Papers

Spatially varying lattice distortions can produce local changes in electronic properties that are obscured by measurements averaging over extended regions. With its atomic-scale spatial resolution, scanning tunneling microscopy (STM) provides, in principle, a means of quantifying such distortions locally. A widely used approach for extracting local strain from STM images is the Lawler–Fujita (LF) algorithm [Lawler et al., Nature 466, 347 (2010)], which determines the displacement field from the phase of the lattice modulation and subsequently obtains the strain by differentiation. However, this approach requires two-dimensional phase unwrapping, which can fail in regions where the phase is ill-defined, particularly near topological defects such as edge dislocations. Here, we introduce an alternative approach that circumvents these limitations by directly measuring the local wave vector in real space. We derive the relationship between the local wave vector, strain, and the phase gradient, and show that the local wave vector mapping (LWM) algorithm remains robust even when the phase itself is ill-defined. This makes the method particularly well suited for imaging the phase-gradient textures associated with vortices and other topological defects. We benchmark the algorithm using simulated data containing known lattice distortions and artificially generated vortices. Finally, we apply the method to experimental STM data and resolve the charge density wave (CDW) incommensurabilities, lattice distortions, and CDW phase vortices in 4Ha-NbSe2 and 2H-NbSe2. The physical implications of these observations were recently reported by S. Sajan, X. Yang, et al. [arXiv:2607.20335 (2026)]. The program package and the example usage is publicly available at the GitHub: https://github.com/jsntxyang/LocalWavevectorMeasuremnt.

Training parameterized quantum circuits requires balancing optimization progress with limited evaluation budgets and the statistical uncertainty of near-term quantum hardware. We present a modular benchmarking framework that separates quantum search-direction estimation from classical parameter-update rules, allowing their interactions and sensitivities to be examined independently. We evaluate a suite of gradient-based, stochastic, and derivative-free optimizers across combinatorial optimization using the Quantum Approximate Optimization Algorithm (QAOA), supervised quantum machine learning using Iris classification and a quantum convolutional neural network (QCNN) for binary MNIST classification, and quantum chemistry using the variational quantum eigensolver (VQE) for molecular hydrogen. By separating objective evaluations from sample-circuit costs under finite-shot simulation and physical execution on a 156-qubit processor, we compare optimizer behavior across workloads with different parameter counts and measurement requirements. Four selected simulator and hardware case studies report terminal and best observed objectives separately, alongside post-training classifier accuracies. The comparisons include selected MaxCut and hydrogen trajectories and illustrate workload-dependent objective changes and evaluation costs, while the simulator benchmark compares both peak and mean performance across three seeds. The hardware runs do not establish an optimizer ranking or isolate a causal effect of device noise.

In the cow-path problem, a cow must find a goal lying at an unknown distance on one of $w$ paths connected only at the origin, and performance is measured by competitive ratio. Kao, Reif and Tate designed an efficient randomized algorithm in which the cow visits the paths in a fixed cyclic order. They proved the algorithm is optimal for $w=2$, and subsequently Kao, Ma, Sipser and Yin proved its optimality for all $w$, with a claim that no algorithm does better than the best cyclic one. This note provides a detailed proof of that claim.

We establish general conditions for the weak convergence of regular one-dimensional diffusion processes when their scale functions, speed measures, state spaces and boundary conditions may substantially vary. Our approach is based on the classical Itô–McKean representation of regular diffusions via time changes and scale transformations of a single Brownian motion. This reduces many stochastic convergence problems to deterministic questions concerning the convergence of integrals, monotone functions, compositions, and generalized inverses. As a main application, we investigate various homogenization regimes for diffusions in media with closely spaced semipermeable membranes, where the resulting limiting processes include not only reflected and skew Brownian motions, but also skew-sticky Brownian motions and snapping-out diffusions. This yields a representation of a snapping-out diffusion as a function of a single Brownian motion, without introducing an auxiliary sequence of independent exponential random variables.

We present the deterministic generation and all-optical switching of ternary topologically distinct vortex states in a triad of trapped exciton-polariton condensates with direct state readout from momentum-space emission distribution. Tuning the non-resonant pump power controls the sign of the dissipative coupling, selecting between a ferromagnetic in-phase state and two frustrated states with global phase winding. The zero-winding ferromagnetic state has unstructured far-field emission at low momenta and forms vortex-antivortex arrays at the Dirac points, reflecting the C_3v global symmetry. The two frustrated states of +-2pi winding can be activated on-demand by a control beam and produce different triangular emission patterns selectively localized at the K or K' points. Although all three states share identical real-space emission distributions, their momentum-space photoluminescence profiles distinguish them unambiguously, revealing the chirality without the need for interferometry. This enables on-demand orbital angular momentum control using a pump independent of resonant phase or OAM imprinting, establishing condensate triads as a compact platform for reconfigurable chiral light generation operating on a ternary logic basis.

We study minimum-variance control of an unknown autoregressive system with exogenous inputs and coefficients that change at unknown times. Under bounded independent disturbances, fixed detection gaps, stability and feasibility conditions, and sufficient time between changes, we prove \(O((C+1)\log((T+1)/\delta))\) regret with probability at least \(1-\delta\), where \(T\) is the horizon and \(C\) the number of changes. Unlike switching bandits, where unselected arms can change unobserved, admissible plant changes provide information during exploitation: the correct feasible controller leaves only the disturbance in the output, whereas a detectable change raises output energy under the old controller. PIECE-CD explores initially and after alarms, then uses gated recursive least squares for control. Its energy test compares windowed output power with a threshold above the noise floor; the extension to unstable controller mismatches also monitors the reference controller's input proposal. We control false alarms across the horizon and prove logarithmic detection delay. Inputs are clipped to prescribed bounds. Logarithmic regret also holds under an explicit condition ensuring that clipping becomes inactive after a finite burn-in. Under the stated feasibility conditions, the extended detector covers destabilizing changes with detectable excess energy over a fixed window.

Generative AI offers new resources for collaborative design ideation, yet how designers work with it as a shared concept develops remains underexplored. We developed DuoSketch, integrating a shared canvas and live transcripts with a separate AI for each member of a pair. An exploratory qualitative study with 12 pairs identified three challenges: unresolved fit between AI proposals and shared design purposes, shared meanings not expressed in AI responses, and agreed changes missing from later outputs. We found that pairs navigated these challenges through bridging work in two directions: AI -> Shared Design (reworking features, developing uses, and assessing fit with shared requirements) and Shared Design -> AI (supplying materials, explaining meanings, and communicating decisions). We discuss how future AI systems can support the joint exploration and articulation of design ideas.

We construct an explicit binary linear system game that separates $C_{qa}$, the closure of the set of finite dimensional quantum correlations, from $C_{qc}$, the set of commuting operator correlations. The game admits a perfect commuting operator strategy, while every correlation in $C_{qa}$ has a success probability strictly less than 1. This provides a concrete counterexample to Tsirelson's problem in its approximation form. The defining system has 1417152 equations in 1889684 variables, with exactly three nonzero coefficients per equation and a single nonzero entry on the right hand side. We also compute the classical value exactly. The complete system is specified in Lean 4, and the separation and exact classical value are formalized using Mathlib.

We study the exact $L^p$ norms for a number of maximal operators, which are prominent in harmonic analysis, semigroup theory, and probability. We prove that a strongly continuous positive symmetric sub-Markovian semigroup $(T_t)_{t\ge0}$ on a nonatomic $\sigma$-finite measure space $(X,\mathcal F,\mu)$ has maximal $L^p$ norm $p'=p/(p-1)$ for every $1<p<\infty$ whenever $f\mapsto T_{t_0}(\mathbf 1_Ef)$ is compact on $L^2(\mu)$ for some $t_0>0$ and some measurable set $E$ of positive measure. The lower bound follows by approximating finite dyadic conditional expectations at suitable semigroup times; the upper bound is Stein's maximal inequality in its sub-Markovian form. Important examples of such maximal operators we discuss include maximal functions of various semigroups (heat, Poisson, Ornstein–Uhlenbeck, Schrödinger, Dunkl, Laguerre, Jacobi) as well as the centered Hardy-Littlewood maximal operator on $\mathbb{R}.$ In particular, the heat and Poisson maximal operators on every complete connected Riemannian manifold of positive dimension without boundary have exact $L^p$ norm $p'$, with no curvature or stochastic completeness assumptions. Combining our approach with previous work of the first and last authors we also prove that $p'$ is the limit, as $d\to \infty,$ of the $L^p$ norms $B(p,d)$ of the centered Hardy-Littlewood maximal operators on $\mathbb{R}^d.$ On the other hand, we show that the norm of this maximal operator is strictly larger than $p',$ in dimensions $d\ge2.$

Constant-step stochastic approximation generally has a nonzero stationary mean error that persists under time averaging. This paper studies that error for nonlinear two-timescale recursions driven by an exogenous finite-state Markov chain. Under stated smoothness assumptions and conditions on the stationary distribution, we derive a first-order bias expansion whose error bound remains uniform as the slow step size becomes much smaller than the fast step size. Fast-manifold coordinates keep the associated covariance equation regular in this limit. For fast step $\eta$ and slow step $\varepsilon$, the expansion reveals a mixed contribution $\varepsilon^2/\eta$ alongside terms linear in each step size. This dependence matters for bias reduction: along power-law step-size paths, the bias exponents need not be integers, so Richardson–Romberg extrapolation requires weights matched to the path. An exactly solvable nonlinear Markov example verifies the coefficients. We verify localization for temporal-difference learning and compare finite-run extrapolation at equal update budgets. For finite runs, we bound the initialization error of tail averages on both timescales under an additional coupling assumption. In the special case of additive independent noise, signed third-moment cancellation yields a sharper remainder.

Despite the ubiquity of sensors in wearable and mobile devices and the abundance of human movement data they generate, translating unlabeled recordings into foundational motion models remains an open challenge. Self-supervised learning (SSL) has alleviated the need for costly annotations, yet existing approaches leave the global structure of large-scale motion data largely untapped, relying on randomly sampled batches and local comparisons that become particularly problematic for in-the-wild inertial data dominated by stationary, low-variance behaviors. Here we introduce Morphological Contrastive Learning (MorphCL), a self-supervised pretraining framework that uses structure-aware grouping to inject explicit modeling of global structure into inertial-based SSL approaches. Building on two well-established pillars of motion analysis, the discovery of motion primitives, or motifs, and domain-specific feature descriptors, we show that MorphCL substantially improves linear probing and finetuning results of learned encoders by up to 15 percentage points in F1-score. In a comparison with existing foundation models, we demonstrate that MorphCL-pretrained encoders match or surpass them models in linear probing performance while trained on $4600\times$ less data. Qualitative analysis of the resulting embedding spaces further reveals morphologically meaningful cluster structure, with improved separation of kinematically similar activity classes.

Strong gravitational lensing provides a unique tool of studying intrinsically faint high-redshift active galactic nuclei (AGN). In particular, low-luminosity AGN (LLAGN) can probe black-hole growth in low-mass galaxies that current X-ray surveys cannot otherwise access. We investigate the nature and physical properties of a faint X-ray source triply imaged by the galaxy cluster SDSS J2243$-$0935, and determine whether the source hosts an AGN. We combine archival Hubble Space Telescope imaging and Chandra X-ray observations with the published strong-lensing models of SDSS J2243$-$0935. We constrain the source redshift geometrically, derive magnification maps, and infer the physical properties of the host galaxy. The geometric lensing analysis gives a source redshift of $z=2.14^{+0.12}_{-0.17}$, consistent with the photometric redshift estimate, and a total magnification of $\mu=78\pm7$. The source is triply lensed and detected only in the HST F160W band, with an intrinsic magnitude of $m_{\rm AB}=28.2\pm0.2$. The rest-frame optical emission implies a stellar mass of $8.6\lesssim\log(M_\star/M_\odot)\lesssim9.4$. The unobscured star formation rate is constrained to $\mathrm{SFR}_{\rm UV}<0.009\,M_\odot\,\mathrm{yr}^{-1}$, although substantial dust obscuration remains possible. Chandra detects X-ray point-source counterparts to all three HST-detected lensed images. The combined X-ray spectrum contains $100\pm11$ net counts and is consistent with a moderately absorbed power law, with $\Gamma \simeq 1.48$ and $N_{\rm H}=(5.5\pm4.4)\times10^{22}\,\mathrm{cm}^{-2}$. After correcting for gravitational magnification, we obtain an intrinsic rest-frame $2$–$10$ keV luminosity of $L_{\rm X}=1.2\pm0.3\times10^{42}\,\mathrm{erg\,s^{-1}}$. The X-ray emission is likely dominated by an AGN hosted by a low-mass galaxy at $z\simeq2.14$, making this system a strongly lensed LLAGN at cosmic noon (abridged).

For elliptic curves $E/\mathbb{Q}$ in short Weierstrass form \[ E=E(a_4,a_6): y^2=x^3+a_4x+a_6, \] we derive a multivariable Euler product generating function which encodes the Tamagawa product $Tam(E)=\prod_p c_p(E)$. Using this generating function, we compute limiting distributions, exact covariances, and moment and tail bounds for four important statistics on the Tamagawa number; for example we show that more than half of all curves in this height ordering have trivial Tamagawa product, about $42.2\%$ have exactly one prime with nontrivial local Tamagawa number, and only about $6.8\%$ have exactly two such primes. The product is obtained by specializing an Euler product indexed by local reduction data that we derive from Tate's algorithm. The results in this paper were autoformalized in Lean by AxiomProver.

SWE-agent ensembles improve issue resolution by combining candidate patches from different agents with complementary strengths. The central problem is therefore test-based selection: generate tests, execute candidate patches, and identify the best patch. We formulate this process as test-space optimization: evolving an executable repository test suite until it distinguishes competing patches. Existing test-generation methods are limited optimizers. They usually lack an explicit loss for ensemble selection, optimize through incomplete directions that mostly create new tests or delete old ones, and perform one-off generation without feedback from repeated failures. Inspired by gradient descent with momentum, we introduce TestGRAD, a framework for automatic test optimization. TestGRAD centers on three concepts. Differential loss gives the optimizer an explicit execution-defined target: useful tests should separate candidate patches by behavior. Full CRUD gradients expand the update direction from merely creating or deleting tests to reading existing test infrastructure, creating new tests, updating stale assertions, and deleting only obsolete tests. Failure Pattern Momentum mines frequent failure sequences from memory, allowing the optimizer to avoid repeated non-discriminative directions while compressing the failure-history context. On SWE-bench Verified, TestGRAD achieves 84.2% Pass@1 with a 4-agent ensemble, outperforming the strongest baseline (80.6%) by an absolute improvement of 3.6 percentage points, while compressing failure-history context by over $100\times$.

AI could change which parts of science impede progress. Consider a world in which machine systems are better, faster, and cheaper than people at most scientific work that can be done through a computer. Our question is what would limit science in that world. Literature synthesis, hypothesis generation, software development, simulation, and analysis could become abundant, while experiments, observations, well-supported conclusions, and accountable institutional au- thority remain scarce. Science would then be constrained by a different set of resources. In this paper, we call this change the scarcity inversion and consider four parts of it: selection, physical access, validation, and organizational choice. This change is arriving first in mathematics and coding/software/algorithm design, where the whole scientific loop can run inside computation. For national laboratories, the change could be striking. Their distinctive role is to turn abundant machine reasoning into trustworthy results by combining controlled experiments, protected data, expert judgment, and accountable authority. The practical question is how facilities, verification, provenance, resource allocation, and scientific governance should change when reasoning is plentiful and trustworthy evidence is scarce.

In this work, we present a comprehensive analysis of classification performance for distinguishing child-directed speech (CDS) from adult-directed speech (ADS) using speech data from corpora containing natural in-lab and in-the-wild CDS and ADS. We establish classification benchmarks for these datasets using self-supervised learning (SSL) representations, along with a range of time-pooled representations that go beyond first- and second-order statistics by incorporating cross-channel covariances in high-dimensional embeddings. In addition, we probe these representations to examine how different pooling methods capture prosodic information using linear probes. Overall, SSL-based representations prove particularly effective, achieving the best performance, while different pooling methods offer complementary advantages for the task.

Scientific dynamics forecasting is often framed as an architecture choice, although deployment is also determined by observed history, rollout feedback, compute budget, physical objective, and test distribution. We formulate protocol-dependent model selection and introduce ProtocolMatch, a compute-matched, validation-selected, and failure-preserving evaluation framework. On driven quantum-spin dynamics, we compare recurrent, patched-attention, causal-attention, and low-rank linear predictors across three independently generated datasets. The causal-attention–recurrence ordering reverses as the training set grows within a fixed two-spin task, while a linear predictor has the lowest mean error in the six-spin local-observable comparison. Restricting observed history worsens every refreshed-history view but improves every closed-loop view in the four-spin study. A latest-state MLP has lower error than persistence on every dataset under state refresh across all five cells, yet its closed-loop rank varies by system and includes finite explosive errors. Physical penalties improve targeted consistency without reliably improving prediction error, and in-distribution intervals lose most coverage after a driving-frequency shift. Thus scientific model selection should return a predictor with its protocol and report accuracy, physical validity, and shifted-distribution reliability separately.

Medical vision-language models (VLMs) have shown increasing potential for clinical image interpretation. However, these models still struggle to interpret multi-phenotype lesions whose diagnosis requires the joint assessment of multiple pathological phenotypes. Existing vision-language alignment methods produce visual representations that fail to preserve anatomical hierarchies and relationships among phenotypic subclasses. This stems from their reliance on semantic supervision, which lacks geometric constraints to preserve these relationships in the visual embedding space. Moreover, the sparsity of lesion-related anatomical and phenotypic representations makes it difficult for medical VLMs to capture important diagnostic evidence. To address these limitations, we propose PureVision, a geometry-supervised visual representation learning framework for multi-phenotype lesion interpretation in medical VLMs. It combines a geometry-supervised representation learning module, PureEyes, and an anatomy-guided evidence aggregation module, PureNeurons. PureEyes provides geometric supervision through ideal spatial distributions that encode anatomical hierarchies and phenotypic subclass relationships. PureNeurons projects visual representations into the learned latent space, using their positions to selectively aggregate lesion-specific anatomical and phenotypic evidence. Experiments on LIDC-IDRI, CBIS-DDSM, and 3DReasonKnee demonstrate that PureVision improves lesion grounding and phenotype characterization in visual question answering and radiology report generation. Code is available at: https://anonymous.4open.science/r/purevision-06C2.

An optimal stopping problem is monotone when its stopping region is preserved under subsequent transitions, so that the one-step look-ahead stopping time is optimal. We develop a finite-horizon discrete-time theory of multiple stopping that separates general recursive dynamic programming from this stronger monotone structure. Extending the classical martingale-system approach, we construct a recursively defined optimal vector of stopping times without assuming monotonicity. If monotonicity holds at every exercise level, this vector coincides with the recursive vector of one-step look-ahead stopping times. A bounded counterexample shows that monotonicity of the original single-stopping problem alone is insufficient, whereas monotonicity propagates to every level when the level-one one-step look-ahead function is deterministic. We apply the theory to multiple-selection last-success problems. For the classical independent odds setting, a continuous-time Poisson embedding gives an alternative proof of the multiple-stopping lower bound and clarifies the associated threshold constants. We also examine random horizons, Markov-dependent trials, and an unknown success probability

We develop q-ary Guessing Random Additive Noise Decoding (GRAND) for linear codes over a q-ary alphabet. The decoder works on a sorted symbol-likelihood array and uses three local child generation rules to generate symbol deviation patterns. We then prove that these rules induce a monotone rooted spanning tree of the full row-index space, so best-first traversal gives maximum-likelihood (ML) decoding under unlimited search. Reed-Solomon simulations verify ML agreement and show the finite-budget performance-complexity tradeoff.

Fluid antenna systems (FASs) offer significant spatial flexibility, yet securing them against eavesdropping is critical for practical FAS deployment in military, satellite, and internet-of-things networks. Although large language model (LLM)-assisted genetic algorithms (LLM-GAs) can address this secure FAS port selection problem, whether further algorithmic improvement is possible warrants deeper investigation. To this end, we propose a memetic algorithm based on reflective evolution (ReEvo). Unlike the state-of-the-art LLM-GAs, which design only crossover or mutation operators with an LLM, our algorithm leverages an LLM to evolve dedicated crossover, mutation, and local-search operators offline. These operators are then embedded into a memetic search framework, thereby obviating any online LLM queries during execution. Simulation results at equal generation counts demonstrate that our proposed algorithm achieves a higher secure sum-rate than the conventional GA and the state-of-the-art LLM-GAs.

This article generalizes the de-initialization framework, proposed by (Roberts and Rosenthal, 2001) for total variation distance, to Wasserstein distances. In essence, de-initialization captures the phenomenon that in the presence of certain structural features, the convergence behaviour of a Markov chain can be reduced to that of a "simpler" stochastic process. Our results allow to measure this in terms of a general Wasserstein distance. We apply them to analyse the Wasserstein convergence behaviour of Gibbs sampling, linchpin variable sampling and slice sampling.

We introduce general methods to derive lower tail large deviation principles, for a variety of nonlinear problems of combinatorial interest. These methods are especially effective in "critical" regimes, where the nonlinearities are not sparse enough for Poissonian lower tails (e.g., Janson's inequality does not provide a sharp tail bound), but the situation is not so dense that combinatorial effects dominate (e.g., we cannot directly apply the theory of hypergraph containers or the relative entropy framework of Kozma-Samotij). In particular, these are regimes where one expects interesting phase transitions to occur. Our results have a number of consequences related to subgraphs in random graphs, answering various questions of Warnke and Jenssen-Perkins-Potukuchi-Simkin. For example, consider a random graph $G\sim \mathbb G(n,p)$, and let $L_\triangle(n,p)=\log\Pr[G\text{ is triangle-free}]$. The first-order asymptotics of $L_\triangle(n,p)$ have long been known in all parameter ranges except the critical regime where $p$ has order of magnitude $1/\sqrt n$. We are able to fill this gap: for any fixed $c>0$, we show that $n^{-3/2}L_\triangle(n,c/\sqrt n)$ converges to a limit expressed in terms of a two-parameter variational problem, which has a single phase transition at $c\approx 4.341$. In contrast, we show that there is no such phase transition for the probability that $G$ is $C_{2\ell}$-free, for any fixed even cycle $C_{2\ell}$. We also obtain some applications in combinatorial design theory. Addressing conjectures of Glock-Kühn-Lo-Osthus, Kelly, and Kwan-Sah-Sawhney-Simkin, we obtain new estimates on the number of order-$n$ Steiner triple systems with no Pasch configuration and the number of order-$n$ Latin squares with no $2\times 2$ Latin subsquare (both tight up to a factor of $\exp(o(n^2))$).

Large language models (LLMs) have already been shown to match or exceed human experts in persuasion. While their persuasive capabilities hold promise for beneficial uses such as education and health communication, they can also be used to manipulate and misinform, making their evaluation a growing priority for developers and regulators. That evaluation, however, remains fragmented: studies differ in what they treat as persuasion, and broad claims often rest on narrow, situation-specific assessments. Automated methods, often modelled on human studies, offer a way to compare such assessments directly, as they can be run on the same models at scale and can include high-risk forms of persuasion that would be difficult or unethical to test on people. In this study, we adapt nine published automated methods to a shared setup, run them on the same fifteen LLMs, and ask whether their rankings agree and why. We find that the methods agree only weakly (mean Spearman $\rho = 0.25$). Our analyses point to two contributing factors. Models that refuse some tasks but not others, directly or indirectly, lower agreement by about a quarter, and these refusals fall mostly on manipulation tasks. General capability also plays a part: most rational persuasion (non-manipulative) methods track it, whereas most manipulation methods do not. Together, these findings suggest that agreement depends more on the task a method sets than on how it scores persuasion, although this pattern is only indicative given the eight methods available for analysis. More broadly, our results suggest that persuasion scores combine a model's ability to persuade with its willingness to do so. A single score is therefore informative about its own setting, but says little about a model's persuasiveness across tasks.

Photometric decomposition is often used to study galaxy structure by modelling surface brightness distribution in its components. However, most of the relevant codes are designed for single-band analysis. I present IMFIT-MW: a lightweight, open source Python tool that extends IMFIT capabilities to simultaneously model multiwavelength images. IMFIT-MW is able to set shared parameters between filters and to impose smooth variation of them with wavelength. IMFIT-MW achieves this by using IMFIT to produce model brightness distributions for given parameters, whereas hyperparameters defining their wavelength dependence are fitted by an external optimizer. The code is available on GitHub (https://github.com/IVChugunov/IMFIT-MW). I provide a few examples of use, showing that IMFIT-MW matches GALFITM in single-thread performance and can have advantage of multithreading. In a broader sense, I demonstrate the possibility to extend IMFIT features even without modifying its source code.

Let $X_\sigma$ be the affine normal toric variety associated with a full-dimensional strongly convex rational polyhedral cone $\sigma$, and let $m$ be a primitive degree with slice $P_m=\sigma\cap[m=1]$. We construct an explicit flat algebraic family whose completion is universal for equivariant deformations in all degrees $-jm$, $j\ge1$, simultaneously. We prove that the irreducible components of the reduced base correspond bijectively to maximal lattice-friendly Minkowski decompositions of $P_m$, and describe the induced family on each component. When $m\in\sigma^\vee$, we also obtain a formally universal equivariant family for the pair $(X_\sigma,V(\chi^m))$. These results extend earlier miniversality and component theorems to arbitrary affine normal toric varieties and arbitrary primitive degrees.

For a convex body $K$ its parallel section function in the direction $\theta\in S^{n-1}$ is defined by $$A_{K,\theta}(t) = \mathrm{vol}_{n-1} \left( K \cap \{ x \in \mathbb{R}^n ~ | ~ \langle x, \theta \rangle = t \} \right) .$$ We study to what extent the body $K$ is determined by partial information about these functions. As applications, we characterize convex bodies with locally separable section functions, establish partial results on the homothety conjecture for bodies of flotation, and answer a question of Barker and Larman concerning the determination of convex bodies from section functions at infinitely many distances from the origin.

Let $C_1,\ldots,C_k$ be the convex hulls of independent partial-sum processes in $\mathbb{R}^d$ whose increments are exchangeable within each process. We express the expected mixed volume $\mathbb{E} V_d(C_1[m_1],\ldots,C_k[m_k])$, $m_1+\cdots+m_k=d$, through mean absolute determinants of disjoint block sums of the increments; no general-position assumption is needed. For random walks with i.i.d. integrable increments the block sums are independent, and the formula extends the expected-volume formula of Barndorff-Nielsen and Baxter and of Vysotsky and Zaporozhets to mixed volumes. We then prove a continuous-time counterpart: for independent Lévy processes with finite first moments, the expected mixed volume of the closed convex hulls of their paths is an explicit integral of mean absolute determinants over a product of simplices. For symmetric stable processes the integral is evaluated in terms of the associated zonoids. As a geometric application, we compute the mean mixed volume of random projections of mutually orthogonal canonical orthoschemes.

While Large Language Models (LLMs) possess rich world knowledge and impressive generalization capabilities, their direct application to tabular data classification is hindered by high inference costs and limited interpretability. In contrast, decision trees are fast and transparent but often underperform in low-data regimes. In this work, we propose a novel framework that bridges these paradigms by distilling LLM knowledge into interpretable decision trees under a few-shot learning setting. Instead of directly prompting the LLM to generate full trees, which is often unstable and inefficient, we develop a three-stage paradigm that prompts the LLM to generate rules and organize the rules into a tree. Experiments on multiple real-world tabular datasets demonstrate that our method achieves superior accuracy and interpretability with significantly lower prompting overhead compared to existing baselines.

Can AI make Software Engineering (SE) – the discipline – obsolete? And can it make software engineers – the professionals – redundant? This paper argues that the rise of capable AI coding agents makes SE and software engineers essential, not obsolete: the missing foundation without which AI-assisted development produces misleadingly plausible, unverifiable, and ultimately untrustworthy software. Three structural properties of large language models (probabilistic generation, agnosticism, and semantic statelessness) create a structural vacuum that no amount of training can eliminate. Filling it requires four knowledge levers: methodological knowledge, domain knowledge, design choices, and process choices. All four must be reified as persistent artifacts, and each requires the software engineer as methodologist, mediator, and custodian.