Papers

We consider a closed macroscopic quantum system in a pure state $\psi$ evolving unitarily with Hamiltonian $H$ in a finite-dimensional Hilbert space $\mathscr{H}$. Following von Neumann, we assume a decomposition of $\mathscr{H}$ into mutually orthogonal subspaces $\mathscr{H}_{\nu}$, where each $\nu$ represents a macro state, and $\mathscr{H}_\nu$ contains the pure states compatible with $\nu$. We define the quantum Boltzmann entropy for a macro state $\nu$ as $S_\mathrm{qB}(\nu) = k_{\mathrm{B}} \log \dim \mathscr{H}_{\nu}$, where $k_\mathrm{B}$ is the Boltzmann constant. Goldstein et al. [http://arxiv.org/abs/1903.11870] have suggested, as a version of the second law of thermodynamics, that $S_\mathrm{qB}$ predominantly increases with time for realistic macroscopic systems. Here, we discuss precise versions of this statement and confirm them for several simple models of $H$ and $\mathscr{H}_\nu$.

Teacher-free one-step generative models such as W-Flow learn directly from distributional objectives, with the Sinkhorn divergence providing the strongest empirical performance among the energies studied in W-Flow. However, Sinkhorn divergence is based on balanced optimal transport, whose exact marginal constraints can induce undesirable transport under finite minibatch fluctuations. We introduce UW-Flow, which replaces the balanced Sinkhorn energy with a debiased unbalanced Sinkhorn divergence and derives the corresponding Wasserstein velocity. The resulting dynamics retain W-Flow's cross-minus-self structure, recover the balanced limit, and preserve target identifiability under suitable assumptions. We further characterize the finite-batch velocity error of balanced and unbalanced transport by comparing each minibatch field with its population counterpart. Our analysis identifies conditions under which marginal relaxation yields smaller minibatch-to-population velocity error, and shows that for separated matched modes, balanced sensitivity grows with inter-mode separation while the corresponding Unbalanced Optimal Transport (UOT) sensitivity remains controlled by local transport geometry. Controlled experiments on fixed-support and continuous two-mode distributions support these predictions. On MNIST, CIFAR-10, and ImageNet-100, UW-Flow improves generation under matched training budgets, with the largest gains in unconditional generation and consistent improvements in the class-conditional setting.

CMinHash promises a new direction for designing MinHash algorithms with reduced variance and storage space via circulant permutations. However, the original variance analysis is difficult to extend to other MinHash estimators. We present a new approach that gives explicit and easily comparable approximate variance expressions, and use it to analyze the Minner estimator. Numerical experiments show that our variance expression matches the empirical mean square error (MSE) of estimates. Our work thereby demonstrates how to apply circulant permutation to other MinHash estimators and compute their approximate variance.

Mixed-anion semiconductors offer opportunities for bandgap engineering beyond conventional multication alloys, but competing phases and disorder complicate their synthesis and the identification of their electronic properties. Gallium oxynitride (GaON) could connect the technologically important GaN and Ga2O3 systems, yet establishing a clear relation between its crystal structure, composition and bandgap remains challenging. Here we report the direct growth of crystalline spinel gamma-GaON films by nitrogen-plasma-assisted pulsed-laser deposition. Varying the nitrogen and oxygen supply yields monoclinic beta-Ga2O3, predominantly disordered films and crystalline spinel GaON. Complementary diffraction, atomic-resolution microscopy and elemental analysis establish the spinel structure and nitrogen incorporation. Temperature-dependent photocurrent spectroscopy reveals a clear redshift of the crystalline gamma-GaON response, supporting assignment of a band-edge transition near 4.3 eV at room temperature. These results open opportunities to investigate mixed-anion wide-bandgap semiconductor physics and develop heterostructures.

Generative Artificial Intelligence (GenAI) tools are increasingly integrated into everyday life, including sensitive domains such as religion. Prior works examined alignment with religious values and discussed the potential risks. However, to gain empirical insights into the adoption of these tools in users' religious lives, in this study, we conducted an online survey with 178 Muslim participants in Bangladesh and collected 726 prompts on religious use cases, along with open-ended responses about using these tools in religious contexts. Using reflexive thematic analysis, we identify three key roles these tools play in users' religious lives: (1) religious assistants, (2) perceived religious scholars, and (3) faith counselors. However, participants expressed concerns about misguidance, overreliance, and violations of Islamic tradition, and reported adopting protective strategies to avoid these pitfalls. Based on these, we propose a set of design guidelines for faith-sensitive GenAI systems that would preserve religious boundaries while supporting religious practices.

Language and vision language models can support autonomous systems, but changing operating conditions may compromise the reliability of their semantic outputs. This work introduces the Attribute Reliability Monitor (ARM), a framework that accumulates evidence of suspicion over time to assess individual semantic attributes. ARM accommodates different detectors and temporal monitoring methods, while leaving intervention policies to the consuming system. An initial implementation was evaluated using a specialized vision language model developed for satellite thermal infrared imagery. Monitoring followed a black box approach, using only observable responses without access to internal model states, confidence scores, or external reference answers. Under controlled input degradation, disagreement between responses provided evidence for selective suspension of attributes. The results illustrate how external behavioral monitoring can inform continued use of model outputs, while distinguishing operational suspicion from semantic correctness.

People often tackle difficult problems by thinking more deeply or exploring more possibilities. Similarly, recursive reasoners, which reason by repeatedly applying a learned update rule to latent states, can also scale inference through more updates (depth) or additional trajectories generated by perturbing latent states (width). When people hit a dead end in their thinking, they may step back and approach the problem from a fresh angle. However, neither depth scaling nor perturbation-based width scaling of recursive reasoners can reliably achieve this: both may converge to incorrect attractors. To address this, we propose to explore diverse reasoning traces by applying different update maps to the same latent state, without any additional training. We realize this idea through Input-Conditioned Search (ICS), which searches over input conditions to induce alternative update maps from the pretrained reasoner. Across six reasoning tasks, ICS achieves the highest mean accuracy, while requiring up to two orders of magnitude fewer model FLOPs than the evaluated scaling baselines. Controlled experiments confirm that ICS can redirect trajectories from the same state. Our theoretical analysis and rigorous numerical verification also confirm that ICS can locally eliminate incorrect attracting fixed points. Together, these findings establish ICS as an effective axis for efficient test-time scaling and motivate further research on shaping recursive reasoning during training and at test time.

We prove that the maximum modulus of every nonzero second Neumann eigenfunction on a bounded convex planar domain $\Omega$ is attained only on the boundary, without boundary smoothness or eigenvalue simplicity. The key is a rigidity theorem for any positive Neumann eigenvalue $\mu$: an interior maximum-modulus point $p$ with $\sqrt{\mu}\,\operatorname{dist}(p,\partial\Omega)<\pi/\sqrt{3}$ forces the domain to be a rectangle and the eigenfunction to be a cosine of one coordinate. In particular, $\mu\ge4\mu_2(\Omega)$, and the factor $4$ is sharp. This result reveals how a local distance condition at a single interior maximum-modulus point can enforce global geometric rigidity, thereby linking extremum localization to the classification of domains and eigenfunctions.

Cu-C materials and interfaces are central to diamond thermal management, metallization, graphene growth and carbon-coating technologies, but atomistic simulations across these applications require a potential that describes metallic Cu, diverse carbon bonding, heterogeneous Cu-C interactions and close atomic encounters. Here we develop NEP-CuC, a transferable neuroevolution potential trained on a dataset of 2,727 uniformly relabelled configurations selected from a heterogeneous Cu-C candidate pool. Disagreement within an ensemble of five independently trained models provides an internal indicator of model sensitivity. A Ziegler-Biersack-Littmark (ZBL) correction describes short-range nuclear repulsion, while a runtime D3(BJ) correction supplements dispersion-sensitive calculations. Tests on source-derived held-out and independent external datasets, together with structural, elastic, defect, interfacial, vibrational and finite-temperature transport benchmarks, establish broad but property-dependent physical fidelity across elemental Cu, diverse carbon environments and heterogeneous Cu-C configurations. Large-scale simulations of amorphous-carbon formation and orientation-dependent shock compression of Cu further test transferability under strongly non-equilibrium conditions. For Cu/diamond interfaces, the same potential describes deposition, annealing and tensile response: preparation history leaves persistent differences in interphase structure, while the mean peak nominal tensile stress at 500 K decreases from $12.35\pm0.71$ to $6.65\pm0.98$ GPa as the incident Cu energy during the initial deposition stage increases from 1 to 50 eV. NEP-CuC provides an atomistic framework for linking processing, interfacial structure and physical response across Cu-C systems, with direct relevance to Cu/diamond thermal-management interfaces, diamond metallization and non-equilibrium Cu-C processing.

In this paper we present partial progress towards two conjectures on the variation of maximal operators of convolution type $\mathrm{M}_{\varphi}$, associated to non-negative and radial decreasing kernels $\varphi$ of integral $1$. In dimension $d=1$ we prove that $\mathrm{var}(\mathrm{M}_{\varphi} f) \leq \mathrm{var}(f)$ when $f$ is a characteristic function. In higher dimensions we prove that $\mathrm{var}(\mathrm{M}_{\varphi} f) \leq C_d \mathrm{var}(f)$ when $f$ is a radial characteristic function, with a constant $C_d$ depending only on the dimension $d$ but not on the kernel $\varphi$. We further show that $C_d \sim \sqrt{d}\log d $ is the sharp dependence on $d$ for such an inequality. This last claim relies on a novel explicit asymptotic estimate for the Hardy-Littlewood maximal function of the unit ball in $\mathbb R^d$, a result of independent interest.

Foundation models are increasingly used as feature extractors for downstream tasks with limited data, typically through representations obtained from their encoders. However, foundation models trained for structured tasks such as segmentation also form object-specific representations during task execution, whose transferability remains largely unexplored. We investigate Segment Anything Model 2 (SAM2) object pointers, compact object-level representations produced through the segmentation pathway, as transferable representations for downstream tumour prediction. Through experiments on two PET/CT tumour prediction tasks, we find that object pointers learned using segmentation supervision alone already encode tumour-discriminative information, despite never being optimised for the downstream labels. We further introduce task-guided object pointers, where an auxiliary objective jointly shapes the object pointer space for the downstream prediction together with segmentation. Task-guided pointers improve cancer-type classification, outperforming conventional encoder-based transfer and radiomics, while segmentation-only pointers perform best on the more challenging recurrence prediction task. Representation analyses reveal associations between object pointers and interpretable radiomics features, class-dependent organisation of the learned representation space, and explicit spatial grounding around the segmented tumour. Together, these findings suggest that useful transferable representations can emerge through the execution of a foundation model's primary task, providing an alternative to conventional encoder-based transfer.

The work presents a canonical Hamiltonian system in a classical scenario and further extend the analysis to a quantum mechanical framework for a homogeneous and isotropic universe governed by Bumblebee gravity, in which spontaneous Lorentz symmetry breaking is triggered by a vector field acquiring a vacuum expectation value (VEV). Starting from the full four-dimensional covariant action, the derivation has been carried out for the one-dimensional mini-superspace Lagrangian density for a flat Friedmann-Lemaître-Robertson-Walker (FLRW) background with a purely temporal bumblebee field configuration. Further, the work focuses on to construct the classical Hamiltonian system through a canonical Legendre transformation, identifying the lapse function $N(t)$ as a Lagrange multiplier enforcing a primary Hamiltonian constraint. Apart form that, the conservation of phase-space volume via Liouville's theorem, locate and classify the critical points of the resulting dynamical system, and examine the stability of the Lorentz-violating vacuum has been carried to test the classical Hamiltonian system in the work. Furthermore, the modified first and second Friedmann equations and discuss the resulting time-dependence of the effective gravitational coupling, illustrating the results with numerical phase-space has been discussed in this work. Finally, the work extended to construct the Wheeler-DeWitt equation through the canonical quantization and obtain the approximate solutions in the deep-quantum ($a\to0$) and near-vacuum regimes for it.

Language models are increasingly expected to support users in stressful or uncertain situations, where warmth and empathy are desirable. While recent work associates warmer models with reduced reliability, we show that this conclusion is incomplete. Prior studies often train primarily on empathetic dialogue, making it difficult to separate tone effects from SFT-induced distribution shift. We revisit this issue at both the model and evaluation levels. We study prompting, SFT, and RL for inducing warm behavior, and comprehensively evaluate capability preservation using an 11-dataset benchmark suite and EMOFACT under emotional contexts. Across multiple model families, we find that warmth effects are task- and model-dependent, while task-aligned adaptive tone can improve empathy and targeted reliability, although some capability trade-offs remain. We further introduce human-validated empathy evaluators specifically developed to assess empathetic model responses. Our results suggest that models can be more supportive in social interactions while remaining reliable.

The Boolean Satisfiability (SAT) problem is a remarkably expressive computational problem, serving both as a cornerstone of complexity theory and as a unifying language for a wide range of search and reasoning tasks. While $\mathrm{NP}$-complete in general, SAT is well known to be linear-time solvable on instances with bounded treewidth. This motivates the search for "hidden" tree-like structures in formulas—for instance by detecting so-called weak $t$-treewidth backdoors. Such backdoors are sets of $k$ variables that can be assigned in such a way that the remaining instance has treewidth at most $t$ and is satisfiable whenever the original instance is. Ever since Gaspers and Szeider [ICALP 2012] showed that finding a weak $t$-treewidth backdoor for a SAT formula is $\mathrm{W}[2]$-hard with respect to $k$, tractability results have been restricted to structurally sparse formulas—most notably, a linear-time algorithm for $d$-SAT (bounded clause-length SAT) by Fomin, Lokshtanov, Misra, Ramanujan, and Saurabh [SODA 2015] and a non-uniform polynomial-time algorithm when the incidence graph of the input formula belongs to a nowhere-dense graph class (or more generally is $K_{d,d}$-free) by Lokshtanov, Panolan, and Ramanujan [ICALP 2022]. In this paper, we push the tractability boundaries of finding weak $t$-treewidth backdoors beyond sparsity and into the regime of "well-behaved" dense formulas. Aiming towards relaxing structural input restrictions from nowhere-dense to edge-stable graph classes, as featured in the theory of beyond sparsity for FO model checking, or to bounded VC-dimension set systems, we identify signed semi-ladder-freeness as the correct dense structural restriction that establishes the boundary between tractability and intractability along this line.

Kagome lattices provide a simple geometry for engineering electronic states that exhibit novel quantum behavior. In conventional kagome materials, however, the relevant states are often embedded in chemically complex three-dimensional crystals and are difficult to isolate or tune. Molecule-based kagome networks offer structural precision and tunability, but have largely been confined to metallic substrates incompatible with 2D devices. Here we report growth of single-layer Ni$_3$(HITP)$_2$ metal-organic frameworks (MOFs) on van der Waals substrates including graphene-based field-effect devices using metal-salt molecular beam epitaxy. Low-temperature scanning tunneling spectroscopy resolves kagome-derived flat-band and van Hove singularity-based spectral features consistent with first-principles calculations. Gating hybrid 2D MOF devices allows us to energetically shift kagome spectral features, thus establishing electrostatic control of chemically programmable 2D kagome electronic lattices.

Understanding agent heterogeneity can inform the design of shared and agent-specific policy representations, however existing criteria provide no unified structural test. We study finite-horizon POSGs and define a type-compatible transformation as a policy-evaluation symmetry when complete policy evaluation commutes with it for every decentralized behavioral policy. Policy-evaluation symmetries form a group for which agent orbits define structural agent types. This classification does not depend on the particular policy chosen or the distribution of initial states. We prove that every componentwise environment automorphism is a policy-evaluation symmetry, but that the converse does not hold. Applying the framework to cooperative Dec-POMDPs shows that agents can belong to different structural types even when they share a common reward. For prescribed weighted reward aggregation, we derive an exact value identity and sufficient conditions for the resulting Dec-POMDP to inherit the policy-evaluation symmetries of the source POSG. Finally, we give sufficient conditions under which repeated model factorizations induce policy-evaluation symmetry, and show that preserving an optimal Boltzmann policy set neither implies nor follows from this symmetry in general.

In the Hunters and Rabbit game, hunters shoot at a set of vertices at each round, while an invisible rabbit must move along an edge. We prove that deciding whether a given number of hunters can guarantee capture is PSPACE-complete, even on simple bipartite graphs. The reduction uses intersection nonemptiness for inverse finite automata: one common word must put all automata in accepting states simultaneously. A clique acting as a safe haven and two families of guards force every winning strategy to simulate common transitions. Existing transformations remove the restriction on the rabbit's initial positions and yield PSPACE-completeness for the directed game with one hunter, even on oriented graphs with positive minimum outdegree. Adjusting the guard sizes also proves PSPACE-completeness of the firefighting game, in which the rabbit may remain at its current vertex.

Let $F:M^n\to\mathbb{S}^{2n+1}$ be a minimal Legendrian immersion of a connected closed manifold, where $n\ge3$, and let $B$ be its second fundamental form. We prove that if $|B|^2\le(n-1)(n+2)/n$, then, up to a unitary transformation, $F$ parametrizes a totally geodesic sphere or is a finite covering of the image of the minimal Calabi product of a point and a totally geodesic Legendrian sphere. This proves Conjecture 1 of Luo and Sun [Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), p. 548].

Video understanding benchmarks are the primary yardstick for measuring progress in video-language modeling, and a growing body of work relies on them to claim improvements. The reported benchmark scores in these works, however, are almost always produced by frameworks (e.g., lmms eval) used as black boxes: certain evaluation choices like frame sampling, preprocessing, and input formatting are inherited from preset defaults and seldom inspected carefully. We show these unexamined choices are a major source of variance. Even for the same model checkpoint, evaluation configuration alone can shift accuracy by up to 19.5pp. Hence, it is unclear whether reported gains reflect genuine improvements in capability or differences in evaluation setup. Moreover, many benchmark samples offer little evidence of visual capability, since they can be answered without the video, and the full suites are expensive to evaluate. We address these issues in two parts. First, we curate existing suites for discriminability and efficiency, producing Meta-Video, a balanced 1,000-question benchmark over 791 videos spanning diverse tasks and durations, which largely preserves model rankings, at 5 times inference speedup for a full-frame count sweep experiment (4-128 frames). Second, we establish a standardized evaluation protocol by sweeping the configuration space to isolate choices that measurably affect results, distill them into best practices and reporting standards, and release reproducible reference scores across five models and seven benchmarks. We hope that the proposed evaluation framework and the Meta-Video benchmark will support video understanding research by providing a reproducible and efficient foundation to measure progress in a reliable way.

Flow Matching has recently become popular as a technique for generative modeling with applications in many domains. In the standard Flow Matching formulation, we interpret the vector field as the solution of a least-squares regression problem, which is known to be sensitive to contamination. Therefore we leverage the viewpoint of robust statistics and replace the squared Euclidean distance with the Euclidean distance as a regression objective. This seemingly minor change strongly affects the properties of Flow Matching. We provide a probabilistic interpretation of the new regression target as a conditional geometric median and use this characterization to study distributional robustness of the Euclidean Distance-based Flow Matching. We analyze the mathematical properties of this changed objective and the effect of a contaminated target distribution on the involved conditional velocity distributions. We discuss the challenge of directly characterizing the distribution that our method induces. In specific contamination settings we show theoretically that our method yields an improvement for severe contamination. Finally, we present numerical experiments conducted on the CIFAR-10 image dataset where a proportion of the samples has been contaminated. Our Euclidean distance-based Flow Matching shows a clear advantage in scenarios of severe contamination of the target distribution.

In many applications one object is chosen to represent a sample and the least typical objects are set aside, usually without any measure of uncertainty. We measure how central an object is by its reference depth, the probability that a random reference point is closer to it than to a random opponent. Using only pairwise dissimilarities, we construct simultaneous intervals for all ranks, a centre set that contains the most central observation under stated conditions, and outlier sets that contain only objects among the least central ones, each with asymptotic probability at least a prescribed level. These guarantees hold even though the ranked objects are themselves random. For symmetric distributions on the real line, the centre set has the smallest possible order of size up to a logarithmic factor, and in dimension three or more it reduces to the most central observation with probability tending to one. For typical meteorological years the centre set usually contains several candidate years, so the data do not determine the year on which building-energy simulations rest; for three collections of gene trees it contains one tree, whose features can be reported as those of the most central tree.

Function-as-a-Service (FaaS) increases developer productivity by removing operational concerns such as managing hardware and software runtimes. Because function instances are started on demand, some requests must wait for a new instance, called a cold start. These are a common obstacle to more widespread adoption of FaaS, as they lead to unpredictable latency spikes that are not acceptable for some use cases. To solve this, many cold start optimizations have been implemented. However, comparatively little attention has been directed towards the slowest warm starts. With improvements in cold start times, slow warm starts emerge as a new problem for extreme tail events. By analyzing a dataset of several million requests across seven FaaS platforms, we show that the long tail of slow warm starts can be hundreds of times slower than their median performance, while a virtual machine we use as a baseline is substantially more consistent. We confirm this across four independent public datasets spanning 2020 to 2025, so the effect is neither new nor specific to one platform. Moreover, the warm share among the slowest calls starts to rise again at the highest percentiles: the most extreme latencies are not only cold starts, and cold-start mitigation alone cannot remove them.

We investigate the cosmological dynamics of gravitational particle creation in a spatially flat FLRW Universe comprising radiation, baryons, dark matter (DM), and a separately conserved dark-energy component. The DM sector is described as an effective open thermodynamic fluid incorporating intrinsic pressure and particle-creation-induced negative pressure. Introducing the compact variable $u=H_0/(H_0+H)$, we construct a four-dimensional autonomous system and systematically investigate its critical points, dynamical stability, and physical admissibility. The model is constrained using three observational combinations: PPS+BAO+OHD, Union3+BAO+OHD, and DES-DOV+BAO+OHD. The observationally preferred parameter regions exclude the nontrivial DM-dominated de Sitter-like branches from the physical phase space, while identifying a stable, physically admissible, dark-energy-dominated accelerating attractor. The reconstructed cosmological evolution consistently reproduces the radiation–matter–acceleration sequence, with a transition redshift $z_{\rm tr}\simeq0.71$–$0.73$. Posterior-level analysis reveals that approximately $92\%$–$97\%$ of the samples support this accelerating attractor. The asymptotic evolution approaches a quintessence-like power-law expansion with $H\rightarrow0$, rather than a de Sitter state. These findings establish a connection between observational constraints, gravitational particle creation, and the global dynamical structure governing the distant cosmic future.

We consider the image of the comparison homomorphism from bounded cohomology to ordinary cohomology. Its dimension is called the bounded Betti number. In our earlier work we proved that the total bounded Betti number of a closed Riemannian $n$-manifold satisfying $\Ric\geq -(n-1)$ and $\Diam\leq D$ is bounded by a constant depending only on $n$ and $D$. The original proof used the amenability of the images of sufficiently small local fundamental groups. Since that work was written, Kapovitch and Wilking proved Gromov's generalized Margulis lemma under a lower Ricci curvature bound. Their theorem gives the stronger conclusion that these local groups are uniformly virtually nilpotent. We explain how their result supplies a direct modern input for the bounded Betti number theorem. We also discuss the topological meaning of the bounded part of ordinary cohomology and its relation to the $\ell^1$-seminorm on homology.

Jev has attracted considerable attention for its rapid, generalizable decision-making, yet its medical reliability remains underexplored. To address this gap, our study compares various types of system-one decision models across 15 medical datasets comprising 8,742 instances under a unified protocol. Beyond accuracy, we assess decision stability and the ability to recognize unreliable decisions. Our results show that Jev rivals leading models on primarily text-driven tasks but is less effective on numerically driven tasks, revealing a distinct capability boundary. Under question-format variations, Jev maintains strong decision consistency and small accuracy fluctuations. StartLux-27B exhibits comparable consistency, while D3-27B achieves higher overall accuracy and a lower mean decision reversal rate. But these strengths do not extend to reliable abstention awareness: most evaluated models show limitations in both recognizing missing correct answers and using optional abstention. Moreover, we find that Jev's probability-based answer selection can be improved by changing how decisions are scored: normalizing option-wise binary scores yields higher retained accuracy than choice scoring. Taken together, our findings provide early guidance for improving and applying decision models in medicine.

We characterize several properties of higher-arity function classes, specifically finite $\VC_k$-dimension, slicewise finite $\VC$-dimension, and finite slicewise Littlestone dimension, in terms of generalizations of Rademacher complexity. These characterizations sets up simple new proofs of several results on finite $\VC_k$-dimension and slicewise finite $\VC$-dimension. These include the preservation of finite $\VC_k$-dimension under taking expectations and uniform law of large numbers statements for both properties. All of these results apply naturally to real-valued function classes as well as boolean-valued classes. We also prove a new adversarial uniform law of large numbers for classes with finite slicewise Littlestone dimension.

We investigate the rotational properties of a zero-temperature Bose-Einstein condensed gas, which is confined in a hard-wall potential. For weak interactions the gas carries its angular momentum via vortices of multiple quantization. As the coupling increases and for sufficiently slow rotational frequencies of the trap, the multiply-quantized vortex states become unstable against a mixed state. For stronger couplings, as the angular frequency of the trap increases, there is a transition from a vortex lattice, to a vortex lattice with a hole at the trap center, and finally to a giant vortex. For weak effective attractive interactions we still have vortices of multiple quantization. These states become unstable towards states which break the axial symmetry of the problem and resemble center-of-mass excitation, as the coupling becomes more negative. Finally, for even more attractive forces, the system becomes unstable against collapse.

Multilingual dense retrievers place all languages in one shared space so that a question finds relevant documents, regardless of which language they are written in. We show that the same question, asked in different languages, retrieves substantially different documents, and this disagreement is not because of weak language understanding. Each language performs well on its own and ranks candidates consistently with the other languages; divergence appears only when languages share an index. We trace the mechanism: each language keeps its own ranking, but in the shared index, its correct answers are pushed far down, buried under non-relevant documents that are either in the query's own language or in English, depending on the model. We introduce disagreement-guided steering: for each language, we contrast the documents that other languages agree on for a question against those that language retrieves idiosyncratically, and take the difference of their representations as a language-specific direction, added to that language's queries at inference. Against baselines of language-mean removal, subspace removal, and per-language rotation, it most strongly reduces same-language crowding while remaining competitive on recall, trading off against mean removal, which better preserves ranking quality.

Strain gradients, unlike homogeneous strain, are odd under space inversion and unlock couplings that are strictly forbidden in the parent crystal. At the nanoscale, and in curved low-dimensional systems in particular, such gradients are readily attainable and often unavoidable, so that curvature acts as a symmetry-breaking field written into the lattice itself. This chapter reviews the resulting couplings to electric polarization and magnetism. After establishing the geometric relation between strain gradients and curvature, we review flexoelectricity as the better-established reference case: the bulk, surface and dynamic contributions that complicate its definition, its microscopic mechanisms, its manifestation in two-dimensional materials, and the first-principles methodology now available for its calculation. Using the magnetoelectric effect as a conceptual bridge, we then develop flexomagnetism in a broad sense, covering any coupling phenomena between magnetism and strain gradients, and specifically discuss the linear magnetization response to curvature and curvature-induced Dzyaloshinskii-Moriya interactions, beyond the established curvilinear treatment. Finally, we briefly describe some additional effects fitting under this broad umbrella of flexomagnetism, including nonlinear ones, inhomogeneous magnetoelectric and Rashba-type phenomena, and connections to altermagnetism.

Finite-horizon optimal control becomes difficult when a dynamical model can be simulated reliably, but its derivatives are unavailable, discontinuous, or unreliable. Hamiltonian Primal-Dual Policy Optimization (HPDPO) is introduced as a derivative-free, model-based reinforcement learning framework for this setting. The control is represented as a time-indexed finite-horizon policy. Candidate actions are evaluated by deterministic suffix rollouts of the supplied transition model, while a regularized Hamiltonian dual formulation supplies structured future-cost information through block-coordinate multiplier updates. No differentiation of the state dynamics or learned critic is required. The method is validated on nonsmooth problems with exact solutions, an analytical and learned ReLU biomarker cascade, an amyloid Neural ODE trained from longitudinal ADNI data, and three multidimensional-control problems with up to five simultaneous controls. Across these examples, HPDPO closely reproduces exact or reference objective values, trajectories, and control structures. The results show that forward simulation and Hamiltonian dual structure can be combined to solve deterministic finite-horizon control problems involving nonsmooth, learned, or black-box dynamics.