Papers

Agent skills are Markdown documents that a tool-using agent reads as instructions and then acts on with the privileges of the session that loaded them. Installing one grants authority (the standing to direct the agent, not new privilege), and the size of that grant is what we measure. We present a two-phase attack-surface profiling methodology that combines whole-bundle static analysis and dependency provenance checking with instrumented sandbox execution and separate model-based assessment of the resulting traces. We apply the methodology to 6,309 listings from two marketplaces, selecting listings for execution without conditioning that decision on a prior threat or risk signal. Half the corpus carries evidence of shell-execution capability; the corresponding shares are 35% for filesystem read, 31% each for filesystem write and network egress, and 13% for credential access; seven listings in ten carry at least one elevated capability. That is the surface a user accepts at install time, and almost none of it is visible then. A permission model exists but is close to inert. Authors may list the tools a skill pre-approves in its own frontmatter, which removes confirmation prompts rather than restricting what the skill can reach; of the 6,061 listings whose frontmatter parses, 770 declare anything at all, the vocabulary cannot express credential or environment access, and nothing checks what it does express against the bundle. Declaring is also not declaring accurately: 59% of declaring listings understate what their bundle can do. ClawHub declares roughly twice as often as skillsmp, yet under-declares on 88% of its declarations against 31%. Triage at the marketplace boundary therefore cannot rest on what a listing declares. It must rest on what the bundle can do. These results position capability-oriented attack-surface profiling as a necessary complement to maliciousness detection.

We give an infinite family of non-diffeomorphic exotic $\mathbb{R}^4$'s, each of which is the union of two copies of a standard open tubular neighborhood of the circle, which intersect in a standard open tubular neighborhood of the wedge of two circles. This gives the simplest known decomposition of an exotic $\mathbb{R}^4$ into two open sets, making progress on Question 4.43 (A) of Kirby's 1995 problem list. The decompositions are obtained by studying fibered slice disk complements in certain Casson-Gordon homotopy 4-balls, for which we provide similar decompositions. We distinguish the $\mathbb{R}^4$'s using end Floer homology.

Robots following persistent language instructions must decide which cached room judgments remain valid as their environment changes. We present MIND (Maintenance via INstruction-conditioned Dependencies), a framework for maintaining target-room decisions without re-evaluating every room after each event. A language model compiles each instruction into dependencies for semantic qualification and target selection. The controller intersects semantic dependencies with event footprints, refreshes affected judgments, and updates reachability and selection separately. For fixed deterministic evaluation, planning, and selection, conservative dependency and footprint coverage preserves the decisions of full recomputation. On 240 five-event ProcTHOR sequences with an exact benchmark predicate, MIND reduces mean semantic evaluations from 30.6 to 11.2 while preserving all 1,200 post-event decisions. Frozen-evaluator experiments with Phi-4 and Qwen3-14B also preserve all 1,200 decisions, without improving the evaluators' underlying task accuracy. In 32 closed-loop AI2-THOR episodes using exact-predicate qualification and metadata-derived footprints, MIND matches the final target decision of full recomputation in every episode. Unitree G1 experiments demonstrate target reuse and reselection across room transitions without additional semantic evaluations under MIND, and abstention after a barrier and an object removal at 4 evaluations against 15 to 18. Additional materials are available at https://mind-icra27.github.io/.

Generators of continuous-time quantum Markovian evolution, known as Lindblad operators, play a key role in the theory of open quantum systems. The question of how to define and characterize random Lindblad operators has attracted considerable attention in recent years. Purely dissipative random Lindbladians have been extensively studied using toolboxes of free probability and non-Hermitian random matrix theories, which helped, among other results, to uncover a characteristic lemon-like spectral support. In this work, we focus on random Lindbladians with a Hamiltonian component. We analyze three ensembles of increasing complexity, designed to illustrate the joint effects of dissipative and Hamiltonian dynamics. While the first two ensembles allow for an exact analytic treatment with standard probabilistic and free-probability methods, the generic case of independently random Hamiltonian and dissipative components requires a novel framework that combines operator-valued free probability with sequential noise averaging. For this ensemble, our approach preserves the underlying tensor-product structure of the Lindblad operator and derives a general equation for the spectral boundary. We characterize the effect of Hamiltonian-induced spectral squeezing and explain how the spectral support changes with the relative Hamiltonian strength and the rank of the dissipative component.

We study the Ahlfors-regular conformal dimension of spaces with locally expanding self-covering maps and, more generally, limit spaces of contracting self-similar groups. We show that the conformal dimension is characterized by combinatorial moduli associated with the Schreier graphs of these groups. We compare these moduli with contraction coefficients of self-similar group actions and with algebras of functions on the limit spaces satisfying asymptotic $p$-energy conditions. We also give conditions on the group when the conformal dimension is equal to the infimum of visual metrics on the self-similarity complex.

Phononic integrated circuits enable the development of compact systems to manipulate radio frequency signals through interconnected networks of acoustic waveguides. Introducing topologically protected robustness in these circuits to counter fabrication imperfections and local perturbations, remains a challenge. This work presents a validated modeling framework for topologically protected edge states using surface acoustic waves in etch-defined waveguides consisting of scandium aluminum nitride on silicon carbide, directly comparing analytical, full-wave numerical simulation, and laser Doppler vibrometer measured experimental approaches and their agreement. This work represents the first experimental demonstration of topological edge states on a ScAlN on SiC platform, validated through direct quantitative comparison with analytical and numerical analyses, with future implications for topological protection in phononic and optomechanical integrated circuit platforms.

We develop a systematic, Whitham-based construction of perturbative/non-perturbative (P/NP) relations for one-dimensional quantum mechanical systems admitting a genus-one WKB period description, and derive a higher-genus formulation under explicit regularity assumptions. We select external deformations that preserve the independent quantum residues and lift them by varying the internal moduli so that the independent perturbative quantum periods remain fixed. The Riemann bilinear identity then determines the induced evolution of the non-perturbative periods through local residue pairings and the inverse period Jacobian. At genus one, this reduces to a relation between the flow of the non-perturbative quantum period and the energy derivative with respect to the perturbative action, providing a constructive connection between the Rayleigh-Schrödinger expansion and fluctuations around non-perturbative saddles. We recover known P/NP relations and derive extensions for explicitly coupling-dependent potentials, which we verify at the computed orders through independent perturbative and non-perturbative period calculations for quantum-corrected quartic and Jacobi elliptic examples.

Memory for long-horizon video generation must satisfy two competing requirements: it must remain bounded so generation can continue indefinitely, while still recovering what was observed at a place when the camera returns, regardless of how long ago it was seen. Reconstruction-based memories provide spatial recall by caching and reprojecting explicit geometry, but require growing storage and depend on estimated 3D structure. Recurrent memories keep a constant-size state, but retention is governed by temporal recurrence, causing old observations to fade with time. We introduce SMART, a slot memory addressed by region tokens, which combines bounded storage with spatially addressed recall. SMART stores content in a constant-size slot recurrence whose writes are addressed by the observing camera and whose reads are queried by the rendering camera, without constructing an explicit 3D representation or maintaining a growing cache. Consequently, recall is conditioned on inter-view geometry rather than on recency. Each read additionally provides a confidence estimate that conditions a chunk-causal diffusion transformer, allowing it to selectively trust retrieved content while relying on its generative prior elsewhere. On long trajectories in Memory Maze and Minecraft, SMART maintains high-fidelity recall far beyond its training horizon while using constant memory and per-step cost.

Constituent mass predictions for bottom-charm $[bc]$ color-antitriplet diquarks are generated using QCD Laplace sum rules. We calculate the diquark correlator within the operator product expansion to next-to-leading-order (NLO), including terms proportional to the four- and six-dimensional gluon and six-dimensional quark condensates. Existing analytic expressions for the resulting multi-scale master integrals are insufficient for these calculations, so we utilize numerical $D$-dimensional loop integration techniques. The sum rule analyses stabilize in all four channels yielding constituent masses in GeV of: scalar ($0^+$) $(5.60\pm0.56)$, pseudoscalar ($0^-$) $(6.04\pm0.60)$, axial vector ($1^+$) $(5.60\pm0.56)$ and vector ($1^-$) $(5.99\pm0.60)$. These constituent mass results are discussed in comparison with values used in diquark models of tetraquark hadronic states. We also summarize QCD sum rule predictions for diquark masses for all possible quark-flavor combinations, providing QCD-based criteria for assessing the diquark mass parameters used in constituent diquark models of hadrons.

Quantum-walk algorithms can quadratically reduce the dependence of spatial search on the classical hitting time $\mathrm{HT}$. Realising this speedup using standard quantum-walk algorithms, however, requires deep coherent circuits, limiting their applicability on near-term and early fault-tolerant quantum devices. We derive a parameterised family of hybrid quantum-classical spatial search algorithms that continuously interpolates between the classical and fully coherent quantum regimes, trading off circuit depth for total query complexity. For any $\gamma\in[0,1]$, our algorithms use circuits of depth $\widetilde{O}(\mathrm{HT}^{\gamma/2})$ and find a marked vertex with constant success probability at total cost $\widetilde{O}(\mathrm{HT}^{1-\gamma}\mathsf{S} + \mathrm{HT}^{1-\gamma/2}(\mathsf{U} + \mathsf{C}))$, where $\mathsf{S}$, $\mathsf{U}$, and $\mathsf{C}$ denote the setup, update, and checking costs, respectively. We also derive an optimal continuous depth-query trade-off for fixed-point amplitude amplification, providing a potential building block for other hybrid designs.

Physics-informed neural networks provide a flexible framework for learning parametric PDE solution families without labeled data, yet preserving accurate derivatives across heterogeneous regimes remains challenging. Structured neural solvers offer compact representations, but restrictive conditional parameterizations can limit adaptation across the solution family. We introduce BeTINN, a structured conditional architecture extending time-induced modulation through multi-directional time-parameter conditioning and a joint residual pathway. We evaluate BeTINN on the Heston stochastic-volatility PDE, a demanding parametric problem with mixed derivatives and economically relevant first- and second-order sensitivities. BeTINN improves joint accuracy across normalized option value and key sensitivity metrics relative to our baselines. Controlled ablations characterize the roles of its architectural components. We further find that validation physics loss is informative but does not fully determine differential fidelity, and that accurate solution values can coexist with localized derivative errors. These results support structured conditional adaptation and motivate derivative-aware evaluation when sensitivities are downstream quantities of interest.

The aim of this note is to introduce natural geometry, in which the rectilinear plane is a primitive. This plane is admitted together with its rectitude, and its rectitude states that a Lambert quadrilateral has its fourth angle right. The fourth angle of a Lambert quadrilateral may be right, acute or obtuse, and this trichotomy distinguishes three planes. Starting from the rectitude, we prove the uniqueness of the parallel by a chain of neutral steps: the rectangle, the triangle whose angles sum to $\pi$, and Legendre's theorem \cite{legendre}, all borrowed from the classical literature. That uniqueness is equivalent to the fifth postulate, which we thus reach in Euclid's own wording: the equivalence is read in the natural direction, and it is not a derivation of the fifth postulate from the other four, since the independence of that postulate is not in question. In the rectilinear plane the parallels are characterised as exactly the images of a line under a point-reflection, and equivalently as the pairs of equidistant lines. The other two planes are then placed beside the rectilinear one, and the same definitions serve for all three.

Fairness audits with several sensitive attributes are often performed one coordinate at a time. This can miss purely intersectional disparities: a score may be independent of each sensitive coordinate while depending on their joint profile. We study distributional demographic parity for a real-valued score using one-dimensional Wasserstein geometry. We distinguish two complementary constructions. Conditional and sequential audits characterize joint parity through their zero sets, but their positive values do not generally decompose a fixed joint Wasserstein disparity. By contrast, full-cell conditional quantile functions define a Hilbert-valued random variable whose variance is the joint disparity. Conditional expectations of this quantile process induce a monotone information game: every reveal order yields an exact martingale decomposition of the same joint disparity, while Shapley values provide an order-free attribution across sensitive attributes. Under independent sensitive attributes, the game reduces to a Hoeffding-ANOVA decomposition and Shapley splits each interaction energy equally among its members; under dependence, orthogonality is lost and explicit formulas reveal suppression and redundancy effects that marginal audits can hide. We give fixed-grid estimators and permutation calibration, and illustrate the framework on synthetic examples and ACSIncome. The resulting framework separates three questions that are often conflated: whether joint distributional parity holds, how a fixed joint disparity decomposes along an information path, and how that disparity can be attributed across sensitive coordinates without privileging an ordering.

Let $G$ be a finite group, $M$ a $G$-module, and $F$ a field. A continuous group homomorphism $i : \Gamma_L=\text{Gal}(L_{\text{sep}}/L) \rightarrow G$ yields a homomorphism $i^* : H^d(G,M) \rightarrow H^d(L,M)$ for every $d\in \mathbb{N}$. An element $u \in H^d(G,M)$ is negligible over $F$ if $u\in\ker(i^*)$ for every $i$ and every field extension $L/F$. If $M=\mathbb{F}_p$ for a prime $p$ with a trivial $G$-action, an element $u$ is eventually negligible over $F$ if $u^k$ is negligible over $F$ for some $k\geq1$. The (eventually) negligible elements over $F$ form an ideal of the cohomology ring $H^*(G,\mathbb{F}_p)$, called the (eventually) negligible ideal. We prove an analog of Quillen-Venkov for eventually negligible elements. An element $u\in H^*(G,\mathbb{F}_p)$ is eventually negligible over $F$ if and only if the restriction of $u$ to the cohomology of each elementary abelian $p$-subgroup of $G$ is eventually negligible over $F$. Paired with applications of a Gysin-type long exact sequence, we identify generators of the negligible ideal over $\mathbb{Q}$ of the mod-2 cohomology of dihedral groups, cyclic $2$-groups, and generalized quaternion groups. We prove that the negligible ideal over $\mathbb{Q}$ of the mod-2 cohomology of every finite irreducible Coxeter group is radical. The result was previously known for Weyl groups; we prove it for the remaining non-Weyl types $I_2(m)$, $H_3$, and $H_4$.

Reduced-order models (ROMs) make model predictive control (MPC) of high-dimensional nonlinear systems tractable, but the resulting modeling error can lead to constraint violations on the real system. We introduce CoRE-MPC, a robust MPC formulation that plans with a nonlinear ROM and guarantees recursive feasibility and constraint satisfaction for the full-order system. It can be applied with a broad class of nonlinear model reduction techniques and the resulting MPC does not require evaluating the full-order model online, thus ensuring computational efficiency. The key insight is in how the ROM affects the dynamics of the prediction error. We account for this with two error radii that are propagated alongside the ROM and provide a guaranteed bound on the prediction error. The approach is validated in simulation on an 80-state oscillator chain, a Burgers equation, and a 9,768-state finite-element soft robot, where a two-state ROM with data-driven bounds keeps all 1,628 nodes clear of a wall at a median solve time below 1 ms for a 20 ms control period. Across these examples, CoRE-MPC is 13 to 61 times faster than full-order robust MPC at only 27 to 35% higher closed-loop cost, whereas existing certified schemes are either infeasible or at least twice as conservative. Code: [released upon publication].

We investigate Conjecture 7.4 of Guo and Krattenthaler \cite{GK}, which predicts that for every positive integer $n$, the polynomial \[ P_n(q) = \frac{(1-q)^2}{(1-q^{10n-1})(1-q^{15n-1})}\qbin{30n}{5n} \] has non-negative coefficients, except for $q^1$ and $q^{125n^2-25n+3}$, which equal $-1$. We study this problem through three main perspectives: First, by linking the boundary coefficients to the second differences of the partition function, we settle the first and last $25n+1$ coefficients via a novel partition injection. Second, by introducing a “defect” generalization of Andrews' classical lemma, we reduce the conjecture to an almost-unimodality property and employ a uniform saddle-point analysis to prove it analytically for all large $n \ge 2\cdot 10^{29}$. Finally, we connect these quotients to graph binomial coefficients, providing a natural orbit-counting interpretation via Burnside's lemma on directed cycles and paths.

Robotic disassembly of end-of-life (EoL) products requires robots to accommodate product variation, follow prerequisite sequences, and approach components precisely for contact-rich manipulation. Supervised fine-tuning (SFT) provides a practical path for adaptation, but limited and costly disassembly data cannot capture the distribution shifts arising from changing environments and product uncertainties. The policy's own rollouts record its approach attempts, and human corrections demonstrate recovery from its errors, but turning these experiences into better policy performance remains challenging. We introduce PACER, a Process-Aware Correction and Evidence Reweighting framework that learns from these experiences through SFT, using progress toward the target and each trace's source and outcome to weight valid action chunks in the native supervised action loss. PACER trains one policy per weighting setting and selects one of the resulting policies through held-out validation. We evaluate PACER on UR5e motherboard disassembly across five component classes, where it achieves 87/100 successful approach-and-handoff trials on the held-out benchmark, compared with 54/100 for vanilla post-SFT.

Transformers are intuitively considered to be efficient at copying from their inputs. We actually show that the ability of LLMs to copy in context can be improved through explicit modeling. We propose Copy-Paste Language Models (CPLMs), which add a lightweight Copy-Paste head to the conventional language modeling (LM) head. CPLMs improve training efficiency, downstream performance, and offer new operating points in long-context and retrieval setups. We validate our approach by pretraining models from 135M to 1.7B parameters, showing improvements across all scales and backbones with negligible training and inference overheads. We explain the success of CPLMs by showing that they partially mitigate the LM head gradient bottleneck through an alternative gradient path.

Equivariant structure-prediction pipelines alternate coordinate updates with the construction of a k-nearest-neighbour (kNN) graph whose membership is a discrete, piecewise-constant function of the coordinates. Rebuilding that graph at every block boundary (PB) is the expensive part of an epoch, whereas reusing one frozen graph (F) is fast but leaves a topology gap. We study a two-pass design in which the cheap pass records a small amount of state (margins, kNN radii, measured per-boundary displacements) and a second pass compensates conservatively. We formulate the choice of the graph-construction basis as a damped fixed-point iteration whose unique fixed point is exactly PB, and derive (i) exact non-flip certificates with the two-sided constant $m_i/4$ and one-sided constant $m_i/2$, (ii) the minimum-variance (BLUE) replay gain $\gamma^*$ and the resulting feasibility crossover, (iii) an $\ell_\infty$ trust-region projection that preserves convergence, and (iv) an analytic near-tie spacing law $E[m] \approx r_k/(3k)$ that matches measurements to 1%. Three compensation levels are implemented in C++ (L1: frozen set with live geometry; L2: margin-gated local repair; L3: damped, projected basis) and validated under a matched-conditions protocol (identical seed, MSA depth and trajectory family, only the topology mode differs). L2 reproduces the PB edge set exactly ($D = 0.0000$ at every boundary), L1 stays within 0.6% and L3 interpolates monotonically: L2 reproduces the PB edge set exactly when the trust region is removed, and L3 interpolates between the frozen and the PB topology. The recorded trajectory itself yields a measured $1.5\times$ Pass-1 and $1.76\times$ epoch speed-up with no measurable loss gap. An open-source implementation of the runtime and of all the experiments reported here is available at https://github.com/AlbertHuang96/ProteinPredictMachineLearning.

High-resolution optical imaging of the human retina is essential for diagnosing ophthalmic and neurodegenerative diseases. While adaptive optics combined with coherence gating can resolve retinal structures at the cellular scale, it requires complex, costly, and cumbersome hardware to physically correct for ocular aberrations and motion. Matrix imaging offers a powerful digital alternative by leveraging wave linearity to undo aberrations and multiple scattering; however, its in vivo deployment has been hindered by continuous, involuntary eye movements that destroy phase coherence during acquisition. Here, we overcome this fundamental barrier and demonstrate the first in vivo reflection matrix imaging of the human retina using an ultra-simplified optical setup without physical correctors. We introduce a fully computational framework that tracks and compensates for multi-axial ocular motion directly from raw interferometric data, thereby restoring the phase coherence of a multi-spectral distortion matrix. This enables local, high-order aberration correction to achieve diffraction-limited, volumetric imaging of retinal tissues. By shifting the hardware burden to physics-guided numerical computation, this paradigm opens the way for accessible, non-invasive, and high-resolution digital ophthalmology.

We present a systematic spectroscopic study of hydrogen Brackett-alpha (Br-alpha) and carbon monoxide (CO) fundamental-band (4.3-5.2 micron) emission from a homogeneous sample of 0.2-1.5 Msun pre-main-sequence (PMS) stars in the Orion Nebula observed with JWST/NIRSpec-MOS as part of our Chandra-JWST project. Using 73 Br-alpha and 46 CO spectra, we investigate correlations between warm-gas emission, stellar properties, accretion, and environment with censored correlation and regression analyses accounting for multiple testing. Br-alpha luminosity correlates strongly with accretion luminosity (Lacc), even after excluding sources within 2' of Theta1 Ori C, the nebula's main ionizing O-type star, whose radiation may bias proplyd Lacc estimates. This supports Br-alpha as a robust empirical tracer of accretion in PMS stars. Br-alpha also correlates with visual extinction (Av) and more weakly with stellar bolometric luminosity and mass. CO fundamental-band luminosities likewise correlate positively with Lacc and Av, relationships that persist in a larger combined JWST/NIRSpec+MIRI sample spanning multiple star-forming regions. In contrast to Br-alpha, CO luminosity shows no significant dependence on basic stellar properties, suggesting that warm molecular gas is governed primarily by accretion and local disk conditions. The weaker correlations of the Spitzer/IRAC spectral slope with Lacc and Av suggest that warm inner-disk gas may evolve or disperse faster than optically thick dust. Overall, the CO results tentatively indicate declining warm-gas emission as disks evolve, modulated by accretion-powered thermal and ultraviolet excitation, although variations in emitting area inherited from natal core conditions may also contribute.

The Briot–Bouquet theorem describes the meromorphic solutions of autonomous first-order algebraic differential equations. We prove Eremenko's conjecture extending it to all orders: if $f$ is meromorphic in $\mathbb{C}$ and $P(f,f^{(n)})=0$ for a nonzero polynomial $P$, then $f$ is rational, elliptic, a rational function of $e^{cz}$ with $c\neq 0$, or satisfies $f^{(n)}=af+b$. In particular, every meromorphic solution with $P$ irreducible of degree at least two belongs to the Weierstrass class. Solutions with poles were classified by Eremenko, Liao and Ng. We settle the remaining entire case with a rigidity theorem for polynomial compositions: if $h$ is nonconstant entire and $D^nR(h)=S(h)$, where $D=d/dz$ and $\deg R=\deg S\ge 2$, then $h(z)=Ae^{bz}+Be^{-bz}+C$. No growth assumption is needed. The proof turns the cancellation of leading terms into a contraction of indicator diagrams and obtains incompatible zero counts near an exposed edge. As an arithmetic application, an algebraic Laurent expansion at one algebraic point and an algebraic value at another force a solution to lie in $\overline{\mathbb{Q}}(z)$. Hence any two distinct derivatives of a function in Mahler's class (transcendental meromorphic functions with algebraic Laurent coefficients at every algebraic point) are algebraically independent over $\mathbb{C}$.

Saturn's moon Enceladus has active plumes suggestive of locally high heat fluxes at present, and the present-day heating is reasonably well characterized. However, the moon's thermal history is not well understood. In this study, we model flexural deformation of chasmata and dorsae in various regions of Enceladus to find the effective elastic thickness of the ice shell and constrain temporal and spatial variations in heat flux. We find paleo-heat fluxes of 60-$\geq$350 mWm$^{-2}$, with most clustered around 125-200 mWm$^{-2}$. Present day measured heat fluxes at the South Polar Terrain (SPT) are comparable to our estimated paleo-heat fluxes in non-SPT regions. Our paleo-heat fluxes, however, are significantly higher than the corresponding present day heat fluxes for the same locations inferred from shell thickness maps. This suggests that regions outside the SPT experienced a past episode of heating (heat fluxes in excess of 100mWm$^{-2}$) followed by a cooling and thickening ice shell. A relatively constant paleo-heat flux for terrains of different ages (determined from crater densities) supports the "resonance locking" hypothesis of dissipation in Saturn and might imply a long-lived resonance between Enceladus and Dione. Our results suggest that Enceladus's tidal heating appears to have been consistent over time. Consistent peak heating at Enceladus could support a long-lived subsurface ocean, which has important consequences for astrobiology and the habitability of the outer solar system.

Given a complete and non-compact Riemannian manifold of dimension $3\le n\le 7$, we find conditions for the existence of compact exhaustions $\{U_i\}_{i=1}^\infty$ such that all boundaries $\partial U_i$ have the same sign of mean curvature. In particular, we show that in these dimensions, a manifold with one end satisfying a no tentacle condition always has a weakly mean convex, weakly mean concave, or minimal exhaustion, according to the behaviour of the area profile of separating hypersurfaces. This is a version of a result sketched by Gromov, under a condition which is weaker than his thickness at infinity. In more general situations, we study the existence of exhaustions which are weakly mean concave or weakly mean convex apart from an arbitrarily small $L^1$ error. That is, we find sufficient conditions for the existence of exhaustions $\{U_i\}_{i=1}^\infty$ which satisfy $\int_{\partial U_i}H^+\to0$, $\int_{\partial U_i}H^-\to0$, or $\int_{\partial U_i}|H|\to0$, where $H^-$, $H^+$, and $|H|$ are the negative part, positive part, and the absolute value of the mean curvature respectively. As an application of our methods, we study the existence of almost $c$-convex exhaustions, that is, exhaustions whose boundaries satisfy $\int_{\partial U_i}(H-c)^-\to0$ for some $c>0$.

Financial disclosures express the underlying relationships of double-entry accounting through unstructured text. While financial LLM benchmarks predominantly emphasize question answering and classification, schema-guided evidence discovery remains relatively under-explored. We investigate whether open-weight LLMs can extract and classify evidence corresponding to predefined double-entry accounting relationships underlying the text. Using a financial accounting graph comprising 11 nodes (accounts) and 11 edges (transaction types), we evaluate mid-sized models from 5 open-weight families across 1,300 SEC 10-K Item 7 filings spanning 26 years, comparing their extraction performance against three proprietary models. Results reveal substantial differences in extraction volume, edge distributions, and cross-model agreement. Building on this study, we introduce a human-annotated relation classification benchmark comprising 483 passages across three financial accounting categories, evaluating 34 open-weight and 10 proprietary models. The strongest open-weight and proprietary models achieve 84.1% and 92.8% accuracy, respectively, with considerable variation across accounting categories. Our findings highlight the importance of evaluating evidence discovery and relation understanding separately, while investigating locally deployable models for scalable, privacy-sensitive financial analysis.

In this note we explore the following simple question: How does the classic gambler's ruin phenomenon change if we allow the gambler financial leverage? We show that, with a moderate amount of leverage, it is possible for an (idealized) gambler to sustain seemingly improbable profits over a remarkably long time horizon without ever going bankrupt, even if the gambler makes only fair bets. With a leverage ratio of $L = 10$ (less than what Lehman Brothers Holdings had from 1995–2008), the probability of multiplying one's money by $1000\times$ without going bankrupt is roughly $50\%$. More generally, given $L$-fold leverage, and given the goal of multiplying one's money by a factor of $r > 1$, the optimal gambling strategy has probability roughly $r^{-1/L}$ of success. The results in this note are designed as part of the course development for the CMU randomized algorithms course.

When writing code, the same computation can be expressed in different programming languages and through different implementation strategies. We show that the activations of LLMs capture this shared structure in reusable and composable abstract representations when they generate code. These representations emerge in the middle layers, are reusable across programming languages, and afford the model an impressive amount of generalization ability when generating code under various interventions. Using activation steering, we show that function, programming language, and implementation strategy can be independently manipulated. The independent interventions on language and function can be flexibly composed to control code generation, even in cases when the prompt contains no information about language and function. We additionally find that finetuning on examples of arbitrarily defined functions in one programming language transfers to other languages, consistent with the models reusing abstract features of code generation that are shared across different examples. Finally, to study how such abstract representations come to be, we implement a minimal model of shared, reusable, and composable structure in a synthetic setting, and show that transformers trained on such data learn representations that reuse these parts flexibly across contexts.

Patient-specific 3-dimensional (3D) models are increasingly central to planning surgery and catheter-based intervention in children with congenital heart disease (CHD), whose anatomy is often atypical and variable. These models inform care only when the surgeons and interventional cardiologists who make treatment decisions can rotate, crop, and measure them directly. Desktop modeling applications require local installation, capable hardware, and training. Commercial web-based viewers are portable but closed, and, in our experience, cannot display volume rendering, time-resolved (e.g., 4-dimensional) data, or the custom workflows centers build for CHD. Because CHD populations are small and heterogeneous, such workflows are unlikely to be prioritized commercially, and end users have no means to add them. To address these limitations, we developed the SlicerHeart Viewer, an open-source module that presents a simplified interface to a complete 3D Slicer/SlicerHeart session running on a server. Clinicians access models created by clinical modeling services from any device through a web browser. Because 3D Slicer runs underneath, volume rendering, time-resolved data, and any SlicerHeart workflow, including those developed for CHD, are available without modification. The module is independent of the hosting environment; we chose internal servers behind institutional single sign-on, and a cloud-hosted option is in development. Over one year at a single pediatric center, nearly all of 82 modeled cases were shared through the Viewer with clinicians from five specialties, with cases loading in 5 to 10 seconds. The approach is not specific to CHD and is deployable at other centers.

As its 40th anniversary nears, we present a history of the International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC), through interviews with figures from its past, present, and future.

Automatic stuttering systems primarily detect dysfluencies after they have occurred, limiting their use for anticipatory intervention. We address this gap by formulating stuttering prediction as continuous causal time-to-event forecasting, where a model estimates the risk of an upcoming stuttering onset every 100 ms using only preceding speech. To support this task, we augment FluencyBank Timestamped for 36 speakers with manually reviewed repetitions, prolongations, and blocks, yielding 2,872 forecasting events. We then model the temporal evolution of frozen Whisper-large-v3 speech representations using a compact causal Conformer with 3.67M trainable parameters. The proposed approach achieves an AUROC of 0.63. Temporal analysis further shows that explicitly modeling preceding speech states outperforms simply increasing the static acoustic window, while a discrete-time survival formulation provides coherent forecasts from 300 ms to 1.5 s using a single model. These results provide initial evidence that stuttering onset is preceded by measurable temporal information in speech and establish a framework for future anticipatory stuttering systems.