Papers

Let $f(X)$ be a polynomial over a finite field, and let its defining function record the multiplicities of its roots. In this paper, we introduce the affine stabilizer group and the affine stabilizer group scheme of the defining function of $f(X)$. These objects characterize the affine symmetries of the roots of $f(X)$ together with their multiplicities. We establish structural descriptions of these affine stabilizers and relate them to the coefficient structure of $f(X)$. In particular, the affine stabilizer group determines quotient structures of the defining function, which yield canonical descriptions of the corresponding semi-invariant polynomial spaces. The resulting coefficient constraints provide explicit restrictions on the possible supports of polynomials with prescribed affine symmetries. The affine stabilizer group scheme further captures infinitesimal symmetries arising from multiple roots and gives refined coefficient constraints beyond those obtained from ordinary affine transformations. We apply this framework to generator polynomials of constacyclic codes over finite fields. The affine stabilizer structure of a generator polynomial induces subcodes defined by semi-invariant conditions and provides constraints on their dimensions and Hamming distances. In the special case of multiplicative symmetries, the construction specializes to the support-primitive decomposition of constacyclic codes.

We construct the Super-Composite Higgs, a scenario designed to stabilize the Higgs mass and generate the flavor structure within a single dynamical framework. The basic idea is to combine supersymmetry and compositeness, where the Higgs mass is protected by supersymmetry and Yukawa couplings are generated through partial compositeness. This combination offers mutual advantages and ameliorates several drawbacks of each individual theory. The Super-Composite Higgs makes experimentally testable predictions across three different areas of fundamental physics. At the precision frontier, the electron EDM is predicted to be within reach of future experiments. At the observational cosmology frontier, measurable effects in CMB and large-scale structures are expected from an ultra-light gravitino. At the high-energy frontier, an upper bound on supersymmetric particle masses, derived from cosmological considerations, indicates that squarks should be lighter than about 10 TeV, which is within reach of the FCC-hh.

We establish the large-$n$ asymptotics of Hankel determinants for unitary random matrix ensembles possessing a higher-order edge singularity. We focus on ensembles where the equilibrium measure is supported on a single interval and the limiting eigenvalue density vanishes to order $2k+\frac{1}{2}$ for $k \in \mathbb{N}$. Notably, we explicitly evaluate the constant term in the asymptotic expansion, which involves a regularized integral of the Hamiltonian associated with the Painlevé I ($P_{\rm I}^{2k}$) hierarchy. As a by-product, we also prove the universality of the eigenvalue correlation kernel near this singular edge and derive a limiting kernel expressed through functions related to a special solution of the $P_{\rm I}^{2k}$ equation. Our method relies on the Deift-Zhou nonlinear steepest descent analysis for the Riemann-Hilbert problem of orthogonal polynomials.

This paper investigates the impact of in-phase/quadrature (IQ) imbalance on signal detection in affine frequency-division multiplexing (AFDM) systems over doubly dispersive Rayleigh fading channels. The bit error rate (BER) performance of AFDM under various IQ mismatch conditions is evaluated and compared with that of an ideal transceiver. Simulation results show that, under ideal conditions, BER decreases rapidly with increasing signal-to-noise ratio (SNR). However, IQ imbalance introduces image interference that spreads across the affine domain, causing significant performance degradation. The results further highlight the impact of practical IQ imbalance on AFDM systems across different quadrature amplitude modulation (QAM) orders, channel scenarios, and comparisons with orthogonal frequency-division multiplexing (OFDM) and orthogonal time frequency space (OTFS) systems. In particular, higher-order QAM performance degrades significantly, indicating the need for IQ imbalance compensation techniques or impairment-aware detection algorithms to ensure robust operation in practical AFDM systems.

A metric space is called all-set-homogeneous if every isometry between two of its subsets extends to an isometry of the whole space. We classify all-set-homogeneous length spaces: besides the classical examples, they include the universal metric trees of finite valence.

Let $\Sigma$ be a compact oriented surface. We investigate the graded Lie algebra of homology cylinders over $\Sigma$, as introduced by M. Goussarov and K. Habiro. To analyze its torsion, we refine a strategy initiated by Y. Nozaki, M. Suzuki, and the third author, which relies on the reduction modulo $1$ of the LMO functor and on clasper calculus. Specifically, we develop general tools for studying, under the standard action of the symplectic group, the Sp-module structure of the torsion in the odd-degree component of this Lie algebra. We show that this torsion part surjects onto an $2$-torsion Sp-module, which is explicitly described in terms of Jacobi diagrams. As an application, we provide an intrinsic description of the Sp-module given by the degree-three component of the Lie algebra of homology cylinders. A further motivation is to understand torsion phenomena in the associated graded of the lower central series of the Torelli group of $\Sigma$: in this direction, we exhibit an explicit Sp-module onto which the torsion of the Torelli Lie algebra surjects in degree three.

Map-based vision-language navigation (VLN) relies on persistent spatial representations to connect language understanding with geometric planning. However, acquiring semantics beyond the needs of the current instruction can introduce unnecessary perception cost and irrelevant annotations. Continuously accumulating unrelated objects may not only waste computation, but also clutter the visual-spatial representation consumed by the vision-language model (VLM) planner. To address this problem, we present SparseNav, a training-free framework that follows a less-is-more principle for semantic navigation. SparseNav persistently maintains a lightweight geometric bird's-eye-view (BEV) map and sparse landmark memory, acquiring new semantics on demand using the active sub-instruction to decide what is worth grounding. An instruction manager first tracks navigation progress and identifies the active landmark query. An instruction-conditioned perception mechanism then invokes open-vocabulary segmentation when the queried landmark is visible and its metric location can inform the next decision. The resulting landmark memory supports VLM selection among hybrid frontier and local directional waypoint candidates. Without any additional training, SparseNav achieves success rates of 42.8% on R2R-CE and 40.7% on RxR-CE, both on the Val-Unseen splits. Controlled ablations examine semantic perception strategies and the contributions of individual framework components. Furthermore, we successfully deployed SparseNav on a Unitree Go2 quadruped equipped with an Intel RealSense D455 RGB-D camera for geometric mapping and landmark grounding and a Livox MID-360 LiDAR for localization, without a prebuilt map. We validated its effectiveness across multiple indoor environments using instruction-conditioned waypoint navigation.

The Nernst effect, which converts a heat current into a charge current in the orthogonal direction, is generally classified into the ordinary effect due to an external magnetic field applied to conductors and the anomalous effect due to static magnetization in magnetic materials. Thus, one expects that the anomalous Nernst effect disappears at the Curie temperature following the magnetization. Here, we observe the Nernst effect which rather manifests around the Curie temperature. In a series of ferromagnetic (Mn,Cr)Sb samples, the Nernst coefficients exhibit finite values even across the Curie temperatures, whereas the anomalous Hall resistivity disappears, suggesting the breakdown of the Mott relation. These surviving Nernst and vanishing Hall effects result in a peak behavior of the transverse thermoelectric conductivity around the Curie temperature, which we theoretically reproduce by introducing longitudinal fluctuations of spins without accounting for an exotic spin texture, a non-trivial electronic structure, or magnon- and phonon-drag effects. The Nernst effect driven by the longitudinal spin fluctuations reveals that not only the static magnetic order but also the spin fluctuations can be a driving force of the Nernst effect in magnetic materials, opening another way for boosting transverse thermoelectric conversion.

Principal component analysis (PCA) is a fundamental tool to reduce the dimensionality of the data in many applications. PCA finds a few signal directions that contain most of the variability of the data by computing the eigenvectors of the sample covariance matrix. In this work, we focus on the spiked covariance model, in which the data vectors are defined by a few orthogonal signals plus an isotropic Gaussian noise, and our goal is to estimate one or more of the leading signals. Our main theoretical finding is that the subspace spanned by several leading eigenvectors of the sample covariance matrix contains significant information about the desired signals long before the individual eigenvectors converge to the population principal components. To prove this, we derive a posteriori bounds for the angle between the subspace spanned by the desired population signals and the subspace obtained from the sample using perturbation theory for singular vectors. This leads to a new algorithm, SuperPCA (SUbsPace subsamplER PCA), which capitalizes on an approximate eigenspace of the sample covariance matrix to find the leading signals far more efficiently and accurately than classical PCA in the high-dimensional, multi-signal setting. SuperPCA exploits only a small number of subsampled coordinates of the data, which can lead to tremendous savings in data acquisition cost, especially when the signals are approximately sparse. For the same number of measurements, SuperPCA can offer a factor $10$ improvement in accuracy compared to the classical PCA method.

Longitudinal data, often in the form of panel data, are commonly encountered in the pharmaceutical industry. However it is not always possible to fit an unstructured covariance matrix when using the Mixed Model for Repeated Measures (MMRM). Here we argue that high-order autoregressive models provide a useful framework when more general correlation structures, such as the unstructured and/or Toeplitz, are unfeasible. The autoregressive model of order one is often suggested but, in our experience, higher orders are seldom considered. We present the case for using autoregressive models, with potentially high orders, as a flexible framework to model correlations in panel data, when more general structures are hard to identify.

We study the linearization of the random $p$-adic orthogonal matrix model. Let $A_n$ and $B_n$ be Haar-random $n\times n$ special orthogonal and alternating matrices over $\mathbb{Z}_p$, respectively. For an odd prime $p$, we prove that $\mathrm{cok}(A_n-I_n)$ and $\mathrm{cok}(B_n)$ have the same limiting distribution as $n \to \infty$ through integers of the same parity. The two parity-dependent limits are interchanged for orthogonal matrices of determinant $-1$, while the full orthogonal group gives their equal mixture. For $p=2$, linearization does not preserve the limiting cokernel distribution, and we explicitly determine the limiting distributions for the special orthogonal model in terms of the spinor norm, which controls the parity of the $2$-adic valuation of the order of the torsion subgroup.

AI assistants that support email composition may shift cultural communication norms, such as the directness typical of low-context cultures like the US versus the indirectness and contextual sensitivity central to high-context cultures like Japan. Yet it remains unknown to what extent people adopt and edit AI drafts inconsistent with their cultural communication norms. We address this through a preregistered within-subject experiment in which Japanese and American participants wrote workplace emails in their native language without AI, with a low-context AI, and with a high-context AI. We found that Japanese participants wrote emails with significantly more high-context markers (politeness, apologies) than Americans. But AI drafts shifted participants' emails toward the draft's style, with larger shifts when the draft was culturally misaligned: Japanese drifted most under low-context drafts, Americans most under high-context drafts. These findings suggest AI drafts risk overwriting cultural communication norms unless they adapt to users' communication styles.

The discovery of novel inorganic materials drives technological breakthroughs in critical fields such as computing and energy storage. Generative AI has promised to accelerate the materials discovery pipeline, but state-of-the-art flow and diffusion models remain bottlenecked by the cost of proposing candidate materials. To address this, we introduce OMatG-flash, an all-atom flow map for inorganic crystal structure prediction (CSP) and de novo generation (DNG). OMatG-flash is a Pareto-optimal inference engine for materials, sampling candidate materials with an order of magnitude fewer inference steps and less wall-clock time than existing flow and diffusion models while demonstrating benchmark performance on par with the state-of-the-art. To enable post-training fine-tuning we apply Reinforce Adjoint Matching to flow maps, further improving match rates and RMSE on the unconditional CSP task. OMatG-flash showcases the potential of flow maps to accelerate generation of high-quality candidate inorganic materials and demonstrates a step forward in sample throughput necessary for data-hungry materials discovery workflows.

A central theme in extremal graph theory is to understand the relationship between the density of a graph and the richness of its cycle length spectrum, which is the set of distinct cycle lengths occurring in the graph. In 1966, Erdős and Hajnal suggested studying $s(G):=\sum_{\ell\in{C}(G)}1/\ell$ as a measure of the richness of the cycle length spectrum ${C}(G)$ of a graph $G$. Through a series of increasingly strong conjectures, Erdős suggested that the complete bipartite graphs minimise $s(G)$ among all graphs $G$ with the same average degree. The sharpest such conjecture, from 1981, states that the graph $K_{k, n-k}$ minimises $s(G)$ among all $n$-vertex graphs with at least $k(n-k)$ edges (where $k\leq n/2$). We prove this conjecture for all sufficiently large $k$, by showing the stronger statement that any $n$-vertex graph $G$ with $e(G)>(k-1)(n-k+1)$ and $n\geq 2k$ satisfies $s(G)\geq\sum_{\ell=2}^{k}1/(2\ell)$. Moreover, we show that the complete bipartite graph $K_{k,n-k}$ is the unique graph with at least $k(n-k)$ edges that achieves equality here.

Mass-conserving reaction-diffusion systems with two species correspond to a seemingly simple case where pattern formation occurs under the influence of a conservation law. Here, we first revisit their linear stability behavior and point out that generically two instabilities can occur: a stationary large-scale mass-conserving (Cahn-Hilliard) instability and an instability that combines features of a stationary large-scale non-mass-conserving (Allen-Cahn) instability and a oscillatory large-scale mass-conserving (conserved-Hopf) instability. We term it an Allen-Cahn-Hopf instability. Second, we investigate the nonlinear dynamics for a specific model related to the formation of cell polarization where only a Cahn-Hilliard instability can occur, i.e., all primary bifurcations are stationary. We analyze how secondary and further bifurcations subsequently give rise to various oscillatory states. The emerging rich spectrum of spatiotemporal behavior includes several period-doubling cascades related to different forms of spatial and temporal symmetry breaking. Beside regular states, three types of low-dimensional spatiotemporal chaos occur and involve transitions like fusion and an outer crises. Our results demonstrate the importance of nonlinear interactions in the dynamics of mass-conserving reaction-diffusion systems, and show that even a simple two-species system with primary bifurcations of Cahn-Hilliard type can show complex spatiotemporal behavior.

Scene safety understanding plays a life-or-death role in situational awareness in various critical domains. Traditional methods that rely on learning direct mappings between scenes and safety levels often lack interpretability, limiting their reliability in critical applications. An effective approach to overcoming this challenge lies in interpreting human cognitive processes and equipping machine models with analogous cognitive capabilities. This work explores an effective way of integrating scene safety cognitive process modeling and process supervision. Specifically, we first construct a hierarchical cognitive safety structure, which motivates the development of a novel, high-quality scene safety understanding dataset based on multi-step reasoning with process labels. This dataset serves both as a benchmark and a resource to improve the safety reasoning capabilities of Large Language Models (LLMs), while also enabling a granular analysis of intermediate reasoning steps through information flow and saliency-based techniques. Building upon this foundation, we introduce a modular and flexible process supervision framework that reflects the hierarchical nature of human cognition. This framework leverages LLMs as the core architecture and incorporates Low-Rank Adaptation(LoRA) and Mixture-of-Experts (MoE) strategies to enable specialization and collaboration among expert modules, each tasked with specific sub-processes of the overall reasoning chain. Systematic experimental evaluations and analyses confirm that our framework exhibits superior interpretability and performance characteristics compared to traditional approaches.

The concept of weighted universal Value-at-Risk superadditivity (WUVS) was recently introduced by Chen et al. (2026) as a generalization of the question whether for some infinite mean distributions convex combinations of i.i.d. random variables can stochastically dominate the parent distribution. In this short note we prove that the property WUVS can basically never hold for discrete distributions except for the case of comonotonicity. This implies as a corollary that for discrete distributions with infinite mean it can also never hold that convex combinations of i.i.d. random variables can stochastically dominate the parent distribution. This settles an open problem mentioned in Müller (2025).

In this paper we study whether there is a global to local principle for the homology of order complexes of finite lattices. We explore whether the non-vanishing of the homology of a lattice in degree $a+b-2$ forces the existence of a pair of lattice-complements whose open intervals below have non-vanishing homology in degrees $a$ and $b$ respectively. We consider the same question also in the language of multigraded free resolutions. We provide positive evidence for the existence of such pairs and give counterexamples to strengthenings of the question.

We study additively idempotent semirings obtained from commutative words by equipping their subwords with the divisibility order. Every finite semiring associated with a power of one letter is finitely based, whereas one associated with a linear word is finitely based exactly when the word has length at most two. A hypergraph preservation lemma yields the nonfinite basis result and extends it to intervals of varieties. We establish a sharp containment criterion between the power and linear families, and determine the finite basis property of every join of two varieties generated by one member of each family. The unrestricted power family generates the nonfinitely based max-plus variety. In contrast, the unrestricted linear family has a finite basis, as does its join with each finite power member. We also realize a previously known six-element limit semiring as a quotient of a subsemiring of the eight-element linear-word semiring. This gives a proper nonfinitely based subvariety and resolves the corresponding minimality question.

We reexamine F. Wiener's classical proof that the Bohr radius of $H^\infty$ is $1/3$. We replace $H^\infty$ by the multiplier algebras of complete Pick spaces and describe the Bohr set for a wide class of such algebras. In particular, we show that the Bohr set of the multiplier algebra of the Drury–Arveson space $H^2_d$ is the closed $\ell^2$-ball of radius $1/3$. Other cases covered by our description of Bohr sets include Dirichlet-type spaces of analytic functions in the unit disc and a class of Hilbert spaces of ordinary Dirichlet series, studied earlier by McCarthy and Shalit.

These lecture notes are intended as an introduction to the theory of rational Dunkl operators, the associated special functions and related Markov processes with an emphasis on examples which are related to Riemannian symmetric spaces of Euclidean type and Bessel hypergroups on the matrix cones of positive semidefinite matrices. We start with a comprehensive introduction into Dunkl theory: Dunkl operators, the intertwining operator and its positivity, the Dunkl kernel and the Dunkl transform, the Dunkl Laplacian and the associated heat semigroup. We further give an outline of the connection with Calogero-Moser-Sutherland models and generalized Hermite polynomials. Moreover, of central interest will be product formulas, generalized translations and associated commutative hypergroup structures on closed Weyl chambers. In particular, we explain how Dunkl theory for particular multiplicities is related to Riemannian symmetric spaces of Euclidean type and Bessel hypergroups on the matrix cones, and how this leads to a bunch of multiplicities for which the Weyl-group invariant Dunkl theory admits a probability preserving translation and an associated commutative hypergroup structure on the closed Weyl chamber. We finally discuss Markov processes on R^N which are related with Dunkl theory with an emphasis on connections to random walk on groups and hypergroups. In particular associated martingales, martingale characterizations, moment functions and Appell characters are studied, i.e. diffusion-reflection processes with the Dunkl Laplacians as generators.

We present an extension of the Frenkel exciton model to incorporate exchange interactions between monomers in molecular aggregates in conjunction with {a multi-reference electronic structure approach}. Our derivation, {which combines the Frenkel exciton Hamiltonian and the single-electron pair exchange approximation}, yields a non-perturbative, variational expression for the exchange coupling that naturally excludes any basis set superposition error. The method has been implemented in OpenMolcas and enables combination with multi-reference electronic structure techniques. The main objective of the present study is to assess the role of exchange in systems with strong multi-configurational character. Illustrative examples demonstrate how the inclusion of exchange at different levels of approximation can substantially alter the magnitude and sign of intermonomer couplings and thus, for instance, potentially converting the predicted classification of the aggregate from H-type to J-type. Comparison with TDDFT-based couplings highlights significant discrepancies arising from multi-reference effects, double excitations, and Rydberg transitions. Overall, this approach advances the predictive modeling of photophysical and photochemical processes in aggregates of polyacenes, carotenoids, and other systems where multi-configurational and Rydberg states are essential.

Conventional wisdom in deep learning holds that overparameterization—having more parameters $p$ than training samples $n$—is benign: larger models generalize better and, even without regularization, interpolating models generalize well, the test error following a double-descent curve. One might expect the same benign overfitting for diffusion models, whose training reduces to regression, i.e. to minimizing a quadratic score-matching loss. Yet the opposite is observed: overfitting here is catastrophic, driving the model into a memorization regime. We resolve this paradox by combining experiments on U-Nets trained on CelebA with a random-features model for which we derive closed-form learning curves. We show that with a fixed number $m$ of noise realizations per training sample, an interpolation peak does occur, but at $p\sim nm$ rather than at $p\sim n$ as in standard regression. The rise of the test loss, however, sets in much earlier, at $p\sim n$, independently of $m$. This overfitting is malign because, although the implicit regularization of training is fully at work, it drives the model toward the empirical score, which memorizes the training set, rather than toward the true score. A bias-variance decomposition pinpoints the mechanism: the bias of the score estimator starts to grow at $p\sim n$; past the peak the variance decays, as in regression, whereas the bias keeps growing and both saturate at a large value. Since diffusion models are trained with $m\gg1$, the peak is pushed to very large model sizes, and therefore sit on the rising branch that precedes it, where malign overfitting is already in play. Nevertheless, overparameterization remains beneficial when paired with regularization: in the random-features theory and in U-Net experiments, optimally regularized large models—via a ridge penalty or early stopping, respectively—outperform any unregularized models.

We analyze stationary, axisymmetric Einstein-Maxwell-dilaton theory with zero cosmological constant in the conformal Carter class, employing a generalized Ernst approach on the space of potentials. We begin by establishing a complete local correspondence between Plebański-Demiański principal coordinates and Weyl data, covering the electric, magnetic, and twist potentials as well as the 1-forms \(A\), \(B\), and \(C\). In the source-free Weyl reduction, requiring the invariant area density to be harmonic singles out the \(\Lambda=0\) sector, and the principal chiral splitting isolates both electromagnetic misalignment and the part due to acceleration. Next, we obtain the Einstein reconstruction equations for general local EMD configurations in this setting. In a physical Carter domain with an ordinary scalar, the bilinear factor \(\Om=1-\acc pq\) enforces \(C=0\) and \(B_-=0\). With fixed \(\alphao\neq0\), the scalar field equation further rules out the charged PD Coulomb-type 1-form. Lastly, assuming an exact quadratic Stäckel scalar 1-form, the conditions of exactness together with reconstruction restrict the system to a single aligned branch, but consistency with the full potential equations removes this branch for arbitrary smooth Carter structure functions. Consequently, within the quadratic Stäckel class studied here, there is no local nonconstant solution with an ordinary dilaton. This conclusion does not extend to phantom scalars, conformal factors of higher degree, or non-Stäckel 1-forms.

We demonstrate a laser-wakefield accelerator driven by a high-repetition-rate optical parametric chirped-pulse amplification (OPCPA) laser, producing stable, quasi-monoenergetic electron beams at 100 Hz. Using 70 mJ, 8.1 fs laser pulses, we obtain 47 MeV electron beams with 15 pC charge and an intrinsic energy spread below 10%. Measurements over 10,000 consecutive shots show rms fluctuations of only 1.4% in peak energy and 12% in charge, together with low pointing jitter. Principal-component analysis reveals that the beam fluctuations are described by five physically interpretable modes, dominated by slow variations below 1 Hz. Particle-in-cell simulations reproduce the measured spectrum and show that nonlinear self-focusing localizes ionization injection, resulting in the observed narrow energy distribution. The demonstrated combination of electron energy, charge, beam quality, and stability represents a significant step toward high-average-power laser-plasma accelerators.

Real-world time series evolve continuously, with meaningful changes potentially emerging at any moment. However, existing time-series language models (TSLMs) remain inherently static. They either receive complete sequences for offline processing or alternate between streaming input and response generation, which prevents processing of new observations during interaction. We introduce a new regime, Time-Series Interaction: a model continuously perceives incoming time-series observations and user intent, autonomously decides when to remain silent or respond, and continues processing new observations during response generation. To realize this, we develop TimeInteract with three key designs: a dual-view streaming TS encoder that captures local variations and historical dynamics, a response control mechanism that learns when to trigger a response, and a decoupled streaming inference mechanism that separates control from response generation to avoid blocking subsequent observations. We further formulate a hierarchy of interaction capabilities, progressing from Understanding to Adaptivity. Based on this hierarchy, we construct StreamTSI-34K, a large-scale streaming TS interaction dataset with 34,588 episodes and 77,505 responses across synthetic and real-world time series in single- and multi-turn settings. Across all four interaction levels, TimeInteract consistently outperforms existing LLMs, VLMs, and TSLMs, with gains of up to 23.92 points on challenging tasks. It also improves response triggering while achieving near-zero stream stall and up to $2.15\times$ inference speedup.

Recent advances in large language models (LLMs) have made them widely used for code-related tasks. Identifier names are statistically informative in naturally occurring code, but their information is not always reliable. We investigate whether current LLMs assign disproportionate weight to lexical cues when renaming preserves program structure. We introduce Face/Off, a semantics-preserving identifier-renaming framework, and evaluate progressive naming conditions across multiple models and code-comprehension tasks. Within this framework, lexical overemphasis is pervasive across the evaluated models and primary tasks: performance generally decreases as identifier information is removed or made misleading, and outputs are often directed toward the meanings suggested by misleading names. The pattern persists under representative prompt- and fine-tuning-based interventions, suggesting that lexical overemphasis is an entrenched problem. A type-inference control confirms a boundary: naming effects are smaller when the answer is locally recoverable without the target name. These results do not imply that identifiers are unhelpful; rather, they reveal a systematic vulnerability in how current LLMs balance lexical cues against program structure. Our findings motivate evaluations and modeling methods that preserve the benefits of natural code regularities while keeping conclusions grounded in accurate, formalized code semantics.

We provide the first compact examples of type-changing generalized Calabi–Yau manifolds using Fei's construction of non-Kähler Calabi–Yau manifolds on branched covers of twistor spaces \cite{Fei16}. The structure, which is generically of type 1 and changes to type 3, is constructed on the product of a hyperelliptic surface of genus three and a four-torus, $\Sigma_3 \times \mathbb{T}^4$, with starting point a non-Kähler Calabi–Yau structure.

We study infinite-dimensional Dyson Brownian motions obtained as limits of finite systems without rescaling the actual stochastic dynamics. For $\beta=2$, we construct determinantal processes on an extended space of initial data and prove convergence of their finite-dimensional distributions under essentially optimal conditions. This extends the seminal results of Katori and Tanemura. The additional parameters record information at infinity and enter through an associated Laguerre-Pólya entire function. Moreover, for explicit classes of configurations, we establish convergence on path space and the Markov property. We prove rescaled long-time convergence, in finite-dimensional distributions, to the stationary extended $\mathsf{Sine}$ process from arbitrary symmetric initial configurations with power-law counting exponent $q\in(0,2)$. This extends the integer lattice relaxation result of Katori and Tanemura which was the only such result for explicit deterministic initial conditions. For $\beta\geq1$, we prove convergence of finite particle systems from regular initial data to the unique strong solution of an infinite-dimensional stochastic differential equation in a certain rigid-path-regularity class. This extends seminal works of Tsai and Osada. We finally derive a stochastic partial differential equation of Burgers-type for the Stieltjes transform of the dynamics.

Floquet bundle theory serves as a natural extension of the eigenvalue theory for elliptic and periodic parabolic operators to the spectral theory of non-autonomous, non-periodic parabolic operators, which provides a useful spectral tool for studying threshold dynamics in more general non-autonomous systems. However, current understanding and applications of the normalized principal Floquet bundle remain rather limited compared with the eigenvalue theory for elliptic or periodic parabolic problems. In this work, we first extend several fundamental properties of the principal eigenvalue to the normalized principal Floquet bundle of non-autonomous, non-periodic parabolic operators, and obtain a series of parallel results. Furthermore, we focus on the asymptotic behavior of the principal Floquet exponent with respect to the generalized frequency and diffusion rate under various typical parameter limits (including both independent and coupled parameter regimes) for the operator under homogeneous Neumann boundary conditions. The main tools are a parabolic comparison principle developed in this work for long-time limit processes, which is used to derive bounds for the principal Floquet exponent via suitably constructed sub- and super-solutions, supplemented by Harnack's inequality and long-time averaging techniques. Finally, we employ the normalized principal Floquet bundle to define two critical numbers (as extensions of the basic reproduction number) for a non-autonomous spatially diffusive SIS epidemic model, examine the extinction and weak persistence of the disease, and investigate the limiting behavior of the basic critical numbers with respect to the generalized frequency and diffusion rate.