We consider the refined $\mathrm{SU}(r)$ Vafa-Witten partition function of a smooth projective surface with non-zero holomorphic 2-form. This partition function has a vertical contribution, expressible in terms of nested Hilbert schemes. First, we write the vertical contribution in terms of $χ_y$-genera of moduli spaces of framed sheaves on ${\mathbb P}^2$. Then, we state two wall-crossing identities for moduli spaces of framed sheaves: a blow-up formula due to Kuhn-Leigh-Tanaka and a new stable/co-stable wall-crossing formula. We prove the latter using the theory of mixed Hodge modules. We apply these identities to obtain constraints on Vafa-Witten invariants predicted by conjectures of Göttsche and the second- and third-named authors. For $r=2$, we obtain a proof of the vertical part of a celebrated formula by Vafa-Witten.

This paper describes the wave-front correction and image stabilisation system (CWS) developed for the Sunrise III balloon-borne telescope, and provides information about its performance as measured during the integration into the telescope and during the 2024 science flight. The fast image stabilisation is done by a correlation tracker (CT) and a fast tip-tilt mirror, low order aberrations such as defocus and coma are measured by a six-element Shack-Hartmann wavefront sensor (WFS) and corrected by an active telescope secondary mirror for automated focus and manual coma correction. The CWS is specified to deliver a stabilised image with a precision of 0.005 arcsec (rms). The autofocus adjustment is specified to maintain a focus stability of 0.01 waves in the focal plane of the CWS.

Traditional coding theory guarantees valid decoding only if a minority of symbols are adversarially manipulated. In contrast, the game of coding framework ensures reliable decoding, even in the presence of an adversarial majority. This formulation is motivated by emerging permissionless applications, particularly decentralized machine learning (DeML), where computation tasks are outsourced to external volunteer nodes that are predominantly rational and reward-seeking. Prior investigations have analyzed the game of coding in the scalar setting. Since the results of most major computations in machine learning are vectors (e.g., computing the gradient of the loss for a machine learning model), we extend the framework in this paper to the general multi-dimensional Euclidean space. As a first, yet fundamental step, in this paper, we study a two-repetition code in which at least one node is controlled by a rational adversary, and we fully characterize the equilibrium and the optimal strategies of the players. Similar to the scalar case, this result serves as a cornerstone for addressing more general scenarios.

We investigate a scenario where Strongly Interacting Massive Particle (SIMP) dark matter interacts with an axion-like particle (ALP) that couples exclusively to electrons. This minimal setup provides interactions which enforce thermal equilibrium between dark matter and the SM in the early Universe. We analyze the cosmological evolution of the dark sector and the constraints arising from dark matter annihilations, ALP laboratory searches and astrophysical observations. Our results show that the allowed parameter space is wider than previous studies and an ALP with mass $m_a \sim {\cal O}(10)~\text{MeV}$ can act as a viable portal between the visible and dark sectors. Interestingly, this mass range overlaps with the parameter space suggested by the reported $X_{17}$ anomaly. Furthermore, the introduction of non-vanishing $θ$ angle in the dark sector of the model opens up the parameter space to heavy ALP masses.

We provide a bicategorical generalization of Barr's landmark 1970 paper, in which he describes how to extend Set-monads to relations and uses this to characterize topological spaces as the relational algebras of the ultrafilter monad. With two-sided discrete fibrations playing the role of relations in a bicategory, we first describe how to extend pseudomonads on a bicategory to skew monads on its bicategory of two-sided discrete fibrations, and we characterize in terms of exact squares when these extensions are themselves pseudomonads. As a wide class of examples, we show that every Set-monad induces a pseudomonad on the 2-category of categories admitting a skew extension to profunctors, and in a few relevant cases we introduce suitable quotients also extending to profunctors. Among the latter, we then focus on the ultracompletion pseudomonad, whose pseudoalgebras are ultracategories: we characterize the normalized lax algebras of its profunctorial extension as ultraconvergence spaces, a recently-introduced categorification of topological spaces.

Classical mechanics admits multiple equivalent formulations, from Newton's equations to the variational Lagrange-Hamilton framework and the scalar Hamilton-Jacobi (HJ) theory. In the HJ formulation, classical ensembles evolve through the continuity equation for a real density $ρ= R^{2}$ coupled to Hamilton's principal function $S$. Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair $(R,S)$ into a single complex field. Starting from a completely general complex ansatz $ψ= f(R,S)\, e^{i g(R,S)},$ and imposing two minimal structural requirements, we obtain a unique map $ψ= R\, e^{iS/κ}\, $ together with a linear HJS equation whose $|κ| \to 0$ limit reproduces the HJ formulation exactly. Remarkably, when $\mathrm{Re}(κ)\neq 0$, essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.

We show that the complex-plane structure of light hadron resonances is governed by a unified geometric framework where the threshold position plays a decisive role. By applying this framework to $ππ$, $πK$, and $πN$ phase shifts, we show that the residue phase $θ$ is primarily determined by the geometric phase $δ_0$ (the angle between pole and real axis seen from the threshold). While vector resonances exhibit excellent alignment with this geometric baseline, scalar resonances show systematic deviations of $10^\circ$--$15^\circ$, which we identify as the dynamical imprint of Adler zeros.

In downlink massive random access (DMRA), a base station transmits messages to a typically small subset of active users, selected randomly from a massive number of total users. Explicitly encoding the identities of active users would incur a significant overhead scaling logarithmically with the number of total users. Recently, via a random coding argument, Song, Attiah and Yu have shown that the overhead can be reduced to within some upper bound irrespective of the number of total users. In this remark, recognizing that the code design for DMRA is an instance of covering arrays in combinatorics, we show that there exists deterministic construction of variable-length codes that incur an overhead no greater than $1 + log_2 e$ bits.

GitHub natively supports workflow automation through GitHub Actions. Yet, workflow maintenance is often considered a burden for software developers, who frequently face difficulties in writing, testing, debugging, and maintaining workflows. Little knowledge exists concerning the automation and reuse practices favoured by workflow practitioners. We therefore surveyed 419 practitioners to elucidate good and bad workflow development practices and to identify opportunities for supporting workflow maintenance. Specifically, we investigate the tasks that practitioners tend to automate using GitHub Actions, their preferred workflow creation mechanisms, and the non-functional characteristics they prioritise. We also examine the practices and challenges associated with GitHub's workflow reuse mechanisms. We observe a tendency to focus automation efforts on core CI/CD tasks, with less emphasis on crucial areas like security analysis and performance monitoring. Practitioners strongly rely on reusable Actions, but reusable workflows see less frequent adoption. Furthermore, we observed challenges with Action versioning and maintenance. Copy-pasting remains a common practice to have more control and avoid the complexity of depending on reusable components. These insights suggest the need for improved tooling, enhanced support for a wide range of automation tasks, and better mechanisms for discovering, managing, and trusting reusable workflow components.

The modern integrated circuit ecosystem is increasingly reliant on third-party intellectual property integration, which introduces security risks, including hardware Trojans and security vulnerabilities. Addressing the resulting trust deadlock between IP vendors and system integrators without exposing proprietary designs requires novel privacy-preserving verification techniques. However, existing privacy-preserving hardware verification methods are all simulation-based and fail to offer formal guarantees. In this paper, we propose ZK-CEC, the first privacy-preserving framework for hardware formal verification. By combining formal verification and zero-knowledge proof (ZKP), ZK-CEC establishes a foundation for formally verifying IP correctness and security without compromising the confidentiality of the designs. We observe that existing zero-knowledge protocols for formal verification are designed to prove statements of public formulas. However, in a privacy-preserving verification context where the formula is secret, these protocols cannot prevent a malicious prover from forging the formula, thereby compromising the soundness of the verification. To address these gaps, we first propose a blueprint for proving the unsatisfiability of a secret design against a public constraint, which is widely applicable to proving properties in software, hardware, and cyber-physical systems. Based on the proposed blueprint, we construct ZK-CEC, which enables a prover to convince the verifier that a secret IP's functionality aligns perfectly with the public specification in zero knowledge, revealing only the length and width of the proof. We implement ZK-CEC and evaluate its performance across various circuits, including arithmetic units and cryptographic components. Experimental results show that ZK-CEC successfully verifies practical designs, such as the AES S-Box, within practical time limits.

We presents a systematic analysis of the fall-apart decays for the $1S$, $1P$, and $2S$-wave fully-strange tetraquark states. It shows that most of the fully-strange tetraquark states have a relatively narrow fall-apart decay width of $\mathcal{O}(10)$ MeV. The newly observed axial-vector state $X(2300)$ at BESIII may favor the low-lying $1S$-wave $1^{+-}$ state $T_{(4s)1^{+-}}(2323)$, while the $X(2500)$ resonance observed in the earlier BESIII experiment may favor the low-lying $1P$-wave $0^{-+}$ state $T_{(4s)0^{-+}}(2481)$. Some fully-strange tetraquark states predicted in theory can be searched for in their dominant fall-apart decay channels in experiment, such as $ϕϕ$, $ϕϕ(1680)$, $η^{(\prime)}ϕ$, $η^{(\prime)}h_1(1415)$, and $ϕf_2^{\prime}(1525)$, to which they have relatively large couplings.

The opioid epidemic remains a major public health challenge in the United States, requiring a multi-pronged intervention approach to mitigate harms to communities. Given the heterogeneity of the epidemic across the country, it is crucial for policymakers to understand localized treatment effects of different intervention components and utilize limited resources efficiently. While locally calibrated simulation models offer a useful computational tool to project the epidemic outcomes for any given intervention policy, collecting simulation results for all intervention combinations to estimate localized treatment effects for each community is impractical because the number of possible intervention combinations grows exponentially with the number of interventions and levels at which they are applied. To tackle this, we develop a bi-level metamodel framework with a two-stage sequential design for efficient sampling. The metamodel consists of a response function linking health outcomes to each intervention component's treatment effect, and a Gaussian process regression to learn spatial and socio-economic structures of the treatment effects based on locally-contextualized covariates. With two-stage sequential sampling, we leverage spatial correlations and posterior uncertainty to sequentially sample the most informative counties and treatment conditions. We apply this framework to estimate treatment effects of buprenorphine dispensing and naloxone distribution on overdose mortality rates using a calibrated agent-based opioid epidemic model in PA counties. Our approach achieves approximately 5% average relative error using one-tenth the number of runs required for an exhaustive simulation. Our bi-level framework provides a computationally efficient approach to support policymakers, in evaluating resource-allocation strategies to mitigate the opioid epidemic in local communities.

We validate Leonid Space's satellite lifetime prediction pipeline through comprehensive backtesting against 934 non-maneuvering satellites that deorbited from LEO between 1961 and 2024. This represents the first large-scale validation of lifetime prediction tooling using forecasted space weather conditions rather than historical hindsight. Our toolchain combines ballistic coefficient estimation from on-orbit data with probabilistic orbit propagation under varying environmental conditions. Using TLE data and space weather records spanning six solar cycles, our three-stage validation approach progressively removes hindsight bias to arrive at fully predictive operational conditions. We achieve 1-year prediction accuracy (median continuously ranked probability score) of 6.0 days (1.6%) under perfect knowledge conditions, 18.6 days (5.1%) with estimated ballistic coefficients and known space weather, and 45.5 days (12.4%) under fully predictive conditions. Comparison against ESA's standard DRAMA & DISCOS toolchain demonstrates a 4x improvement in state-of-the-art accuracy for well-characterized satellites, and an 8x improvement over NASA's DAS software. A custom semianalytic propagator provides a 340x speedup over Orekit and 55x speedup over DRAMA, enabling rapid Monte Carlo analysis across large satellite populations. Our analysis reveals that solar cycle forecasting dominates error budgets after ballistic coefficient estimation, with higher-fidelity propagators and atmosphere models providing marginal benefit. These results establish a validated performance baseline for operational lifetime prediction services supporting LEO mission planning and regulatory compliance.

In this work, we investigate the existence and orbital (in)stability of several branches of standing--wave solutions for the cubic nonlinear Schrödinger equation (NLS) posed on a looping--edge graph $\mathcal{G}$, consisting of a circle and a finite number $N$ of infinite half--lines attached to a common vertex. The model is endowed with $δ'$--type interaction boundary conditions at the vertex, which enforce continuity of the derivatives of the wave functions, while continuity of the wave function itself is not required. By means of the Implicit Function Theorem, we establish the existence of families of standing--wave profiles that converge, on the circular component of the graph, to Jacobi elliptic solutions of dnoidal type, coupled with soliton--type tail profiles on the half--lines. Tools from perturbation theory and Kre\uın--von Neumann extension theory for symmetric operators play a central role in the (in)stability analysis of such standing wave solutions. Our approach may be extended to other bound states for the NLS on looping graphs or more general non--compact metric graphs.

We investigate uncertainty relations for quantum observables evolving under non-Hermitian Hamiltonians, with particular emphasis on the role of metric operators. By constructing appropriate metrics in each dynamical regime, namely the unbroken-symmetry phase, the spontaneously broken-symmetry phase, and at exceptional points, we provide a consistent definition of expectation values, variances, and time evolution within a Krein-space framework. Within this approach, we derive a generalized Heisenberg-Robertson uncertainty inequality which is valid across all spectral regimes. As an application, we analyze a spin model with parity-time reversal symmetry and show that, while the uncertainty measure exhibits oscillatory behavior in the unbroken phase, it evolves towards a minimum-uncertainty steady state in the spontaneously broken-symmetry phase and at exceptional points. We further compare our metric-based description with a Lindblad master-equation approach and show their agreement in the steady state. Our results highlight the necessity of incorporating appropriate metric structures to extract physically meaningful predictions from non-Hermitian quantum dynamics.

This paper explores computational methods for solving the Longest Vector Problem (LVP) and Closest Vector Problem (CVP) in $p$-adic fields. Leveraging the non-Archimedean property of $p$-adic norms, we propose a polynomial time algorithm to compute orthogonal bases for $p$-adic lattices when the $p$-adic field is given by a minimal polynomial. The method utilizes the structure of maximal orders and $p$-radicals in extension fields of $\mathbb{Q}_{p}$ to efficiently construct uniformizers and residue field bases, enabling rapid solutions for the LVP and CVP. In addition, we introduce the characterization of norms on vector spaces over $\mathbb{Q}_p$.

Chaotic motion near black holes has recently been examined through the lens of the Maldacena-Shenker-Stanford (MSS) chaos-bound, but reported violations remain contradictory. A significant source of ambiguity stems from treating the particle angular momentum as an independently adjustable parameter instead of as a quantity fixed by the circular-orbit conditions. We develop a constrained framework in which the angular momentum is determined self-consistently from the geometry. Applied to the charged Kiselev black hole, this framework shows that certain previously reported violations of the chaos bound can be attributed to inconsistent parameter choices rather than to intrinsic curvature effects. By extending the analysis to geometries containing higher-order curvature terms, we find genuine chaos-bound violations at large charge-to-mass ratios, originating from curvature corrections rather than orbital parameters. Our approach, therefore, provides a systematic means to distinguish between parameter-induced (apparent) and curvature-induced (physical) violations in Einstein gravity and its extensions.

Sensing biological forces with microscopic lasers is an emerging technique that offers significant advantages over conventional fluorescent probes and imaging-based techniques. However, the limited availability of suitable deformable or elastic microlaser materials is restricting the scale of forces that can be detected which strongly narrows their overall applicability. Here, we describe the synthesis of spherical whispering gallery mode microbead lasers from a commercially available elastomer material in a microfluidic system with high viscosity. Upon doping with an organic dye, the microbeads show multimode lasing with thresholds in the range of 2-11 nJ. Measurements of the mechanical properties reveal that the width of the laser modes is directly proportional to the applied external force. The measured mean Young's modulus is 36 kPa, comparable to the stiffness of single cells and soft tissues. We also demonstrate that elastomer microlasers are stable under cell culture conditions for several days and observe splitting of the laser modes for intracellular microlasers. The observed properties render elastomer microlasers as a robust material platform for biointegrated lasers that also allows further tuning of the mechanical and optical properties for tailored force sensing inside cells, tissues, and small animals.

A scale mixture of normals is a distribution formed by mixing a collection of normal distributions with fixed mean but different variances. A generalized gamma scale mixture draws the variances from a generalized gamma distribution. Generalized gamma scale mixtures of normals have been proposed as an attractive class of parametric priors for Bayesian inference in inverse imaging problems. Generalized gamma scale mixtures have two shape parameters, one that controls the behavior of the distribution about its mode, and the other that controls its tail decay. In this paper, we provide the first demonstration that the prior model is realistic for multiple large imaging data sets. We draw data from remote sensing, medical imaging, and image classification applications. We study the realism of the prior when applied to Fourier and wavelet (Haar and Gabor) transformations of the images, as well as to the coefficients produced by convolving the images against the filters used in the first layer of AlexNet, a popular convolutional neural network trained for image classification. We discuss data augmentation procedures that improve the fit of the model, procedures for identifying approximately exchangeable coefficients, and characterize the parameter regions that best describe the observed data sets. These regions are significantly broader than the region of primary focus in computational work. We show that this prior family provides a substantially better fit to each data set than any of the standard priors it contains. These include Gaussian, Laplace, $\ell_p$, and Student's $t$ priors. Finally, we identify cases where the prior is unrealistic and highlight characteristic features of images that suggest the model will fit poorly.

Ultralight bosons such as axions and dark photons are well-motivated hypothetical particles, whose couplings to ordinary matter can be effectively constrained by stellar cooling. Limits on these interactions can be obtained by demanding that their emission from the stellar interior does not lead to excessive energy loss. An intriguing question is whether the same microphysical couplings can also be probed through neutron star superradiance, in which gravitationally bound bosonic modes grow exponentially by extracting rotational energy from the star. Although both processes originate from boson-matter interactions, they probe very different kinematic regimes. Stellar cooling probes boson emission at thermal wavelengths, while superradiance is governed by modes whose wavelength is comparable to or greater than the size of the star. Previous work has attempted to relate the microphysical neutron-nucleon scattering and inverse-bremsstrahlung absorption rates directly to the macroscopic growth rate of superradiant bound states. In this work, we re-examine this connection and show that a naive extrapolation of the microphysical absorption rate to the superradiant regime would imply superradiant rates comparable to astrophysical timescales characterised by pulsar spindown. These naive rates are especially high for vector fields. However, we demonstrate that this conclusion changes once collective multiple-scattering effects in dense nuclear matter are taken into account. Repeated nucleon collisions modify the effective low-energy absorption experienced by the bosonic bound state, strongly suppressing the rate relevant for superradiance.

The rare decays $η^{(\prime)}\to\ell^+\ell^-$, $\ell\in\{e,μ\}$, are highly suppressed in the Standard Model, both by their chirality structure and the required loop attaching the lepton line to the $η^{(\prime)}\toγ^*γ^*$ matrix element. The latter is described by a single scalar function, the transition form factor, which has recently been studied in great detail for $η^{(\prime)}$ in the context of the pseudoscalar-pole contributions to hadronic light-by-light scattering in the anomalous magnetic moment of the muon. Based on these results, we evaluate the corresponding prediction for the $η^{(\prime)}$ dilepton decays, supplemented by an improved evaluation of the asymptotic contributions including pseudoscalar mass effects. In particular, the dispersive representation for the $η^{(\prime)}$ transition form factors allows us, for the first time, to perform a robust evaluation of the imaginary parts due to subleading channels besides the dominant two-photon cut. Our final results are $\text{Br}[η\to e^+e^-]=5.37(4)(2)[4]\times 10^{-9}$, $\text{Br}[η\to μ^+μ^-]=4.54(4)(2)[4]\times 10^{-6}$, $\text{Br}[η'\to e^+e^-]=1.80(2)(3)[3]\times 10^{-10}$, and $\text{Br}[η'\to μ^+μ^-]=1.22(2)(2)[3]\times 10^{-7}$, where the errors refer to the uncertainty in the normalized branching fraction, the one propagated from $\text{Br}[η^{(\prime)}\toγγ]$, and the total uncertainty, respectively. The branching fraction for $η\toμ^+μ^-$ exhibits a mild $1.6σ$ tension with experiment, and we explore the bounds that can be derived on physics beyond the Standard Model.

We consider the Lotka-Volterra system and provide necessary conditions for an equilibrium to be stable. Our results naturally complement earlier fundamental results by N. Adachi, Y. Takeuchi, and H. Tokumaru, who, in a series of papers, give sufficient (and for some cases necessary) conditions for the existence of a stable equilibrium point.

We propose a quark-lepton symmetric Pati-Salam (PS) model based on the gauge group $SU(2)_L \times SU(2)_R \times SU(4)_C$ with an $E_6$-inspired particle spectrum which naturally accommodates a multi-TeV leptoquark gauge boson $X_μ(3,1,\frac{2}{3})$. A softly broken $Z_2$ symmetry plays a crucial role in realizing this scenario, under which the Standard Model (SM) fermions are even, while new vector-like fermions present in the model are odd. A notable feature of this model is that the PS gauge boson $X_μ$ itself is $Z_2$-odd, causing it to couple exclusively between SM fermions and the vector-like fermions, except in the right-handed down-quark sector, where mixing is induced by the soft breaking of $Z_2$. This structure leads to helicity suppression of tree-level meson decays mediated by $X_μ$, with helicity-unsuppressed contributions arising only via one-loop diagrams. We show that $X_μ$ can be as light as 1.1 TeV while being compatible with all flavor-violating constraints. However, mass relations among the $(W'^\pm_μ, Z'_μ, X_μ)$ gauge bosons push the $X_μ$ mass limit up to 4.3 TeV from the stringent LHC bounds on the $Z^\prime$ mass. This comparatively low PS-breaking scale opens up promising collider opportunities for probing the leptoquark gauge boson, as well as the distinctive signature of vector-like down-type quarks carrying an unusual baryon number of $2/3$. The model can be further tested via lepton flavor-violating processes induced by the leptoquark gauge boson, such as $μ\to e γ$, $μ\to e e e$, and $μ$-$e$ conversion in nuclei. Neutrino masses arise in the model through dimension-seven operators at tree-level, as well as from dimension-five operators via one-loop diagrams.

We introduce a driven-dissipative Floquet model in which a single harmonic oscillator, with both frequency and decay rate modulated, realizes a non-Hermitian synthetic lattice with an effective electric-field gradient in frequency space. Using the Floquet-Green's function and the doubled Hamiltonian representation of non-Hermitian matrices, we show that the linear response of this system is characterized by a local winding number. Nontrivial values of the winding number induce directional amplification in the synthetic dimension, thereby converting input signals to different frequencies. The underlying mode structure is well described by a Jackiw-Rebbi-like continuum theory with Dirac cones and solitonic topological zero modes in synthetic frequency. Our results establish a simple and experimentally feasible route to non-Hermitian topological amplification, naturally implementable in current quantum technologies such as superconducting circuits.

We present the constraint on the growth rate of structure from the combination of DESI DR1 BGS sample, Fundamental Plane, and Tully-Fisher peculiar velocity catalogues using the maximum likelihood fields method. The combined catalogue contains 415,523 galaxy redshifts and 76,616 peculiar velocity measurements. To handle the large amount of data in the DESI DR1 peculiar velocity catalogue, we significantly improve the computational efficiency by rewriting the algorithm with JAX. After removing outliers and Tully-Fisher galaxies that are affected by systematics, we find $fσ_8 = 0.483_{-0.043}^{+0.080}(\mathrm{stat}) \pm 0.018(\mathrm{sys})$, consistent within $1σ$ with the power spectrum and correlation function analysis using the same dataset. Combining all three measurements with appropriate correlations, the consensus measurement is $fσ_8 (z_{\mathrm{eff}}=0.07) = 0.450\pm0.055$, consistent with Planck $+Λ$CDM cosmology $(fσ_8 = 0.449 \pm 0.008)$. Combining with the high redshift growth rate of structure measurements from DESI ShapeFit, the constraint on the growth index is $γ= 0.58\pm0.11$, consistent with GR.

Sub-Neptunes represent the largest exoplanet demographic, yet their bulk compositions remain poorly understood. Recent studies suggested that only very cold planets, such as Uranus and Neptune, could experience stratification of volatiles in their envelopes. Transiting warm sub-Neptunes, with $10^3$ to $10^4$ times more stellar irradiation, were therefore believed to have fully-miscible compositions. Here, we present ATHENAIA, an interior-atmosphere composition inference framework we leverage to assess the potential for water-hydrogen demixing on warm sub-Neptunes and for the 350 K planet TOI-270 d as a case study, using radiative-convective atmosphere models coupled to interior models. We find that the higher temperatures at which hydrogen and water demix in water-rich environments open a window for demixing on sub-Neptunes with bulk envelope metallicities of $\sim 150$ to $700\times$ solar, compatible with TOI-270 d. Demixing is easier to achieve on more massive and colder planets, but still broadly affects warm ($\simeq $330 to 450 K) metal-rich sub-Neptunes. Therefore, combining atmosphere metallicities with models of fully-miscible envelopes may lead to underestimated bulk envelope metallicities and mass fractions. Further, we find that considering the increased greenhouse effect in metal-rich atmospheres in concert with the composition-dependent adiabatic gradient in the convective envelope increases the range of compositions under which molten mantle conditions should be expected on sub-Neptunes. This work encourages a reconsideration of the current paradigm for linking sub-Neptune atmospheres to their interiors and motivates evolutionary modeling describing the onset of metallicity gradients in sub-Neptune envelopes.

We study a class of random homogeneous systems. Our main result says that under suitable general assumptions, these systems converge weakly, upon a suitable normalization, to the probability distribution with density $\frac34 \, (1-x^2) \, {\bf 1}_{\{ x\in (-1, \, 1)\} }$. Two special cases are of particular interest: for the effective resistance of the critical random series-parallel graph, our result gives an affirmative answer to a conjecture of Hambly and Jordan (Adv. Appl. Probab. 2004) and further conjectures of Addario-Berry et al. (Probab.Theory Related Fields 2020) and Derrida whereas for the hipster random walk, we recover a previous result of Addario-Berry et al.~(Probab. Theory Related Fields 2020).

The fully-connected tensor network (FCTN) decomposition has recently exhibited strong modeling capabilities by connecting every pair of tensor factors, thereby capturing rich cross-mode correlations. However, this advantage comes with an inherent limitation: updating the factors typically requires reconstructing auxiliary sub-networks, which entails extensive and cumbersome (un)folding. In this study, we propose the intra-block FCTN (iFCTN) decomposition, a novel (un)folding-free variant of FCTN decomposition designed to enhance computational efficiency. We parameterize each FCTN factor through Khatri-Rao products, which significantly reduces the complexity of reconstructing intermediate sub-networks and yields subproblems with well-structured coefficient matrices. Furthermore, we deploy the proposed iFCTN decomposition on the representative task of tensor completion and design an efficient proximal alternating minimization algorithm. Theoretically, we establish its global convergence to a critical point. Extensive experiments demonstrate that iFCTN outperforms state-of-the-art methods with a lower computational overhead.

We study feasibility guarantees for safety filters developed using Control Barrier Functions (CBFs) when a safe set is defined using the pointwise minimum of continuously differentiable functions, a construction that is common for the backup CBF (BCBF) method and typically nonsmooth. We replace the minimum by its log-sum-exp (soft-min) smoothing and show that, under a strict safety condition, the smooth function becomes a CBF (or extended CBF) for a range of the smoothing parameter. For compact safe sets, we derive an explicit lower bound on the smoothing parameter that makes the smooth function a CBF and hence renders the corresponding safety constraint feasible. For unbounded sets, we introduce tail conditions under which the smooth function satisfies an extended CBF condition uniformly. Finally, we apply these results to BCBFs. We show that safety of a compact (terminal) backup set under a backup controller, together with a condition ensuring safety of the backup trajectories on the relevant boundary of the safe set, is sufficient for constraint feasibility for BCBFs. These results provide a recipe for a priori feasibility guarantees for smooth inner approximations of nonsmooth safe sets without the need for additional online certification.

We prove that in all dimensions at least 3 and for any Hörmander-type oscillatory integral operator satisfying Bourgain's condition, the sticky case of the corresponding curved Kakeya conjecture reduces to the sticky case of the classical Kakeya conjecture. This supports a conjecture of Guo-Wang-Zhang, that an operator satisfies the same $L^p$ bounds as in the restriction conjecture exactly when it satisfies Bourgain's condition. Our result follows from a new geometric characterization of Bourgain's condition based on the structure of curved $δ$-tubes in a $δ^{1/2}$-tube. We find examples in all dimensions at least 3 which show this property does not persist in a larger tube, and in particular these are the first operators satisfying Bourgain's condition for which there is no diffeomorphism taking the corresponding families of curves to lines. This suggests that a general to sticky reduction in the spirit of Wang-Zahl needs substantial new ideas. We expect these examples to provide a good starting point.