrability conditions, the criterion is uniform boundedness of averages over Carleson squares in the unit disc. We obtain norm estimates independent of dimension. The proof uses a separated kernel expansion and a layer decomposition of the tent tree. No reverse Hölder assumption is needed. We also give a finer matrix estimate with a dimension-dependent constant. Boundedness is unchanged by compactly supported changes of the weight that preserve the integrability conditions.

Before acting upon an unobservable physical system, an autonomous agent must determine which latent distinctions govern downstream tasks, how many active interventions are necessary to certify them, and when to abstain to prevent catastrophic errors. Classical observability treats state reconstruction as an unconditioned binary predicate, failing when passive observations cannot break latent degeneracies without perturbation, full microscopic inversion is prohibitively costly, and distinguishing task-irrelevant degrees of freedom wastes interaction budgets. We formalize task-conditioned active observability complexity: the minimum worst-case expected interaction cost required to identify task-relevant states under certified error and safe abstention guarantees. We prove that task-predictive equivalence induces the unique minimal sufficient quotient $\mathcal{H}/\!\sim_\tau$, leaving active observability complexity strictly invariant while eliminating superfluous distinctions. In deterministic regimes, this complexity is characterized by an optimal adaptive distinguishing tree and Bellman recursion; in noisy regimes, it obeys a stopped-transcript relative-entropy lower bound and adaptive martingale certificates that compose without independence assumptions. We instantiate a prospective certified observer with staged recovery: a nominal verifier defers candidate compilation, triggering active probing only upon evidence, while a history-measurable score shell prunes hypotheses without sacrificing risk bounds. Stress audits across high-dimensional physical systems and thousands of operational trials demonstrate certified state recovery with zero false acceptances and substantial reductions in sensor reads and model steps.

Repeated two-dimensional discrete cosine transforms (DCTs) require implementations that remain fast across several array sizes while preserving the mathematical transform. Fewer arithmetic operations do not guarantee a faster DCT: execution time also depends on how the computation uses the CPU's single-instruction, multiple-data (SIMD) units, registers and memory hierarchy. We implement established DCT algorithms in SIMD and compare execution strategies for forward, inverse, and normalized round-trip transforms on float32 arrays from $8\times8$ to $256\times256$. The resulting library, BRiDCT, combines the Shao–Johnson factorization with SIMD lane organization, banded intermediate storage, padding, and register blocking in C11/ARM NEON. On one Apple M3 Max, the final native library has lower execution times than the tested general-purpose routes—Apple's Accelerate framework (vDSP), FFTW and Ooura—across their available cases from $8\times8$ to $256\times256$. It also outperforms the tested libjxl and floating-point libjpeg-turbo routes. The compiled NumPy interface outperforms OpenCV in all 130 tested contracts per session, including squares up to $1024\times1024$ and two rectangles, with call and allocation costs included. Complementary comparisons evaluate SciPy, DUCC and cached/planned pyFFTW. The measured gains come from organizing SIMD execution and intermediate storage, rather than introducing a new arithmetic factorization. A reproducible 474-input verification corpus probes cancellation, neighboring floats, dynamic range, and extreme amplitudes: all 1 260 core checks of the final configuration pass, while boundary diagnostics reveal overflow and weak-component limitations. Larger native transforms show size-dependent limits. Code, versioned releases, dataset and recorded measurements: https://github.com/antmoev/bridct.

We establish a cutoff criterion for finite Markov chains with non-negative Bakry–Émery curvature without assuming that the support of the transition matrix is symmetric. This extends a criterion of Salez based on a refined product condition. For each allowed transition, we consider a shortest directed path from its endpoint back to its starting point. Our cutoff criterion depends on the maximum length of these paths. We also obtain a discrete-time counterpart for lazy chains. For random walks on finite Abelian groups, we prove non-negative Bakry–Émery curvature for arbitrary generating sets. As an application, we establish almost-sure cutoff for random walks on products of cyclic groups of order three.

Post-hoc malware calibrators can condition confidence on graph structure, but their structural inputs may leave the support represented by validation data under malware-type shift. We study this risk on MalNet-Tiny by holding out each of four malware types across three seeds, freezing a graph isomorphism network, and fitting post-hoc mappings only on known-type data. On known-type tests, generic and combined structural calibrators have negative log likelihood (NLL) of 0.4341 and 0.4628. Their mean held-out-type gap widens to 0.1369, with a four-type 95% interval of [-0.0125, 0.2864] and improvement in one of four types. Eight label-independent nuisance variables produce a smaller mean NLL increase of 0.0415, but the paired combined-versus-nuisance interval includes zero. We then use calibration data to define a community-support guard: predictions outside the 95th percentile of calibration-standardized community displacement revert to the generic calibrator. The guard reduces combined NLL by 0.1133 and high-confidence errors from 43.25 to 33.25 per 400-sample cell. It also outperforms matched-rate random fallback on the original folds, while generic and uncalibrated probabilities retain lower held-out-type NLL. This exploratory recovery localizes harmful combined predictions; the stress test exposes heterogeneous calibration failures without establishing a general structural-calibration penalty or a deployment gain over simpler alternatives.

Contact-rich aerial manipulation remains fundamentally challenging because interaction forces are directly transmitted to the aerial platform, often leading to instability and degraded task performance. While compliant manipulators can mitigate these effects, existing aerial manipulation systems typically struggle to reconcile interaction compliance with manipulation precision. To this end, this article presents an aerial rigid-soft integrated manipulator (AeRSoM) robot that realizes embodied compliance for aerial manipulation. The proposed system integrates a fully actuated aerial platform, a rigid-soft manipulator, and variable-stiffness regulation to simultaneously achieve stable flight, compliant interaction, and precise manipulation. By distributing compliance throughout the manipulation system, the proposed design leverages distributed embodied compliance to passively absorb contact disturbances while preserving sufficient stiffness for task execution. To fully exploit the mechanical design, a composite control framework is developed for precise end-effector trajectory tracking in the presence of uncertainties and external disturbances. Extensive real-world experiments are conducted in representative contact-rich aerial manipulation tasks, including dynamic transmission-line grasping, physical interaction with a wind turbine blade, peg-in-hole, and screwing operations. The results demonstrate that the proposed rigid-soft integration significantly improves interaction robustness and task adaptability while maintaining manipulation accuracy, highlighting that embodied compliance provides a promising design paradigm for enhancing the safety, robustness, and versatility of aerial manipulation.

Action-chunked visuomotor policies predict overlapping trajectories, so every executed action is covered by several predictions. Temporal ensembling smooths execution by combining these predictions with an exponentially weighted mean. One corrupted prediction can move the aggregate without bound: its breakdown point is 0. We use adversarial corruption to stress this deployed aggregator and to compare two kinds of guarantee. A metric guarantee bounds the response to a perturbation of a given size. A combinatorial guarantee instead bounds the damage when at most q of the M candidates covering a timestep are corrupted, whatever their size. Encoder adversarial fine-tuning recovers 44% of the loss under the published patch attack, but only 7.3% after the attacker's step size is increased. By contrast, the coordinate-wise median of the same candidate set keeps its recovered fraction flat as attack optimisation increases. Median temporal ensembling costs one line and requires no retraining. Across 25 (configuration, corruption-level) combinations it is never worse than the mean and is significantly better in 15. It also transfers to a second policy class, and it recovers performance under a failure with no attacker in the loop at all: camera frames that arrive blank. Its effect on clean data is configuration-dependent, from -0.04 to +0.07. We also give the boundary: corruption that shifts every covering prediction by the same amount is invisible to this whole family of statistics, and no equivariant aggregator can remove it.

Capability and efficiency are two key dimensions of reasoning in large language models (LLMs). Capability refers to the ability to solve a given problem correctly, whereas efficiency refers to the ability to do so with limited resources. When LLMs use Chain-of-Thought (CoT) reasoning to solve problems of controlled hardness, both the number of problems solved correctly and the number of tokens required to reach a correct answer depend on problem hardness and model size. However, how these factors jointly shape capability and efficiency remains poorly understood. Here, we use hierarchical Bayesian models to evaluate the capability and efficiency of LLMs from the DeepSeek-R1-Distill model family across four classes of arithmetic and algorithmic reasoning problems. At a fixed model size, the probability of correctly solving an instance decays approximately exponentially with instance size, our proxy for problem hardness. The decay scale grows sublinearly with model size, indicating that larger models are more capable, but that capability gains diminish with scale. Output length grows as a power law with instance size, which serves as a proxy for difficulty. However, the parameters of this power law do not vary systematically with model size, suggesting that larger models do not become more efficient. Together, these findings reveal potential limitations of naive scaling as a strategy for developing more capable AI systems: capability improves with diminishing returns, while efficiency shows little to no improvement.

Large language models (LLMs) are increasingly used to measure public value orientations from long social media posts, yet such posts often mix background, quotations, concessions, and only a few stance-bearing sentences. Existing approaches either ask the model to predict a document-level label directly, which can be overconfident, or aggregate sentence-level predictions by majority or soft voting, which treat uncertain and decisive sentences as equally informative. We formulate long-text value measurement as a decision-fusion problem and propose Tempered Evidence Fusion (TEF), a training-free rule that weights each sentence's log-odds by its normalized information gain, as derived from a generalized Bayesian posterior. This makes the fused score nearly vanish for uncertain sentences while preserving the Bayes-optimal weight of decisive evidence. We further introduce Multi-event Insight Network Dimensions (MIND), a benchmark of 8,358 Chinese and English posts spanning five years of public events and six value dimensions. On MIND, TEF outperforms the strongest baseline among Direct, Majority Vote, and Soft Vote by an average of 4.5 accuracy points and 4.6 macro-F1 points across five LLMs and two languages. MIND dataset and code are available at https://github.com/Kzczc/ICASSP2027-TEF.

In the pursuit of steady-state fusion energy, Advanced Tokamak regimes rely on Internal Transport Barriers (ITBs) with reversed magnetic shear. While Resonant Magnetic Perturbations (RMPs) are commonly applied to control Edge Localized Modes (ELMs) by intentionally making the plasma edge chaotic, the RMPs can also affect the reversed-shear cores. To address this compatibility, we propose a new, adjustable analytical model for plasma current that explicitly incorporates the core, ITB, and edge pedestal components. By applying this current formulation to a Hamiltonian map, we compare how easily magnetic chaos reaches the core in ITB-driven reversed-shear (Non-Twist) plasmas. We find that the localized ITB current creates "twin" inner resonances and pushes outer magnetic island chains closer to the edge. These results highlight a critical operational trade-off: while the Non-Twist topology makes ELM control easier at lower coil currents, it severely shrinks the safety margin against a complete, RMP-induced loss of global confinement.

In this paper, we establish a Bourgain–Brezis–Mironescu formula for BV functions on arbitrary open sets for a general class of radial mollifiers. Building on this formula, we identify the \(\Gamma\)-limit of the improved nonlocal energies with respect to \(L^1\)-convergence as a constant multiple of the total variation. As a consequence, we obtain a characterization of BV functions in terms of the finiteness of the corresponding nonlocal energies. We also establish local \(L^1\)-compactness on arbitrary open sets and global \(L^1\)-compactness on bounded open sets under a natural uniform integrability assumption. In particular, except for the global compactness result, all these results hold on arbitrary open sets, with no assumptions on boundedness, connectedness, or boundary regularity. Counterexamples show that for global compactness, both the boundedness of the domain and the uniform integrability assumption are essential. The proofs rely primarily on one-dimensional restrictions of BV functions and a concentration argument for radial mollifiers.

IC3 is a state-of-the-art algorithm for hardware model checking that proves safety properties by incrementally constructing an inductive invariant consisting of a set of lemmas. Its effectiveness depends on generalization heuristics that identify useful lemmas and guide proof search. However, many leading IC3 hardware model checkers operate on lowered, bit-level representations, where high-level design relationships are difficult to exploit for generalization. Those operating at a higher level remain limited in exploiting high-level design structure and semantics. We present Agentic-IC3, built on Pono's word-level model-checking infrastructure, which integrates a language-model agent into IC3 to guide semantic proof search using register-transfer-level (RTL) design information. The framework exposes an agent-oriented interface to a persistent IC3 backend, allowing the agent to interact with an explicit, evolving proof state throughout verification. Across successive proof obligations, the agent relates intermediate proof states and solver feedback to the RTL and proposes high-level lemmas through both SAT and UNSAT generalization. Beyond generalization, the agent can introduce derived observation signals to express design relationships succinctly and obtain more informative feedback, and backtrack to revise proposals that lead to unproductive proof branches. The backend checks proposals before updating the proof state, preserving soundness and providing feedback for further reasoning. On a suite of 14 benchmarks spanning security information-flow verification and functional verification of communication protocols, processors, and functional units, Agentic-IC3 solves 10 cases within a one-hour timeout, including 4 unsolved by all three evaluated baselines: rIC3, Pono-IC3Bits, and A-IC3.

Resource allocation systems often restrict each request to a short list of options before coordinating assignments globally. We study this separation in stochastic bipartite matching under independent vertex arrivals. Each request draws a state from its own known distribution, determining its compatible resources, and independently retains a menu of at most $k$ edges. A maximum matching is then computed on the retained graph. We show that bounded local menus universally suffice for near-optimal matching. For every $\varepsilon>0$, there is a menu size $k_\varepsilon$ depending only on $\varepsilon$ that preserves at least a $(1-\varepsilon)$ fraction of the expected maximum-matching size of the full realized graph. Earlier guarantees required additional assumptions on how matching mass is distributed across edges; our result resolves the unrestricted case. Moreover, the menus are simple to generate from any benchmark matching rule, either by weighted sampling according to the benchmark's edge marginals, or by applying the benchmark to sampled realizations and retaining the resulting partners. Our proof constructs a near-optimal certificate inside the sparsifier by combining a locally computable surrogate for the large-marginal edges with a fractional completion from sampled light edges. The surrogate nearly preserves the benchmark's value and endpoint loads while controlling dependencies, which makes the statistical light-edge completion possible.

Unlike most intensely physical pursuits, surgical robotic teleoperation training focuses primarily on task outcomes rather than surgeon body posture or biomechanics during task completion. Toward the question of the role of biomechanics in surgical expertise, we sought to characterize the articular motion of expert teleoperators as compared to novice users. Twenty-seven novices and nine experts completed a non-medical cylinder-on-peg transfer task while their upper limb biomechanics were recorded via motion trackers. During more difficult motions, experts stabilized their wrist motion more than novices, while maintaining adequate range of motion in their shoulder and elbow and completing the task significantly faster than novices. This marker of expertise suggests the importance of attention to user biomechanics during teleoperation of surgical robots.

We investigate the impact of axions on the period-luminosity relation (PLR) of RR Lyrae stars. During the red giant branch phase, emission through the electron coupling cools the core, increasing its mass at ignition. The larger core produces more luminous horizontal-branch stars, systematically shifting the zero point of the RR Lyrae PLR toward brighter magnitudes. We model the RR Lyrae population of the globular cluster M3 using stellar evolution and pulsation calculations tailored to its age and composition, and compare the resulting synthetic $K_s$-band PLR with observations. For $\alpha_{26}\equiv 10^{26}g_{ae}^2/4\pi=0.5$, the axion-induced shift in the predicted PLR is comparable to the current uncertainty in the absolute M3 zero-point calibration.

To make claims about representations that generalize beyond a particular trained model, we need to specify when two representations should count as equivalent. The Linear Representation Hypothesis is often discussed without making this equivalence explicit. Different notions of equivalence preserve different structures, so metrics, probes, and interventions that appear to study the same representation may in fact correspond to different hypotheses. We therefore argue that the Linear Representation Hypothesis is not one hypothesis but a family of claims distinguished by representation equivalence. We formalize this idea using group actions, specifying the representation object, the procedure that produces it, and the property ultimately asserted, while accounting for equivalences imposed by the model architecture. This framework clarifies how assumptions can change across metrics, reading points, and analysis stages, and we use it to audit common representation quantities and recent interpretability analyses.

Recently, Heath, McCourt, Parker, Schwieder, and Zerbib initiated the systematic study of the $r$-Kneser Ramsey number $R_r^{KG}(s,t)$ and investigated related Ramsey-type problems. A central motivation for their work comes from a question of Holmsen, Hrusak, and Roldán-Pensado, who asked whether, for $n=2k-1$ and sufficiently large $k$, every red/blue edge-coloring of the complete graph on the vertex set $V = \binom{[n]}{k}$ necessarily contains a monochromatic triangle $ABC$ with $A,B,C \in V$ and $A \cap B \cap C = \emptyset$. Heath, McCourt, Parker, Schwieder, and Zerbib established that this conclusion holds when $n \ge \frac{7k}{3}$ and $k\ge 12$. We make substantial progress toward the problem of Holmsen, Hrusak, and Roldán-Pensado by proving that the conclusion already holds for every $k\ge 2$ whenever $n\ge 2k+1$. In addition, we obtain improved lower bounds for $R_r^{KG}(s,t)$ when $s$ and $t$ are fixed and $r$ is sufficiently large.

Large reasoning models can produce correct yet unnecessarily long reasoning traces. Existing methods improve reasoning efficiency with trajectory-level objectives or local token- and step-level signals, but rarely model inter-step semantic dependencies. This limits their ability to distinguish redundant steps from those that support later deductions, making it harder to shorten reasoning without sacrificing accuracy. We introduce RECAP (REdundancy-aware Credit Assignment via Propagation), which addresses this limitation by assigning credit where it is due based on both a step's downstream role in the reasoning structure and its contribution to solving the problem correctly. We define structural responsibility to capture the step's downstream role by measuring how strongly later reasoning depends on it, using credit propagated backward from the final-answer node through an outcome-independent, LLM-annotated semantic dependency graph. However, a step can have high structural responsibility yet steer the reasoning away from the correct solution. RECAP therefore introduces step efficacy to measure answer-directed progress through changes in gold-answer log-likelihood as each step is added. Together, these signals reshape rollout-level GRPO advantages into step-specific updates. RECAP requires neither a separately trained process reward model nor preconstructed concise trajectories. Across two 7B models and four mathematical reasoning benchmarks, RECAP improves the accuracy-efficiency trade-off. On Qwen2.5-Math-7B, it improves pass@1 by 2.0-3.7 percentage points while reducing reasoning tokens by 8%-31% relative to GRPO across all four benchmarks. Analysis suggests these savings reflect fewer reasoning operations and less dead-end reasoning, rather than more compact expression.

With recent advancements in large language models (LLMs) and LLM-based agents, these agents are becoming increasingly autonomous and gaining broader access to act on users' behalf on the internet. However, the vulnerability of automated agents deployed on social media platforms (e.g., for managing a user's personal account) remains underexplored. Existing studies on agent poisoning typically assume that the adversary can expose poisoned content to the agent. Although such an attack is direct and effective, it is more easily detected and mitigated. In the context of social media platforms, this leaves open whether the recommendation system itself would surface such content to the agent in a more subtle manner. Through theoretical analysis, we show that the like-score mechanism used in OASIS can be exploited, and we characterize the conditions under which a multi-stage chain of poisoned posts can steer the agent's feed. Based on these insights, we further develop an algorithm that crafts realistic poisoned posts. Experiments support our theoretical findings and demonstrate the effectiveness of the proposed algorithm. Notably, by exploiting the like-score feedback loop, the attack causes the recommendation system to select poisoned posts even when their user-post similarity falls below the retrieval threshold.

In human-robot interaction, traditional interfaces such as joysticks and handheld tablets introduce latency into navigation tasks and require the operator's explicit attention on the device, instead of the robot. We propose a new human-robot interaction framework in which a human communicates intent directly through sparse multimodal signals such as physical pushes and spoken commands. Human intent is represented as a parameterized linear dynamical system (LDS) that encodes the desired goal and motion behavior. The robot estimates this intent (parameters) online using a particle filter, where each particle represents a candidate LDS hypothesis and is reweighted online as new information becomes available. We validate this framework on a robotic blimp, whose inherent compliance and collision tolerance make it well-suited for repeated physical interaction. Experiments with multiple participants across 300 trials show that combining pushes and voice commands identifies the intended goal in 86% of trials within at most five interactions, with most trials resolved in two. The inferred dynamical systems can also produce curved trajectories that avoid obstacles known only to the human.

We study the Ooms multiplicities of Frobenius maximal parabolics $W(a,b)$ of type~$A$. For $a=mb+r$ with $1\le r<b$, the Euclidean algorithm relates their potential histograms to those of the smaller algebra $W(r,b-r)$. We prove strict unimodality for every positive coprime pair: the multiplicities increase strictly to the equal central values at eigenvalues $0$ and $1$, then decrease strictly. This strengthens the recent maximal-parabolic unimodality theorem of Giaquinto, Irving, Lauve, and Mastnak. We also prove log-concavity for the unbounded families $W(mb\pm t,b)$, where $t\in\{1,2,3\}$, $b>t$, $\gcd(b,t)=1$, and $m\ge1$. These results cover every Frobenius case with smaller block at most $8$, and also smaller block $10$. For each fixed residue class, we then prove that log-concavity of the whole family is determined by finitely many initial spectra. The finite cutoff is obtained symbolically. Exact integer verification of the resulting finite lists proves strict internal log-concavity whenever the smaller block is at most $128$, with no bound on the larger block. The unrestricted log-concavity conjecture is not proved here.

We prove a sparse version of the four-colour theorem of Brandt and Thomassé, answering a question of Allen, Böttcher, Kohayakawa and Roberts. For every fixed $0<\gamma\le1/10$ and every $p=p(n)\in(0,1]$, asymptotically almost surely every spanning triangle-free $H\subseteq G(n,p)$ with $\delta(H)\ge(1/3+\gamma)pn$ can be made four-partite by deleting at most $\min\{C_\gamma n/p,(1/8+\gamma)pn^2\}$ edges. In fact, deleting at most $C_\gamma n/p$ edges yields a graph that admits a homomorphism to an Andrásfai or Vega graph with certificate complexity at most $1/(3\gamma)$. Together with matching lower bounds from random blow-ups, this structural result determines, uniformly in $p$, the minimum-degree thresholds for $q$-partiteness with $O(n/p)$ edge deletions: $2/5$ for $q=2$, $10/29$ for $q=3$, and $1/3$ for every fixed $q\ge4$. For every fixed $q\ge2$ and $\log n/n\ll p\ll n^{-1/2}$, asymptotically almost surely $G(n,p)$ contains a spanning triangle-free subgraph with minimum degree $(1-o(1))pn$ that requires $(1/(2q)+o(1))pn^2$ edge deletions to become $q$-partite, showing that the coefficient $1/(2q)$ cannot be improved even under this stronger degree condition.

The principal commutation formula underlying the paper by Deift [24], is of the type \[ - z (AB - z I)^{-1} + A (BA - z I)^{-1} B = I, \quad z \in [\rho(AB) \cap \rho(BA)]\backslash\{0\}, \] where, temporarily, and for simplicity only, $A,B$ represent bounded operators in a Banach space. A fundamental consequence of this commutation formula is the fact that $AB$ and $BA$ are essentially isospecral in the sense that \[ \sigma(AB) \backslash \{0\} = \sigma(BA) \backslash \{0\}. \] Moreover, under appropriate trace class hypotheses, this implies the fundamental identity for (Fredholm) determinants \[ \det(I - AB) = \det(I - BA), \] which extends to modified (or regularized) Fredholm determinants. In this paper we briefly recall some of the applications of these commutation formulas to isospectrality, integrable systems, regularization of PDEs, and estimates in Statistical Mechanics and Quantum Field Theory discussed in [24]. Subsequently, we review some of the ramifications of commutation formulas that were established over the past fifty years, including determinants and dimension reduction, random matrix theory and dimension expansion, integrable operators, Poncelet's porism, interpolation, eigenvalue computation, KPZ, Pfaffians, the focusing Gross-Pitaevskii equation, one-dimensional scattering theory, Floquet theory, and the Miura transformation connecting the KdV and mKdV hierarchies.

Multi-agent debate (MAD) has emerged as a promising paradigm for improving the reasoning accuracy of large language models (LLMs) through iterative peer interaction. Communication topology plays a central role in this process, motivating increasingly sophisticated mechanisms that learn, adapt, or dynamically reconfigure agent interactions to improve accuracy or reasoning reliability. Meanwhile, prior studies suggest that much simpler sparse communication can already achieve competitive performance at substantially lower cost. In this work, we take a closer look at sparse MAD and ask whether complex topology control is actually necessary to improve collective reasoning. We find that a simple random-without-replacement routing policy, which lets each agent debate with two distinct and newly sampled peers at every round, provides a surprisingly strong baseline and consistently improves the accuracy-cost trade-off of sparse MAD. Building on this observation, we further study deliberation stopping and show that lightweight stopping can substantially reduce inference cost while preserving competitive accuracy. Our results suggest that sophisticated topology control such as learned topology adaption should be evaluated against strong simple routing and stopping baselines before its additional complexity is justified.

Change Detection (CD) in remote sensing requires jointly modeling bitemporal context and fine-grained spatial details to accurately localize changed regions. While dense attention effectively captures bitemporal relationships, its computational cost limits scalability to high-resolution imagery. In contrast, conventional feature fusion and coarse decoding often struggle to distinguish true changes from appearance variations and preserve object boundaries. To overcome these limitations, we propose Bitemporal Mamba-Diffusion for Change Detection (BMD-CD), a framework that integrates temporally structured state-space modeling with logit-space diffusion refinement. BMD-CD first converts deep bitemporal features into region tokens and organizes them into explicit temporal partitions for bidirectional state-space propagation. It then employs a Learnable Trigonometric Fusion (LTF) module to integrate temporal information and a Conditional Diffusion Decoder that refines coarse predictions in logit space with a short DDIM schedule to generate fine-grained, high-resolution change maps. Extensive experiments on LEVIR-CD, WHU-CD, DSIFN-CD, CDD, and S2Looking demonstrate the effectiveness of the proposed approach. BMD-CD achieves IoU scores of 0.923, 0.957, 0.882, and 0.979 on WHU-CD, DSIFN-CD, LEVIR-CD, and CDD, respectively, and further exhibits strong zero-shot robustness on ValaisCD and B-FLAIR-test.

The classical $A_2$-frame construction of the Coxeter–Todd lattice uses the hexacode as its glue. Here we replace the inert Eisenstein reduction at $2$ by the ramified Gaussian chain ring $S=\mathbb{Z}[i]/(1+i)^3$. For odd $\ell$, a one-generator cyclic graph code over $S$ lifts from an $A_2(4)^{2\ell}$-frame to an even strongly $6$-modular lattice $L_\ell$ of rank $4\ell$. In the real frame ordering the same glue is a one-generator negacyclic code. The cases $L_5$ and $L_7$ have minima $6$ and $8$ and give examples for the type-$6\mathrm{a}$ entries in dimensions $20$ and $28$, which are listed as open in the Nebe–Sloane catalogue. After completing the square, the minimum calculation becomes a four-symbol carry problem; for $L_7$ the key relation is the boundary map of a $7$-cycle. For $\ell=3$, $L_3$ has an index-$8$ overlattice isometric to the Coxeter–Todd lattice.

Sky-projected mass distributions of galaxy clusters are widely used to constrain particle properties of dark matter, to magnify in flux and angular extent high redshift background galaxies, and to construct their luminosity function down to faint absolute magnitudes that are otherwise hard or impossible to reach. Obtaining accurate and precise mass maps of clusters is therefore of central importance for these studies. Yet, a visual inspection of any two mass maps of the same cluster shows that the shapes of their isodensity contours are often dissimilar in detail on <30kpc scales, and there is no simple relation connecting the two. These differences are due to systematic uncertainties in reconstructions, which are in turn due to lensing degeneracies. However, commonly known degeneracies, like the mass sheet degeneracy or the source position transformation cannot account for most of these differences, implying that more generalized shape degeneracies are responsible. In this paper we study shape degeneracies using an analytical approach. We conclude with three main takeaways: (i) Shape degeneracies are the likely cause of differences in the mass maps of cluster lenses recovered by different techniques. (ii) Shape degeneracies are significantly suppressed in regions of the lens plane that have tight groupings of lensed images. (iii) In clusters where image numbers are high, such as MACS J0416, their effect on time delays between images from the same source are generally small, of order 1%, but can reach ~10% for some multiply imaged background sources, affecting the determination of H_0.