We decompose the superfluid weight D_s of magic-angle twisted bilayer graphene (MATBG) into conventional (band-velocity) and geometric (interband-coherence) contributions using a band-basis current operator splitting applied to the Bistritzer-MacDonald continuum model. In the flat-band subspace, quantum geometry accounts for 22-26% of D_s at charge neutrality depending on pairing symmetry, with cross terms vanishing to machine precision. Including remote bands raises the geometric fraction to ~55-58%, while D_s^conv converges to within 2% -- demonstrating that remote bands contribute exclusively through interband coherence. The geometric fraction peaks at ~27-33% near the nu = +/- 2 fillings where superconductivity is strongest, and is insensitive to gap magnitude in the experimentally relevant range.

Non-Maxwellian electron velocity distributions (eVDFs) are ubiquitous in collisionless plasmas. For example, various types of non-Maxwellian eVDFs exist in magnetic reconnection jets in the Earth's magnetotail. At thermal energies, eVDF can be flat-topped due to electron trapping associated with magnetic reconnection. However, the occurrence of such eVDFs in magnetotail reconnection remains largely unconstrained. Here, we statistically investigate flat-top eVDFs in fast plasma jets in the magnetotail using a new method for classifying eVDFs. We show that only $\sim 7\%$ of the eVDFs in the jets are flat-tops. Nevertheless, we find that most jets exhibit flat-top eVDFs, indicating that this signature of parallel acceleration and electron streaming is characteristic of the jets. We find that these flat-top eVDFs are localized within an ion-inertial-length-scale region near the edges of the current sheet and close to the reconnection region. Our results highlight the importance of flat-top eVDFs in non-local thermodynamic equilibrium collisionless plasmas.

Realistic modeling of scattering from curved metallic bodies - such as vehicles and roadside structures - is essential for cellular and vehicular channel modeling as well as radar applications. A practical approach is to approximate curved surfaces with planar facets and apply ray-tracing with diffraction methods; however, accuracy depends critically on both geometric discretization and diffraction modeling. This work investigates ray-tracing-based modeling of near-field scattering from curved bodies, including the forward (shadow) region, using the Uniform Theory of Diffraction (UTD), extended with vertex diffraction and double-bounce interactions. A discretization strategy linking facet size to local curvature and wavelength is proposed to balance geometric fidelity, computational accuracy and efficiency. Validation is performed against analytical solutions and full-wave simulations for canonical geometries (sphere and circular cylinder), as well as a realistic vehicle model to demonstrate the method's practical relevance. Results show that appropriate discretization combined with extended diffraction modeling significantly improves scattering prediction from curved bodies, providing a computationally efficient framework for vehicular propagation and integrated sensing and communication (ISAC) channel modeling.

Direct Air Carbon Capture and Storage (DACCS) can mitigate hard-to-abate emissions, e.g. from transport or industry. However, there is a wide variety of cost estimates for DACCS, driven, to a significant extent, by differences in electricity cost. At the same time, there is a notable gap in research that integrates direct air capturing systems into long-term energy system models. We separate direct air capturing, carbon transport, and carbon storage and integrate them into a European capacity expansion model for a fully decarbonised electricity system in 2050. We explore how two dimensions affect the total system costs of DACCS. The first dimension is the availability of CO2 storage locations: In one analysis, storage locations are restricted to offshore storage locations in the North Sea only, i.e. depleted natural gas fields. The alternative analysis comprises suitable storage locations distributed across Europe, including onshore. We find that limiting CO2 storage to North Sea sites increases overall capture costs by approximately 10 %. The second dimension is whether DACCS is analysed as stand-alone or integrated into the electricity system. We differentiate between three alternatives: fully isolated, fully integrated, and retrospectively added to an existing system. We find that neglecting system integration - i.e. treating direct air capture system as a stand-alone technology - increases capture costs by up to 30 %.

Entanglement is a key quantity for characterizing quantum correlations in particle scattering processes, but its direct evaluation is computationally demanding on quantum hardware. In this work, we investigate whether fermion density profiles, which are easier to access, can serve as proxies for entanglement by framing the problem as a classification task across multiple entanglement thresholds. Using the fermion scattering in the Thirring model as a test bed, we compare Quantum Convolutional Neural Networks (QCNNs) with classical CNNs of comparable parameter counts, and find that QCNNs achieve consistently competitive or superior accuracy with faster convergence and lower variance. Notably, we observe that increasing the model size does not improve the performance within the architectures studied here, and larger models appear to be more sensitive to the choice of encoding. Instead, a compact 4-qubits QCNN provides the best results, suggesting the importance of trainability and encoding choices over model scaling. These findings demonstrate the potential of quantum and quantum-inspired machine learning models for extracting nontrivial quantum information from accessible observables, with implications for high-energy physics and quantum many-body systems.

The classical tail dependence coefficient (TDC) may fail to capture non-exchangeable features of tail dependence due to its restrictive focus on the diagonal of the underlying copula. To address this limitation, the framework of path-based maximal tail dependence has been proposed, where a path of maximal dependence is derived to capture the most pronounced feature of dependence over all possible paths, and the path-based maximal TDC serves as a natural analogue of the classical TDC along this path. However, the theoretical foundations of path-based tail analyses, in particular the existence and analytical tractability, have remained limited. This paper addresses this issue in several ways. First, we prove the existence of a path of maximal dependence and the path-based maximal TDC when the underlying copula admits a non-degenerate tail copula. Second, we obtain an explicit characterization of the maximal TDC in terms of the tail copula. Third, we show that the first-order asymptotics of a path of maximal dependence is characterized by a one-dimensional optimization involving the tail copula. These results improve the analytical and computational tractability of path-based tail analyses. As an application, we derive the asymptotic behavior of a path of maximal dependence for the bivariate t-copula and the survival Marshall--Olkin copula.

We present a formalization in Lean of the core interior De Giorgi--Nash--Moser theory for uniformly elliptic divergence-form equations with bounded measurable coefficients. The formalized results include local boundedness of weak subsolutions, the weak Harnack inequality for positive weak supersolutions, Moser's Harnack inequality for positive weak solutions, and interior Hölder regularity. This is, to our knowledge, the first machine-checked formalization of a major theorem in modern PDE theory. The development also required substantial new infrastructure for Sobolev spaces on bounded domains, weak solutions of elliptic equations, and quantitative regularity estimates. More broadly, it suggests that large-scale autoformalization of hard analysis in Lean is now within reach.

Graphics Processing Units (GPUs) excel at regular data-parallel workloads where massive hardware parallelism can be readily exploited. In contrast, many important irregular applications are naturally expressed as task parallelism with a fork-join control structure. While CPU runtimes for fork-join task parallelism are mature, it remains challenging to efficiently support it on GPUs. We propose GTaP, a GPU-resident runtime that supports fork-join task parallelism. GTaP is based on the persistent kernel model, and supports two worker granularities: thread blocks and individual threads. To realize fork-join on GPUs, GTaP represents joins as continuations and executes each task as a state machine that can be split into multiple execution segments. We also extend Clang's frontend with a pragma-based programming model that enables programmers to express fork-join without exposing low-level mechanisms. GTaP employs work stealing for load balancing, providing better scalability than a global-queue approach. For thread-level workers, we further introduce Execution-Path-Aware Queueing (EPAQ), which allows programmers to partition task queues using user-defined criteria, reducing warp divergence caused by mixing heterogeneous control flows within a warp. Across representative irregular applications, GTaP outperforms OpenMP task-parallel execution on a 72-core CPU in many cases, especially for large problem sizes with compute-intensive tasks. We also show that GTaP's design choices outperform naive GPU alternatives. The benefit of EPAQ is workload-dependent: it can improve performance for some benchmarks while having little effect on others; on Fibonacci, EPAQ achieves up to a 1.8$\times$ speedup.

In this paper, we show that if the holomorphic tangent bundle $TX$ of a compact Kähler manifold $X$ is uniformly weakly RC-positive, then $X$ is projective and rationally connected. This result is previously established by Xiaokui Yang under the stronger assumption that $TX$ is uniformly RC-positive. The result we obtain is, in fact, more general. If a holomorphic vector bundle $E$ is uniformly weakly RC-positive, then $E$ admits a Hermitian metric whose mean curvature is positive. A quasi-positive version is also proved in this paper.

We establish the dynamical instability of a static, spherically symmetric, and infinitesimally thin shell in general relativity. The shell is made up of a perfect fluid with a barotropic equation of state, and it produces a Schwarzschild spacetime in its exterior and a Minkowski spacetime in its interior. We reveal the existence of two modes with a purely imaginary frequency, one negative (which describes stable oscillations), the other positive (which describes an exponential growth); these modes occur for all sampled values of the shell's compactness and adiabatic index, and all sampled values of the multipolar order $\ell \geq 2$, in the even-parity sector of the perturbation. All other quasinormal modes describe damped oscillations. This study complements a recent analysis by Yang, Bonga, and Pen, which also concluded in a dynamical instability, but was limited by an eikonal approximation to small angular scales ($\ell \gg 1$); our treatment applies to all angular scales. The eigenvalue problem for the mode frequencies is formulated by introducing a perturbation of Minkowski spacetime, a perturbation of Schwarzschild spacetime, and a perturbation of the shell matter. The metric perturbations are governed by the Einstein field equations, and they are matched across the shell with the help of Israel's junction conditions. The matter perturbation is governed by the equations of fluid mechanics, and it produces a source term in the junction conditions. All calculations are carried out in full general relativity, but we also examine a nonrelativistic formulation of the problem; we show that a Newtonian shell also is necessarily unstable to a time-dependent perturbation. Our conclusion suggests that a compact object that features a thin shell at its surface will be dynamically unstable; this makes it nonviable as a model of black-hole mimicker.

This paper addresses the problem of tracking control for robotic vehicles equipped with pivoted unidirectional actuators. Starting from a baseline robust controller that assumes unconstrained inputs, we redesign the control law to be compatible with the pivoted actuator. This is accomplished by driving the output of the pivoted actuator to a ball centered at the target input value. The guarantees for the baseline controller are recovered in a practical sense. The theory is illustrated with simulation examples.

Incentive design constitutes a foundational paradigm for influencing the behavior of strategic agents, wherein a system planner (principal) publicly commits to an incentive mechanism designed to align individual objectives with collective social welfare. This paper introduces the Regret-Minimizing Adaptive Incentive Design (RAID) problem, which aims to synthesize incentive laws under information asymmetry and achieve asymptotically minimal regret compared to an oracle with full information. To this end, we develop the RAID algorithm, which employs a switching policy alternating between probing (exploration) and estimate-based incentivization (exploitation). The associated type estimator relies only on a weaker excitation condition required for strong consistency in least squares estimation, substantially relaxing the persistence-of-excitation assumptions previously used in adaptive incentive design. In addition, we establish the strong consistency of the proposed type estimator and prove that the incentive obtained asymptotically minimizes the planner's average regret almost surely. Numerical experiments illustrate the convergence rate of the proposed methodology.

We analyze a weighted convolution of Catalan numbers $$ \sum_{k=0}^{n} \binom{2k}{k}\binom{2(n-k)}{n-k} a^k = \sum_{k=0}^{n} (k+1)(n-k+1) C_k C_{n-k} a^k, $$ emphasizing its combinatorial, analytic, and probabilistic aspects. We derive a compact closed form in terms of the Gauss hypergeometric function ${}_2F_1(-n,1/2;1;1-a)$, valid for all complex values of the parameter $a$. The sum admits a natural interpretation in terms of return probabilities of independent simple random walks, linking weighted convolutions of central binomial coefficients to classical probability theory. Furthermore, a refinement via Narayana numbers highlights the contribution of peak distributions in pairs of Dyck paths, providing a finer combinatorial perspective. An integral representation is also proposed, suggesting a connection with orthogonal polynomials and spectral measures. Our approach illustrates how analytic and probabilistic techniques complement combinatorial reasoning in evaluating complex sums.

In this paper, we study the computational question of whether the Steklov spectrum of smooth simply connected planar domains can be approximated accurately by a boundary-only formulation based on harmonic conjugation. For the unit disk, the Dirichlet-to-Neumann operator can be written explicitly in terms of the classical conjugation operator. We show how this viewpoint extends to general bounded and unbounded simply connected domains through the generalized conjugation operator defined through the boundary integral equation with the generalized Neumann kernel. Combined with Fourier differentiation on an equidistant boundary grid, this leads to a dense algebraic eigenvalue problem for the boundary traces of Steklov eigenfunctions. The resulting method uses only boundary data, treats interior and exterior problems in a unified way, and reconstructs eigenfunctions in the domain by harmonic extension. Numerical experiments on benchmark domains and on parameter-dependent smooth families, including ellipses and star-like curves, show high accuracy for smooth boundaries and illustrate how the Steklov spectrum changes with geometry.

In ecological studies niche overlap is often used to quantify species interaction and dynamics. This paper develops a robust, nonparametric statistical framework for quantifying and analyzing multivariate niche overlap. Parametric methods are often constrained by restrictive assumptions and tend to underperform in complex multivariate settings. We introduce a nonparametric overlap index and propose estimators for it. Further, we investigate asymptotic properties of the estimators. We also propose bootstrap-based inference procedures that enable statistical testing and simultaneous confidence intervals in small sample settings. Extensive numerical examples demonstrate that our proposed methods maintain correct size and exhibit robust power across various scenarios. We illustrate the practical utility of our methodology using stable isotope measurements from multiple fish species and provide distinct ecological insights regarding species niche differentiation.

Expectation values of measurement operators, interpreted as measurement probabilities, arise frequently throughout quantum algorithms. When quantum states are randomly distributed, their expectation values are also randomly distributed. In this work, with the goal of understanding non-unitary dynamics, we generalize previous derivations for distributions of expectation values (Campos Venuti and Zanardi, Physics Letters A (377), 2013) to the case of sets of measurement operators and random mixed quantum states within variable sized environments. Using combinatorics approaches, we derive expressions for their moments. We proceed to construct empirical distributions of simulated Haar random brickwork quantum circuits with local depolarizing noise, and compare their form to a proposed effective global-depolarizing-like model with variable effective noise scales and environment dimensions. The fitted effective distributions reproduce peak behaviour across circuit depths, noise scales, and system sizes, while deviations in the distribution tails arise from local noise effects. The fit effective model parameters are also shown to vary smoothly and consistently with circuit depth and noise scale. Finally, sets of non-symmetric measurement operators are shown to exhibit distinct multi-modal distributions relative to uni-modal distributions for symmetric measurement operators, opening up questions about their simulability.

We apply artificial neural networks to the holographic inverse problem, reconstructing bulk geometry from boundary entanglement entropy by using the Ryu--Takayanagi area functional as a differentiable loss. Validated on the AdS-Schwarzschild background, this approach recovers the blackening factor to 1.7% accuracy. For finite-density backgrounds like the Gubser--Rocha model, we demonstrate that strip entanglement entropy determines only the spatial metric. We resolve this exact one-function degeneracy by incorporating holographic Wilson loop data, which couples to the timelike metric. We present a semi-analytical inversion combining Bilson's and Hashimoto's formulas, alongside a general three-network variational method minimizing the combined area and Nambu--Goto actions. The neural network achieves sub-0.2% accuracy for both metric functions without closed-form derivative relations, establishing a flexible framework for integrating multiple holographic observables.

Refractory compositionally complex alloys (RCCAs) are considered the next generation high-temperature materials. However, their high-dimensional composition spaces are too large to explore by traditional density functional theory or experimental means, making new RCCA discovery slow and cumbersome. This work has addressed these challenges with an integrated composition design framework that can efficiently and exhaustively explore the relationship between the compositions and two fundamental aspects: 1) the phase stability, including the target body-centered cubic (BCC) phase and its competing phases (hexagonal closed-pack (HCP) structures, Laves and B2 intermetallic phases), and 2) the mechanical properties. This framework is demonstrated with RCCAs within nine refractory metals (Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, and W). Theory-guided machine learning (ML) models were employed to find the composition-mechanical property relationship of RCCAs, where the established theory is used to supplement the yield strength data at ultra-high temperature, and a forward sequential feature selection (SFS) is used to determine feature selection. The resulting ML model for temperature-dependent yield strength was found to have an R_squared value of 0.98 over the entire temperature range (from 0 to 2000 K). The impact of each constituent element on the six key properties is evaluated. The addition of Nb tends to stabilize the BCC phase and the addition of Ti improves the ductility of RCCAs. Combined with all methods involved in this framework, the on-demand designer allows the alloy designers to have all properties for any RCCA compositions and narrow down the composition space by applying custom screening criteria. The output from the predictor and screener provides valuable guidance for our experimental study of RCCAs and accelerates the pace of materials discovery.

We provide a practical relaxation of Willems' fundamental lemma for discrete-time linear time-invariant (single-input-single-output) systems. Instead of maintaining conventional Willems' persistency of excitation condition in the behavioral theory, we reformulate the problem in terms of signal generators, hence going back to the dynamical systems theory. We discuss the relationship between the persistency of excitation order and the dimension of the signal generator. Furthermore, we identify a necessary and sufficient condition on the signal generator that can generate informative input--output data for almost all systems and initial conditions. This even includes inputs outside the class originally suggested by Willems' fundamental lemma, for example, sinusoidal sequences with fewer frequencies. Finally, the signal generator perspective allows a natural extension to continuous-time systems.

We introduce a framework for distributed quantum inference under communication constraints. In our model, $m$ distributed nodes each receive one copy of an unknown $d$-dimensional quantum state $ρ$, before communicating via a constrained one-way communication channel with a central node, which aims to infer some property of $ρ$. This framework generalizes the classical distributed inference framework introduced by Acharya, Canonne, and Tyagi [COLT2019], by allowing quantum resources such as quantum communication and shared entanglement. Within this setting, we focus on the fundamental problem of quantum state certification: Given a complete description of some state $σ$, decide whether $ρ=σ$ or $\|ρ-σ\|_1\geq ε$. Additionally, we focus on the case of limited quantum communication between distributed nodes and the central node. We show that when each communication channel is limited to only $n_q\leq \log d$ qubits, then the sample complexity of distributed state certification is $\mathcal{O}(\frac{d^2}{2^{n_q}ε^2})$ when public randomness is available to all nodes. Moreover, under the assumption that the channels used by the distributed nodes are mixedness-preserving, we prove a matching lower bound. We further demonstrate that shared randomness is necessary to achieve the above complexity, by proving an $Ω(\frac{d^3}{4^{n_q} ε^2})$ lower bound in the private-coin setting under the same assumption as above. Our lower bounds leverage a recently introduced quantum analogue of the celebrated Ingster-Suslina method and generalize arguments from the classical setting. Together, our work provides the first characterization of distributed quantum state certification in the regime of limited quantum communication and establishes a general framework for distributed quantum inference with communication constraints.

Speckle metrology exploits the high sensitivity of scattered fields to parameters of interest, yet this also leaves measurements vulnerable to unintended perturbations. Here we employ transmission matrix formalism to engineer light fields that produce speckle correlation "revivals", selectively reducing response to a chosen parameter. In a multimode fiber scattering system, we suppress bending-induced correlation changes over a limited curvature range without strongly degrading wavelength sensitivity, opening a route to tailored, parameter-specific sensitivities of speckle-based measurements.

We determine the irreducible ${\rm SL}(2,\mathbb{C})$-character variety of the 3-chain link exterior which is called the `magic $3$-manifold', and deduce a formula for the twisted Alexander polynomial associated to each ${\rm SL}(2,\mathbb{C})$-representation.

We report on the lattice location of the short-lived ion implanted nuclear probe 6He (t1/2=807 ms) in diamond, which was performed using the beta emission channeling method at CERN's ISOLDE facility. 6He was implanted with 30 keV into a single-crystalline artificial diamond sample kept at a temperature ranging from 30 deg C up to 800 deg C. By means of comparing the measured emission channeling patterns along different crystallographic directions with simulated yields for a variety of possible sites, we conclude that the implanted 6He occupies tetrahedral (T) interstitial sites, in agreement with theoretical predictions that T sites should be the preferred position of He in diamond. Implantation at 800 deg C resulted in a drop in the tetrahedral interstitial fraction by 20%, which we interpret as the onset of diffusion, 6He thus being able to change to lattice sites of low crystallographic symmetry, or reach the surface of the sample or escape to the bulk during its lifetime. We estimate the activation energy for interstitial migration of He to be around 1.63-2.89 eV, which agrees with theoretical predictions of 1.41 eV, 1.97 eV, 2.35 eV and 2.36 eV from the literature. Activation energies around 2 eV would mean that simple interstitial He cannot be stable in diamond on geological time scales, thus to remain inside, it should be bound to some defect in the material or exist in another form such as within inclusions of other minerals or liquids, or possibly small He bubbles.

Previously, we proposed an infrared vacuum wave functional for $SU(N)$ Yang-Mills theory peaked at thin center vortices and showed that it yields an area law for the Wilson loop. In this work, we use this wave functional to calculate the spatial 't Hooft loop, for which we find a perimeter law, in accordance with 't Hooft's criterion for confinement.

Let c denote the cardinality of the continuum. Let L denote the family of all Hausdorff topologies on the real line coarser than the natural topology. We construct 2^c pairwise non-homeomorphic completely normal topologies in L among which 2^c are Baire and 2^c are of first category. We also construct c pairwise non-homeomorphic completely metrizable topologies in L. Furthermore, we investigate complete lattices of topologies in L and construct extremely long chains of homeomorphic topologies in L.

The goal of ensemble forecasting is to maximise sharpness subject to reliability. Marginal reliability means that, over all cases, the ensemble is statistically consistent with reality: the ensemble mean is unbiased, the expected ensemble variance equals the expected mean-squared error of the ensemble mean, and the variance of the ensemble members matches the variance of the truth. Equivalently, forecasts that assign probability $p$ to an event verify with relative frequency $p$. However, climatological consistency is not sufficient for users acting on individual forecasts. A natural extension is to assess reliability conditional on the forecast itself, by examining whether, on average, larger ensemble means imply larger observed values, larger spreads imply larger forecast errors, or higher probabilities imply higher event frequencies. This motivates conditional reliability diagnostics such as reliability diagrams and spread-error relationships. Here we show that conditional reliability diagnostics are systematically biased for finite ensemble sizes. We present a unified framework for slope attenuation caused by finite-ensemble sampling noise, which affects conditional diagnostics for ensemble means, spreads, and probabilities. Using synthetic forecasts that are perfectly reliable by construction, we isolate finite-ensemble effects. We derive analytical expressions for the expected attenuation and propose practical estimators computable directly from ensemble data. The framework is illustrated using 2-metre temperature sub-seasonal ensemble forecasts from ECMWF, where finite-ensemble slope attenuation substantially affects the spread-error relationship and tercile-based reliability diagrams. These results demonstrate that attenuated conditional slopes should not be interpreted as evidence of forecast deficiencies unless finite-ensemble effects are explicitly taken into account.

We calculate the effects of polariton bi- and multistability in a semiconductor Bragg microcavity with multiple quantum wells and a chiral photonic crystal on the upper mirror for resonant coherent pumping normal to the structure. Even if the system is not optimized for obtaining photoluminescence with a high degree of circular polarization in the spontaneous mode, it is shown that linear-polarized pumping can cause nonlinear switching to states with a degree of circular polarization of polaritons up to 90%. Calculations were performed in both the mean-field and self-consistent approximations, accounting for the difference in exciton density among the microcavity's quantum wells.

We study pathwise $p$-th variation of continuous paths on a compact interval along a fixed partition sequence. Although the class of continuous paths with finite $p$-th variation is generally not linear, we develop a coefficient-based approach via Faber-Schauder expansions that, for any $p>1$, enables the construction of paths with prescribed $p$-th variation while preserving useful linear structures and Hölder regularity. We first construct continuous paths with linear $p$-th variation from suitable conditions on their Faber-Schauder coefficients. We then prescribe nonlinear $p$-th variation through a multiplicative transformation and show that, whenever nonempty, the class of Hölder continuous paths with a given $p$-th variation is dense in $C([0,1])$. Next, we introduce a transport procedure that turns a Banach subspace of continuous functions into a Banach subspace of paths with explicitly controlled $p$-th variation. We also prove stability of the associated pathwise Föllmer-Itô map on these transported subspaces. Finally, via time-changes, we show that this constructive framework extends from $q$-adic partition sequences to broader classes of dense $q$-refining partition sequences.

We study relaxation-based approaches for conserving mass and energy in the numerical solution of Schrödinger-Poisson (SP) type systems. Relaxation-based methods offer a general approach that can be applied as post-time step processing to achieve conservation with any time-stepping scheme. Here we study two types of relaxation techniques applied to implicit-explicit Runge-Kutta schemes, with Fourier collocation in space. We also study SP equations with time-varying coefficients (which appear naturally in cosmology) where energy is not conserved but satisfies a balance equation. We show that the fully-discrete system conserves both mass and energy (or satisfies the balance equation in case of time-varying coefficients), up to rounding errors. The effectiveness of these methods is demonstrated via numerical examples, including a three-dimensional cosmological simulation.

The three dimensional cubic defocusing nonlinear wave equation is known to be ill-posed for general low regularity initial data. However, well-posedness can be recovered globally in time on a probabilistic level when considering random Gaussian initial data approximated by truncation of Fourier modes. These fine behaviors of nonlinear wave equations have not yet been observed numerically . In this article we perform numerical simulations of the one dimensional fractional cubic defocusing wave equation in a periodic setting. This allows us to explore energy subcritical and supercritial regimes. Our numerical results suggest that both norm inflation and probabilistic well-posedness can be observed numerically in energy sub-critical and super-critical regimes.