This paper presents a novel space-filling polyhedron (SFPH), here named the Josehedron, derived from the extremal points of the Fischer-Koch S triply periodic minimal surface (TPMS). The Josehedron is a plesiohedron with 12 faces (4 isosceles triangles and 8 mirror-symmetric quadrilaterals), 12 vertices, and 22 edges. It tiles three-dimensional space with 12 instances per cubic unit cell in 6 distinct orientations. The generating point set exhibits a remarkable connection to the pentagonal Cairo tiling when projected onto any coordinate plane. Several additional geometric properties are described, including integer vertex coordinates, interwoven labyrinths, and chiral symmetry between the polyhedra obtained from the combined minima and maxima of the function. Finally, the paper presents a general method for finding novel SFPHs based on any periodic function, TPMS, or other functions. The described method is applied to a selection of TPMS, and 7 additional, previously undocumented SFPH are shown in the Appendix.

Randomized sketching is currently introduced into every area of numerical linear algebra. In Krylov subspace methods, it allows runtime savings at the cost of small accuracy reductions. This work offers a different view on sketching in Krylov methods by analyzing what subspace embeddings are obtained by arbitrary sketching matrices. The analysis gives rise to a deterministic sketching approach leveraging row subset selection techniques that yield subspace embeddings holding with probability 1. We propose deterministically sketched Krylov methods for matrix functions, linear systems, and eigenvalue problems that show a similar performance to their randomly sketched counterparts.

In this work we construct new multidimensional families of complete minimal submanifolds, of the classical non-compact Riemannian symmetric spaces SL_n(R)/SO(n), Sp(n,R)/U(n), SO*(2n)/U(n) and SU*(2n)/Sp(n), of codimension two.

Grouping-based measurement strategies are widely used to reduce measurement complexity in near-term quantum algorithms. While these schemes have typically produced disjoint groups, recently this has been relaxed in what is known as overlapped grouping or coefficient splitting where operators may appear in more than one compatible group. In recent work, it has been numerically shown that this strategy can reduce the variance of energy estimates on small benchmark problems, motivating both the application and further analysis of the method. Here we prove that overlapped grouping for energy estimation can lead to a maximal variance reduction that is linear in the number of Hamiltonian terms. We introduce a new algorithm which we call repacking to transform existing groups into overlapped groups, and we show this repacking procedure iteratively reduces variance under mild assumptions. We also perform numerical simulations with Hamiltonians up to $44$ qubits and $575 \cdot 10^{3}$ terms, assessing overlapped grouping at scale on problems of practical importance. Our numerics show that the variance reduction relative to state-of-the-art (disjoint) grouping increases linearly with the problem size, suggesting that overlapped grouping methods can be a powerful strategy for quantum energy estimation at the scale of Megaquop computers and beyond.

Isogeometric analysis was proposed to bridge the gap between computer-aided design and numerical discretization. However, standard multi-patch isogeometric analysis mandates conformal discretizations across patch interfaces, posing challenges for multi-material domain problems. In the context of electric machines, this requirement becomes evident in a large number of patches needed to represent machines consisting of several domains and materials. In this work, we adopt, extend, and evaluate three non-conformal discretization strategies for magnetostatic problems: a fully immersed approach, the union with non-conformal patches, and the union with conformal layers. In all three methods, boundary-conformal high-order quadrature rules are employed for integration over trimmed boundary and interface elements. In the two union approaches, material regions are, as far as possible, represented by independent non-conformal spline patches that are embedded within a background patch and coupled weakly through Nitsche's method. In the latter framework, critical interfaces are additionally surrounded by conformal layers that enable the strong imposition of boundary conditions and improved resolution of interface behavior. The proposed approaches are assessed through several magnetostatic benchmark problems and an industrial application. The numerical results show that the union methods achieve highly accurate solutions, while the fully immersed approach struggles with discontinuities in field gradients across material interfaces. Nevertheless, these methods significantly reduce the geometric preprocessing effort compared to conventional, conformal multi-patch analysis and require substantially fewer patches. Overall, this demonstrates that our immersed boundary-conformal isogeometric framework possesses great potential for efficient simulation of complex electromagnetic devices.

Kernel methods provide a flexible and powerful framework for nonparametric statistical testing by embedding probability distributions into a reproducing kernel Hilbert space (RKHS). In this work, we study the kernel two-sample testing problem and focus on a normalized version of the Maximum Mean Discrepancy (MMD) as a test statistic, which scales the discrepancy by the within-group covariance operator to account for data variability. This normalization has been shown to improve test power in both theoretical and empirical settings. Because this normalization requires regularization, we study the non-asymptotic properties of the spectrally truncated normalized MMD (st-nMMD) and derive an exponential upper bound under the null hypothesis. Thanks to this result we propose a sharp and explicit upper bound for the corresponding non-asymptotic quantile, along with a data-adaptive estimator. We further propose an algorithm to tune the hyperparameters involved in the quantile estimation, including the truncation level, without requiring data splitting. We demonstrate the performance of the st-nMMD through numerical experiments under both the null and alternative hypotheses.

We show that, in certain circumstances, a Boolean ample monoid may be fully embedded into a Boolean inverse monoid in a way that generalizes how right reversible cancellative monoids may be embedded into groups. We use groupoids of fractions and non-commutative Stone duality to prove the result.

This paper studies the transmit waveform optimization for a quantized multiple-input multiple-output (MIMO) integrated sensing and communication (ISAC) system, where one-bit analog-to-digital converters (ADCs) are employed to enable a low-cost and power-efficient hardware implementation. Focusing on the parameter estimation task, we propose two novel Cramér-Rao bounds (CRBs) for both point-like target (PT) and extended target (ET) to characterize the impact of quantization distortion on the estimation accuracy, where associated estimation methods are also developed to approach these theoretical CRBs. Moreover, with the goal of jointly enhancing the sensing and communication performances, we formulate the bi-criterion ISAC waveform optimization problem by minimizing the derived CRB objectives subject to a communication symbol error probability (SEP) constraint and a total power constraint, which, due to the high nonlinearity of the one-bit CRBs, are extremely nonconvex. To yield a high-quality suboptimal solution, we develop an efficient alternating direction method of multipliers (ADMM) framework which exploits the majorization-minimization (MM) technique to address the nonconvex issue. Simulation results verify that the one-bit CRBs are tight for characterizing the quantized estimation performance and the proposed estimation methods also show clear performance advantages over the existing benchmark schemes. Furthermore, a flexible trade-off between the CRB and the SEP performance can be achieved by the developed ADMM framework, demonstrating the effectiveness of the optimized ISAC waveform.

We introduce a Bergman-space framework for the study of boundary-forced heat equations and show that, in the one-dimensional case, boundary white noise gives rise to a sharp holomorphic regularity phenomenon. More precisely, we consider the heat equation on a bounded interval with Dirichlet or Neumann boundary conditions driven by independent white noises at the endpoints, and we prove that for every positive time the corresponding state extends holomorphically to a rhombus in the complex plane having the original interval as one of its diagonals. Moreover, the resulting process admits a continuous version with values in a scale of weighted Bergman spaces on that rhombus, depending on two parameters $δ\in(0,1)$ and $Θ\in\left(0,\fracπ{4}\right)$. To our knowledge, this is the first systematic use of Bergman spaces as state spaces for parabolic equations with stochastic boundary forcing. We also prove that the result is optimal, in the sense that the conclusion fails at the critical values $δ=0$ and $Θ=\fracπ{4}$.

Cellular-connected unmanned aerial vehicles (UAVs) operating in 5G New Radio (NR) macro networks experience severe and spatially non-uniform downlink interference. This is primarily caused by the interference from the sidelobes of downtilted base station (BS) antennas serving terrestrial users, which limits the ability of the network to provide uniform and high-quality coverage to aerial users. Supporting aerial users requires boosting the coverage of certain cells or sectors, which can further exacerbate inter-cell interference in dense macro deployments. This motivates the need for inter-cell interference coordination (ICIC) in multi-cell 5G NR networks serving both aerial and terrestrial users. In this work, we propose an ICIC framework that jointly optimizes antenna-domain coordination through BS uptilt angle optimization and time-domain interference coordination (TDIC) through NR-compliant scheduling. The framework is formulated as a multi-cell NR macro deployment problem that maximizes the minimum UAV signal-to-interference ratio (SIR) over a spatial grid of UAV locations while maintaining acceptable performance for ground user equipment (GUEs). The resulting optimization problem is non-convex and is solved using bio-inspired optimization techniques, including particle swarm optimization (PSO) and genetic algorithm (GA). Simulation results demonstrate that coordinated uptilt optimization with the booster-cell architecture significantly improves worst-case UAV SIR and downlink reliability in multi-cell 5G NR networks. booster-cell architecture significantly improves worst-case UAV SIR and downlink reliability in multi-cell 5G NR networks.

Modern Large Language Model (LLM) serving operates in highly volatile environments characterized by severe runtime dynamics, such as workload fluctuations and elastic cluster autoscaling. Traditional serving systems rely on static, human-engineered serving policies (e.g., scheduling algorithms and rescheduling strategies) to manage these dynamics. However, these policies must navigate deeply intertwined runtime trade-offs (e.g., scheduling overhead vs. execution efficiency, rescheduling frequency vs. reconfiguration overhead), whose optimal balance is workload-specific and shifts continuously as runtime conditions evolve, rendering any fixed policy fundamentally unable to adapt. We propose Autopoiesis, a novel online self-evolving system that shifts LLM serving from static policy deployment to continuous online policy evolution. First, Autopoiesis introduces an LLM-driven program synthesis workflow to evolve serving policies with respect to real-time observed dynamics, where the evolved policies reflect the optimal decision in navigating the complex, multi-dimensional trade-off space. Second, Autopoiesis enables this synthesis process to operate continuously during serving, observing real-world system behavior, and rewriting the policy code as runtime trade-offs shift, thereby transforming policy design from a one-time offline endeavor into an ongoing system component, enabling autonomous adaptation to evolving runtime conditions. Together, we establish a new paradigm: Serving policies are no longer static artifacts designed by humans before deployment, but living code that LLMs continuously evolve throughout deployment to navigate runtime trade-offs beyond human design. We evaluate Autopoiesis across diverse runtime dynamics and show up to 53% and on average 34% improvements over state-of-the-art LLM serving systems.

Gardiner, Lazarus, Metropolis, and Ulam introduced a variation of the sieve of Eratosthenes that (instead of producing the sequence of prime numbers) produces the sequence of "lucky numbers". The distribution of lucky numbers has a striking similarity to that of prime numbers. In particular, Hawkins and Briggs proved that if $\ell_n$ denotes the $n$th lucky number then $\ell_n \sim n \log n$, which is analogous to the prime number theorem. This work provides explicit upper and lower bounds on $\ell_n$.

The realization of robust quantum anomalous Hall (QAH) phases at elevated temperatures remains a central challenge in condensed matter physics. While quadratic band crossing points (QBCP) provide a promising route towards QAH states, existing proposals are largely confined to idealized models or hindered by interaction-driven competing orders. Here, we demonstrate that these limitations are not intrinsic to QBCP but arise from their specific implementation. We propose a general mechanism where band inversion between a symmetry-protected orbital doublet (e.g. $d_{xz},d_{yz}$) and an isolated orbital (e.g. $d_{z^2}$)-generically generates a QBCP with opposite curvature. This crossing is directly gapped at the single-particle level by intrinsic atomic spin-orbit coupling, while the underlying band inversion naturally shields the resulting topological gap against other interaction-driven instabilities. We further suggest monolayer compounds $MNX_2$ ($M$= Ni, Pd, Pt; $N$= Nb, Ta; $X$= S, Se, Te) as a realistic material class that intrinsically realizes this mechanism. These findings provide a concrete pathway toward robust QAH phases in correlated materials.

We propose a computational framework for computing low-rank approximations to the ensemble of solutions of a parametrized system of the form $A(ξ)x(ξ)+g(x(ξ))=b(ξ)$ for multiple parameter values. The central idea is to reinterpret the parametrized system as the first-order optimality condition of an optimization problem set over the space of real matrices, which is then minimized over the manifold of fixed-rank matrices. This formulation enables the use of Riemannian optimization techniques, including conjugate gradient and trust-region methods, and covers both linear and nonlinear instances under mild assumptions on the structure of the parametrized system. We further provide a theoretical analysis establishing conditions under which the solution matrix admits accurate low-rank approximations, extending existing results from linear to nonlinear problems. To enhance computational efficiency and robustness, we discuss tailored preconditioning strategies and a rank-compression mechanism to control the rank growth induced by nonlinearities. Numerical experiments demonstrate that the proposed approach achieves significant computational savings compared to solving each system independently, as well as highlight the potential of Riemannian optimization methods for low-rank approximations in large-scale parametrized nonlinear problems.

In the era of big data, leveraging information from multiple clients while preserving data privacy has emerged as a critical challenge in modern statistical modeling and forecasting. This paper introduces a privacy-preserving federated learning framework for high-dimensional vector autoregressive models, where each client's dynamics are characterized by a common low-rank structure augmented with sparse client-specific deviations. We develop a two-stage estimation procedure that integrates differentially private representation learning for the shared component with local personalization for client-specific adjustments, enabling effective information pooling under selective privacy constraints. Non-asymptotic error bounds are established for both the single-client and federated estimators to characterize the inherent privacy-utility trade-off, and consistency of a ridge-type rank selection criterion is proved. Simulation studies demonstrate that federation substantially improves estimation accuracy when local sample sizes are limited. Two empirical applications to analyzing electricity-economy linkages across U.S. states and conducting multi-task macroeconomic forecasting across countries, highlight the superior predictive accuracy of the proposed method over existing single-client benchmarks.

We present LightCurveLynx, a flexible and extensible software framework for end-to-end forward modeling time-domain light curves. Given the growing need for realistic simulations in the time-domain astronomy community, LightCurveLynx is designed to support a wide range of applications, including the development and validation of analysis pipelines, the optimization of survey strategies, and simulation-based inference studies. Realistic simulations can be generated from real survey metadata, forecasted survey plans, or user-defined mock survey strategies. We demonstrate the functionality of LightCurveLynx by generating a realistic simulation of Type Ia supernovae that is representative of the ZTF SN Ia Data Release 2 dataset and perform extensive comparisons between the simulated and observed samples to validate the software. The simulation shows excellent agreement with the data in parameter distributions (with the Kullback-Leibler divergence values around 0.01-0.02) and in noise properties. The Hubble diagram generated from the simulation also indicates that the sample is complete up to redshift 0.06, which is consistent with previous studies. Our results confirm that LightCurveLynx is robust, accurate, and ready for community use and contribution.

Under heterogeneous treatment effects, the GMM weighting matrix in overidentified IV models dictates the estimand. We show that efficient GMM downeights high-variance instruments and frequently assigning negative weights that undermine causal interpretation. Moreover, GMM cannot simultaneously achieve efficiency and accommodate researcher-specified weights. We resolve this trade-off by developing the Representative Targeting (RT) estimator. By averaging instrument-specific Wald estimators under Positive Regression Dependence, RT ensures non-negative weights while achieving the semiparametric efficiency bound for its targeted estimand. We demonstrate the heterogeneity penalty empirically in a class-size experiment and apply RT to recover the Policy-Relevant Treatment Effect within a patent leniency design.

Dyson [Commun. Math. Phys. 12, 91 (1969)] rigorously proved the existence of a phase transition in the one-dimensional Ising model with long-range interactions of the form $r^{-α}$ for $1 < α< 2$. In the present study, we extend this result to the Ising spin glass model with Gaussian disorder on the Nishimori line. Following Dyson's method, we first prove the existence of long-range order at finite low temperatures in the Dyson hierarchical Ising spin glass model on the Nishimori line, with power-law-like interactions $J(r) \sim r^{-α}$ for $1 < α< 3/2$. The key ingredients of the proof are the interpolation method developed in the rigorous analysis of mean-field spin glass models, the Gibbs--Bogoliubov inequality on the Nishimori line, and the Tsirelson--Ibragimov--Sudakov inequality (Gaussian concentration inequality). We then use the Griffiths inequality on the Nishimori line to rigorously establish the existence of a phase transition in the one-dimensional Ising spin glass model with long-range interactions on the Nishimori line for $1 < α< 3/2$. For $α\ge 3/2$, the existence of a phase transition remains an open problem.

Let $S_α= k\langle x_1,\dots,x_n\rangle /(x_i x_j - α_{ij} x_j x_i)$ be a standard graded skew polynomial algebra over an algebraically closed field $k$ of characteristic not equal to $2$. We show the following results. When $n$ is odd and $f = x_1x_2 + \cdots + x_{n-2}x_{n-1} + x_n^2$ is a normal element of $S_α$, the even Clifford algebra of the skew quadric hypersurface $S_α/(f)$ is isomorphic to a full matrix algebra $M_{2^{(n-1)/2}}(k)$, and the stable category $\underline{\mathsf{CM}}^{\mathbb Z}(S_α/(f))$ of graded maximal Cohen-Macaulay modules over $S_α/(f)$ is triangle equivalent to the derived category $\mathsf{D}^b(\mathsf{mod}\,k)$. When $n$ is even and $f = x_1x_2 + \cdots + x_{n-1}x_n$ is a normal element of $S_α$, the even Clifford algebra of $S_α/(f)$ is isomorphic to $M_{2^{(n-2)/2}}(k)^2$, and the stable category $\underline{\mathsf{CM}}^{\mathbb Z}(S_α/(f))$ of graded maximal Cohen-Macaulay modules over $S_α/(f)$ is triangle equivalent to the derived category $\mathsf{D}^b(\mathsf{mod}\,k^2)$. As a consequence, $S_α/(f)$ is of finite Cohen-Macaulay representation type in both cases. These results demonstrate that $S_α/(f)$ is a natural noncommutative generalization of the homogeneous coordinate ring of a smooth quadric hypersurface.

Systems of differential equations have been used to model biological systems such as gene and neural networks. A problem of particular interest is to understand the number of stable steady states. Here we propose conjunctive networks (systems of differential equations equations created using AND gates) to achieve any desired number of stable steady states. Our approach uses combinatorial tools to predict the number of stable steady states from the structure of the wiring diagram. Furthermore, AND gates have been successfully engineered by experimentalists for gene networks, so our results provide a modular approach to design gene networks that achieve arbitrary number of phenotypes.

Generative AI's emphasis on automation and efficiency challenges design education, where learning is grounded in exploration, reflection, and responsibility. This work introduces AI Craftsmanship, a value-oriented framework drawing on craftsmanship traditions that emphasize risk, rhythm, and care as central to learning through making. Through a Research through Design (RtD) approach, we designed an AI-integrated creative coding tool embedding these values into interactions and interface rather than outcomes. The tool supports designers learning generative pattern-making with p5.js by constraining AI, encouraging iterative experimentation, and foregrounding reflection. We studied the tool with five design practitioners through one-hour sessions and semi-structured interviews. Findings show craft values manifest unevenly: risk and rhythm shape early sense-making, while care emerges through reflective practices. Emergent values -- such as aesthetic judgment and confidence -- also motivated learning. AI Craftsmanship mediates values, tools, and materials, offering a value-driven perspective on designing AI systems for reflective, responsible, craft-informed learning in design education.

Cr$_{\frac{1}{4}}$NbSe$_2$ is a triangular lattice magnet in which magnetic Cr$^{3+}$ ions are intercalated to form triangular lattices between NbSe$_2$ van der Waals layers stacked along the c axis. By unpolarized and polarized neutron scattering experiments, we have revealed that the magnetic ground state of this system is a 120$^{\circ}$-type antiferromagnetic order characterized by the magnetic propagation wave vector of $q=(\frac{1}{3}, \frac{1}{3}, 0)$. We also performed inelastic neutron scattering measurements using co-aligned single crystals, and determined dispersion relations of magnetic excitations at low temperatures. Comparing the observed spectra with calculations based on the linear spin-wave theory, we revealed that the out-of-plane ferromagnetic interaction is fairly strong as compared to the in-plane nearest neighbor antiferromagnetic interaction. Although the crystal structure of this system is composed of two-dimensional van der Waals layers, the magnetic order has a three dimensional character, which would be attributed to long-range magnetic interactions mediated by conduction electrons.

The notion of smoothness was introduced originally in the context of step systems on connected graphs. Smoothness turns out to be a very general property of metrics defined by a five-point condition. Restricted to graphs, it is closely related to the convexity of point-shadows. We show that smoothness is preserved by isometric subgraphs, both Cartesian and strong graph products, and gated amalgams. As a consequence, median graphs and many of their generalizations are smooth. We also show that l1-graphs are smooth. On the other hand, an induced K2,3 or K1,1,3 is incompatible with smoothness. Finally, we characterize smooth graphs among the Ptolemaic graphs as precisely the K1,1,3-free Ptolemaic graphs.

The behaviour of excess charges in ionic lattices, such as the formation of polarons and charge trapping at defect sites, influences the physical and chemical properties of materials and translates into applications in electronics, optics, photovoltaics, and catalysis. Here we show that the bulk-terminated SrTiO3(001) surface accumulates photoexcited charges and keeps the associated photovoltage for many days at cryogenic temperatures. A combination of scanning tunneling microscopy, atomic force microscopy (STM/AFM) and Kelvin probe force microscopy (KPFM) was used to measure this photovoltage and to localize the photoexcited charges with atomic precision down to the single-quasiparticle limit. Density functional theory (DFT) shows that holes favor localization at oxygen 2p orbitals adjacent to Sr vacancies, creating long-lived trapped states. The methodology presented here provides guidelines for imaging of charges trapped in the crystal lattice using noncontact AFM.

Inverse beta decay (IBD), $\overlineν_e p \to e^+ n \left( γ\right)$, is the main detection channel for reactor and supernova antineutrinos. To provide precise IBD cross sections at antineutrino energies $E_{\overlineν_e} \gtrsim 10~\mathrm{MeV}$, we evaluate radiative corrections from virtual pions within the framework of heavy baryon chiral perturbation theory. At leading order, only the pion isospin-splitting contributions are not suppressed by the electron mass. At next-to-leading order, besides recoil effects, only the Wilson coefficient $c_4$ contributes to the kinematic dependence. However, its precise value is not relevant for IBD at relatively low energies since all next-to-leading order radiative corrections are relatively small. We find the kinematic dependence of the pion-induced QED radiative corrections at the level and below the uncertainty from the momentum dependence of the nucleon form factors. Our results enable sub-permille theoretical precision of charged-current elastic (anti)neutrino-nucleon scattering at antineutrino energies $E_{\overlineν_e} \gtrsim 10~\mathrm{MeV}$.

We propose alterelectricity, an electrical analogue of altermagnetism, in which two switchable states possess alternating band structures. Such alterelectric states arise when a switchable sublattice-selective structural change connects two configurations related by a non-inversion symmetry. Using an anisotropic Lieb-lattice model, we establish a general symmetry framework for identifying alterelectricity. We further identify two material realizations of alterelectricity: (i) interlayer sliding in bilayers, as exemplified by tetragonal Ag2N and hexagonal FeHfI6; and (ii) ferroelectrically switchable Ti-adsorbed SnP2S6. We also propose an alterelectric tunnel junction that exploits switchable anisotropic Fermi surfaces to achieve a sizable tunneling electroresistance of 120%. This work establishes the foundational concept of alterelectricity and expands the material landscape of ferroic electronics.

The photon density of states (pDOS) governs fundamental light matter interactions and is a critical parameter for designing next generation light driven technologies such as photocatalysis and solar energy harvesting. Achieving a target pDOS in 3D nanoarchitected structures remains challenging due to the nonlinear and non unique relationship between geometry and spectral response. Here, we present an end to end inverse design framework for tailoring the pDOS of 3D photonic metamaterials fabricated via the scalable nanofabrication approach of metasurface-based holographic lithography. A data driven forward surrogate model is constructed to predict frequency resolved pDOS spectra from metasurface diffraction parameters and lithographic thresholds. Inverse design is performed using a conditional generative adversarial network (cGAN) that generates candidate metasurface diffraction parameters for target pDOS features. 3D structures featuring high local pDOS were obtained across a broad normalized frequency range and consistently outperformed those in the original dataset. Structural analysis revealed that these high pDOS architectures fall into two predominant structural categories with similar rotational symmetry characteristics. Our work establishes the first inverse design strategy for 3D photonic metamaterials fabricated via holographic lithography.

Hydrogen technology is set to be a key energy alternative for mitigating pollution and reducing CO$_2$ emissions. However, the current storage mechanism of hydrogen molecules in carbon fibre tanks detracts from the fuel economy of hydrogen in mobile applications, necessitating the development of alternative storage mechanisms. Adsorbing hydrogen in its molecular form (H$_2$) at typical operating conditions of proton exchange membranes can potentially meet storage requirements. However, H$_2$ is the smallest molecule with only two electrons and therefore it has very limited propensity to physisorb in a material within the binding energy window of $-0.2$ to $-0.4$ eV that is suitable for storage. Calcium atom decorators on graphene have previously shown promise for tunable H$_2$ binding, but the system is thermodynamically unstable toward the formation of calcium hydride. Moreover, the absolute adsorption of H$_2$ is challenging to predict accurately and is typically overestimated with van der Waals inclusive density functional approximations. In this work, we perform state-of-the-art fixed-node diffusion Monte Carlo alongside a selection of density functional approximations for two strategies of anchoring Ca: (i) Ca on boron doped graphene and (ii) Ca inside carbon nanotubes. We predict reliable Ca and H$_2$ binding energies, and establish that Ca is anchored inside carbon nanotubes and on boron doped graphene, while boosting the H$_2$ adsorption energy. Importantly, the H$_2$ adsorption energy is found to be improved by the anchoring strategies, with the energy inside a Ca decorated carbon nanotube reaching the viable storage window. The reference DMC binding energies provide much-needed benchmarks for developing data-driven methods and guiding experiment in the systematic design of hydrogen storage materials.

Given two hypergraphs $G$ and $H$, the weak saturation number $\operatorname{\mathrm{wsat}}(G,H)$ is the minimum number of edges in a spanning subhypergraph $F$ of $G$ such that the missing edges of $F$ can be added one at a time so that each added edge creates a copy of $H$. In this work, we determine weak saturation numbers for the case when $G$ and $H$ are tensor product of cliques, generalizing a result of Moshkovitz and Shapira (Journal of Combinatorial Theory, Series B, 2015), who found the exact values of $\operatorname{\mathrm{wsat}}(K^d_{n_1,\ldots,n_d},\ K^d_{r_1,\ldots,r_d})$. The proof also yields results for colored weak saturation numbers $\operatorname{\mathrm{c-wsat}}(G,H)$ of colored hypergraphs $G$ and $H$, where the colorings of the copies of $H$ must be compatible with the coloring of $G$. We determine these numbers when $G$ and $H$ are unions of tensor product of cliques, generalizing a result of Bulavka, Tancer, and Tyomkyn (Combinatorica, 2023), who determined $\operatorname{\mathrm{c-wsat}}(K^q_{n_1,\ldots,n_d}, K^q_{r_1,\ldots,r_d})$. Moreover, our proof allows us to generalize a result of Balogh, Bollobás, Morris, and Riordan (Journal of Combinatorial Theory, Series A, 2012) by determining colored weak saturation numbers $\operatorname{\mathrm{c-wsat}}(K^d_{n_1,\ldots,n_d},\{K^d_{r_1,\ldots,r_d}\}_{\mathbf{r}\in \mathcal{R}})$ for an arbitrary family $\mathcal{R}$. The quantity $\operatorname{\mathrm{c-wsat}}(G,\mathcal{H})$ extends colored weak saturation by allowing, at each step, the creation of a colored copy of any hypergraph in the fixed family of hypergraphs $\mathcal{H}$.

We demonstrate the experimental realization of two-dimensional, continuous variable (CV) cluster states between 191 microwave frequency modes. This result is obtained by exposing vacuum fluctuations to the input of a Josephson Parametric Amplifier, parametrically pumped by a sum of coherent tones around twice its resonant frequency. By carefully tuning pump frequencies, amplitudes, and phases we engineer the interference between mixing products and realize honeycomb and square lattice CV cluster states with three and four pump tones respectively. We prove the presence of the cluster states with a suitable nullifier test, reaching up to $-1.2$ dB of squeezing of the cluster state's nullifiers. We study hidden entanglement (HE) and show no hidden entanglement up to $\sim -1$ dB of squeezing and negligible HE at optimal squeezing.