Consider a sequence of Markov processes $X^1, X^2,...$ with state space $E$, where $X^N$ has a strong drift to $D \subseteq E$, such that $Φ(X^N)$ is slow for some appropriate $Φ: E\to D$. Using the method of martingale problems, we give a limit result, such that $Φ(X^N) \xRightarrow{N\to\infty} Z$ in the space of càdlàg paths, and $X^N \xRightarrow{N\to\infty} X$ in measure. \\ We apply the general limit result to models for copy number variation of genetic elements in a diploid Moran model of size $N$. The population by time $t$ is described by $X^N \in \mathcal P(\mathbb N_0)$, where $X^N_k$ is the frequency of individuals with copy number $k$, and $Φ: \mathcal P(\mathbb

In this paper, we study large-time asymptotics for heat and fractional heat equations in two discrete settings: the full lattice \(\mathbb Z^d\) and finite connected subgraphs with Dirichlet boundary condition. These results provide a unified discrete theory of long-time asymptotics for local and nonlocal diffusions. For \(d\ge1\) and \(s\in(0,1]\), we consider on \(\mathbb Z^d\) the Cauchy problem \[ \partial_t u+(-Δ)^s u=0,\qquad u(0)=u_0\in \ell^1(\mathbb Z^d), \] and derive a precise first-order asymptotic expansion toward the lattice fractional heat kernel \(G_t^{(s)}\). The main technical input is a pair of sharp translation-increment bounds for \(G_t^{(s)}\): a pointwise estimate and an \(\ell^1\)-estimate. As consequences, under finite first moment we obtain the optimal decay rate \(t^{-1/(2s)}\) in \(\ell^p\)-asymptotics (\(1\le p\le\infty\)), and we prove sharpness by explicit shifted-kernel examples. Without moment assumptions, we still establish convergence in the full \(\ell^1\)-class, and we show that no universal quantitative rate can hold in general. We also analyze fractional Dirichlet diffusion on finite connected subgraphs (restricted fractional setting, including \(s=1\) as the local case). In this finite-dimensional framework, solutions admit spectral decomposition and exhibit exponential large-time behavior governed by the principal eigenvalue and the spectral gap. In addition, we study positivity improving properties of the associated semigroups for both the lattice and Dirichlet evolutions.

Zero-knowledge proof generation imposes stringent timing and reliability constraints on blockchain systems. For ZK-rollups, delayed proofs cause finality lag and economic loss; for Ethereum's emerging L1 zkEVM, proofs must complete within the 12-second slot window to enable stateless validation. The Ethereum Foundation's Ethproofs initiative coordinates multiple independent zkVMs across proving clusters to achieve real-time block proving, yet no principled orchestration framework addresses the joint challenges of (i) strict head-of-chain ordering, (ii) sub-slot latency bounds, (iii) fault-tolerant task reassignment, and (iv) prover-agnostic workflow composition. We present push0, a cloud-native proof orchestration system that decouples prover binaries from scheduling infrastructure. push0 employs an event-driven dispatcher--collector architecture over persistent priority queues, enforcing block-sequential proving while exploiting intra-block parallelism. We formalize requirements drawn from production ZK-rollup operations and the Ethereum real-time proving specification, then demonstrate via production Kubernetes cluster experiments that push0 achieves 5 ms median orchestration overhead with 99--100% scaling efficiency at 32 dispatchers for realistic workloads--overhead negligible (less than 0.1%) relative to typical proof computation times of 7+ seconds. Controlled Docker experiments validate these results, showing comparable performance (3--10 ms P50) when network variance is eliminated. Production deployment on the Zircuit zkrollup (14+ million mainnet blocks since March 2025) provides ecological validity for these controlled experiments. Our design enables seamless integration of heterogeneous zkVMs, supports automatic task recovery via message persistence, and provides the scheduling primitives necessary for both centralized rollup operators and decentralized multi-prover networks.

The combination of uninterpreted function symbols and universal quantification occurs in many applications of automated reasoning, for example, due to their ability to reason about arrays. Yet the satisfiability of such formulas is, in general, undecidable. In practice, SMT solvers are often successful in the unsatisfiable case, using heuristics. However, in the satisfiable case, they rely on explicit model construction, which fails for formulas whose smallest model is not small enough. We introduce an alternative approach that certifies satisfiability using induction arguments, and apply it to the case of linear integer arithmetic. The resulting algorithm is able to prove satisfiability of formulas that are out of reach for current SMT solvers.

One of the most well-known conjectures concerning Hamiltonicity in graphs asserts that any sufficiently large connected vertex transitive graph contains a Hamilton cycle. In this form, it was first written down by Thomassen in 1978, inspired by a closely related conjecture due to Lovász from 1969. It has been attributed to several other authors in a survey on the topic by Witte and Gallian in 1984. The analogous question for vertex transitive digraphs has an even longer history, having been first considered by Rankin in 1946. It is arguably more natural from the group-theoretic perspective underlying this problem in both settings. Trotter and Erdős proved in 1978 that there are infinitely many connected vertex transitive digraphs which are not Hamiltonian. This left open the very natural question of how long a directed cycle one can guarantee in a connected vertex transitive digraph on $n$ vertices. In 1981, Alspach asked if the maximum perimeter gap (the gap between the circumference and the order of the digraph) is a growing function in $n$. We answer this question in the affirmative, showing that it grows at least as fast as $(1-o(1)) \ln n$. On the other hand, we prove that one can always find a directed cycle of length at least $Ω(n^{1/3})$, establishing the first lower bound growing with $n$, providing a directed analogue of a famous result of Babai from 1979 in the undirected setting.

We show that the Auslander-Reiten Formula for a finite dimensional hereditary algebra is invariant under the Auslander-Reiten translate.

We uncover a "junction law" for genuine multipartite entanglement, suggesting that in gapped local systems multipartite entanglement is controlled and effectively localized near junctions where subsystem boundaries meet. Using the Rényi-2 genuine multi-entropy $\mathrm{GM}^{(\mathtt{q})}_2$ as a diagnostic of genuine $\mathtt{q}$-partite entanglement, we establish this behavior in $(2+1)$-dimensional gapped free-fermion lattices with correlation length $ξ$. For partitions with a single junction, $\mathrm{GM}^{(\mathtt{q})}_2$ exhibits a universal scaling crossover in $L/ξ$, growing for $L\llξ$ and saturating to a $ξ$-dependent constant for $L\ggξ$, up to $\mathcal{O}(e^{-L/ξ})$ corrections. In sharp contrast, for partitions without a junction, $\mathrm{GM}^{(\mathtt{q})}_2$ is exponentially suppressed in $L/ξ$ and drops below numerical resolution once $L\ggξ$. We observe the same pattern for $\mathtt{q}=3$ (tripartite) and $\mathtt{q}=4$ (quadripartite) cases, and further corroborate this localization by translating the junction at fixed system size. We also provide a geometric explanation of the junction law in holography. Altogether, these results show that in this gapped free-fermion setting genuine multipartite entanglement is localized within a correlation-length neighborhood of junctions.

We show that a family of quantum Ornstein-Uhlenbeck semigroups is hypercontractive. We also obtain the optimal order of the optimal time up to a constant for those elements whose Gibbs state is zero. The main ingredient of our proof is Meixiner polynomials.

Computer experiments involving both qualitative and quantitative (QQ) factors have attracted increasing attention. Gaussian process (GP) models have proven effective in this context by choosing specialized covariance functions for QQ factors. In this work, we extend the latent variable-based GP approach, which maps qualitative factors into a continuous latent space, by establishing a general framework to apply standard kernel functions to continuous latent variables. This approach provides a novel perspective for interpreting some existing GP models for QQ factors and introduces new covariance structures in some situations. The ordinal structure can be incorporated naturally and seamlessly in this framework. Furthermore, the Bayesian information criterion and leave-one-out cross-validation are employed for model selection and model averaging. The performance of the proposed method is comprehensively studied on several examples.

Community detection is a fundamental task in complex network analysis. Fairness-aware community detection seeks to prevent biased node partitions, typically framed in terms of individual fairness, which requires similar nodes to be treated similarly, and group fairness, which aims to avoid disadvantaging specific groups of nodes. While existing literature on fair community detection has primarily focused on group fairness, we introduce a novel measure to quantify individual fairness in community detection methods. The proposed measure captures unfairness as the vectorial distance between a node's true and predicted community representations, computed using the community co-occurrence matrix. We provide a comprehensive empirical investigation of a broad set of community detection algorithms from the literature on both synthetic networks, with varying levels of community explicitness, and real-world networks. We particularly investigate the fairness-performance trade-off using standard quality metrics and compare individual fairness outcomes with existing group fairness measures. The results show that individual unfairness can occur even when group fairness or clustering accuracy is high, underscoring that individual and group fairness are not interchangeable. Moreover, fairness depends critically on the detectability of community structure. However, we find that Significance and Surprise for denser graphs, and Combo, Leiden, and SBMDL for sparser graphs result in a better trade-off between individual fairness and community quality. Overall, our findings, together with the fact that community detection is an important step in many network analysis downstream tasks, highlight the necessity of developing fairness-aware community detection methods.

Automated theorem provers (ATPs) can disprove conjectures by saturating a set of clauses, but the resulting saturated sets are opaque certificates. In the unit equational fragment, a saturated set can in fact be read as a convergent rewrite system defining an explicit, possibly infinite, model -- but this is not widely known, even amongst frequent users of ATPs. Moreover, ATPs do not emit these explicit certificates for infinite (counter-)models. We present such a certificate construction in full, implement it in Vampire and E, and apply it to the recent Equational Theories Project, where hundreds of implications do not admit finite countermodels. The resulting rewrite systems can be checked for confluence and termination by existing certified tools, yielding trustworthy countermodels.

Young adults often take breaks from screen-intensive work by consuming digital content on mobile phones, which undermines rest through visual fatigue and inactivity. We introduce a design framework that embeds light break activities into media content on AR smart glasses, balancing engagement and recovery. The framework employs three strategies: (1) seamlessly guiding users by embedding activity cues aligned with media elements; (2) transitioning to audio-centric formats to reduce visual load while sustaining immersion; and (3) structuring sessions with "rise-peak-closure" pacing for smooth transitions. In a within-subjects study (N = 16) comparing passive viewing, reminder-based breaks, and non-narrative activities, InteractiveBreak instantiated from our framework seamlessly guided activities, sustained engagement, and enhanced break quality. These findings demonstrate wearable AR's potential to support restorative relaxation by transforming breaks into engaging and meaningful experiences.

Multi-objective optimisation using BRIGHT has proven insightful and effective in prostate cancer brachytherapy treatment planning. BRachytherapy via artificially Intelligent GOMEA-Heuristic based Treatment planning (BRIGHT) generates multiple treatment plans, each with a different trade-off between tumour coverage and organs-at-risk sparing. BRIGHT was recently extended to cervical cancer brachytherapy. In this study, we present a novel, custom-developed graphical user interface (GUI) that enables plan navigation, pairwise comparisons, dose distribution visualisation, and possibility for adjustments - essential for efficient clinical use of BRIGHT. End-user validation of BRIGHT with the dedicated GUI was conducted for cervical cancer brachytherapy by emulating clinical practice in ten previously treated patients. A multidisciplinary brachytherapy team used BRIGHT to create new treatment plans. GUI usability was assessed using the System Usability Scale (SUS). BRIGHT plan quality was compared to clinical practice via blinded one-on-one comparisons. The GUI offered helpful features for plan navigation and evaluation, giving users quick insight into whether planning aims are achievable and what treatment options are available. The overall SUS score was 83.3, indicating an 'excellent' system. BRIGHT outperformed clinical practice in five out of ten patients regarding the coverage-sparing trade-off and performed equally well in the remaining five. The BRIGHT plan was preferred over the clinical plan in eight out of ten patients, four of which showed clinically relevant differences. The clinical plan was preferred in two patients, neither with clinically relevant differences. In conclusion, BRIGHT, with its dedicated GUI, is a clinically viable and user-friendly tool for treatment planning in cervical cancer brachytherapy.

The present paper considers multipartite graphs from the perspective of design theory and coding theory. A one-factor $F$ of the complete multipartite graph $K_{n\times g}$ (with $n$ parts of size $g$) gives rise to a $(g+1)$-ary code ${\cal C}$ of length $n$ and constant weight two. Furthermore, if the one-factor $F$ meets a certain constraint, then ${\cal C}$ becomes an optimal code with minimum distance three. We initiate the study of one-factorizations of complete multipartite graphs subject to distance constraints. The problem of decomposing $K_{n\times g}$ into the largest subgraphs with minimum distance three is investigated. It is proved that, for $n\le g$, the complete multipartite graph $K_{n\times g}$ can be decomposed into $g^2$ copies of the largest subgraphs with minimum distance three. For even $gn$ with $n>g$, it is proved that the complete multipartite graph $K_{n\times g}$ can be decomposed into $g(n-1)$ one-factors with minimum distance three, leaving a small gap of $n$ (in terms of $g$) to be resolved (If $gn$ is odd when $n>g$, no such decomposition of $K_{n\times g}$ exists).

This chapter provides a comprehensive overview of proof-theoretic methods for establishing interpolation properties across a range of logics, including classical, intuitionistic, modal, and substructural logics. Central to the discussion are two foundational techniques: Maehara's method for Craig interpolation and Pitts' method for uniform interpolation. The chapter demonstrates how these methods lead to results on the existence of well-behaved proof systems in the contemporary framework of universal proof theory and how they provide a road map for constructing interpolation proofs using modern proof formalisms. The emphasis of the chapter is on constructive, modular, and syntax-driven techniques that illuminate deeper connections between interpolation properties and proof systems.

Computer-Aided Design (CAD) delivers rapid, editable modeling for engineering and manufacturing. Recent AI progress now makes full automation feasible for various CAD tasks. However, progress is bottlenecked by data: public corpora mostly contain sketch-extrude sequences, lack complex operations, multi-operation composition and design intent, and thus hinder effective fine-tuning. Attempts to bypass this with frozen VLMs often yield simple or invalid programs due to limited 3D grounding in current foundation models. We present CADEvolve, an evolution-based pipeline and dataset that starts from simple primitives and, via VLM-guided edits and validations, incrementally grows CAD programs toward industrial-grade complexity. The result is 8k complex parts expressed as executable CadQuery parametric generators. After multi-stage post-processing and augmentation, we obtain a unified dataset of 1.3m scripts paired with rendered geometry and exercising the full CadQuery operation set. A VLM fine-tuned on CADEvolve achieves state-of-the-art results on the Image2CAD task across the DeepCAD, Fusion 360, and MCB benchmarks.

Consider a finite ground set $E$, a set of feasible solutions $X \subseteq \mathbb{R}^{E}$, and a class of objective functions $\mathcal{C}$ defined on $X$. We are interested in subsets $S$ of $E$ that control $X$ in the sense that we can induce any given solution $x \in X$ as an optimum for any given objective function $c \in \mathcal{C}$ by adding linear terms to $c$ on the coordinates corresponding to $S$. This problem has many applications, e.g., when $X$ corresponds to the set of all traffic flows, the ability to control implies that one is able to induce all target flows by imposing tolls on the edges in $S$. Our first result shows the equivalence between controllability and identifiability. If $X$ is convex, or if $X$ consists of binary vectors, then $S$ controls $X$ if and only if the restriction of $x$ to $S$ uniquely determines $x$ among all solutions in $X$. In the convex case, we further prove that the family of controlling sets forms a matroid. This structural insight yields an efficient algorithm for computing minimum-weight controlling sets from a description of the affine hull of $X$. While the equivalence extends to matroid base families, the picture changes sharply for other discrete domains. We show that when $X$ is equal to the set of $s$-$t$-paths in a directed graph, deciding whether an identifying set of a given cardinality exists is $Σ\mathsf{_2^P}$-complete. The problem remains $\mathsf{NP}$-hard even on acyclic graphs. For acyclic instances, however, we obtain an approximation guarantee by proving a tight bound on the gap between the smallest identifying sets for $X$ and its convex hull, where the latter corresponds to the $s$-$t$-flow polyhedron.

In empirical research, when we have multiple estimators for the same parameter of interest, a central question arises: how do we combine unbiased but less precise estimators with biased but more precise ones to improve the inference? Under this setting, the point estimation problem has attracted considerable attention. In this paper, we focus on a less studied inference question: how can we conduct valid statistical inference in such settings with unknown bias? We propose a strategy to combine unbiased and biased estimators from a sensitivity analysis perspective. We derive a sequence of confidence intervals indexed by the magnitude of the bias, which enable researchers to assess how conclusions vary with the bias levels. Importantly, we introduce the notion of the b-value, a critical value of the unknown maximum relative bias at which combining estimators does not yield a significant result. We apply this strategy to three canonical combined estimators: the precision-weighted estimator, the pretest estimator, and the soft-thresholding estimator. For each estimator, we characterize the sequence of confidence intervals and determine the bias threshold at which the conclusion changes. Based on the theory, we recommend reporting the b-value based on the soft-thresholding estimator and its associated confidence intervals, which are robust to unknown bias and achieve the lowest worst-case risk among the alternatives.

The union volume estimation problem asks to $(1\pm\varepsilon)$-approximate the volume of the union of $n$ given objects $X_1,\ldots,X_n \subset \mathbb{R}^d$. In their seminal work in 1989, Karp, Luby, and Madras solved this problem in time $O(n/\varepsilon^2)$ in an oracle model where each object $X_i$ can be accessed via three types of queries: obtain the volume of $X_i$, sample a random point from $X_i$, and test whether $X_i$ contains a given point $x$. This running time was recently shown to be optimal [Bringmann, Larsen, Nusser, Rotenberg, and Wang, SoCG'25]. In another line of work, Meel, Vinodchandran, and Chakraborty [PODS'21] designed algorithms that read the objects in one pass using polylogarithmic time per object and polylogarithmic space; this can be phrased as a dynamic algorithm supporting insertions of objects for union volume estimation in the oracle model. In this paper, we study algorithms for union volume estimation in the oracle model that support both insertions and deletions of objects. We obtain the following results: - an algorithm supporting insertions and deletions in polylogarithmic update and query time and linear space (this is the first such dynamic algorithm, even for 2D triangles); - an algorithm supporting insertions and suffix queries (which generalizes the sliding window setting) in polylogarithmic update and query time and space; - an algorithm supporting insertions and deletions of convex bodies of constant dimension in polylogarithmic update and query time and space.

This article investigates a space-time differential model related to the degradation of stone artifacts caused by exposure to air and atmospheric agents, which specifically lead to the accumulation of salt crystals in the material. A numerical method based on finite-element space discretization and implicit-explicit time marching is proposed as an extension of a one-dimensional finite-difference framework introduced in the literature. Within the same one-dimensional setting, a sensitivity analysis is performed, based on the techniques developed therein. They are also used as a comparison tool for the finite-element formulation, here introduced for more realistic simulations in higher space dimensions. Considerations about stability will be provided, together with an experimental convergence analysis highlighting the performance of the proposed approach. Numerical results in two and three space dimensions, obtained by an efficient code implementation, will be presented and discussed.

Although deceptive design patterns are subject to growing regulatory oversight, enforcement races to keep up with the scale of the problem. One promising solution is automated detection tools, many of which are developed within academia. We interviewed nine experienced practitioners working within or alongside regulatory bodies to understand their work against deceptive design patterns, including the use of supporting tools and the prospect of automation. Computing technologies have their place in regulatory practice, but not as envisioned in research. For example, investigations require utmost transparency and accountability in all the activities we identify as accompanying dark pattern detection, which many existing tools cannot provide. Moreover, tools need to map interfaces to legal violations to be of use. We thus recommend conducting user requirement research to maximize research impact, supporting ancillary activities beyond detection, and establishing practical tech adoption pathways that account for the needs of both scientific and regulatory activities.

We introduce randomized zero forcing (RZF), a stochastic color-change process on directed graphs in which a white vertex turns blue with probability equal to the fraction of its incoming neighbors that are blue. Unlike probabilistic zero forcing, RZF is governed by in-neighborhood structure and can fail to propagate globally due to directionality. The model extends naturally to weighted directed graphs by replacing neighbor counts with incoming weight proportions. We study the expected propagation time of RZF, establishing monotonicity properties with respect to enlarging the initial blue set and increasing weights on edges out of initially blue vertices, as well as invariances that relate weighted and unweighted dynamics. Exact values and sharp asymptotics are obtained for several families of directed graphs, including arborescences, stars, paths, cycles, and spiders, and we derive tight extremal bounds for unweighted directed graphs in terms of basic parameters such as order, degree, and radius. We conclude with an application to an empirical input-output network, illustrating how expected propagation time under RZF yields a dynamic, process-based notion of centrality in directed weighted systems.

We report on singular jets in a free-falling liquid tin droplet following nanosecond laser-pulse impact. Following impact, the droplet (with diameter $D_0=50$ or 70\,$μ$m) undergoes rapid radial expansion and subsequent retraction, resulting in the formation of an axisymmetric jet. Using numerical simulations in tandem with our experiments, we reveal that a delicate interplay between radial flow and the curvature of the retracting droplet governs jet formation. The resulting dynamics is characterized using the impact Weber number, $\We$ (in the experiments $2 \lesssim \We \lesssim 16$), and a pressure width, W (typically $1 \lesssim \W \lesssim 2$), which describes the angular distribution over the droplet surface of the instantaneous pressure impulse exerted by the transient laser-produced plasma. %, within the range $0-20$. For values $\We<10$, the droplet presents a pronounced forward curvature during the retraction, leading to the formation of a cavity. The collapse of such a cavity leads to a singular jet that greatly enhances the jetting velocity up to ten times the impact propulsion velocity, an effect that narrowly peaks around $\We\sim6-8$, reminiscent of singular jets in droplet-solid impact. We identify a further sensitivity of the jet velocity enhancement on the pressure width W and capture the dynamics in a phase diagram connecting the various deformation morphologies with jet velocity.

We present an in-depth Chandra X-ray analysis of the galaxy cluster SPT-CL J0417$-$4748 (hereafter SPT J0417), at z = 0.58, with a focus on its thermodynamic properties and the apparent absence of central star formation. Utilizing a total Chandra exposure of 103 ks, we find that the large-scale X-ray morphology is consistent with a dynamically relaxed, cool-core system. The intracluster medium (ICM) shows a central density of 0.08+/-0.01 cm^{-3}, a central pseudo-entropy of 26^{+6}_{-5} keVcm^{2} and a central cooling time of 515^{+96}_{-75} Myr, values typical of massive cool-core clusters. Despite these conditions, no evidence of recent or ongoing star formation is detected in the brightest cluster galaxy (BCG). Spectral energy distribution (SED) fitting of DES photometry indicates that the bulk of the stellar population formed at z~1.25, with no significant star formation over the past ~3 Gyr, while optical spectra from Magellan show no [O II] emission. Complementary ASKAP radio and Spitzer infrared data indicate a lack of strong current AGN activity in the BCG. SPT J0417 exemplifies massive, relaxed, cool-core clusters in which cooling and star formation appear almost completely quenched, providing valuable insights into how AGN feedback regulates the long-term thermal balance of the intracluster medium.

Deviations from the area law of the horizon entropy, in the cosmological setup, are known to lead to modified Friedmann equations governing the evolution of the universe. In this work, we propose that such modifications need not be introduced phenomenologically but can emerge dynamically from stochastic fluctuations of the spacetime metric. We consider a Friedmann-Robertson-Walker (FRW) universe perturbed by a conformal, time-dependent noise factor, whose ensemble average vanishes, leaving the mean background geometry unchanged. By averaging the Einstein equations to second order in the fluctuation amplitude, we derive a modified Friedmann equation that includes an effective correction term. This correction is shown to be equivalent to the general expression obtained from an arbitrary deformation of the entropy-area relation. By specifying the statistical properties, particularly the variance of the conformal noise, we successfully reproduce the Friedmann equation corrections associated with several well-known generalized entropy frameworks, including Rényi, (dual) Kaniadakis, Barrow, logarithmic, and MOND inspired hypergeometric entropies. Our results suggest that deviations from the area law can be interpreted as the macroscopic, coarse-grained imprint of unresolved, microscopic stochastic degrees of freedom in spacetime.

A transient violation of the null energy condition (NEC) during inflation provides a novel mechanism for producing primordial black holes (PBHs) and stochastic gravitational wave (GW) backgrounds. In this work, we extend previous studies by computing the GW contributions from both the ringdown phase of PBH formation and subsequent binary mergers. Our results show that this scenario produces a rich, multi-component GW spectrum consisting of primordial GWs, scalar-induced GWs, and GW emissions from PBH ringdown and binary mergers. We demonstrate that these correlated signatures across different frequency bands provide a novel and powerful avenue to probe or constrain NEC violation during inflation through future multi-band GW observations.

We study matching problems in which agents form one side of a bipartite graph and have preferences over objects on the other side. A central solution concept in this setting is popularity: a matching is popular if it is a (weak) Condorcet winner, meaning that no other matching is preferred by a strict majority of agents. It is well known, however, that Condorcet winners need not exist. We therefore turn to a natural and prominent relaxation. A set of matchings is a Condorcet-winning set if, for every competing matching, a majority of agents prefers their favorite matching in the set over the competitor. The Condorcet dimension is the smallest cardinality of a Condorcet-winning set. Our main results reveal a connection between Condorcet-winning sets and Pareto optimality. We show that any Pareto-optimal set of two matchings is, in particular, a Condorcet-winning set. This implication continues to hold when we impose matroid constraints on the set of matched objects, and even when agents' valuations are given as partial orders. The existence picture, however, changes sharply with partial orders. While for weak orders a Pareto-optimal set of two matchings always exists, this is -- surprisingly -- not the case under partial orders. Consequently, although the Condorcet dimension for matchings is 2 under weak orders (even under matroid constraints), this guarantee fails for partial orders: we prove that the Condorcet dimension is $Θ(\sqrt{n})$, and rises further to $Θ(n)$ when matroid constraints are added. On the computational side, we show that, under partial orders, deciding whether there exists a Condorcet -- winning set of a given fixed size is NP-hard. The same holds for deciding the existence of a Pareto-optimal matching, which we believe to be of independent interest. Finally, we also show that the Condorcet dimension for a related problem on arborescences is also 2.

We study Higgs-boson pair production in the Standard Model Effective Field Theory (SMEFT) up to dimension six and in the Higgs Effective Field Theory (HEFT) at leading order in the effective theory expansion, and assess which description is appropriate in concrete UV scenarios. Motivated by "Loryon"-inspired models, we compare the Higgs pair production cross sections predicted by the full models to their SMEFT and HEFT counterparts. We identify regimes in which the two EFTs provide comparable descriptions, and clarify the limits required for their couplings to match. We also find that, for parts of parameter space in some of these models, HEFT can reproduce Higgs pair production more accurately than SMEFT, highlighting di-Higgs measurements as a potential probe of non-linear electroweak dynamics.

We present numerical results on dynamo action in a flow driven by an azimuthal body force localized near the end of an elongated cylindrical container. The analysis focuses on the central region of the cylinder, where axial variations in the flow are relatively weak, allowing the magnetic field to be represented as a helically traveling wave. Four magnetic impeller configurations and multiple forcing intensities are examined. In all cases, the velocity profiles in the central region display a similar \propto r^{-2} dependence across a wide range of Reynolds numbers and forcing region widths. The magnetic field is found to start growing under conditions similar to those of the Riga dynamo. However, the growing modes exhibit a substantial nonzero group velocity, indicating that the associated instability is convective: the flow can amplify an externally applied magnetic field but cannot sustain it autonomously. We outline several approaches for overcoming this limitation in order to realize a working laboratory dynamo based on an internally unconstrained swirling jet-type flow.

Darwinian evolution requires (i) heritable records, (ii) repeatable copying with variation, and (iii) routine irreversibility. Categorical quantum mechanics (CQM) makes precise why ``copy'' and ``delete'' are not generic quantum operations: they exist only for a realized \emph{classical data} sector (a preferred basis/observable; a commutative structure). Decoherence explains how a pointer basis can be selected dynamically, but it does not by itself select a unique outcome. This motivates a neutral presentation of the main ontological options (unique-history, decohered multiplicity, agent-relative facticity, and a stochastic foundation with variable diffusion). We also note the relevance of the ``agency constraint'' argued by Adlam-McQueen-Waegell: in a strictly coherent, basis-unselected ``purely quantum'' regime, minimal agency fails due to no-cloning and linearity, which sharpens the role of classical resources for record-based processes. Extended Wigner's Friend scenarios then serve as a stress test, since they treat ``friends'' simultaneously as coherent quantum systems and as agents possessing stable records. Finally, a stochastic-mechanics foundation (with variable diffusion) offers a continuous bridge between quantum and classical regimes, and suggests a principled way to implement measurement update as conditioning plus a time-symmetric minimal-change rule.