Feedback from artificial intelligence (AI) is increasingly easy to access and research has already established that people learn from it. But individuals choose when and how to seek such feedback, and more engaged and motivated individuals may seek it more, creating an illusion of effectiveness that masks self-selection. We investigate how the endogenous choice to seek AI feedback shapes both individual learning and collective outcomes. Using data from over five years and 52,000 individuals on an online chess platform, we show that motivated and higher-skilled individuals self-select into AI feedback use-and use it more productively. This self-selection creates an illusion of AI effectiveness: apparent learning gains disappear once endogenous motivation is accounted for. This same selection mechanism drives two population-level consequences. Because motivated, higher-skilled individuals benefit disproportionately, AI access widens the skill gap. And because individuals exposed to centralized AI feedback converge on common input from a centralized AI source, intellectual diversity declines. Leveraging 42 platform-level natural experiments, we show this diversity reduction is causal. Self-selection into AI use thus connects individual-level learning dynamics to collective-level consequences-a micro-macro linkage with implications for organizational learning, human capital development, and the design of AI-augmented work.

In the rapidly evolving Internet of Things (IoT) domain, we concentrate on enhancing energy efficiency in Deep Learning accelerators on FPGA-based heterogeneous platforms, aligning with the principles of sustainable computing. Instead of focusing on the inference phase, we introduce innovative optimizations to minimize the overhead of the FPGA configuration phase. By fine-tuning configuration parameters correctly, we achieved a 40.13-fold reduction in configuration energy. Moreover, augmented with power-saving methods, our Idle-Waiting strategy outperformed the traditional On-Off strategy in duty-cycle mode for request periods up to 499.06 ms. Specifically, at a 40 ms request period within a 4147 J energy budget, this strategy extends the system lifetime to approximately 12.39x that of the On-Off strategy. Empirically validated through hardware measurements and simulations, these optimizations provide valuable insights and practical methods for achieving energy-efficient and sustainable deployments in IoT.

Executing flow estimation using Deep Learning (DL)-based soft sensors on resource-limited IoT devices has demonstrated promise in terms of reliability and energy efficiency. However, its application in the field of wastewater flow estimation remains underexplored due to: (1) a lack of available datasets, (2) inconvenient toolchains for on-device AI model development and deployment, and (3) hardware platforms designed for general DL purposes rather than being optimized for energy-efficient soft sensor applications. This study addresses these gaps by proposing an automated, end-to-end solution for wastewater flow estimation using a prototype IoT device.

This paper develops a functional-analytic framework for approximating the push-forward induced by an analytic map from finitely many samples. Instead of working directly with the map, we study the push-forward on the space of locally analytic functionals and identify it, via the Fourier--Borel transform, with an operator on the space of entire functions of exponential type. This yields finite-dimensional approximations of the push-forward together with explicit error bounds expressed in terms of the smallest eigenvalues of certain Hankel moment matrices. Moreover, we obtain sample complexity bounds for the approximation from i.i.d.~sampled data. As a consequence, we show that linear algebraic operations on the finite-dimensional approximations can be used to reconstruct analytic vector fields from discrete trajectory data. In particular, we prove convergence of a data-driven method for recovering the vector field of an ordinary differential equation from finite-time flow map data under fairly general conditions.

This paper revisits a classical challenge in the design of stabilizing controllers for nonlinear systems with a norm-bounded input constraint. By extending Lin-Sontag's universal formula and introducing a generic (state-dependent) scaling term, a unifying controller design method is proposed. The incorporation of this generic scaling term gives a unified controller and enables the derivation of alternative universal formulas with various favorable properties, which makes it suitable for tailored control designs to meet specific requirements and provides versatility across different control scenarios. Additionally, we present a constructive approach to determine the optimal scaling term, leading to an explicit solution to an optimization problem, named optimization-based universal formula. The resulting controller ensures asymptotic stability, satisfies a norm-bounded input constraint, and optimizes a predefined cost function. Finally, the essential properties of the unified controllers are analyzed, including smoothness, continuity at the origin, stability margin, and inverse optimality. Simulations validate the approach, showcasing its effectiveness in addressing a challenging stabilizing control problem of a nonlinear system.

Conversational agents are increasingly used to address emotional needs on top of information needs. One use case of increasing interest are counselling-style mental health and behaviour change interventions, with large language model (LLM)-based approaches becoming more popular. Research in this context so far has been largely system-focused, foregoing the aspect of user behaviour and the impact this can have on LLM-generated texts. To address this issue, we share a dataset containing text-based user interactions related to behaviour change with two GPT-4-based conversational agents collected in a preregistered user study. This dataset includes conversation data, user language analysis, perception measures, and user feedback for LLM-generated turns, and can offer valuable insights to inform the design of such systems based on real interactions.

We present the first high-significance spectroscopic stacked kinetic Sunyaev-Zel'dovich (kSZ) measurements of circumgalactic gas profiles for both Bright Galaxy Survey (BGS) and Emission Line Galaxy (ELG) tracers, combining DESI Data Release 2 with ACT Data Release 6. Using reconstructed line-of-sight velocities from the DESI galaxies and high-resolution ACT temperature maps, we detect the kSZ signal at high significance, reaching signal-to-noise ratios of up to $\sim$9 for BGS and $\sim$7.5 for ELGs in optimal stellar-mass selections. Together with the LRG measurements presented in Paper I, these constitute the most significant kSZ detections from any spectroscopic survey to date. We perform the analysis in both real and harmonic space, obtaining consistent results. By splitting both tracers into stellar-mass bins, we study the scaling of the kSZ amplitude with galaxy properties. Combining the kSZ measurements with ACT Data Release 6 (DR6) CMB lensing maps enables a joint calibration of the galaxy-halo connection and the gas fractions of host halos. For the BGS galaxies, we observe low gas fractions around the virial radius relative to standard expectations, likely attributable to active galactic nuclei (AGN) activity. We find some evidence for higher-mass halos retaining a larger fraction of their baryons, consistent with more efficient feedback in lower-mass systems. For the ELG sample, dominated by blue, star-forming galaxies, we provide the first detection of the gas distribution in ELG host halos. The ELGs appear to exhibit relatively high gas fractions, which points to the possibility of weaker feedback (due to e.g. low AGN and supernova feedback activity) at their mass scale. Finally, we present generalized Navarro-Frenk-White (GNFW) fits to the harmonic-space measurements, providing a compact parametrization of gas profiles for forward modeling in large-scale structure analyses.

We present the most precise measurements of the kinetic Sunyaev-Zel'dovich (kSZ) effect around luminous red galaxies to date, detecting the signal at $18σ$ significance in both harmonic and configuration space. Our analysis cross-correlates 2.4 million spectroscopic LRGs from the Dark Energy Spectroscopic Instrument (DESI) DR2 sample with Data Release 6 (DR6) of the Atacama Cosmology Telescope (ACT). We develop a novel harmonic-space cross-correlation approach using momentum-weighted kSZ templates, yielding nearly uncorrelated bandpowers within a framework consistent with other large-scale structure analyses. By incorporating the LRG halo occupation distribution (HOD) and its uncertainty, we convert measured galaxy gas profiles into halo gas profiles and provide generalized Navarro-Frenk-White (GNFW) fitting profiles, providing empirical targets for tuning feedback efficiency in hydrodynamical simulations and for baryonic modeling in large-scale structure analyses. We find strong evidence that gas profiles do not trace dark matter, providing direct evidence for gas redistribution beyond gravitational collapse. Comparing to hydrodynamical simulations, our measurements favor feedback efficiencies exceeding those in the Battaglia profile, suggesting more efficient gas ejection in group-scale halos than previously predicted. Splitting by redshift, we detect the kSZ signal at SNR $\approx 5$--$10$ in each of four bins and find amplitude evolution consistent with the expected decline in mean halo mass at fixed comoving number density. Splitting by stellar mass, we study the scaling of kSZ amplitude with galaxy properties. Together with BGS and ELG measurements in Paper II, these results span $0.1 \lesssim z \lesssim 1.6$ across three galaxy populations, demonstrating the potential of spectroscopic kSZ to map circumgalactic gas and constrain baryonic feedback.

Disks (Keplerian or otherwise, particulate or fluid) are often assumed to have densities that drop off vertically as Gaussians. Recent mm-wave imaging of circumstellar debris disks contradicts this assumption, revealing vertical profiles in dust that resemble Lorentzians. As part of the ARKS ALMA Large Program, we calculate how Lorentzians and Gaussians define an evolutionary sequence for disks of gravitationally scattering (viscously stirring) particles. When orbits are crossing and eccentricities $e \gg$ inclinations $i$, each scattering changes a particle's inclination by $\pm \,Δi \propto i$. A random walk with fixed steps in $Δi/i = Δ\ln i$ produces a log normal $i$ distribution, whose thick tail at large $i$ leads to thick Lorentzian tails in density. This result holds independent of the origin of the large eccentricities; what matters is that relative motions parallel to the disk midplane are faster than perpendicular motions. After enough scatterings, $i$ comes into equipartition with $e$, $Δi$ stops exponentiating, and the vertical density profile relaxes to a Gaussian. We estimate the numbers and masses of perturbers needed to stir themselves and observable dust grains in Lorentzian and Gaussian debris disks imaged by ARKS. The big bodies may be sufficiently few in number as to be collisionless, in which case their masses range from the Moon to several Earths. But if Pluto-sized or smaller, the big body stirrers may be so numerous and collide so frequently that they can source the collisional cascades that produce observable dust.

Large language models (LLMs) have achieved strong results in code generation, but their ability to generate GUI applications, especially games, remains insufficiently studied. Existing benchmarks mainly evaluate correctness through test cases, which are inadequate for GUI applications because these systems are interactive, event-driven, and require correct state transitions across sequences of user actions. Their evaluation therefore should consider interaction flows and UI logic rather than only pass/fail outcomes. To study this problem, we introduce PlayEval, a repository-aware benchmark built from 43 multilingual GUI applications in Python, TypeScript, and JavaScript. Unlike prior GUI benchmarks that are difficult to adapt to desktop environments, PlayEval covers six major GUI application categories and directly supports code-generation evaluation. We further propose Play@k, a metric that measures whether at least one of *k* generated candidates can be played end-to-end without logical errors. To support reliable evaluation, we develop PlayTester, an LLM-based agent that performs task-oriented GUI playthroughs and detects logic violations automatically. Experiments on 10 state-of-the-art code LLMs show that, despite high compilation rates, they achieve near-zero Play@3, revealing major weaknesses in generating logically correct GUI applications. To address this limitation, we present PlayCoder, a multi-agent, repository-aware framework that generates, evaluates, and iteratively repairs GUI application code in a closed loop. PlayCoder substantially improves both functional correctness and semantic alignment for open-source and closed-source models, reaching up to 38.1% Exec@3 and 20.3% Play@3. Case studies further show that it can uncover silent logic bugs missed by traditional metrics and fix them through targeted edits.

We introduce a natural bilinear fractional integral type operator induced by a third order hypermetric on Ahlfors regular quasi-metric spaces. Given a quasi-metric space $(X,d)$ the function $ρ(x,y,z)$, defined as the distance, in $X^3$, of $(x,y,z)$ to the diagonal $\bigtriangleup_3=\{(x,x,x)\in X^3:x\in X\}$ is said to be a third order hypermetric in $X$. When $(X,d)$ is a Euclidean space or, more generally, when $(X,d,μ)$ is $η$-Ahlfors regular for some $η$ positive, the function $ρ(x,y,z)$ generates kernels for bilinear operators of the type $T^γ(f,g)(x)=\iint_{X\times X}ρ(x,y,z)^{-γ}f(y)g(z)dμ(y)dμ(z)$, for a given positive $γ$. In the setting of $η$-Ahlfors regular space, the power $-γ=-2η$ of $ρ(x,\cdot,\cdot)$ provides the natural singularity for this family of kernels. In this paper we consider the fractional integral rank $0<γ<2η$. We prove boundedness properties of the type $\|T^γ(f,g)\|_{p_3}\leq C\|f\|_{p_1}\|g\|_{p_2}$ for adequate values of the exponents $p_1,p_2$ and $p_3$. The proof is based on three upper bounds for $T^γ(f,g)$ in terms of the classical linear fractional Riesz operators $I_{η-\fracγ{2}}$, using the linear Hardy-Littlewood-Sobolev inequality.

Recent advancements in neutral atom platforms have enabled exploration of early fault-tolerant (FT) architectures for applications with quantum advantage, such as quantum dynamics simulations. An efficient fault-tolerant architecture has both spatially efficient quantum error correction codes (low qubit overhead), and efficient methodologies (transversal based gates, extractor based gates, etc.) for logical computation, to minimize overall execution time. Achieving the right balance between space and time can be critical for enabling early FT demonstrations of quantum advantage. In this work, we identify bottlenecks in existing spatially efficient schemes, which tend to be very serial, and do not take advantage of unutilized space. We introduce a teleportation-based scheme that leverages the reconfigurable connectivity of neutral atoms to parallelize logical operations. Our approach achieves up to \textbf{$\mathbf{\sim 3 \times}$ speedup} over extractor architectures at no extra space cost and achieves the best spacetime performance among other viable architectures before accounting for external \textit{resource-states}. To rigorously evaluate performance, we construct explicit quantum advantage benchmarks and \textit{simulate} compilation to a fault-tolerant instruction set, including low-level gate scheduling and shuttling patterns, and resource-state nondeterminism. We find that our speedups still apply and report exact space-time cost along with success probabilities, identifying architectures capable of achieving quantum advantage \textbf{with as little as $\mathbf{11,495}$ atoms and a runtime of $\mathbf{\sim 15}$ hours}.

We analyze greedy routing in a random graph G_n constructed on the vertex set V = {1, 2, ..., n} embedded in Z. Vertices are inserted according to a uniform random permutation pi, and each newly inserted vertex connects to its nearest already-inserted neighbors on the left and right (if they exist). This work addresses a conjecture originating from empirical studies (Ponomarenko et al., 2011; Malkov et al., 2012), which observed through simulations that greedy search in sequentially grown graphs exhibits logarithmic routing complexity across various dimensions. While the original claim was based on experiments and geometric intuition, a rigorous mathematical foundation remained open. Here, we formalize and resolve this conjecture for the one-dimensional case. For a greedy walk GW starting at vertex 1 targeting vertex n -- which at each step moves to the neighbor closest to n -- we prove that the number of steps S_n required to reach n satisfies S_n = Theta(log n) with high probability. Precisely, S_n = L_n + R_n - 2, where L_n and R_n are the numbers of left-to-right and right-to-left minima in the insertion-time permutation. Consequently, E[S_n] = 2H_n - 2 ~ 2 log n and P(S_n >= (2+c) log n) <= n^(-h(c/2) + o(1)) for any constant c > 0, with an analogous lower tail bound for 0 < c < 2, where h(u) = (1+u) ln(1+u) - u is the Bennett rate function. Furthermore, we establish that this logarithmic scaling is robust: for arbitrary or uniformly random start-target pairs, the expected routing complexity remains E[S_{s,t}] = 2 log n + O(1), closely mirroring decentralized routing scenarios in real-world networks where endpoints are chosen dynamically rather than fixed a priori.

We consider the dissipative generalized Surface Quasi-Geostrophic equation with dissipation given by any fractional power of the Laplacian. In the inviscid limit, it is proved that anomalous dissipation of the Hamiltonian is prevented by the strong compactness of the solutions in the lowest norm that makes the nonlinearity well-defined. In fact, only the dynamics at certain frequencies matters. The argument is quite robust as it applies regardless of the criticality regime and of the presence of a, possibly noncompact, external forcing. This reveals a more general mechanism behind some recent results obtained for the Navier-Stokes and the critical dissipative Surface Quasi-Geostrophic equations. Because of nonuniqueness issues, in our broader context it is important to work with Leray solutions enjoying suitable higher-order bounds. The existence of such solutions is shown and it might be of independent interest. Finally, we prove that the strong compactness is guaranteed for any initial datum with critical integrability, from which global existence of conservative, although Onsager's supercritical, weak solutions of the inviscid problem is deduced. This offers the largest class of initial data for which global existence is known so far, matching with the one considered by Delort at the endpoint.

For a graph $G$, let $f_o(G)$ denote the maximum order of an induced subgraph of $G$ all of whose vertices have odd degree, and let $χ(G)$ denote the chromatic number of $G$. Scott (CPC, 1992) proved that $f_o(G) \ge |V(G)|/(2χ(G))$ for every graph without isolated vertices, and conjectured that the factor $2$ can be removed. Wang and Wu (JGT, 2024) showed that this conjecture fails for bipartite graphs, but holds for line graphs. In this article, we confirm Scott's conjecture for claw-free graphs without isolated vertices, thereby strengthening the result of Wang and Wu. We also construct $K_{1,r}$-free graphs of arbitrarily large order to show that the conjecture fails for this broader class, for every integer $r \ge 4$. Wang and Wu also asked whether $f_o(L(G)) \ge n/2$ holds for every connected regular graph $G$ of order $n \ge 3$. We show that $C_5$ is the smallest counterexample to this problem. On the positive side, we prove that if $G$ is a connected $k$-regular $C_5$-free graph on $n$ vertices with $k \ge 2$, then $f_o(L(G)) \ge n/2$.

Self-interacting dark matter (SIDM) is the leading framework resolving small-scale cold dark matter (CDM) crises, yet high-redshift SIDM constraints are fundamentally limited by degeneracies between dark matter microphysics and galaxy formation astrophysics. We demonstrate that the UV luminosity function alone cannot constrain SIDM: star formation suppression from SIDM halo core formation is fully absorbed by modest adjustments to standard astrophysical parameters. We show that 21 cm reionization topology breaks this degeneracy completely, providing a nuisance-immune probe: the SIDM-enhanced duty cycle of ionizing photon escape leaves a morphological signature fully independent of star formation efficiency. Combining JWST UVLF measurements with SKA1-Low forecasts, constant-cross-section SIDM with $σ/m \gtrsim 1$--$2\ \mathrm{cm^2/g}$ is either excluded or detectable across all physically motivated star formation coupling strengths. Our results establish a robust new avenue to probe dark matter microphysics in the early Universe.

Juniper Green is a simple combinatorial game invented by Rob Porteous and popularized by Ian Stewart. It was originally designed to familiarize school children with the concepts of multiplication and division. We analyze this elementary game through a completely different lens and show that it recovers the Gallai-Edmonds decomposition of the divisibility graph on the vertex set $V= \left\{1,2,\dots, n\right\}$. This characterizes the winning moves of the game; as a byproduct, we show that this decomposition seems to have many interesting and curious patterns that are currently unexplained.

In this work, we introduce a new notion for representing graph classes with formal languages. In contrast to the seminal work by Kitaev and Pyatkin to represent graphs by words, we use formal binary languages in order to have a set of patterns (given by the languages' words) defining the edges in the graph. In particular, we investigate famous languages like the palindromes, copy-words, Lyndon words, and Dyck words to represent all graphs or specific graph classes by restricting these languages.

Nucleic acid sequence design via codon optimization is a fundamental task with applications across synthetic biology, mRNA therapeutics, and vaccine design. Given a target protein, it is a major open challenge to navigate the combinatorially large design space of codon sequences mapping to its amino acid sequence. Computational approaches generally seek to optimize simple objectives based on the codon sequence, possibly together with more complicated contributions based on secondary structure analysis. In this work, we demonstrate a direct and efficient algorithm to sample sequences from a suitable Boltzmann distribution defined in terms of the codon sequence and a fully detailed secondary structure free energy model, as well as related algorithms for exact computation of statistical quantities such as free energies, base pairing probabilities, and base and codon marginals. These algorithms draw upon a recently developed tensor-based formulation of secondary structure thermodynamics and demonstrate, for the first time, that global sequence design can be accomplished with respect to a highly accurate free energy model. Moreover, the algorithms can leverage any available CPU and GPU resources in parallel for massive computational speedups.

In this paper, we give a mathematical proof that bounds the number of CNOT gates required to synthesize an $n$ qubit phase polynomial with $g$ terms to be at least $O(\frac{gn}{\max (\log g, 1)})$ and at most $O(gn)$. However, when targeting restricted hardware, not all CNOTs are allowed. If we were to use SWAP-based methods to route the qubits on the architecture such that the earlier synthesized gates are natively allowed, we increase the number of CNOTs by a routing overhead factor of $O(\log n) \leq α\leq O(n \log^2 n)$. However, if we only synthesize allowed gates, we do not need to route any qubits. Moreover, in that case the routing overhead factor is $1 \leq α\leq 4 \simeq O(1)$. Additionally, since phase polynomials and Hadamard gates together form a universal gate set, we get qubit routing for almost free.

This work proposes a bootstrapping with positivity methodology to study random $U(N)^{D}$ invariant tensors in the large $N$ limit. As has been done for $U(N)$ invariant random matrices, we combine the Dyson-Schwinger equations and positivity constraints of moments to approximate the moments of such tensor models. As examples, we bootstrap the quartic and two hexic rank three tensor models. All models studied converge quickly, and for those which have known analytic formulae, they converge to such solutions. We conjecture new explicit formulae for all moments of the rank three quartic model and support this conjecture using bootstrapped results and explicit double-series computations with 'feyntensor'.

We analyse the 2025 Signalgate leak of sensitive US military information by the Trump administration, addressing why confidentiality was violated (messages leaked to the press) in spite of encryption (Signal), to deepen the socio-technical considerations when designing and deploying encryption. First, we use applied pi-calculus to formally model the boutique secure facility setup requested by the US Defence Secretary, to prove that a leak would not be prevented. We then examine how using a secure channel might still not give overall information security, as, in this case, power imbalances between personnel and officials led to the application of cryptography that compromised their operational security. We look at how cryptographic tools may have instilled a false sense of security, and led officials to "overshare". We then apply this analysis to the Trump administration's general desire to burn through political, legal, and now technical process, and demonstrate geopolitical harms that may arise from such ineffective use of cryptography in a brief use case. We conclude that, even with advancements in usability of cryptographic tools, genuine message security is still out of reach of the "average user".

This paper considers a networked tracking architecture in 6G integrated sensing and communication (ISAC) systems, where multiple base stations (BSs) cooperatively transmit radio signals and process received echo signals to track multiple moving targets. Compared to the single-BS counterpart, networked tracking allows the moving targets to be associated with different BSs over time such that the wireless resources can be dynamically allocated among BSs based on target locations. However, networked tracking imposes new challenges for algorithm design and resource allocation. In this paper, we first design the networked Kalman Filter (NKF) that is suitable for multi-BS based tracking, then characterize the posterior Cramer-Rao bound (PCRB) under this NKF, and last design the beamforming vectors of all the BSs to minimize the tracking PCRB. Numerical results show that our dynamic beamforming design can properly associate the targets to the suitable BSs at various sensing blocks and reduce the tracking mean-squared error (MSE).

The standard cosmological model is challenged by an ever-growing collection of observations, which invites (and stimulates) inquiry into possible additions and/or alterations. One such alteration comes from letting cosmic isotropy -- as demanded by the cosmological principle -- go, whilst maintaining only homogeneity. This study concerns Bianchi models, a class of anisotropic, homogeneous spacetimes, and in particular their perturbations. Knowledge of their properties under perturbations (such as allowed wavemodes) aids in understanding cosmological signatures of such universes, e.g. CMBs, and thus allows for comparsion to observation and the theory of the standard model. This study develops linear perturbation theory of general Bianchi models, by working in a frame such that metric components depend solely on (cosmic) time. Perturbation equations in the Newtonian gauge, but for arbitrary metric, are derived for energy density $ρ$, (relativistic) pressure $p$, momentum density $q$, and anisotropic stress $π$, for the case of scalar and pure tensor perturbations. For the former, the equations for density and pressure are combined to yield the equivalent of the Mukhanov-Sasaki equation for Bianchi models. For a specific choice of metric and fluid flow $u$, the Friedmann equations for Bianchi models are also formulated, as this knowledge is necessary to fully formulate the perturbation equations. Finally, the obtained results are applied to the formulations of density contrasts in an Einstein-de Sitter universe and a Bianchi I universe.

We study the local Lipschitz one subsets of a finite dimensional space, that is, sets for which there exists a continuous function whose local Lipschitz derivative is the characteristic function of said set. We give a characterization of a local Lipschitz one set on the real line in terms of a certain measure-theoretic density condition, which we call quasi-density. We show that any local Lipschitz one set needs to be quasi-dense, but the converse does not hold. Finally, we show that any regular closed subset of a normed space is a local Lipschitz one set, but there exist local Lipschitz one sets that are not regular closed.

The lowest eigenstates of the hopping matrix on the line graph of a cubic lattice with periodic boundary conditions are highly degenerate, they form a lowest flat band. Further, these states are localized. If one considers a repulsive bosonic Hubbard model on this lattice it is possible to construct exact multi-particle ground states simply by putting particles in the localized single particle ground states such that they avoid each other. This can be done up to a certain critical particle number $N_c$. We prove that at this particle number the ground state entropy is subextensive $\propto N_c^{2/3}$. For lower densities the entropy is extensive. We further show that the problem is related to the number of 4-cycle decompositions of the cubic lattice with periodic boundary conditions.

Inspiralling supermassive black-hole binaries (SMBHBs) are expected to be the main source of the nanohertz gravitational-wave background (GWB) targeted by pulsar timing arrays (PTAs). We provide a simple and general analytic expression for the probability distribution function (PDF) of the GWB characteristic strain squared $h_c^2$ in the limit of a large but finite effective number of sources, $N$, relevant for the lowest-frequency bands where PTAs are most sensitive. Explicitly, we show that for $N \gg 1$, the PDF of the rescaled variable $y \equiv h_c^2/\overline{h_c^2}$ takes the universal self-similar form $P(y) \simeq N^{1/3} \mathcal{P}(N^{1/3} (y -1))$, where $\mathcal{P}$ is the reflected map-Airy distribution. The effective number of in-band sources $N$ is fully specified by the mean $\overline{h_c^2}$ and the cubic shot-noise strain scale $\overline{h_0^3}$, a new summary statistic of the GWB that depends only on the local properties of the SMBHB population. This result is universal: it applies to any population of SMBHBs, regardless of whether they are circular or eccentric, and of the mechanism dominating orbital hardening. We explicitly quantify the accuracy of the large-source-count PDF for a simple but physically realistic SMBHB model, and outline its practical application to PTA data analysis.

We study null controllability for the parabolic equation on $\mathbb{S}^{2}$ endowed with its canonical almost-Riemannian structure. For a spherical crown $ω=\{α<x_3<β\}$, where $0\le α<β\le1$, we prove the sharp minimal time formula $T_{\min}(ω)=\ln(1/\sqrt{1-α^{2}})$ for null controllability in $ω$. We also prove that, whenever the control region contains the equator, null controllability holds in every positive time. The proof combines two complementary tools. First, after Fourier decomposition with respect to the periodic variable, we establish observability estimates for a family of one-dimensional singular parabolic equations, with constants uniform with respect to the Fourier mode; the singularities at the poles are handled via a Hardy-Poincaré inequality. Second, for crowns away from the equator, we use the moment method to construct controls on the pole-touching crown $α<x_3< 1$ from sharp weighted lower bounds on associated Legendre functions, and then pass to a general crown $α<x_3<β$ by a cut-off argument on the full domain combined with the arbitrary-time controllability of crowns containing the equator. The result closes the large-time gap left in earlier work and gives the exact null-controllability threshold for the canonical almost-Riemannian heat equation on $\mathbb S^2$.

A recent paper published on Nature [Nature,646,813(2025)] by Aziz and Howl, claims that quantum particles become entangled when they interact gravitationally, even if the gravitational potential is treated classically. We show that the entanglement found by the authors stems from discarding some of the transition amplitudes, which, when kept, guarantee that an initially factorized state remains so over time. Therefore, no entanglement is generated by the classical gravitational interaction in the scenario considered by the authors.

Mixed-effects logistic regression is widely used for binary outcomes in hierarchical data, yet formal goodness-of-fit tests remain limited to random-intercept models and do not address sparse cluster settings. We extend a grouping-based Wald test to mixed-effects logistic models with random slopes. The procedure groups observations by predicted probabilities within clusters, augments the model with pooled group indicators, and tests their joint significance using a Wald statistic. To accommodate small clusters, we introduce a data-driven rule for selecting the number of groups, G=min(10,n_min), where n_min is the smallest cluster size, ensuring feasible estimation. Simulation studies across 24 null scenarios show that the test maintains nominal Type I error in three-level random slope models, including at smaller sample sizes than previously studied. The test exhibits increasing power to detect fixed-effects misspecification: power against omitted nonlinearity rises from 0.07 to 1.00 across effect sizes, and power against omitted interactions reaches 0.87. As expected, the test has no power to detect omission of a clustering level, reflecting its focus on residual structure in predicted probabilities. In sparse balanced designs, fixing G=10 leads to complete test failure, whereas the data-driven rule performs reliably. The method is implemented in the Stata program mlm_gof.