A tournament has Schuttes property $S_k$ if for every set of $k$ vertices, there is a vertex which dominates the set. In 1963, Erdos provided bounds for $f(k)$, the smallest order of an $S_k$ tournament. Schuttes property has various applications, including the design of unfair dice games. A set of dice introduced by James Grime motivates a generalization of Schuttes property to sets of tournaments: a set of tournaments on the same vertex set has property $S_k$ if for every set of $k$ vertices, there is a vertex which dominates the set in at least one of the tournaments. We explore this generalization and provide bounds on the fewest number of vertices needed to have an $S_k$ set of $m$ tournaments. We then apply these results to introduce a few new sets of dice similar to Grimes dice that can be used to play a game that gives one player an advantage.

A new digital optical module (DOM) has been developed for the proposed expansion to the IceCube detector at the South Pole, IceCube-Gen2. The "Gen2-DOM" has 4 times the integrated photon sensitivity of the current IceCube DOMs and has built off the design features of the IceCube Upgrade modules. The Gen2-DOM has up to 18 4" photomultiplier tubes (PMTs) in a borosilicate glass pressure vessel, arranged in a uniform 4$π$ angular distribution. The mechanical design has been optimized to fit into a reduced borehole diameter which, in turn, will reduce drilling costs during installation. Each PMT has a dedicated readout board, designed to increase sensitivity to high-energy events aligned with the science goals of IceCube-Gen2. Internal storage enables multi-level triggering schemes with reduced overall flow of data on the long cables. Twelve prototypes of the Gen2-DOM will be deployed in the IceCube Upgrade in the 2025-2026 austral summer. This article will focus on the current status of design development and initial performance testing results.

In the statistical thermodynamics of classical discrete systems, the map from microscopic interactions to thermodynamic equilibrium configurations generally exhibits complex nonlinearity, known as "canonical nonlinearity" (CN). While conventionally characterized by the Kullback-Leibler (KL) divergence, this approach inevitably misses intrinsic nonlinearities arising from the discretization of continuous Gaussian families themselves. This intrinsic effect of unavoidable CN (UCN), has recently been quantified within a transport-information-geometric framework. However, the UCN is fundamentally limited to evaluating the discretization-induced cost for a single continuous distribution. It therefore does not capture the information-geometric cost between a continuous Gaussian reference and an actual discrete distribution, nor between states with fundamentally different supports, making it conceptually unclear how to decompose the overall CN. To address this limitation, we propose the Path-Integral UCN (PUCN) quantifying the cumulative information-geometric cost between distinct distributions. The PUCN adopts a path by (i) retaining the canonical distribution as an exponential family via the $e$-mixture (geometric mean) of the base measure, leading to an arithmetic mixture of the Fisher metric as the CN standard, and (ii) enforcing covariant changes in the discretization cell through the harmonic mixture of its second-moment matrix $M$, reflecting the uncertainty in parameter variations on the statistical manifold. The resulting PUCN provides a flexible measure of the geometric cost between arbitrary states, including those with essentially different supports. This formulation enables an explicit quantification of CN between different CDOS systems and a natural decomposition of the total CN into the UCN and a residual contribution, which has not been clearly separated in existing approaches.

Convolutional neural networks (CNNs) have emerged as a powerful tool for automatic modulation classification (AMC) by directly extracting discriminative features from raw in-phase and quadrature (I/Q) signals. However, deploying CNN-based AMC models on IoT devices remains challenging because of limited computational resources, energy constraints, and real-time processing requirements. Early-exit (EE) strategies alleviate this burden by allowing qualified samples to terminate inference at an EE branch. However, our empirical analysis reveals a critical limitation of existing confidence-based EE strategies: they predominantly select samples whose early and final predictions are correct and consistent, while failing to capture whether deeper inference can provide a tangible accuracy gain. To address this limitation, we propose BEACON, a Benefit-Aware Early-Exit framework for AMC via recoverability prediction. BEACON introduces a benefit-aware EE criterion that explicitly predicts recoverable errors, defined as instances where the final-exit branch corrects an initial early-branch misclassification. Using only short-branch observables, we design a lightweight benefit-aware predictor (LBAP) to implement this criterion, estimating the likelihood of such recoverable cases and triggering deeper inference only when an accuracy gain is expected. Extensive experiments on ResNet-18-based AMC models demonstrate that the proposed approach consistently outperforms state-of-the-art baselines, achieving a superior accuracy-computation tradeoff across diverse EE threshold settings and signal-to-noise ratio regimes. These findings validate the effectiveness of the benefit-aware criterion and its practicality for energy-efficient on-device AMC under stringent resource constraints.

In AI-rich higher education, polished written mathematics has become easier to produce than trustworthy evidence of understanding. This article develops a human-scale methodology for service mathematics, with informatics as its main running case. Its central move is not prohibition of tools but relocation of evidential trust. Students prepare openly, often with digital assistance, but grade-relevant evidence shifts toward live explanation, contingent questioning, and cumulative observation against course outcomes. The design is guided by Realistic Mathematics Education, question-first task construction, short human-scale mathematical tasks, and instructor synthesis after student attempt. It contributes a weekly operational cycle, a realism framework distinguishing professional, disciplinary, and experiential realism, a middle-out white-box / black-box stance on tools, a bounded role for retrieval-grounded AI assistants for students and teachers, and a cumulative oral-evidence model for small and medium cohorts. The paper also provides concrete implementation artifacts: process figures, an ecology of problem types, time-budget estimates, an evidence hierarchy, and a five-grade oral rubric. This is a methodology paper rather than an effectiveness study. Its claim is that the proposed design is pedagogically coherent, operationally plausible for human-scale teaching settings, and responsive to current concerns about AI, oral evidencing, and active learning in undergraduate mathematics education.

Periodic boundary conditions (PBCs) for computing magnetic fields in repeating magnetic structures, e.g. in micromagnetic simulations, are typically imposed using the quasi periodic macrogeometry approach, where many copies of the simulated domain are introduced. This can be computationally problematic, especially if the simulated domain is incommensurate with the desired sample shape. In this work, we present a formal proof that for sufficiently large magnetic samples, only the average magnetisation gives non-negligible shape effects. Using this insight, we develop a simple, computationally efficient modification of existing implementations which incorporates shape effects in PBC methods.

We give an algorithm to determine factorization types of primes in the number fields generated by a single point of odd order on an elliptic curve. We apply this to compute coefficients of the Dedekind zeta function of the field.

Most scientific research begins in the context of the then-contemporary state of knowledge of the field and moves toward a deeper understanding of the subject. This colloquium presents the Arecibo telescope$^{'}$s contribution to radio astronomy from the point of view of its contemporary impact and relevance and how that evolved over the 57 years of its long life. Sometimes, serendipitous discoveries at Arecibo and elsewhere brought revolutionary changes to the field. Further, significant upgrades to the reflector, optics, receiving, and data-taking systems helped clear a pathway for a leap ahead. Charting these movements through time and without any claim to completeness, this Colloquium presents a progression through Arecibo telescope$^{'}$s role in the history of radio astronomy.

The rapid advancement of neuromorphic technology aims to address the memory wall challenge inherent in conventional von Neumann architectures. This paper critically examines current digital neuromorphic processors and their strategies to mitigate this bottleneck. While designed to bring computation closer to memory through distributed architectures, our findings indicate that on-chip memory systems, including SRAM and emerging technologies like STT-MRAM, have become significant consumers of area and energy, leading to a new memory wall. Through an analysis of energy and area efficiency in various memory technologies, we argue that without a re-evaluation of memory organization, digital neuromorphic processors may struggle to compete effectively in edge and embedded applications. We conclude with potential pathways for future research to overcome the limitations of on-chip memory in neuromorphic systems.

A representation $V$ of an algebraic group $G$ induces a vector bundle $[V/G] \to BG$. The representation $V$ of $G$ is neutral if, for every twisted form $\mathcal{V} \to \mathcal{G}$ of $[V/G] \to BG$ over a field $k$, we have $\mathcal{G}(k) \neq \emptyset$. Twisted forms of representations arise in many ways, for instance as cohomology of families of varieties on residual gerbes of moduli spaces, and from quotient singularities. Moreover, every Tannakian category is the category of vector bundles on some gerbe. Because of this, studying neutral representations yields numerous applications, especially to problems about fields of moduli. The present article has three main results. First, we completely classify neutral, faithful representations of finite groups in dimension $\leq 3$. Second, we give a very general, computation-friendly result for proving that representations of finite abelian groups are neutral, in arbitrary dimensions. Third, we develop the abstract concept of the normalizer $\mathcal{G} \to \mathcal{N} \to \mathcal{H}$ of a morphism of gerbes $\mathcal{G} \to \mathcal{H}$ on an arbitrary site (twisted representations correspond to morphisms of gerbes $\mathcal{G} \to B\mathrm{GL}_{n}$), and show that the normalizer $\mathcal{N}$ only depends on the geometric type of $\mathcal{G} \to \mathcal{H}$.

Predicting group behavior, how individuals coordinate, communicate, and interact during collaborative tasks, is essential for designing systems that can support team performance through real-time prediction and realistic simulation of collaborative scenarios. Large Language Models (LLMs) have shown promise for processing sensor data for human-activity recognition (HAR), yet their capabilities for team dynamics or group-level multimodal sensing remain unexplored. This paper investigates whether LLMs can predict group coordination patterns from multimodal sensor data in collaborative Mixed Reality (MR) environments. We encode hierarchical context -- individual behavioral profiles, group structural properties, and temporal activity context -- as natural language and evaluate three LLM adaptation paradigms (zero-shot, few-shot, and supervised fine-tuning) against statistical baselines. Our evaluation on 16 groups (64 participants, $\sim$25 hours of sensor data) reveals that LLMs achieve 3.2$\times$ improvement over LSTM baselines for linguistically-grounded behaviors, with fine-tuning reaching 96\% accuracy for conversation prediction while maintaining sub-35ms latency. Beyond performance gains, we characterize the boundaries of text-based LLMs for multimodal sensing conversation prediction succeeds because turn-taking maps to linguistic patterns, while shared or joint attention may require spatial and visual reasoning that text only LLMs cannot capture. We further identify simulation mode brittleness (83\% degradation from cascading context errors) and minimal few-shot sensitivity to example selection strategy. These findings establish guidelines when LLMs are appropriate for CPS/IoT sensing for team dynamics and inform the design of future multimodal foundation models.

We present the 1.6$-$28 $μ$m spectra of the young protostar EC 53, obtained with JWST NIRSpec IFU and MIRI MRS during the quiescent and burst phases of its periodic brightness variations. To isolate ice absorption features, we modeled and removed the mid-infrared silicate dust absorption using a dedicated continuum-fitting procedure. In the optical depth spectrum, we first fit the broad H$_2$O ice features and then decomposed the major ice components, including NH$_3$, CO$_2$, CH$_3$OH, CO, and CH$_4$, by matching laboratory profiles for both pure and H$_2$O-mixed ices. The 4.62 $μ$m and 6.85 $μ$m bands are attributed to OCN$^-$ and NH$_4^+$ ions, respectively. Minor or tentative contributions from complex species (HCOOH, H$_2$CO, CH$_3$COOH, CH$_3$CHO, CH$_3$CH$_2$OH, and NH$_2$CHO) are also considered to our global ice analysis. The silicate-corrected spectra reveal no measurable change in any ice absorption band between the two phases, indicating that moderate and short-period accretion bursts in EC~53 do not significantly alter the physical or chemical state of the ices within its envelope. The derived abundances of these major species relative to H$_2$O significantly exceed the values typically observed toward other embedded protostars. Finally, we place the ice inventory of EC~53 in the context of other protostellar systems observed with JWST, highlighting that its chemically rich, thermally quiescent ice reservoir provides a benchmark for studying ice evolution under episodic accretion.

Assuming a large cardinal hypothesis, Laver gave a representation of the monogenerated free left distributive algebra (LDA) using elementary embeddings and used this representation to prove many algebraic results. Some of these results were later proved by Dehornoy in ZFC, without the large cardinal hypotheses. However, there is an important algebraic result whose consistency strength is unknown. (See Laver (1995) and Dougherty & Jech (1997).) Recent results [arXiv:2508.02244] extend the connection between elementary embeddings of set theory and free LDAs to the many-generated case. Assuming large cardinals, we prove two results. First, we prove that finitely-generated free LDAs with distinct numbers of generators are $Σ_1$-elementarily equivalent but not $Σ_2$-elementarily equivalent. We also prove a partial structural analogue to Laver's representation of LDAs. We construct an extension of the monogenerated free LDA where application by any fixed element is an elementary embedding of LDAs. We argue that this extension is canonical by demonstrating homogeneity and universality properties. These results also provide additional examples of algebraic properties provable from large cardinals without known proofs from the standard axioms of set theory.

The O-band (1260-1360 nm), located near the minimum of chromatic dispersion of standard single-mode fiber, is the transmission window of major interest and importance for short-reach data-center interconnects. However, full capacity offered by this spectral band is yet to be unlocked, due to limited availability of scalable multi-wavelength, high-power, low noise O-band light engines. While Kerr microcombs in CMOS-compatible silicon nitride resonators provide mutually coherent wavelength channels with precise spacing and chip-scale footprints, their practical deployment in the O-band has been hindered by limited pump laser power, insufficient per-line power and the lack of flat, wideband amplification technologies to uniformly boost multiple coherent carriers. Here we demonstrate a high-power O-band soliton microcomb architecture that overcomes this bottleneck by combining self-injection-locked (SIL) operation in a Silicon Nitride microring with a single-stage bismuth-doped phosphosilicate fiber amplifier designed for wideband, flat-top gain. The SIL microcomb operates with an 834 GHz free spectral range and spans over 1050-1650 nm. The amplifier simultaneously boosts 21 O-band lines across 100 nm to powers exceeding 0 dBm per carrier without gain flattening or external equalization, while preserving low-noise characteristics. We validate each amplified microcomb line as a carrier across the entire O-band using dual-polarization 32 GBaud 64-QAM coherent transmission. This approach establishes a practical route towards high-power, broadband O-band microcomb engines for next-generation data-center interconnects and scalable photonic systems.

Scanpath prediction models forecast the sequence and timing of human fixations during visual search, driving foveated rendering and attention-based interaction in mobile systems where their integrity is a first-class security concern. We present the first study of backdoor attacks against VLM-based scanpath prediction, evaluated on GazeFormer and COCO-Search18. We show that naive fixed-path attacks, while effective, create detectable clustering in the continuous output space. To overcome this, we design two variable-output attacks: an input-aware spatial attack that redirects predicted fixations toward an attacker-chosen target object, and a scanpath duration attack that inflates fixation durations to delay visual search completion. Both attacks condition their output on the input scene, producing diverse and plausible scanpaths that evade cluster-based detection. We evaluate across three trigger modalities (visual, textual, and multimodal), multiple poisoning ratios, and five post-training defenses, finding that no defense simultaneously suppresses the attacks and preserves clean performance across all configurations. We further demonstrate that backdoor behavior survives quantization and deployment on both flagship and legacy commodity smartphones, confirming practical threat viability for edge-deployed gaze-driven systems.

Let $G=\SL(2,\R)\ltimes(\R^2)^{k}$, let $Γ$ be a congruence subgroup of $\SL(2,\Z)\ltimes(\Z^2)^{k}$, and let $u_{\R}=(u_x)_{x\in\R}$ be the one-parameter subgroup of $G$ given by $u_x=\left(\matr 1x01,0\right)$. We prove polynomially effective asymptotic equidistribution results for expanding translates of $u_{\R}$-orbits and for long pieces of individual $u_{\R}$-orbits in $Γ\backslash G$. An important ingredient of the proof is the delta symbol version of the circle method.

The detection of gravitational waves (GWs) from a binary neutron star (BNS) merger by Advanced LIGO and Advanced Virgo (GW170817), together with its electromagnetic counterpart, the short gamma-ray burst GRB~170817A, heralded the birth of multi-messenger astronomy. The detection of TeV emission from GRBs motivates follow-up observations with the Cherenkov Telescope Array Observatory (CTAO), ideal for detecting such signals due to its unprecedented sensitivity, rapid response, and wide-field survey capabilities. The aim of this work is to evaluate GeV--TeV GW follow-up strategies for CTAO using a multi-step simulation pipeline and to estimate the expected rate of joint GW-GRB detections during observing run O5. Using a simulated sample of BNS systems with corresponding GW detections, gamma-ray emission is simulated through phenomenological prescriptions based on the observed population of short GRBs, including off-axis jet scenarios. CTAO observations are simulated to account for instrument response, sky tiling strategies, integration times, and varying observing conditions. Strategies with variable and constant integration times are investigated. We find that, via an optimized follow-up strategy, about 5% of simulated GW-associated short GRBs produce GeV--TeV radiation detectable by CTAO. Detectability is strongly influenced by the jet opening angle and viewing angle, suggesting that even rough estimates of the viewing angle in GW alerts could enhance targeting. This framework motivates future follow-ups of GW-detectable events, including neutron star-black hole mergers, and further supports the development of advanced strategies incorporating galaxy distributions and synergies with future detectors such as the Einstein Telescope.

Modern software projects depend on third-party dependencies, whose declarations must be maintained as projects evolve. Prior work has focused on dependency version updates, while much less is known about how developers assign dependencies to different roles over time. In this paper, we investigate how developers of JavaScript projects reclassify their dependencies, including removal and role reassignment. Our analysis of 33,087 JavaScript projects with active dependency maintenance reveals that dependency reclassification is a prevalent maintenance activity, occurring in 79.1% of the studied projects. Of these projects, nearly all (97.2%) remove dependencies at some point, while 38.0% undergo role reassignments across Core (runtime), Dev (development-only), and Peer (consumer-provided) roles. These changes are not always final, as 33.1% of projects later reintroduce removed dependencies and 11.2% exhibit repeated role switching. Reclassification practices also tend to unfold over long periods, with a median duration of 408 days. These findings broaden the study of dependency maintenance beyond version updates, and contribute implications for dependency tools, package managers, and research to support correct dependency role declarations.

The Cramér-Lundberg model with exponential claims and proportional investment is solved exactly: the integro-differential equation for the survival probability reduces to a doubly confluent Heun equation, yielding an explicit solution in terms of Heun functions, a verification theorem, and a qualitative analysis of ruin probability versus investment share.

Integrated Sensing and Communications (ISAC) enables trajectory sharing that enhances beamforming, resource allocation, and cooperative perception, yet raises fundamental privacy concerns under the General Data Protection Regulation (GDPR) data minimisation principle. This paper proposes a Fisher Information Density (FID)-constrained trajectory sharing framework that enforces a local lower bound on estimation uncertainty, providing hard, quantifiable privacy guarantees by construction. Unlike fixed-noise approaches, the proposed method bounds the Privacy Leak Ratio (PLR) regardless of sensing power or adversarial post-processing, ensuring that no trajectory segment can be reconstructed beyond a prescribed accuracy threshold. Simulations on the OpenTraj dataset demonstrate that the framework keeps the average PLR below 20-25% and the maximum leakage segment duration under 2-2.5 s, while preserving data utility for downstream tasks such as movement prediction. The resulting criterion is interpretable, model-agnostic, and compatible with GDPR-compliant ISAC system design.

We give a classification of Jordan-Chevalley decompositions of an endomorphism of a finite-dimensional vector space over a not necessarily perfect field, i.e. additive decompositions into commuting semisimple and nilpotent endomorphisms.

Given a closed Riemannian Spin manifold $(M,g)$ of dimension greater or equal than four, we consider a generalized conformally invariant equation involving the Dirac operator with a non-linearity of convolution type. We show that the Aubib-type inequality corresponding to the problem is always strict, unless $(M,g)$ is conformal to the round sphere. In particular, this result provides an existence result for a ground state to the conformal Dirac-Einstein problem in dimension four. We point out that aside from some perturbative or special cases, this presents the first general existence result for the conformal Dirac-Einstein equations in dimension four.

Room-temperature optically active solid-state spin defects are widely known to be useful in quantum sensing applications, however, only a select range of materials have been found to host such systems. Recent measurements in the van der Waals material hexagonal boron nitride (hBN) have shown optically detected magnetic resonance (ODMR) with spin-1/2-like signatures can be explained by a charge transfer mechanism where charges move between adjacent defects forming weakly coupled spin pairs. Interestingly, these ODMR signatures have been reported in a variety of materials aside from hBN, suggesting the spin pair model provides a potentially material agnostic approach for enabling ODMR. Here, we test whether the charge transfer mechanism is supported in a different crystal phase, and report on ODMR signatures in cubic boron nitride (cBN), showing all the characteristic properties identified in hBN are preserved. We consider a selection of different cBN samples of varying size and observe ODMR from a single sub-micron cBN particle, paving the way towards sensing applications. This work further expands understanding of the ubiquity of optical spin defect pairs, and establishes the potential for exploring quantum technologies with a wider range of materials.

We study the duality of lattice Maxwell theory in a modified Villain formulation and employ an ultra-local action with a theta term. Although this action is known to become non-local under Poisson resummation, we show that this non-locality can be removed in the absence of monopoles by incorporating a non-local transformation procedure into the definition of the S-transformation. As a result, the ultra-local action with a theta term exhibits an exact SL(2,Z)-duality. We further analyze the SL(2,Z)-structure of Wilson loops, demonstrating that they transform properly up to a nontrivial phase factor arising from the nontrivial self-linking of magnetic Wilson loops. This effect originates from the non-local transformation procedure in the S-transformation. Remarkably, the resulting SL(2,Z)-structure closely resembles that of non-spin Maxwell theory.

NbRe, a non-centrosymmetric superconductor with a transition temperature $T_\mathrm{c}$ up to 9\,K, attracts interest for its strong antisymmetric spin-orbit coupling and suitability for single-photon detection. While bulk and thin-film polycrystalline NbRe are well studied, how superconductivity and vortex dynamics evolve with increasing grain size in thin films is largely unknown. Here, we investigate as-grown and annealed 20\,nm-thick NbRe films, where annealing increases the average crystallite size from approximately $2$\,nm to $8$\,nm, and study vortex dynamics via current-voltage ($I$-$V$) measurements over a broad temperature and magnetic field range. In contrast to as-grown films, where the low-resistive state breaks down due to flux-flow instability, annealed films exhibit multiple voltage kinks in the $I$-$V$ curves. We attribute these kinks to the nucleation and growth of normal domains, as further suggested by time-dependent Ginzburg-Landau simulations. Overall, the annealed films form superconducting networks with vortex-channeling paths along the grain boundaries, while localized heating and voltage kinks could be harnessed for discrete-resistance switching and sensing.

Reliable positioning is essential for Uncrewed Aerial Vehicles (UAVs) in safety-critical urban operations, yet achieving sub-meter accuracy under stringent latency constraints remains challenging. While 3rd Generation Partnership Project (3GPP) specifies repeated Positioning Reference Signals (PRS) transmissions for accurate Time Difference of Arrival (TDoA) measurements, denoising techniques specifically tailored for extremely limited measurement sequences within 3GPP frameworks remain underexplored. We propose Adaptive Gain Exponential Smoother (AGES), a lightweight filter combining exponentially weighted averaging with adaptive gains informed by 3GPP measurement quality reports. Simulations demonstrate AGES achieves 30-40% reduction in positioning error with only 3-5 repeated measurements while maintaining Fifth Generation New Radio (5G-NR) infrastructure compatibility.

In this paper we consider a quasilinear singular anisotropic elliptic problem with a p-sublinear perturbation. We prove uniqueness of weak solutions and we provide necessary and sufficient conditions for the existence of weak solutions in the same spirit of the celebrated paper by Lazer and McKenna.

For an arbitrary family of predicates $\mathcal{F} \subseteq \{0,1\}^{[q]^k}$ and any $ε> 0$, we prove a single-pass, linear-space streaming lower bound against the gap promise problem of distinguishing instances of Max-CSP$({\mathcal{F}})$ with at most $β+ε$ fraction of satisfiable constraints from instances of with at least $γ-ε$ fraction of satisfiable constraints, whenever Max-CSP$({\mathcal{F}})$ admits a $(γ,β)$-integrality gap instance for the basic LP. This subsumes the linear-space lower bound of Chou, Golovnev, Sudan, Velingker, and Velusamy (STOC 2022), which applies only to a special subclass of CSPs with linear-algebraic structure. (Their result itself generalizes work of Kapralov and Krachun (STOC 2019) for Max-CUT.) Our approach identifies the right ``analytic'' analogues of previously-used linear-algebraic conditions; this yields substantial simplifications while capturing a much larger class of problems. Our lower bound is essentially optimal for single-pass streaming, since: (1) All CSPs admit $(1-ε)$-approximations in quasilinear space, and (2) sublinear-space streaming algorithms can simulate the LP (on bounded-degree instances), giving approximation algorithms when integrality gap instances do not exist. The starting point for our lower bound is a reduction from a "distributional implicit hidden partition'' problem defined by Fei, Minzer, and Wang (STOC 2026) in the context of multi-pass streaming. Our result is an analogue of theirs in the single-pass setting, where we obtain a much stronger (and tight) space lower bound.

Winner-take-all (WTA)--type selection is a fundamental mechanism in networked competition, yet its dependence on higher-order interactions remains insufficiently understood. We study a Lotka--Volterra competitive dynamics on higher-order networks, where classical pairwise inhibition is augmented by multi-way interaction terms induced by hyperedges of uniform hypergraphs. The proposed model shows multiple competitive outcomes, including WTA, winner-share-all (WSA), and variant winner-take-all (VWTA). The existence, uniqueness and stability of equilibria are rigorously proved through mathematical analysis, which relies on classical stability theory and recent advances in tensor algebra. We show that the eventual selection outcome is relatively insensitive to the hyperedge order and the specific higher-order coupling structure, and is instead determined by a small set of interpretable scalar parameters, such as the ratio between self-inhibition and lateral-inhibition and the external inputs. Numerical experiments support the theory by showing that higher-order interactions affect convergence and steady states, yet yield the similar outcome taxonomy (WTA/WSA/VWTA) as in standard graphs. These results provide a network-scientific explanation of the robustness of WTA-type outcomes under complex group interactions and offer principled guidance for designing selection mechanisms on higher-order networks.

We study triples {a,b,c} of distinct nonzero rational numbers such that a+1,b+1,c+1,ab+1,ac+1,bc+1 and abc+1 are all perfect squares. We prove that there exist infinitely many such triples. In contrast, we show that no triple of positive integers has this property.