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The combination of machine learning and physical laws has shown immense potential for solving scientific problems driven by partial differential equations (PDEs) with the promise of fast inference, zero-shot generalisation, and the ability to discover new physics. Examples include the use of fundamental physical laws as inductive bias to machine learning algorithms, also referred to as physics-driven machine learning, and the application of machine learning to represent features not represented in the differential equations such as closures for unresolved spatiotemporal scales. However, the simulation of complex physical systems by coupling advanced numerics for PDEs with state-of-the-art machine learning demands the composition of specialist PDE solving frameworks with industry-standard machine learning tools. Hand-rolling either the PDE solver or the neural net will not cut it. In this work, we introduce a generic differentiable programming abstraction that provides scientists and engineers with a highly productive way of specifying end-to-end differentiable models coupling machine learning and PDE-based components, while relying on code generation for high performance. Our interface automates the coupling of arbitrary PDE-based systems and machine learning models and unlocks new applications that could not hitherto be tackled, while only requiring trivial changes to existing code. Our framework has been adopted in the Firedrake finite-element library and supports the PyTorch and JAX ecosystems, as well as downstream libraries.

We study simplicity of $C^*$-algebras arising from self-similar groups of $\mathbb{Z}_2$-multispinal type, a generalization of the Grigorchuk case whose simplicity was first proved by L. Clark, R. Exel, E. Pardo, C. Starling, and A. Sims in 2019, and we prove results generalizing theirs. Our first main result is a sufficient condition for simplicity of the Steinberg algebra satisfying conditions modeled on the behavior of the groupoid associated to the first Grigorchuk group. This closely resembles conditions found by B. Steinberg and N. Szak\'acs. As a key ingredient we identify an infinite family of $2-(2q-1,q-1,q/2-1)$-designs, where $q$ is a positive even integer. We then deduce the simplicity of the associated $C^*$-algebra, which is our second main result. Results of similar type were considered by B. Steinberg and N. Szak\'acs in 2021, and later by K. Yoshida, but their methods did not follow the original methods of the five authors.

The capacity of a discrete-time multiple-input-multiple-output channel with correlated phase noises is investigated. In particular, the electro-optic frequency comb system is considered, where the phase noise of each channel is a combination of two independent Wiener phase-noise sources. Capacity upper and lower bounds are derived for this channel and are compared with lower bounds obtained by numerically evaluating the achievable information rates using quadrature amplitude modulation constellations. Capacity upper and lower bounds are provided for the high signal-to-noise ratio (SNR) regime. The multiplexing gain (pre-log) is shown to be $M-1$, where $M$ represents the number of channels. A constant gap between the asymptotic upper and lower bounds is observed, which depends on the number of channels $M$. For the specific case of $M=2$, capacity is characterized up to a term that vanishes as the SNR grows large., Comment: 45 pages, 3 figures, submitted to TIT, single-column

Whilst holding great promise for low noise, ease of operation and networking, useful photonic quantum computing has been precluded by the need for beyond-state-of-the-art components, manufactured by the millions. Here we introduce a manufacturable platform for quantum computing with photons. We benchmark a set of monolithically-integrated silicon photonics-based modules to generate, manipulate, network, and detect photonic qubits, demonstrating dual-rail photonic qubits with $99.98\% \pm 0.01\%$ state preparation and measurement fidelity, Hong-Ou-Mandel quantum interference between independent photon sources with $99.50\%\pm0.25\%$ visibility, two-qubit fusion with $99.22\%\pm0.12\%$ fidelity, and a chip-to-chip qubit interconnect with $99.72\%\pm0.04\%$ fidelity, not accounting for loss. In addition, we preview a selection of next generation technologies, demonstrating low-loss silicon nitride waveguides and components, fabrication-tolerant photon sources, high-efficiency photon-number-resolving detectors, low-loss chip-to-fiber coupling, and barium titanate electro-optic phase shifters., Comment: 8 pages, 5 figures

If two compact quantum metric spaces are close in the metric sense, then how similar are they, as noncommutative spaces? In the classical realm of Riemannian geometry, informally, if two manifolds are close in the Gromov-Hausdorff distance, and belong to a class of manifolds with bounded curvature and diameter, then the spectra of their Laplacian or Dirac operators are also close under many scenari. Of particular interest is the case where a sequence of manifolds converge for the Gromov-Hausdorff distance to a manifold of lower dimension, and the question of the continuity, in some sense, of the spectra of geometrically relevant operators. In this paper, we initiate the study of the continuity of spectra and other properties of metric spectral triples under collapse in the noncommutative realm. As a first step in this study, we work with collapse for the spectral propinquity, an analogue of the Gromov-Hausdorff distance for spectral triples introduced by the second author, i.e. a form of metric for differential structures. Inspired by results from collapse in Riemannian geometry, we begin with the study of spectral triples which decompose, in some sense, in a vertical and a horizontal direction, and we collapse these spectral triples along the vertical direction. We obtain convergence results, and by the work of the second author, we conclude continuity results for the spectra of the Dirac operators of these spectral triples. Examples include collapse of product of spectral triples with one Abelian factor, $U(1)$ principal bundles over Riemannian spin manifolds, and noncommutative principal bundles, including C*-crossed-products and other noncommutative bundles., Comment: 41 pages

We address the natural question: as noncommutative solenoids are inductive limits of quantum tori, do the standard spectral triples on quantum tori converge to some spectral triple on noncommutative solenoid for the spectral propinquity? We answer this question by showing that, using appropriate bounded perturbation of the spectral triples on quantum tori, such a spectral triple on noncommutative solenoid can be constructed., Comment: 15 pages

We give explicit computations of the $\Gamma$-Euler characteristic of several families of orbit space definable translation groupoids. These include the translation groupoids associated to finite-dimensional linear representations of the circle and real and unitary representations of the real $2\times 2$ orthogonal group. In the case of translation groupoids associated to linear symplectic quotients of representations of a arbitrary compact Lie group $G$, we show that unlike the other cases, the $\Gamma$-Euler characteristic depends only on the group and not on the representation., Comment: 24 pages

Nowadays, the usage of Large Language Models (LLMs) has increased, and LLMs have been used to generate texts in different languages and for different tasks. Additionally, due to the participation of remarkable companies such as Google and OpenAI, LLMs are now more accessible, and people can easily use them. However, an important issue is how we can detect AI-generated texts from human-written ones. In this article, we have investigated the problem of AI-generated text detection from two different aspects: semantics and syntax. Finally, we presented an AI model that can distinguish AI-generated texts from human-written ones with high accuracy on both multilingual and monolingual tasks using the M4 dataset. According to our results, using a semantic approach would be more helpful for detection. However, there is a lot of room for improvement in the syntactic approach, and it would be a good approach for future work., Comment: Mohammad Heydari Rad, Farhan Farsi, and Shayan Bali have made equal contributions to this work

We explore the optimal pilot positioning for phase tracking in electro-optic frequency comb setups. We show that, in contrast to previous results for regular multichannel systems, allocating the first and the last channels for pilots is optimal given a fixed pilot overhead, Comment: Presented at European Conference on Optical Communications (ECOC) 2023

The conveyance of employees holds paramount significance for expansive corporations. Employees typically commute to their workplaces either via personal vehicles or through public transit. In this research endeavor, our role is that of a third-party entity entrusted with orchestrating the transportation of employees whose place of employment is situated within the grey zone. This zone exclusively permits the ingress of electric/hybrid vehicles and buses. We advocate for employees to adopt carpooling and furnish bus services for those who abstain from it. The primary objective of this research is to curtail the quantity of vehicles leased by the third-party entity, promote carpooling among employees, amplify employee contentment, and mitigate environmental degradation stemming from vehicular gasoline consumption. To decipher the model delineated in this study, the epsilon constraint method is proffered for petite-scale instances, while NSGA-II is introduced as a potent meta-heuristic technique tailored for large-scale scenarios. Computational trials corroborate that the models posited can be efficaciously harnessed by enterprises to pare down transportation expenditures.

We introduce the universal Euler characteristic of an orbit space definable groupoid, a class of groupoids containing cocompact proper Lie groupoids as well as translation groupoids for semialgebraic group actions. Generalizing results of Gusein-Zade, Luengo, and Melle-Hern\'andez, we show that every additive and multiplicative invariant of orbit space definable groupoids with an additional local triviality hypothesis arises as a ring homomorphism applied to the universal Euler characteristic. This in particular includes the $\Gamma$-orbifold Euler characteristic introduced by the first and third authors when $\Gamma$ is a finitely presented group. For Lie groupoids with Riemannian structures in the sense of del Hoyo-Fernandes, and for Cartan-Lie groupoids with Riemannian structures in the sense of Kotov-Strobl, we study the Riemannian structures induced on the suborbifolds of the inertia space given by images of isotropy groups. We realize the $\mathbb{Z}$-orbifold Euler characteristic as the integral of a differential form defined on the arrow space or the space of composable pairs of arrows of the original groupoid., Comment: 23 pages

We propose a novel framework for designing a resilient Model Predictive Control (MPC) targeting uncertain linear systems under cyber attack. Assuming a periodic attack scenario, we model the system under Denial of Service (DoS) attack, also with measurement noise, as an uncertain linear system with parametric and additive uncertainty. To detect anomalies, we employ a Kalman filter-based approach. Then, through our observations of the intensity of the launched attack, we determine a range of possible values for the system matrices, as well as establish bounds of the additive uncertainty for the equivalent uncertain system. Leveraging a recent constraint tightening robust MPC method, we present an optimization-based resilient algorithm. Accordingly, we compute the uncertainty bounds and corresponding constraints offline for various attack magnitudes. Then, this data can be used efficiently in the MPC computations online. We demonstrate the effectiveness of the developed framework on the Adaptive Cruise Control (ACC) problem., Comment: To Appear in ICINCO 2023

Employing the spherical collapse (SC) formalism, we investigate the linear evolution of the matter over-density for energy-momentum-squared gravity (EMSG), which in practical phenomenological terms, one may imagine as an extension of the {\Lambda}CDM model of cosmology. The underlying model, while still having a cosmological constant, is a non-linear material extension of the general theory of relativity (GTR) and includes correction terms that are dominant in the high-energy regime, the early universe. Considering the Friedman{Robertson{Walker (FRW) background in the presence of a cosmological constant, we find the effects of the modifications arising from EMSG on the growth of perturbations at the early stages of the universe. Considering both possible negative and positive values of the model parameter of EMSG, we discuss its role in the evolution of the matter density contrast and growth function in the level of linear perturbations. While EMSG leaves imprints distinguishable from {\Lambda}CDM, we find that the negative range of the ESMG model parameter is not well-behaved, indicating an anomaly in the parameter space of the model. In this regard, for the evaluation of the galaxy cluster number count in the framework of EMSG, we equivalently provide an analysis of the number count of the gravitationally collapsed objects (or the dark matter halos). We show that the galaxy cluster number count decreases compared to the {\Lambda}CDM model. In agreement with the hierarchical model of structure formation, in EMSG cosmology the more massive structures are less abundant, meaning that form at later times., Comment: 9 pages, 8 figures. v3: version accepted for publication in PRD: the title changed

In this paper we study the groups of isometries and the set of bi-Lipschitz automorphisms of spectral triples from a metric viewpoint, in the propinquity framework of Latremoliere. In particular we prove that these groups and sets are compact in the automorphism group of the spectral triple C*-algebra with respect to the Monge-Kantorovich metric, which induces the topology of pointwise convergence. We then prove a necessary and sufficient condition for the convergence of the actions of various groups of isometries, in the sense of the covariant version of the Gromov-Hausdorff propinquity -- a noncommutative analogue of the Gromov-Hausdorff distance -- when working in the context of inductive limits of quantum compact metric spaces and metric spectral triples. We illustrate our work with examples including AF algebras and noncommutative solenoids., Comment: 38 pages