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Massive stars are a major source of chemical elements in the cosmos, ejecting freshly produced nuclei through winds and core-collapse supernova explosions into the interstellar medium. Among the material ejected, long lived radioisotopes, such as 60Fe (iron) and 26Al (aluminum), offer unique signs of active nucleosynthesis in our galaxy. There is a long-standing discrepancy between the observed 60Fe/26Al ratio by {\gamma}-ray telescopes and predictions from supernova models. This discrepancy has been attributed to uncertainties in the nuclear reaction networks producing 60Fe, and one reaction in particular, the neutron-capture on 59Fe. Here we present experimental results that provide a strong constraint on this reaction. We use these results to show that the production of 60Fe in massive stars is higher than previously thought, further increasing the discrepancy between observed and predicted 60Fe/26Al ratios. The persisting discrepancy can therefore not be attributed to nuclear uncertainties, and points to issues in massive-star models.

Diffusive transport processes in magnetized plasmas are highly anisotropic, with fast parallel transport along the magnetic field lines sometimes faster than perpendicular transport by orders of magnitude. This constitutes a major challenge for describing non-grid-aligned magnetic structures in Eulerian (grid-based) simulations. The present paper describes and validates a new method for parallel diffusion in magnetized plasmas based on the anti-symmetry representation [Halpern and Waltz, Phys. Plasmas 25, 060703 (2018)]. In the anti-symmetry formalism, diffusion manifests as a flow operator involving the logarithmic derivative of the transported quantity. Qualitative plane wave analysis shows that the new operator naturally yields better discrete spectral resolution compared to its conventional counterpart. Numerical simulations comparing the new method against existing finite difference methods are carried out, showing significant improvement. In particular, we find that combining anti-symmetry with finite differences in diagonally staggered grids essentially eliminates the so-called "artificial numerical diffusion" that affects conventional finite difference and finite volume methods., Comment: Submitted to Computer Physics Communication

Utilising classical results on the structure of Hopf algebras, we develop a novel approach for the construction of cubature formulae on Wiener space based on unshuffle expansions. We demonstrate the effectiveness of this approach by constructing the first explicit degree-7 cubature formula on $d$-dimensional Wiener space with drift, in the sense of Lyons and Victoir. The support of our degree-7 formula is significantly smaller than that of currently implemented or proposed constructions.

We present photo-electron paramagnetic resonance (EPR) measurements and first-principles calculations that indicate germanium (Ge) is a DX-center in AlGaN. Our photo-EPR measurements on Ge-doped AlGaN samples show no EPR spectra in the dark, while persistent EPR spectra is observed upon photoexcitation with photon energies greater than ~1.3 eV. Thermally annealing the samples decreased the EPR signal, with the critical temperature to quench the EPR signal being larger in the lower Al-content sample. Using detailed first-principles calculations of Ge in AlGaN, we show all of these observations can be explained by accounting for the DX configuration of Ge in AlGaN., Comment: 6 pages, 4 figures

This paper presents a novel distributed vehicle platooning control and coordination strategy. We propose a distributed predecessor-follower CACC scheme that allows to choose an arbitrarily small inter-vehicle distance while guaranteeing no rear-end collisions occur, even in the presence of undetected cyber-attacks on the communication channels such as false data injection. The safety guarantees of the CACC policy are derived by combing a sensor-based ACC policy that explicitly accounts for actuator saturation, and a communication-based predictive term that has state-dependent limits on its control authority, thus containing the effects of an unreliable communication channel. An undetected attack may still however be able to degrade platooning performance. To mitigate it, we propose a tailored Kalman observer-based attack detection algorithm that initially triggers a switch from the CACC policy to the ACC policy. Subsequently, by relying on a high-level coordinator, our strategy allows to isolate a compromised vehicle from the platoon formation by reconfiguring the platoon topology itself. The coordinator can also handle merging and splitting requests. We compare our algorithm in simulation against a state of the art distributed MPC scheme and we extensively test our full method in practice on a real system, a team of scaled-down car-like robots. Furthermore, we share the code to run both the simulations and robotic experiments.

The development of programmable quantum devices can be measured by the complexity of manybody states that they are able to prepare. Among the most significant are topologically ordered states of matter, which enable robust quantum information storage and processing. While topological orders are more readily accessible with qudits, experimental realisations have thus far been limited to lattice models of qubits. Here, we prepare a ground state of the Z3 toric code state on 24 qutrits in a trapped ion quantum processor with fidelity per qutrit exceeding 96.5(3)%. We manipulate two types of defects which go beyond the conventional qubit toric code: a parafermion, and its bound state which is related to charge conjugation symmetry. We further demonstrate defect fusion and the transfer of entanglement between anyons and defects, which we use to control topological qutrits. Our work opens up the space of long-range entangled states with qudit degrees of freedom for use in quantum simulation and universal error-correcting codes., Comment: 8+20 pages, 15 figures

Creating and manipulating anyons and symmetry defects in topological phases, especially those with a non-Abelian character, constitutes a primitive for topological quantum computation. We provide a physical protocol for implementing the ribbon operators of non-Abelian anyons and symmetry defects. We utilize dualities, in particular the Kramers-Wannier or gauging map, which have previously been used to construct topologically ordered ground states by relating them to simpler states. In this work, ribbon operators are implemented by applying a gauging procedure to a lower-dimensional region of such states. This protocol uses sequential unitary circuits or, in certain cases, constant-depth adaptive circuits. We showcase this for anyons and defects in the $\mathbb{Z}_3$ toric code and $S_3$ quantum double. The general applicability of our method is demonstrated by deriving unitary expressions for ribbon operators of various (twisted) quantum doubles., Comment: 7+18 pages, 3+8 figures

Understanding planetary habitability is one of the major challenges of the current scientific era, particularly given the discovery of a large and diverse terrestrial exoplanet population. Discerning the primary factors that contribute to planetary habitability may be extracted through a detailed examination of the terrestrial planets within the Solar System, most particularly Venus, Earth, and Mars, and the evolution of their interiors and atmospheres through time. Here, we provide a detailed description of the fundamental properties of these three planets, the effects of solar evolution, and the potential contributions of these various aspects toward driving their evolutionary pathways. We argue that evolution of Venus, Earth, and Mars provide essential templates from which a more comprehensive approach toward the study of planetary habitability may be derived., Comment: 45 pages, 3 figures, accepted for publication in "The Archean Earth" (ed.: M. Homan, P. Mason, R. Ernst, T. Lyons, C. Heubeck, D. Papineau, E. Stueeken, R. Mazumder, A. Webb, W. Altermann), Elsevier

The Hadean, once thought to be uninhabitable and tumultuous, has more recently been recontextualized as a clement time in which oceans, land, and life likely appeared on Earth. This non-exhaustive chapter follows multiple threads from planet formation to the origin of life. We place significant emphasis on the solar system context for the Earth, the timing and nature of crustal formation and the evolution of the surface and atmosphere. Several scenarios for prebiotic chemistry are also discussed including atmospheric photochemistry, wet-dry and freeze-thaw cycles, and hydrothermal vent systems. We attempt to draw connections between the large-scale, planetary processes and various origin of life pathways to illustrate possible overlaps and correlations. In detail, we conclude with and discuss the "impact of impacts" to show how asteroid and comet impacts during the Hadean may have affected many of these processes and scenarios, from generating land to altering the chemical composition and oxidation state of the early Earth's atmosphere and surface., Comment: 18 pages, 5 figures, accepted for publication in "The Archean Earth" (ed.: M. Homan, P. Mason, R. Ernst, T. Lyons, C. Heubeck, D. Papineau, E. Stueeken, R. Mazumder, A. Webb, W. Altermann), Elsevier

Many therapies are effective in treating multiple diseases. We present an approach that leverages methods developed in natural language processing and real-world data to prioritize potential, new indications for a mechanism of action (MoA). We specifically use representation learning to generate embeddings of indications and prioritize them based on their proximity to the indications with the strongest available evidence for the MoA. We demonstrate the successful deployment of our approach for anti-IL-17A using embeddings generated with SPPMI and present an evaluation framework to determine the quality of indication finding results and the derived embeddings.

This work presents our team's (SignalSavants) winning contribution to the 2024 George B. Moody PhysioNet Challenge. The Challenge had two goals: reconstruct ECG signals from printouts and classify them for cardiac diseases. Our focus was the first task. Despite many ECGs being digitally recorded today, paper ECGs remain common throughout the world. Digitising them could help build more diverse datasets and enable automated analyses. However, the presence of varying recording standards and poor image quality requires a data-centric approach for developing robust models that can generalise effectively. Our approach combines the creation of a diverse training set, Hough transform to rotate images, a U-Net based segmentation model to identify individual signals, and mask vectorisation to reconstruct the signals. We assessed the performance of our models using the 10-fold stratified cross-validation (CV) split of 21,799 recordings proposed by the PTB-XL dataset. On the digitisation task, our model achieved an average CV signal-to-noise ratio of 17.02 and an official Challenge score of 12.15 on the hidden set, securing first place in the competition. Our study shows the challenges of building robust, generalisable, digitisation approaches. Such models require large amounts of resources (data, time, and computational power) but have great potential in diversifying the data available.

Understanding protein dynamics are essential for deciphering protein functional mechanisms and developing molecular therapies. However, the complex high-dimensional dynamics and interatomic interactions of biological processes pose significant challenge for existing computational techniques. In this paper, we approach this problem for the first time by introducing Deep Signature, a novel computationally tractable framework that characterizes complex dynamics and interatomic interactions based on their evolving trajectories. Specifically, our approach incorporates soft spectral clustering that locally aggregates cooperative dynamics to reduce the size of the system, as well as signature transform that collects iterated integrals to provide a global characterization of the non-smooth interactive dynamics. Theoretical analysis demonstrates that Deep Signature exhibits several desirable properties, including invariance to translation, near invariance to rotation, equivariance to permutation of atomic coordinates, and invariance under time reparameterization. Furthermore, experimental results on three benchmarks of biological processes verify that our approach can achieve superior performance compared to baseline methods., Comment: 17 page, 8 figures

We study lattice polytopes which arise as the convex hull of chip vectors for \textit{self-reachable} chip configurations on a tree $T$. We show that these polytopes always have the integer decomposition property and characterize the vertex sets of these polytopes. Additionally, in the case of self-reachable configurations with the smallest possible number of chips, we show that these polytopes are unimodularly equivalent to a unit cube.

Adjusting the control actions of a wheeled robot to eliminate oscillations and ensure smoother motion is critical in applications requiring accurate and soft movements. Fuzzy controllers enable a robot to operate smoothly while accounting for uncertainties in the system. This work uses fuzzy theories and parallel distributed compensation to establish a robust controller for wheeled mobile robots. The use of fuzzy logic type I and type II controllers are covered in the study, and their performance is compared with a PID controller. Experimental results demonstrate that fuzzy logic type II outperforms type I and the classic controller. Further, we deploy parallel distributed compensation, sector of nonlinearity, and local approximation strategy in our design. These strategies help analyze the stability of each rule of the fuzzy controller separately and map the if-then rules of the fuzzy box into parallel distributed compensation using Linear Matrix Inequalities (LMI) analysis. Also, they help manage the uncertainty flow in the equations that exist in the kinematic model of a robot. Last, we propose a Bezier curve to represent the different pathways for the wheeled mobile robot., Comment: 16 pages, 17 figures

The Fusion Synthesis Engine (FUSE) is a state-of-the-art software suite designed to revolutionize fusion power plant design. FUSE integrates first-principle models, machine learning, and reduced models into a unified framework, enabling comprehensive simulations that go beyond traditional 0D systems studies. FUSE's modular structure supports a hierarchy of model fidelities, from steady-state to time-dependent simulations, allowing for both pre-conceptual design and operational scenario development. This framework accelerates the design process by enabling self-consistent solutions across physics, engineering, and control systems, minimizing the need for iterative expert evaluations. Leveraging modern software practices and parallel computing, FUSE also provides multi-objective optimization, balancing cost, efficiency, and operational constraints. Developed in Julia, FUSE is fully open-source under the Apache 2.0 license, promoting transparency and collaboration within the fusion research community.

In this paper, we explore the notion of a \emph{self-reachable} chip configuration on a simple graph, that is a chip configuration which can be re-obtained from itself after a (nonempty) sequence of vertex firings. In particular, we focus on the case of trees and provide a characterization for such configurations, as well as show that all self-reachable configurations with the same number of chips on a tree are reachable from one another. We conclude with a recursive enumeration formula for the number of self-reachable configurations.

Motivated by the idea of using simple macroscopic examples to illustrate the physics of complex systems, we modify a historic experimental setup in which interacting floating magnets spontaneously self-assemble into ordered clusters. By making the cluster components mechanically stable against their natural tendency to flip and coalesce, we can monitor the torque experienced by individual magnets through the macroscopic tilt angles they acquire when exposed to non-collinear external fields. A mathematical model that reproduces the empirical observations is introduced, enabling us to go beyond the experimental cases considered. The model confirms the existence of alternative orderings as the number of objects increases. Furthermore, a simpler and more mathematically transparent version of our model enables us to establish the conditions under which the cluster structure is stable and when it will collapse. We argue that our simple experimental setup combined with the accompanying models may be useful to describe general features seen in systems composed of mutually repulsive particles in the presence of modulating confining fields., Comment: 14 pages, 8 figures, submitted to PRE

We apply the method of inverse iteration to the Laplace eigenvalue problem with Robin and mixed Dirichlet-Neumann boundary conditions, respectively. For each problem, we prove convergence of the iterates to a non-trivial principal eigenfunction and show that the corresponding Rayleigh quotients converge to the principal eigenvalue. We also propose a related iterative method for an eigenvalue problem arising from a model for optimal insulation and provide some partial results.

This workshop will grow and consolidate a community of interdisciplinary CSCW researchers focusing on the topic of contestable AI. As an outcome of the workshop, we will synthesize the most pressing opportunities and challenges for contestability along AI value chains in the form of a research roadmap. This roadmap will help shape and inspire imminent work in this field. Considering the length and depth of AI value chains, it will especially spur discussions around the contestability of AI systems along various sites of such chains. The workshop will serve as a platform for dialogue and demonstrations of concrete, successful, and unsuccessful examples of AI systems that (could or should) have been contested, to identify requirements, obstacles, and opportunities for designing and deploying contestable AI in various contexts. This will be held primarily as an in-person workshop, with some hybrid accommodation. The day will consist of individual presentations and group activities to stimulate ideation and inspire broad reflections on the field of contestable AI. Our aim is to facilitate interdisciplinary dialogue by bringing together researchers, practitioners, and stakeholders to foster the design and deployment of contestable AI., Comment: 5 pages, 0 figure, to be held as a workshop at CSCW'24

Behaviour change lies at the heart of many observable collective phenomena such as the transmission and control of infectious diseases, adoption of public health policies, and migration of animals to new habitats. Representing the process of individual behaviour change in computer simulations of these phenomena remains an open challenge. Often, computational models use phenomenological implementations with limited support from behavioural data. Without a strong connection to observable quantities, such models have limited utility for simulating observed and counterfactual scenarios of emergent phenomena because they cannot be validated or calibrated. Here, we present a simple stochastic individual-based model of reversal learning that captures fundamental properties of individual behaviour change, namely, the capacity to learn based on accumulated reward signals, and the transient persistence of learned behaviour after rewards are removed or altered. The model has only two parameters, and we use approximate Bayesian computation to demonstrate that they are fully identifiable from empirical reversal learning time series data. Finally, we demonstrate how the model can be extended to account for the increased complexity of behavioural dynamics over longer time scales involving fluctuating stimuli. This work is a step towards the development and evaluation of fully identifiable individual-level behaviour change models that can function as validated submodels for complex simulations of collective behaviour change., Comment: 23 pages, 5 figures

Since most inverse problems arising in scientific and engineering applications are ill-posed, prior information about the solution space is incorporated, typically through regularization, to establish a well-posed problem with a unique solution. Often, this prior information is an assumed statistical distribution of the desired inverse problem solution. Recently, due to the unprecedented success of generative adversarial networks (GANs), the generative network from a GAN has been implemented as the prior information in imaging inverse problems. In this paper, we devise a novel iterative algorithm to solve inverse problems in imaging where a dual-structured prior is imposed by combining a GAN prior with the compound Gaussian (CG) class of distributions. A rigorous computational theory for the convergence of the proposed iterative algorithm, which is based upon the alternating direction method of multipliers, is established. Furthermore, elaborate empirical results for the proposed iterative algorithm are presented. By jointly exploiting the powerful CG and GAN classes of image priors, we find, in compressive sensing and tomographic imaging problems, our proposed algorithm outperforms and provides improved generalizability over competitive prior art approaches while avoiding performance saturation issues in previous GAN prior-based methods., Comment: Submitted to IEEE TSP for publication. 13 pages, 2 figures

In this paper we introduce the Higher Order Lipschitz Greedy Recombination Interpolation Method (HOLGRIM) for finding sparse approximations of Lip$(\gamma)$ functions, in the sense of Stein, given as a linear combination of a (large) number of simpler Lip$(\gamma)$ functions. HOLGRIM is developed as a refinement of the Greedy Recombination Interpolation Method (GRIM) in the setting of Lip$(\gamma)$ functions. HOLGRIM combines dynamic growth-based interpolation techniques with thinning-based reduction techniques in a data-driven fashion. The dynamic growth is driven by a greedy selection algorithm in which multiple new points may be selected at each step. The thinning reduction is carried out by recombination, the linear algebra technique utilised by GRIM. We establish that the number of non-zero weights for the approximation returned by HOLGRIM is controlled by a particular packing number of the data. The level of data concentration required to guarantee that HOLGRIM returns a good sparse approximation is decreasing with respect to the regularity parameter $\gamma > 0$. Further, we establish complexity cost estimates verifying that implementing HOLGRIM is feasible.

The 2024 Index is our most comprehensive to date and arrives at an important moment when AI's influence on society has never been more pronounced. This year, we have broadened our scope to more extensively cover essential trends such as technical advancements in AI, public perceptions of the technology, and the geopolitical dynamics surrounding its development. Featuring more original data than ever before, this edition introduces new estimates on AI training costs, detailed analyses of the responsible AI landscape, and an entirely new chapter dedicated to AI's impact on science and medicine. The AI Index report tracks, collates, distills, and visualizes data related to artificial intelligence (AI). Our mission is to provide unbiased, rigorously vetted, broadly sourced data in order for policymakers, researchers, executives, journalists, and the general public to develop a more thorough and nuanced understanding of the complex field of AI. The AI Index is recognized globally as one of the most credible and authoritative sources for data and insights on artificial intelligence. Previous editions have been cited in major newspapers, including the The New York Times, Bloomberg, and The Guardian, have amassed hundreds of academic citations, and been referenced by high-level policymakers in the United States, the United Kingdom, and the European Union, among other places. This year's edition surpasses all previous ones in size, scale, and scope, reflecting the growing significance that AI is coming to hold in all of our lives.

With the advent of AI technologies, humans and robots are increasingly teaming up to perform collaborative tasks. To enable smooth and effective collaboration, the topic of value alignment (operationalized herein as the degree of dynamic goal alignment within a task) between the robot and the human is gaining increasing research attention. Prior literature on value alignment makes an inherent assumption that aligning the values of the robot with that of the human benefits the team. This assumption, however, has not been empirically verified. Moreover, prior literature does not account for human's trust in the robot when analyzing human-robot value alignment. Thus, a research gap needs to be bridged by answering two questions: How does alignment of values affect trust? Is it always beneficial to align the robot's values with that of the human? We present a simulation study and a human-subject study to answer these questions. Results from the simulation study show that alignment of values is important for trust when the overall risk level of the task is high. We also present an adaptive strategy for the robot that uses Inverse Reinforcement Learning (IRL) to match the values of the robot with those of the human during interaction. Our simulations suggest that such an adaptive strategy is able to maintain trust across the full spectrum of human values. We also present results from an empirical study that validate these findings from simulation. Results indicate that real-time personalized value alignment is beneficial to trust and perceived performance by the human when the robot does not have a good prior on the human's values., Comment: This is a preprint of the following chapter: Bhat et al., Value Alignment and Trust in Human-Robot Interaction: Insights from Simulation and User Study, published in "Emerging Frontiers in Human-Robot Interaction", edited by Ramana Kumar Vinjamuri, 2024, Springer Nature reproduced with permission of Springer Nature. The final authenticated version is available online at: [INSERT LINK HERE]

Inspired by experimental evidence collected when processing thin films from ternary solutions made of two solutes, typically polymers, and one solvent, we computationally study the morphology formation of domains obtained in three-state systems using both a lattice model and a continuum counterpart. The lattice-based approach relies on the Blume-Capel nearest neighbor model with bulk conservative Kawasaki dynamics, whereas as continuum system we consider a coupled system of evolution equations that is derived as hydrodynamic limit when replacing the nearest neighbor interaction in the lattice case by a suitable Kac potential. We explore how the obtained morphology depends on the solvent content in the mixture. In particular, we study how these scenarios change when the solvent is allowed to evaporate.

We use photo-electron paramagnetic resonance (EPR) measurements and first-principles calculations to identify and explain the properties of carbon in AlN. We present clear evidence for carbon substitution on the nitrogen site (C$_{\rm N}$). We also clarify the origin of a widely observed EPR spectra in AlN that, although often attributed to a deep donor defect, we demonstrate is surprisingly due to C$_{\rm N}$. Finally, we show the presence of C$_{\rm N}$ is consistent with the absorption spectra at 4.7 eV observed in AlN., Comment: 5 pages, 3 figures

Skew bracoids have been shown to have applications in Hopf-Galois theory. We show that a certain family of skew bracoids correspond bijectively with left cancellative semibraces. A consequence of this correspondence is that skew bracoids in this family can be used to obtain and study solutions of the set-theoretic Yang--Baxter equation; we study this process and the resulting solutions. We give numerous examples of skew bracoids satisfying our hypothesis, drawing upon a variety of constructions in the literature., Comment: 13 pages; title and sequencing of results changed

Photoluminescence intermittency remains one of the biggest challenges to realizing perovskite quantum dots (QDs) as scalable single photon emitters. We compare CsPbBr3 QDs capped with different ligands, lecithin, and a combination of oleic acid and oleylamine, to elucidate the role of surface chemistry on photoluminescence intermittency. We employ widefield photoluminescence microscopy, sampling the blinking behavior of hundreds of QDs. Using change point analysis, we achieve the robust classification of blinking trajectories, and we analyze representative distributions from large numbers of QDs (Nlecithin = 1308, Noleic acid/oleylamine =1317). We find that lecithin suppresses blinking in CsPbBr3 QDs compared to oleic acid/oleylamine. Under common experimental conditions, lecithin-capped QDs are 7.5 times more likely to be non-blinking and spend 2.5 times longer in their most emissive state, despite both QDs having nearly identical solution photoluminescence quantum yields. We measure photoluminescence as a function of dilution and show that the differences between lecithin and oleic acid/oleylamine capping emerge at low concentrations during preparation for single particle experiments. From experiment and first principles calculations, we attribute the differences in lecithin and oleic acid/oleylamine performance to differences in their ligand binding equilibria. Consistent with our experimental data, density functional theory calculations suggest a stronger binding affinity of lecithin to the QD surface compared to oleic acid/oleylamine, implying a reduced likelihood of ligand desorption during dilution. These results suggest that using more tightly binding ligands is a necessity for surface passivation and consequently, blinking reduction in perovskite QDs used for single particle and quantum light experiments., Comment: 38 pages, 5 figures

We investigate the consequence of two Lip$(\gamma)$ functions, in the sense of Stein, being close throughout a subset of their domain. A particular consequence of our results is the following. Given $K_0 > \varepsilon > 0$ and $\gamma > \eta > 0$ there is a constant $\delta = \delta(\gamma,\eta,\varepsilon,K_0) > 0$ for which the following is true. Let $\Sigma \subset \mathbb{R}^d$ be closed and $f , h : \Sigma \to \mathbb{R}$ be Lip$(\gamma)$ functions whose Lip$(\gamma)$ norms are both bounded above by $K_0$. Suppose $B \subset \Sigma$ is closed and that $f$ and $h$ coincide throughout $B$. Then over the set of points in $\Sigma$ whose distance to $B$ is at most $\delta$ we have that the Lip$(\eta)$ norm of the difference $f-h$ is bounded above by $\varepsilon$. More generally, we establish that this phenomenon remains valid in a less restrictive Banach space setting under the weaker hypothesis that the two Lip$(\gamma)$ functions $f$ and $h$ are only close in a pointwise sense throughout the closed subset $B$. We require only that the subset $\Sigma$ be closed; in particular, the case that $\Sigma$ is finite is covered by our results. The restriction that $\eta < \gamma$ is sharp in the sense that our result is false for $\eta := \gamma$., Comment: Minor refinement of main result statements. arXiv admin note: substantial text overlap with arXiv:2205.07495

We present a finite-volume based numerical scheme for a nonlocal Cahn-Hilliard equation which combines ideas from recent numerical schemes for gradient flow equations and nonlocal Cahn-Hilliard equations. The equation of interest is a special case of a previously derived and studied system of equations which describes phase separation in ternary mixtures. We prove the scheme is both energy stable and respects the analytical bounds of the solution. Furthermore, we present numerical demonstrations of the theoretical results using both the Flory-Huggins (FH) and Ginzburg-Landau (GL) free-energy potentials.

Signature kernels are at the core of several machine learning algorithms for analysing multivariate time series. The kernel of two bounded variation paths (such as piecewise linear interpolations of time series data) is typically computed by solving a Goursat problem for a hyperbolic partial differential equation (PDE) in two independent time variables. However, this approach becomes considerably less practical for highly oscillatory input paths, as they have to be resolved at a fine enough scale to accurately recover their signature kernel, resulting in significant time and memory complexities. To mitigate this issue, we first show that the signature kernel of a broader class of paths, known as \emph{smooth rough paths}, also satisfies a PDE, albeit in the form of a system of coupled equations. We then use this result to introduce new algorithms for the numerical approximation of signature kernels. As bounded variation paths (and more generally geometric $p$-rough paths) can be approximated by piecewise smooth rough paths, one can replace the PDE with rapidly varying coefficients in the original Goursat problem by an explicit system of coupled equations with piecewise constant coefficients derived from the first few iterated integrals of the original input paths. While this approach requires solving more equations, they do not require looking back at the complex and fine structure of the initial paths, which significantly reduces the computational complexity associated with the analysis of highly oscillatory time series.