The visualization of concept lattices is a central problem in the field of Formal Concept Analysis. Force-directed algorithms, as popular in graph drawing, are a promising approach, treating lattice diagrams as physical models, optimizing node positions based on forces derived from the lattice structure. We build on the work of Zschalig, who, however, limited himself to attribute-additive diagrams. We use a more general additivity, in which both the attributes and the objects contribute to the positions of the concept nodes. We replace the planarity enhancer used by Zschalig to obtain a starting diagram for force-directed optimization with the DimDraw algorithm, which generates structured order diagrams on its own. The combination results in DimFlux, an algorithm that leverages the advantages of DimDraw but generates additive diagrams in which readability is increased by maximizing the conflict distance between nodes and non-incident edges.

A degenerate string is a sequence of sets of characters. A generalized degenerate (GD) string extends this notion to the sequence of sets of strings, where strings of the same set are of equal length. Finding an exact match for a pattern string inside a GD string can be done in $O(mn+N)$ time (Ascone et al., WABI 2024), where $m$ is the pattern length, $n$ is the number of strings and $N$ the total length of strings constituting the GD string. This is the best classical algorithm achieved so far, and no matching lower bound, neither unconditional nor conditional, has been shown. We make progress on this problem proposing a quantum algorithm that achieves running time $\tilde{O}(\sqrt{mnN})$, thus beating the current best classical solution. To the best of our knowledge, this is the first quantum algorithm proposed in the context of GD strings. We present our results starting from the framework of classical parallel computing, which we believe makes them intuitive to understand and possibly easy to generalise to other similar structures.

This paper studies the clustering behavior of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behavior is closely related to the presence of discontinuous phase transitions in the mean-field PDE. For local attractive interactions, we employ a new variant of the strict Riesz rearrangement inequality to prove that all global minimizers of the free energy are either uniform or single-cluster states, in the sense that they are symmetrically decreasing. We analyze different timescales for the particle system and the mean-field (McKean-Vlasov) PDE, arguing that while the particle system can exhibit coarsening by both coalescence and diffusive mass exchange between clusters, the clusters in the mean-field PDE are unable to move and coarsening occurs via the mass exchange of clusters. By introducing a new model for this mass exchange, we argue that the PDE exhibits dynamical metastability. We conclude by presenting careful numerical experiments that demonstrate the validity of our model.

In a recent paper, Francis, Illickan, Jose and Rajendraprasad showed that every $n$-vertex plane graph $G$ has (under some natural restrictions) a vertex-partition into two sets $V_1$ and $V_2$ such that each $V_i$ is \emph{dominating} (every vertex of $G$ contains a vertex of $V_i$ in its closed neighbourhood) and \emph{face-hitting} (every face of $G$ is incident to a vertex of $V_i$). Their proof works by considering a supergraph $G'$ of $G$ that has certain properties, and among all such graphs, taking one that has the fewest edges. As such, their proof is not algorithmic. Their proof also relies on the 4-color theorem, for which a quadratic-time algorithm exists, but it would not be easy to implement. In this paper, we give a new proof that every $n$-vertex plane graph $G$ has (under the same restrictions) a vertex-partition into two dominating face-hitting sets. Our proof is constructive, and requires nothing more complicated than splitting a graph into 2-connected components, finding an ear decomposition, and computing a perfect matching in a 3-regular plane graph. For all these problems, linear-time algorithms are known and so we can find the vertex-partition in linear time.

The software $\texttt{history}$ is designed to calculate fully-differential cross sections for colour-singlet production processes in hadronic collision up to next-to-next-to-leading order in QCD. It is based on the fully-local nested soft-collinear subtraction scheme, whose implementation is entirely process independent. This allows the program to be readily applied to arbitrary colour-singlet production processes, provided the corresponding process-dependent matrix elements are supplied. In the current release, we include matrix elements for Higgs production via gluon fusion, $pp\to H+X$, and for associated Higgs production with a heavy electroweak vector boson through the Drell-Yan-like Higgs-Strahlung mechanism, $pp\to V^\ast +X \to VH+X$, with $V\in\{W,Z\}$.

We consider an interacting particle system, which generalizes the classical totally asymmetric simple exclusion process (TASEP), in that each site can contain up to a fixed finite number of particles, and the particle movement is governed by a {\em back-pressure} (BP) algorithm (also often called {\em MaxWeight}). There are $N$ sites (with $N$ finite or infinite), each may contain at most $c$ particles, $1 \le c < \infty$. New particles enter the system at the left-most site $1$ as a Poisson process of rate $α\le 1$, unless site $1$ has $c$ particles. Particles (if any) are removed from the right-most site $N$ as a Poisson process of rate $β\le 1$. The left-to-right movement of particles between neighboring sites is governed by the BP rule: one particle moves from site $n$ to $n+1$ at epochs of a rate $1$ Poisson process, as long as the former site has strictly more particles than the latter. When $c=1$, this is the standard TASEP. Our main results address the asymptotics of the stationary distribution of a finite system, and especially the limit of the flux (current) as $N\to\infty$. In particular, we prove that interesting non-trivial phase transitions take place in a system with $c>1$. For example, if $c>1$ and $1/2 \le β\le 1$, the maximum limiting flux $1/4$ is achieved as long as $α\ge α_c^*$, where $α_c^* < 1/2$ is some non-trivial threshold. (For the standard TASEP the threshold is $1/2$.) We also put forward a general conjecture about the stationary distribution asymptotics under an arbitrary parameter setting. We illustrate our formal results and the conjecture by simulations, and identify interesting directions for further research.

We study a diffuse--interface model for a binary incompressible mixture in a periodically perforated porous medium, described by a time-dependent Navier--Stokes--Cahn--Hilliard (NSCH) system posed on the pore domain $Ω_p^\varepsilon\subset\mathbb{R}^3$. The microscopic model involves a variable viscosity tensor, a non-conservative source term in the Cahn--Hilliard equation, and mixed boundary conditions: no-slip on the outer boundary and Navier slip with zero tangential stress on the surfaces of the solid inclusions. The capillarity strength $λ^\varepsilon>0$ depends on the microscopic scale $\varepsilon>0$. The analysis consists of two main parts. First, for each fixed $\varepsilon>0$ we prove existence of a weak solution on a finite time interval $(0,T)$ and derive a priori estimates that are uniform with respect to $\varepsilon$ (and $λ^\varepsilon$). Second, we perform the periodic homogenization for the perforated setting in the limit $\varepsilon\to0$. Depending on the limit value $λ$ of the capillarity strength $λ^\varepsilon$, we obtain two distinct effective models: (i) in the vanishing capillarity regime $λ=0$, the limit system decouples completely into a standalone linear Stokes system for the velocity--pressure pair and a standalone Cahn--Hilliard system with source term $G$ for the phase field and chemical potential, with no macroscopic convection, advection, or capillary coupling between the two; (ii) in the balanced regime $λ\in(0,+\infty)$, we derive a Navier--Stokes--Cahn--Hilliard system with nonlinear convection and advective transport of the phase field at the macroscopic scale, coupled through a capillary forcing term. Finally, we establish the convergence of the microscopic free energy to a homogenized energy functional satisfying an analogous dissipation law.

We analyze CO observations of molecular clouds in the Andromeda galaxy (M31) with a new method to derive surface-density profiles and compare directly with projected solutions of the isothermal Lane-Emden equation. The observed curvature of the surface-density profiles is consistent with the theoretical solutions. The applicability of the Lane-Emden equation indicates that the time scales of the turbulent processes responsible for an average force balance within the clouds are shorter than the time scales for the evolution of the clouds, for example toward collapse or disruption. The M31 profiles resemble those previously reported in Galactic molecular clouds suggesting similar dynamics across different environments.

The recent evidence for a nanohertz gravitational wave background from Pulsar Timing Arrays highlights the urgent need to identify electromagnetic counterparts to supermassive binary black holes. Here, we perform global 3D general relativistic magnetohydrodynamic (GRMHD) simulations of a secondary black hole (mass ratio $q=0.1$) interacting with a Magnetically Arrested Disk around a primary black hole using a time-dependent superposed Kerr-Schild metric and post-processed general relativistic radiation transfer calculations based on thermal electron distribution function (eDF). We explore three orbital configurations: a vertical impact orbit, a coplanar embedded orbit, and a high-spin, eccentric, inclined scenario. Despite clear orbital periodicity and recurrent shock formation, the thermal synchrotron light curves frequently lack expected shock-induced flares. In vertical impacts, shock brightenings are typically sub-dominant to the stochastic MAD variability of the primary black hole, unless viewed at specific alignment phases. Conversely, coplanar orbits produce distinctive, rapid flares driven by gravitational self-lensing. We identify a frequency-dependent emission hierarchy: the primary black hole dominates sub-millimeter flux, while the secondary dominates near-infrared emission due to higher electron temperatures in thermal eDF. Finally, spin-orbit coupling drives Lense-Thirring precession, yielding twisted, wobbling jets that following the tilt and precession of the primary BH. Crucially, we show that intrinsic MAD turbulence can easily mask shock-induced radio flares, making self-lensing flares a more reliable electromagnetic counterpart candidate for supermassive binary black holes.

Universal machine learned interatomic potentials (uMLIPs) embody a growing area of interest due to their transferability across the periodic table, displaying an error of about 0.6 kcal/mol against the Matbench Discovery test set. However, we show that achieving more accurate predictions on out-of-domain tasks requires fine-tuning. Additionally, we investigate the existence and influence of model biases in molecular dynamics (MD) by examining two approaches for data generation: from multiple MD trajectories in parallel, which we call naive fine-tuning, and from a single MD trajectory with fine-tuning after set intervals, which we call periodic fine-tuning. Our results find that naive fine-tuning generates constrained datasets that fail to represent MD simulations, and thus downstream fine-tuned models fail during extrapolation. In contrast, periodic fine-tuning yields models which are more generalizable and accurate, producing low-error dynamics. These findings indicate the role of uMLIP bias in fine-tuning, and highlights the need for multiple fine-tuning steps. Lastly, we relate unphysical behavior to principal component space, and quantify extrapolations through Q-residual analysis, which are useful as a proxy for epistemic uncertainty for larger simulations.

Event-by-event fluctuations of charged hadron multiplicity, net-baryon, and net-electric charge are investigated in central Au+Au collisions at $\sqrt{s_{NN}} =$ 20-200 GeV using a modified HIJING Monte Carlo generator. The model incorporates QCD-based partonic energy loss formalisms in both hot (quark-gluon plasma) and cold media, eliminating free tuning parameters. At energies above 100 GeV/nucleon, multiplicity and net-charge fluctuations serve as sensitive probes for medium properties, showing strong agreement with STAR data when hot-medium finite-size effects are included. By introducing a probabilistic distribution for a first-order phase transition, we demonstrate that multiplicity cumulant ratios ($σ^2/M$ and $Sσ$) provide distinctive signatures of a binodal transition, whereas net-charge and net-baryon fluctuations remain relatively insensitive. Furthermore, expanding the kinematic acceptance significantly enhances the sensitivity of these multiplicity fluctuations to the phase transition boundary.

Achieving high perceptual quality without hallucination remains a challenge in generative speech enhancement (SE). A representative approach, PASE, is robust to hallucination but has limited perceptual quality under adverse conditions. We propose StuPASE, built upon PASE to achieve studio-level quality while retaining its low-hallucination property. First, we show that finetuning PASE with dry targets rather than targets containing simulated early reflections substantially improves dereverberation. Second, to address performance limitations under strong additive noise, we replace the GAN-based generative module in PASE with a flow-matching module, enabling studio-quality generation even under highly challenging conditions. Experiments demonstrate that StuPASE consistently produces perceptually high-quality speech while maintaining low hallucination, outperforming state-of-the-art SE methods. Audio demos are available at: https://xiaobin-rong.github.io/stupase_demo/.

The Newtonian motion of a macroscopic particle is derived from the linear Schrödinger equation with a Hamiltonian consisting of the free-particle term and a random Hamiltonian drawn from the Gaussian Unitary Ensemble. The random term models interaction with the environment. We show that the parameters governing the resulting state-space random walk, together with the treatment of experimentally indistinguishable states as equivalence classes, explain the contrasting behavior of microscopic and macroscopic systems. The analysis extends previous work deriving the Born rule for microscopic particles when the free-particle term is negligible.

In this work, we study several properties of the normal Lebesgue trace of vector fields introduced by the second and third author in [22] in the context of the energy conservation for the Euler equations in Onsager-critical classes. Among other things, we prove that the normal Lebesgue trace satisfies the Gauss-Green identity and, by providing explicit counterexamples, that it is a notion sitting strictly between the distributional one for measure-divergence vector fields and the strong one for $BV$ functions. These results are then applied to the study of the uniqueness of weak solutions for continuity equations on bounded domains, allowing to remove the assumption in [19] of global $BV$ regularity up to the boundary, at least around the portion of the boundary where the characteristics exit the domain or are tangent. The proof relies on an explicit renormalization formula completely characterized by the boundary datum and the positive part of the normal Lebesgue trace. In the case when the characteristics enter the domain, a counterexample shows that achieving the normal trace in the Lebesgue sense is not enough to prevent non-uniqueness, and thus a $BV$ assumption seems to be necessary to get uniqueness.

Optical satellite uplinks rely on high-power optical amplifiers (HPOAs) to overcome free-space attenuation and enable long-distance transmission. However, at high power levels, fiber Kerr nonlinearity becomes significant and degrades system performance. In this work, we develop a realistic model for optical uplinks that accounts for nonlinear effects and analyze their impact, highlighting key differences from conventional longhaul fiber systems. We then introduce low-complexity digital signal processing techniques for nonlinearity compensation, based on constellation shaping via a look-up table (LUT) and a simple nonlinear phase rotation applied at the transmitter and/or receiver. The LUT also enables adaptive rate tuning according to channel conditions, enhancing robustness against link variations. Simulation results show that the proposed techniques increase the maximum acceptable link loss by up to 6 dB with negligible complexity. Finally, we show that, at the system level, propagation in the HPOA can be modeled as a simple nonlinear phase rotation, equivalent to propagation in a zero-dispersion noiseless fiber link, and fully characterized by a single parameter - the characteristic nonlinear power.

The template-fit approach is often used to separate Galactic dust emission from cosmic infrared background (CIB) anisotropies in low $\text{HI}$ column density regions, under the assumption that gas and dust are tightly correlated. However, this method fails in regions where additional Galactic emission from molecular hydrogen, diffuse ionized gas, and dark gas is present. We develop and test a statistical component-separation method to extract the dust signal from contaminated $\textit{Planck}$ $353\,\rm GHz$ observations using Scattering Covariance (SC) statistics. We first obtain a set of CIB maps over $25$ square patches, each covering an area of $222\,{\rm deg}^{2}$, using the linear correlation between dust and Galactic $21\,\rm cm$ $\text{HI}$ emission, which is valid in low $\text{HI}$ column density regions, through the template-fit approach. We then construct, from these $25$ maps, a generative model of the CIB using SC statistics. Finally, we rely on this contamination model to perform component separation of dust and CIB in the $\textit{Planck}$ data for different sky regions. Applying our algorithm to the $\textit{Planck}$ $353\,\rm GHz$ observations, we recover a dust map for a test sky region that exhibits more structure than the corrected SFD map at $100\,μ\rm m$. The differences observed at the map level can be explained by decomposing the recovered $\textit{Planck}$ dust map into two gas phases: dust associated with $N_{\text{HI}}$ and dust associated with $N_{\text{H}_{2}}$. This work provides a clear pathway for mapping Galactic interstellar reddening over intermediate and high Galactic latitudes.

Channel state information (CSI)-based electromagnetic inverse scattering for material reconstruction in ISAC systems enables physics-grounded, material-aware DT. Yet the resulting CSI-induced scattering operator is often severely ill-conditioned. To understand the origin of the ill-posedness, this paper analyzes the mathematical properties of the electromagnetic inverse problem and investigates the operator structure of the ISAC scattering matrix jointly shaped by in-domain scattering responses and Tx/Rx propagation channels. We show that background-related matrix columns are highly coherent and dominate the near rank deficiency, whereas scatterer-related columns are comparatively weakly correlated; their coherence decreases with the number of probing frequencies and thus contributes to the effective rank. Motivated by this analysis, we prove that restricting the ROI around the true scatterer yields a provable condition-number reduction and a tightened CRLB, and we quantify the impact of ROI mismatch numerically. To operationalize these insights, an ROI-constrained QP framework is adopted, where a linear sampling method delineates a coarse ROI and the QP update is performed in the reduced subspace. Full-wave FDTD simulations over multiple geometries and SNR validate pronounced conditioning improvement, substantial complexity savings, and improved robustness, consistent with the proposed analysis, compared with the full-domain formulation.

Keyword spotting (KWS) identifies words for voice assistants, but environmental noise frequently reduces accuracy. Standard adaptation fixes this issue and strictly requires original or labeled audio. Test-time adaptation (TTA) solves this data constraint using only unlabeled test audio. However, current methods fail to handle the severe imbalance between rare keywords and frequent background sounds. Consequently, standard entropy minimization becomes overconfident and heavily biased toward the frequent background class. To overcome this problem, we propose a TTA method named ImKWS. Our approach splits the entropy process into a reward branch and a penalty branch with separate update strengths. Furthermore, we enforce consistency across multiple audio transformations to ensure stable model updates. Experiments on the Google Speech Commands dataset indicate ImKWS achieves reliable adaptation in realistic imbalanced scenarios. The code is available on GitHub.

Accent variability remains a major source of errors in automatic speech recognition, yet most adaptation methods rely on parameter fine-tuning without understanding where accent information is encoded. We treat accent variation as an interpretable subspace in hidden representations and investigate whether it can be identified and controlled directly in activation space. We extract layer-wise encoder activations and estimate mean-shift directions capturing accent-induced representation shifts. By injecting these directions into individual layers and measuring how they align accented and standard embeddings, we derive a layer-wise accent sensitivity profile, revealing that accent information concentrates in a narrow band of middle encoder layers. Leveraging this structure, we further introduce parameter-free accent steering that modifies representations during inference without updating model weights. Experiments across eight accents show consistent word error rate reductions.

Our understanding of the early Universe has long been limited by biased galaxy samples selected through various color criteria. With deep JWST infrared imaging, mass-complete galaxy samples can now be studied up to $z \sim 8$ for the first time. However, recent work has revealed systematic uncertainties in measuring physical properties of galaxies based solely on JWST/NIRCam and HST photometry, due to their limited wavelength coverage. This highlights the need for supplementary data, particularly in the rest-frame UV and near-infrared. Here we present the ULTIMATE-deblending project, which will eventually deliver self-consistent UV to radio photometry for galaxies detected in deep JWST surveys, including both NIRCam and MIRI data. In this first paper, we release a 50-band photometric catalog spanning CFHT/U to JWST/MIRI F1800W, covering a total of 627.1 arcmin$^2$ across two JWST/PRIMER fields. We detail the reduction of the JWST imaging data, the photometric procedures, and the SED-fitting methodology used to derive the galaxy properties. Compared with photometry including only HST and JWST bands, the inclusion of deblended low-resolution photometry from ground-based telescopes improves the accuracy of photometric redshifts by $\sim$40%, while reducing the outlier fraction by $\sim$60%. This galaxy sample can serve as a key reference for statistical studies of galaxy formation and evolution in the early Universe. The UV-to-MIR catalogs and JWST mosaics from the ULTIMATE-deblending project have been made publicly available.

Understanding how opinions evolve through social interactions is crucial for mitigating polarization. Existing opinion-dynamics models incorporate both attractive and repulsive interactions but typically assume that all issues are equally important. We develop and analyze a stochastic agent-based model where issues carry heterogeneous weights that influence both social affinity and the likelihood of opinion change. Surprisingly, introducing even a single issue with arbitrarily small weight can destabilize otherwise stable states, increasing convergence times by orders of magnitude. To explain these dynamics, we derive a mean-field approach and characterize the equilibrium symmetries governing consensus, polarization, and persistent pluralism. A complete classification of these symmetries for up to five issues reveals that polarization increases when importance is concentrated on a small number of issues. Conversely, distributing importance more broadly across issues promotes diversity of opinions and reduces polarization. Our symmetry-based framework highlights how issue salience and social tolerance jointly shape collective opinion evolution.

The diversity of quantum correlations -- discord, entanglement, steering, and Bell nonlocality -- disappears at the kinematic level of observable second moments. By treating state purity as a finite resource, we introduce a local-unitary-invariant budget that splits these moments into local and nonlocal sectors. This maps quantum systems onto compact, two-dimensional manifolds whose topology is governed by purity and time-reversal symmetry. This dimensional reduction reveals a deep structural link: exceeding classical capacity limits requires the activation of intrinsically time-odd generators, providing a dimension-agnostic guarantee of negative partial transpose (NPT) entanglement. For two qubits, this geometry is analytically solvable; a single boundary isolates classical correlations, while nested regions define thresholds for steering and Bell nonlocality, alongside bounds on non-stabiliser magic. Beyond two qubits, dimensional bottlenecks enforce the kinematic limits on correlations. Because this macroscopic representation is completely determined by global and marginal purities, it bypasses the exponential scaling of full-state tomography. Thus, whenever an $n$-partite state's correlations exceed the classical capacity limits, its NPT entanglement is certified by only $n+1$ purity measurements, with sample complexity independent of Hilbert space dimension. By coarse-graining over gauge-like first moments, this geometry acts as a thermodynamic phase diagram, exposing the hierarchy of quantum resources and their dynamic redistribution under decoherence.

Community detection is a foundational capability in large-scale industrial graph analytics, powering applications such as fraud-ring discovery, recommendation systems, and hierarchical indexing for retrieval-augmented generation. Among modularity-based methods, the Leiden algorithm has been widely adopted in production because it delivers high-quality communities with connectivity guarantees. However, real-world graphs evolve continuously, and timely community updates are needed to keep downstream features and retrieval indices fresh. Meanwhile, existing dynamic Leiden approaches recompute the communities whenever their vertices and edges change, thereby almost degrading to near-full recomputation under frequent updates. To alleviate the efficiency issue, we study the efficient maintenance of Leiden communities in large dynamic graphs and present a novel algorithm, called Hierarchical Incremental Tree Leiden (HIT-Leiden). We first provide a boundedness analysis showing that prior incremental Leiden methods can incur essentially unbounded work even for small updates. Guided by this analysis, we propose HIT-Leiden which effectively reduces the range of affected vertices by maintaining connected components and hierarchical community structures. Extensive experiments on large real-world dynamic graphs demonstrate that HIT-Leiden achieves community quality comparable to the state-of-the-art competitors while delivering speedups of up to five orders of magnitude over existing solutions. The production deployment results show that HIT-Leiden meets stringent latency requirements under high-rate updates at scale.

We consider a game with two players, consisting of a number of rounds, where the first player to win $n$ rounds becomes the overall winner. Who wins each individual round is governed by a certain urn having two types of balls (type 1 and type 2). At each round, we randomly pick a ball from the urn, and its type determines which of the two players wins. We study the game under three regimes. In the first and the third regimes, a ball is taken without replacement, whilst in the second regime, it is returned to the urn with one more ball of the same colour. We study the properties of the random variables equal to the properly defined overall net profits of the players, and the results are drastically different in all three regimes.

We present new long-slit stellar kinematics for a sample of 21 Brightest Cluster Galaxies (BCGs) and triaxial Schwarzschild models for 16 of these objects using our orbit modelling code SMART. The new kinematics obtained with the Large Binocular Telescope (LBT) is complemented with high-resolution photometry from HST or new AO-assisted ground-based observations also obtained at LBT and combined with wide-field imaging from the Wendelstein Observatory. These data enable robust modeling from the innermost regions - where the Supermassive Black Hole dominates the potential - to larger radii, where stars and dark matter (DM) are the primary mass contributors. As already discussed in a companion paper, we discovered 8 Ultramassive Black Holes (UMBHs, with mass $> 10^{10}$ M$_\odot$) in this BCG sample, more than doubling the number of galaxies with dynamically detected UMBHs. We show that the DM halos display a wide variety of geometries. Purely kinematical results include low central velocity dispersion with increasing profiles towards the outskirts, and the discovery of one Kinematically Decoupled Core.

We prove two conjectures of Shteiner and Shteyner stating that for an undirected graph $G=(V,E)$, the number of degree sequences arising from its spanning subgraphs is at least the number of forests in $G$, with equality if and only if $G$ is bipartite. In the process of proving the bipartite case, we provide several equivalent evaluations of the Tutte polynomial $T_G(x,y)$ at $(2,1)$, including interpretations in terms of degree vectors obtained from orientations of $G$. For the non-bipartite case, we prove strict inequality by expressing degree sequences as subset sums of signless incidence vectors and comparing these with linearly independent edge sets, showing that the presence of odd cycles yields additional independent sets beyond forests. We further strengthen this result by introducing odd pseudoforests, showing that their number is bounded above by the number of degree sequences and characterizing the corresponding independent sets accordingly.

We propose a protocol to realize arbitrary state synthesis and unitary operations on a qudit encoded in the collective dressed states of a Rydberg-blockaded array of three-level atoms. This system is isomorphic to the Jaynes-Cummings model and acts as a multilevel Rydberg superatom whose nonlinear spectrum can be precisely controlled through the parameters of the laser driving the intermediate-to-Rydberg transition. Control of the qudit state is possible through pulse sequences of the laser driving the ground-to-intermediate transition. The dimension of the qudit Hilbert space is scalable by adjusting the number of atoms involved in the Rydberg-blockaded array. We estimate the fidelity of our protocol for realizing arbitrary unitaries and discuss the influence of the finite lifetime of the Rydberg state. Our work paves the way for processing quantum information with Rydberg blockaded arrays of atoms as an alternative to atom qubit arrays.

After an exhaustive introduction highlighting the strengths and weaknesses of the non-local models proposed so far as ultraviolet completions of the Starobinsky theory, we propose a new nonlocal completion of a general $f(R)$ theory (in the Einstein's frame) suitable for driving inflation in the early universe consistently with observations. The nonlocal theory shares with $f(R)$ the same background solutions and the same equations of motion for perturbations at linear and nonlinear level. Therefore, the classical cosmological observables are not affected by the nonlocal operators needed for the quantum completion. Our construction applies to any local action written in the Einstein's frame, but we will provide the details only for two explicit examples: the Starobinsky model and a general $f(R)$ theory. The new model overcomes the incompatibility of renormalizability and stability present in the previous proposals. Since the nonlocal theory is at least super-renormalizable, but can also be finite depending on the details of the model, this work shows the consistency of the inflationary paradigm with a well-defined quantum theory of gravity at high energy. We could rephrase the latter statement saying that the success of the $f(R)$ theories in cosmology can be traced back to the existence of an ultraviolet completion that preserves all the classical features. The inflationary paradigm survives, or it is actually insensitive to quantum gravity, because it is an exact solution of quantum gravity, up to perturbative corrections.

Multi-Agent Debate (MAD) has shown promise in improving reasoning and reducing hallucinations, yet it remains unclear how information exchange shapes individual reasoning behavior. Empirically, MAD exhibits paradoxical phenomena, including rising accuracy with increasing token entropy and marked differences between homogeneous and heterogeneous agent combinations. In this paper, we introduce a Bayesian uncertainty analysis framework for MAD, which decomposes answer-level predictive uncertainty into epistemic uncertainty and aleatoric uncertainty, corresponding to the potential gain and cost of debate. Across multiple agent configurations, we find that effective debate depends on achieving high epistemic gain under controlled aleatoric cost. Building on this insight, we design an uncertainty-guided multi-agent reinforcement learning algorithm that encourages lower aleatoric cost and more effective epistemic information utilization. Experiments show that our approach simultaneously enhances each agent's accuracy and promotes a more productive debate process, providing an operational Bayesian perspective for understanding and improving MAD.

We study error-correcting codes for an adversarial nanopore channel, where a $q$-ary string is first transformed by an inter-symbol interference channel with window size $\ell$ into a sequence of overlapping $\ell$-mers, and an adversary then corrupts this $\ell$-mer sequence by introducing at most $t$ edits. For the deletion-only nanopore channel, we show that the optimal redundancy of $t$-deletion-correcting codes of length $n$ lies between $t\log_q n+Ω(1)$ and $2t\log_q n-\log_q\log_2 n+O(1)$. We then give two explicit deletion-correcting constructions in the regime $t\leq \min\{(\ell-1)/2,(\ell+2)/3\}$. The first construction relies on generalized Reed-Solomon codes and has redundancy $2t\log_q n+Θ(\log\log n)$. The second is based on Sidon sets (or rather $B_t$ sequences) and has redundancy $t\log_q n+Θ(\log\log n)$, matching the lower bound to first order. We further extend the $B_t$-based approach to the edit channel, allowing insertions, deletions, and substitutions of $\ell$-mers. In the regime $t\leq \min\{(\ell-1)/4,(\ell+2)/6\}$, this gives explicit $t$-edit-correcting codes with redundancy $t\log_q n+Θ(\log\log n)$, which is first-order optimal.