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In contrast to regular ordinary differential equations, the problem of accurately setting initial conditions just emerges in the context of differential-algebraic equations where the dynamic degree of freedom of the system is smaller than the absolute dimension of the described process, and the actual lower-dimensional configuration space of the system is deeply implicit. For linear higher-index differential-algebraic equations, we develop an appropriate numerical method based on properties of canonical subspaces and on the so-called geometric reduction. Taking into account the fact that higher-index differential-algebraic equations lead to ill-posed problems in naturally given norms, we modify this approach to serve as transfer conditions from one time-window to the next in a time stepping procedure and combine it with window-wise overdetermined least-squares collocation to construct the first fully numerical solvers for higher-index initial-value problems., Comment: 51 pages

Frequency mixers are fundamental components in modern wireless communication and radar systems, responsible for up- and down-conversion of target radio-frequency (RF) signals. Recently, photonic-assisted RF mixers have shown unique advantages over traditional electronic counterparts, including broad operational bandwidth, flat frequency response, and immunity to electromagnetic interference. However, current integrated photonic mixers face significant challenges in achieving efficient conversion at high frequencies, especially in millimeter-wave bands, due to the limitations of existing electro-optic (EO) modulators. Additionally, high-frequency local oscillators in the millimeter-wave range are often difficult to obtain and expensive, leading to unsatisfactory cost and restricted operational bandwidth in practice. In this paper, we harness the exceptional EO property and scalability of thin-film lithium niobate (TFLN) photonic platform to implement a high-performance harmonic reconfigurable millimeter-wave mixer. The TFLN photonic circuit integrates a broadband EO modulator that allows for extensive frequency coverage, and an EO frequency comb source that significantly reduces the required carrier frequency of the local oscillator. We experimentally demonstrate fully reconfigurable frequency down-conversion across a broad operational bandwidth ranging from 20 GHz to 67 GHz, with a large intermediate frequency of 20 GHz, as well as up-conversion to frequencies of up to 110 GHz. Our integrated photonic mixing system shows dramatically improved bandwidth performance, along with competitive indicators of frequency conversion efficiency and spurious suppression ratio, positioning it as a promising solution for future millimeter-wave transceivers in next-generation communication and sensing systems., Comment: 8 pages, 7 figures

Light detection and ranging (LiDAR) is a ubiquitous tool to provide precise spatial awareness in various perception environments. A bionic LiDAR that can mimic human-like vision by adaptively gazing at selected regions of interest within a broad field of view is crucial to achieve high-resolution imaging in an energy-saving and cost-effective manner. However, current LiDARs based on stacking fixed-wavelength laser arrays and inertial scanning have not been able to achieve the desired dynamic focusing patterns and agile scalability simultaneously. Moreover, the ability to synchronously acquire multi-dimensional physical parameters, including distance, direction, Doppler, and color, through seamless fusion between multiple sensors, still remains elusive in LiDAR. Here, we overcome these limitations and demonstrate a bio-inspired frequency-modulated continuous wave (FMCW) LiDAR system with dynamic and scalable gazing capability. Our chip-scale LiDAR system is built using hybrid integrated photonic solutions, where a frequency-chirped external cavity laser provides broad spectral tunability, while on-chip electro-optic combs with elastic channel spacing allow customizable imaging granularity. Using the dynamic zoom-in capability and the coherent FMCW scheme, we achieve a state-of-the-art resolution of 0.012 degrees, providing up to 15 times the resolution of conventional 3D LiDAR sensors, with 115 equivalent scanning lines and 4D parallel imaging. We further demonstrate cooperative sensing between our adaptive coherent LiDAR and a camera to enable high-resolution color-enhanced machine vision.

Microwave photonics, with its advanced high-frequency signal processing capabilities, is expected to play a crucial role in next-generation wireless communications and radar systems. The realization of highly integrated, high-performance, and multifunctional microwave photonic links will pave the way for its widespread deployment in practical applications, which is a significant challenge. Here, leveraging thin-film lithium niobate intensity modulator and programmable cascaded microring resonators, we demonstrate for the first time a tunable microwave photonic notch filter that simultaneously achieves high level of integration along with high dynamic range, high link gain, low noise figure, and ultra-high rejection ratio. Additionally, this programmable on-chip system is multifunctional, allowing for the dual-band notch filter and the suppression of the high-power interference signal. This work demonstrates the potential applications of the thin-film lithium niobate platform in the field of high-performance integrated microwave photonic filtering and signal processing, facilitating the advancement of microwave photonic system towards practical applications., Comment: 18 pages, 8 figures, 1 table

The interaction between atoms and a quantized radiation field is fundamentally important in quantum optics and quantum information science. Due to their unusual properties, Rydberg atoms are promising building blocks for two-qubit gates and atom-light quantum interfaces, exploiting the Rydberg blockade interaction which prevents two atoms at close distance from being simultaneously excited to Rydberg states. Recently, this effect was used to engineer quantum processors based on arrays of interacting Rydberg atoms illuminated by Raman lasers. Motivated by these experiments, we extend the Jaynes-Cummings model to study the interaction between two Rydberg atoms interacting by the Rydberg blockade and a quantized radiation field. We find a rich variety of Rabi oscillations and entanglement between the atoms and the radiation field as a function of initial conditions and interaction time, which may be used to obtain atom-light quantum interfaces as components for future long-distance quantum communication., Comment: 21 pages, 7 figures

In the late stages of accretion leading up to the formation of planetesimals, particles grew to pebbles the size of 1-mm to tens of cm. That is the same size range that dominates the present-day comet mass loss. Meteoroids that size cause visible meteors on Earth. Here, we hypothesize that the size distribution and the physical and chemical properties of young meteoroid streams still contain information about the conditions in the solar nebula during these late stages of accretion. From observations of 47 young meteor showers, we find that freshly ejected meteoroids from long-period comets tend to have low bulk density and are distributed with equal surface area per log-mass interval (magnitude distribution index chi ~ 1.85), suggesting gentle accretion conditions. Jupiter-family comets, on the other hand, mostly produce meteoroids twice as dense and distributed with a steeper chi ~ 2.15 or even chi ~ 2.5, which implies that those pebbles grew from particles fragmenting in a collisional cascade or by catastrophic collisions, respectively. Both comet populations contain an admixture of compact materials that are sometimes sodium-poor, but Jupiter-family comets show a higher percentage (~8% on average) than long-period comet showers (~4%), and a wider range. While there are exceptions in both groups, the implication is that most long-period comets formed under gentle particle growth conditions, possibly near the 30 AU edge of the Trans Neptunian Disk, while most Jupiter family comets formed closer to the Sun where pebbles reached or passed the fragmentation barrier. This is possible if the Scattered Disk represents all objects scattered by Neptune during its migration, while the present-day outer Oort cloud formed only during and after the Sun had moved away from sibling stars., Comment: 82 pages, 14 figures, 3 tables

Optical frequency combs play a crucial role in optical communications, time-frequency metrology, precise ranging, and sensing. Among various generation schemes, resonant electro-optic combs are particularly attractive for its excellent stability, flexibility and broad bandwidths. In this approach, an optical pump undergoes multiple electro-optic modulation processes in a high-Q optical resonator, resulting in cascaded spectral sidebands. However, most resonant electro-optic combs to date make use of lumped-capacitor electrodes with relatively inefficient utilization of the input electrical power. This design also reflects most electrical power back to the driving circuits and necessitates costly RF isolators in between, presenting substantial challenges in practical applications. To address these issues, we present an RF circuit friendly electro-optic frequency comb generator incorporated with on-chip coplanar microwave resonator electrodes, based on a thin-film lithium niobate platform. Our design achieves more than three times electrical power reduction with minimal reflection at the designed comb repetition rate of ~ 25 GHz. We experimentally demonstrate broadband electro-optic frequency comb generation with a comb span of >85 nm at a moderate electrical driving power of 740 mW (28.7 dBm). Our power-efficient and isolator-free electro-optic comb source could offer compact, low-cost and simple-to-design solution for applications in spectroscopy, high-precise metrology, optical communications.

The dominant contribution to the energy relaxation of state-of-the-art superconducting qubits is often attributed to their coupling to an ensemble of material defects which behave as two-level systems. These defects have varying microscopic characteristics which result in a large range of observable defect properties such as resonant frequencies, coherence times and coupling rates to qubits $g$. Here, we investigate strategies to mitigate losses to the family of defects that strongly couple to qubits ($g/2\pi\ge$ 0.5 MHz). Such strongly coupled defects are particularly detrimental to the coherence of qubits and to the fidelities of operations relying on frequency excursions, such as flux-activated two-qubit gates. To assess their impact, we perform swap spectroscopy on 92 frequency-tunable qubits and quantify the spectral density of these strongly coupled modes. We show that the frequency configuration of the defects is rearranged by warming up the sample to room temperature, whereas the total number of defects on a processor tends to remain constant. We then explore methods for fabricating qubits with a reduced number of strongly coupled defect modes by systematically measuring their spectral density for decreasing Josephson junction dimensions and for various surface cleaning methods. Our results provide insights into the properties of strongly coupled defect modes and show the benefits of minimizing Josephson junction dimensions to improve qubit properties.

We present an Alternating Direction Method of Multipliers (ADMM) algorithm designed to solve the Weighted Generalized Fused LASSO Signal Approximator (wFLSA). First, we show that wFLSAs can always be reformulated as a Generalized LASSO problem. With the availability of algorithms tailored to the Generalized LASSO, the issue appears to be, in principle, resolved. However, the computational complexity of these algorithms is high, with a time complexity of $O(p^4)$ for a single iteration, where $p$ represents the number of coefficients. To overcome this limitation, we propose an ADMM algorithm specifically tailored for wFLSA-equivalent problems, significantly reducing the complexity to $O(p^2)$. Our algorithm is publicly accessible through the R package wflsa.

The terahertz (THz) frequency range, bridging the gap between microwave and infrared frequencies, presents unparalleled opportunities for advanced imaging, sensing, communications, and spectroscopy applications. Terahertz photonics, in analogy with microwave photonics, is a promising solution to address the critical challenges in THz technologies through optical methods. Despite its vast potential, key technical challenges remain in effectively interfacing THz signals with the optical domain, especially THz-optic modulation and optical generation of THz waves. Here, we address these challenges using a monolithic integrated photonic chip designed to support efficient bidirectional interaction between THz and optical waves. Leveraging the significant second-order optical nonlinearity and strong optical and THz confinement in a thin-film lithium niobate on quartz platform, the chip supports both efficient THz-optic modulation and continuous THz wave generation at up to 500 GHz. The THz-optic modulator features a radio frequency (RF) half-wave voltage of 8V at 500 GHz, representing more than an order of magnitude reduction in modulation power consumption from previous works. The measured continuous wave THz generation efficiency of 4.8*10-6 /W at 500 GHz also marks a tenfold improvement over existing tunable THz generation devices based on lithium niobate. We further leverage the coherent nature of the optical THz generation process and mature optical modulation techniques to realize high-speed electro-THz modulation at frequencies up to 35 GHz. The chip-scale THz-photonic platform paves the way for more compact, efficient, and cost-effective THz systems with potential applications in THz communications, remote sensing, and spectroscopy.

Optical vector analysis (OVA) is an enabling technology for comprehensively characterizing both amplitude and phase responses of optical devices or systems. Conventional OVA technologies are mostly based on discrete optoelectronic components, leading to unsatisfactory system sizes, complexity, and stability. They also encounter challenges in revealing the on-chip characteristics of integrated photonic devices, which are often overwhelmed by the substantial coupling loss and extra spectral response at chip facets. In this work, we demonstrate a miniaturized OVA system for integrated photonics devices based on broadband single sideband (SSB) modulators on a thin-film lithium niobate (LN) platform. The OVA could provide a direct probe of both amplitude and phase responses of photonic devices with kHz-level resolution and tens of terahertz measurement bandwidth. We perform in-situ characterizations of single and coupled microring resonators fabricated on the same chip as the OVA, unfolding their intrinsic loss and coupling states unambiguously. Furthermore, we achieve the direct measurement of collective phase dynamics and density of states of the Bloch modes in a synthetic frequency crystal, by in-situ OVA of a dynamically modulated microring resonator. Our in-situ OVA system provides a compact, high-precision, and broadband solution for characterizing future integrated photonic devices and circuits, with potential applications ranging from optical communications, biosensing, neuromorphic computing, to quantum information processing.

Transformers have recently gained prominence in long time series forecasting by elevating accuracies in a variety of use cases. Regrettably, in the race for better predictive performance the overhead of model architectures has grown onerous, leading to models with computational demand infeasible for most practical applications. To bridge the gap between high method complexity and realistic computational resources, we introduce the Residual Cyclic Transformer, ReCycle. ReCycle utilizes primary cycle compression to address the computational complexity of the attention mechanism in long time series. By learning residuals from refined smoothing average techniques, ReCycle surpasses state-of-the-art accuracy in a variety of application use cases. The reliable and explainable fallback behavior ensured by simple, yet robust, smoothing average techniques additionally lowers the barrier for user acceptance. At the same time, our approach reduces the run time and energy consumption by more than an order of magnitude, making both training and inference feasible on low-performance, low-power and edge computing devices. Code is available at https://github.com/Helmholtz-AI-Energy/ReCycle, Comment: 9 pages, 3 figures, to be published at IEEE CAI 2024, Associated code available at https://github.com/Helmholtz-AI-Energy/ReCycle

Recent successes in Generative Artificial Intelligence (GenAI) have led to new technologies capable of generating high-quality code, natural language, and images. The next step is to integrate GenAI technology into products, a task typically conducted by software developers. Such product development always comes with a certain risk of liability. Within this article, we want to shed light on the current state of two such risks: data protection and copyright. Both aspects are crucial for GenAI. This technology deals with data for both model training and generated output. We summarize key aspects regarding our current knowledge that every software developer involved in product development using GenAI should be aware of to avoid critical mistakes that may expose them to liability claims., Comment: Submission under review

In 1996 Seo proved that two appropriate pairs of current and voltage data measured on the surface of a planar homogeneous object are sufficient to determine a conductive polygonal inclusion with known deviating conductivity. Here we show that the corresponding linearized forward map is injective, and from this we deduce Lipschitz stability of the solution of the original nonlinear inverse problem. We also treat the case of an insulating polygonal inclusion, in which case a single pair of Cauchy data is already sufficient for the same purpose.

We consider the conductivity problem for a homogeneous body with an inclusion of a different, but known, conductivity. Our interest concerns the associated shape derivative, i.e., the derivative of the corresponding electrostatic potential with respect to the shape of the inclusion. For a smooth inclusion it is known that the shape derivative is the solution of a specific inhomogeneous transmission problem. We show that this characterization of the shape derivative is also valid when the inclusion is a polygonal domain, but due to singularities at the vertices of the polygon, the shape derivative fails to belong to $H^1$ in this case.

The Open Databases Integration for Materials Design (OPTIMADE) application programming interface (API) empowers users with holistic access to a growing federation of databases, enhancing the accessibility and discoverability of materials and chemical data. Since the first release of the OPTIMADE specification (v1.0), the API has undergone significant development, leading to the upcoming v1.2 release, and has underpinned multiple scientific studies. In this work, we highlight the latest features of the API format, accompanying software tools, and provide an update on the implementation of OPTIMADE in contributing materials databases. We end by providing several use cases that demonstrate the utility of the OPTIMADE API in materials research that continue to drive its ongoing development.

The escalating data volume and complexity resulting from the rapid expansion of artificial intelligence (AI), internet of things (IoT) and 5G/6G mobile networks is creating an urgent need for energy-efficient, scalable computing hardware. Here we demonstrate a hypermultiplexed integrated tensor optical processor (HITOP) that performs trillions of operations per second (TeraOPS) at the energy cost of 25 femtojoule per operation (25 fJ/OP). Based on space-time-wavelength three-dimensional (3D) data streaming, HITOP is built with arrays of wafer-fabricated III/V-based micron-scale lasers (spanning ~1 THz) incorporating thin-film Lithium-Niobate electro-optic (EO) photonics. Multidimensional parallelism allows matrix-matrix multiplications ($N^{3}$ operations) using $O(N)$ devices, facilitating scalable on-chip integration. With each device activating 10 billion parameters per second, the HITOP scalability is validated in machine learning models with 405,000 parameters, which is 25,000 times more than previous integrated optical systems. A combination of high clockrates (10 GS/s), parallel processing and real-time reprogrammability unlocks the full potential of light for next-generation AI accelerators in applications ranging from training with trillions of parameters, real-time decision making in autonomous vehicles and robotics, dynamic optimization in smart manufacturing, to complex simulation for climate modeling and drug discovery.

In this study, we propose to adopt a novel framework, Knowledge-integrated Machine Learning, for advancing Proton Exchange Membrane Water Electrolysis (PEMWE) development. Given the significance of PEMWE in green hydrogen production and the inherent challenges in optimizing its performance, our framework aims to meld data-driven models with domain-specific insights systematically to address the domain challenges. We first identify the uncertainties originating from data acquisition conditions, data-driven model mechanisms, and domain expertise, highlighting their complementary characteristics in carrying information from different perspectives. Building upon this foundation, we showcase how to adeptly decompose knowledge and extract unique information to contribute to the data augmentation, modeling process, and knowledge discovery. We demonstrate a hierarchical three-level framework, termed the "Ladder of Knowledge-integrated Machine Learning", in the PEMWE context, applying it to three case studies within a context of cell degradation analysis to affirm its efficacy in interpolation, extrapolation, and information representation. This research lays the groundwork for more knowledge-informed enhancements in ML applications in engineering., Comment: 21 pages, 7 figures

On-chip optical microresonators are essential building blocks in integrated optics. The ability to arbitrarily engineer their resonant frequencies is crucial for exploring novel physics in synthetic frequency dimensions and practical applications like nonlinear optical parametric processes and dispersion-engineered frequency comb generation. Photonic crystal ring (PhCR) resonators are a versatile tool for such arbitrary frequency engineering, by controllably creating mode splitting at selected resonances. To date, these PhCRs have mostly been demonstrated in isotropic photonic materials, while such engineering could be significantly more complicated in anisotropic platforms that often offer more fruitful optical properties. Here, we realize the spectral engineering of chip-scale optical microresonators in the anisotropic lithium niobate (LN) crystal by a gradient design that precisely compensates for variations in both refractive index and perturbation strength. We experimentally demonstrate controllable frequency splitting at single and multiple selected resonances in LN PhCR resonators with different sizes, while maintaining high Q-factors up to 1 million. Moreover, we experimentally construct a sharp boundary in the synthetic frequency dimension based on an actively modulated x-cut LN gradient-PhCR, opening up new paths toward the arbitrary control of electro-optic comb spectral shapes and exploration of novel physics in the frequency degree of freedom., Comment: 19 pages, 5 figures

Austin's moving knife procedure is used to divide a cake equitably between two agents: each agent believes that they received exactly half of the cake. This short note deals with the case when the two agents have unequal (rational) entitlements to the cake and seek a weighted equitable division -- one where each agent believes that they received exactly the share that they are entitled to -- and also minimizes the number of connected components that each agent receives. First, we adapt Austin's moving knife procedure to produce a weighted equitable division of a circular cake that gives exactly one connected piece to each agent (recovering a result of Shishido and Zeng's). Next, we use it to produce a weighted equitable division of a star graph cake that gives at most two connected pieces to each agent -- and show that this bound on the number of connected pieces is tight.

Millimeter-wave (mmWave,>30 GHz) radars are the key enabler in the coming 6G era for high-resolution sensing and detection of targets. Photonic radar provides an effective approach to overcome the limitations of electronic radars thanks to the high frequency, broad bandwidth, and excellent reconfigurability of photonic systems. However, conventional photonic radars are mostly realized in tabletop systems composed of bulky discrete components, whereas the more compact integrated photonic radars are difficult to reach the mmWave bands due to the unsatisfactory bandwidths and signal integrity of the underlining electro-optic modulators. Here, we overcome these challenges and demonstrate a centimeter-resolution integrated photonic radar operating in the mmWave V band (40-50 GHz) based on a 4-inch wafer-scale thin-film lithium niobate (TFLN) technology. The fabricated TFLN mmWave photonic integrated circuit consists of a first electro-optic modulator capable of generating a broadband linear frequency modulated mmWave radar waveform through optical frequency multiplication of a low-frequency input signal, and a second electro-optic modulator responsible for frequency de-chirp of the received reflected echo wave, therefore greatly relieving the bandwidth requirements for the analog-to-digital converter in the receiver. Thanks to the absence of optical and electrical filters in the system, our integrated photonic mmWave radar features continuous on-demand tunability of the center frequency and bandwidth, currently only limited by the bandwidths of electrical amplifiers. We achieve multi-target ranging with a resolution of 1.50 cm and velocity measurement with a resolution of 0.067 m/s. Furthermore, we construct an inverse synthetic aperture radar (ISAR) and successfully demonstrate the imaging of targets with various shapes and postures with a two-dimensional resolution of 1.50 cm * 1.06 cm.

We report the first-ever experimental observation of backward stimulated Brillouin scattering (SBS) in thin-film lithium niobate (TFLN) waveguides. The peak Brillouin gain coefficient of the z-cut LN waveguide with a crystal rotation angle of 20$^{\circ}$ is as high as 84.9m$^{-1}$W$^{-1}$, facilitated by surface acoustic waves (SAW) at 8.06GHz.

The rise of populism concerns many political scientists and practitioners, yet the detection of its underlying language remains fragmentary. This paper aims to provide a reliable, valid, and scalable approach to measure populist stances. For that purpose, we created an annotated dataset based on parliamentary speeches of the German Bundestag (2013 to 2021). Following the ideational definition of populism, we label moralizing references to the virtuous people or the corrupt elite as core dimensions of populist language. To identify, in addition, how the thin ideology of populism is thickened, we annotate how populist statements are attached to left-wing or right-wing host ideologies. We then train a transformer-based model (PopBERT) as a multilabel classifier to detect and quantify each dimension. A battery of validation checks reveals that the model has a strong predictive accuracy, provides high qualitative face validity, matches party rankings of expert surveys, and detects out-of-sample text snippets correctly. PopBERT enables dynamic analyses of how German-speaking politicians and parties use populist language as a strategic device. Furthermore, the annotator-level data may also be applied in cross-domain applications or to develop related classifiers.

We extend the K-cowaist inequality to generalized Dirac operators in the sense of Gromov and Lawson and study applications to manifolds with boundary., Comment: 6 pages

Machine learning (ML) is nowadays widely used for different purposes and in several disciplines. From self-driving cars to automated medical diagnosis, machine learning models extensively support users' daily activities, and software engineering tasks are no exception. Not embracing good ML practices may lead to pitfalls that hinder the performance of an ML system and potentially lead to unexpected results. Despite the existence of documentation and literature about ML best practices, many non-ML experts turn towards gray literature like blogs and Q&A systems when looking for help and guidance when implementing ML systems. To better aid users in distilling relevant knowledge from such sources, we propose a recommender system that recommends ML practices based on the user's context. As a first step in creating a recommender system for machine learning practices, we implemented Idaka. A tool that provides two different approaches for retrieving/generating ML best practices: i) an information retrieval (IR) engine and ii) a large language model. The IR-engine uses BM25 as the algorithm for retrieving the practices, and a large language model, in our case Alpaca. The platform has been designed to allow comparative studies of best practices retrieval tools. Idaka is publicly available at GitHub: https://bit.ly/idaka. Video: https://youtu.be/cEb-AhIPxnM., Comment: Accepted for Publication at ESEC/FSE demonstrations track

We report on computations of the cohomology of GL_2(O_D) and SL_2(O_D), where D<0 is a fundamental discriminant. These computations go well beyond earlier results of Vogtmann and Scheutzow. We use the technique of homology of Voronoi complexes, and our computations recover the integral cohomology away from the primes 2, 3. We observed exponential growth in the torsion subgroup of H^2 as $D$ increases, and compared our data to bounds of Rohlfs.

In this paper, we investigate three-term linear relations among theta series of positive-definite integral binary quadratic forms. We extend Schiemann's methods to characterize all possible three-term linear relations among theta series of such forms, providing necessary and sufficient conditions for such relations to exist. To accomplish this, we develop, implement, and execute a novel extended refinement algorithm on polyhedral cones. We show that there is exactly one non-trivial three-term linear relation: it involves quadratic forms with discriminants $-3, -12, -48$, all in the same rational squareclass $-3(\mathbb{Q}^\times)^2$., Comment: 30 pages

Integrated microwave photonics is an intriguing field that leverages integrated photonic technologies for the generation, transmission, and manipulation of microwave signals in chip-scale optical systems. In particular, ultrafast processing and computation of analog electronic signals in the optical domain with high fidelity and low latency could enable a variety of applications such as MWP filters, microwave signal processing, and image recognition. An ideal photonic platform for achieving these integrated MWP processing tasks shall simultaneously offer an efficient, linear and high-speed electro-optic modulation block to faithfully perform microwave-optic conversion at low power, and a low-loss functional photonic network that can be configured for a variety of signal processing tasks, as well as large-scale, low-cost manufacturability to monolithically integrate the two building blocks on the same chip. In this work, we demonstrate such an integrated MWP processing engine based on a thin-film lithium niobate platform capable of performing multi-purpose processing and computation tasks of analog signals up to 92 giga samples per second at CMOS-compatible voltages. We demonstrate high-speed analog computation, i.e., first- and second-order temporal integration and differentiation with computing accuracies up to 98.1 %, and deploy these functions to showcase three proof-of-concept applications, namely, ordinary differential equation solving, ultra-wideband signal generation and high-speed edge detection of images. We further leverage the image edge detector to enable a photonic-assisted image segmentation model that could effectively outline the boundaries of melanoma lesion in medical diagnostic images, achieving orders of magnitude faster processing speed and lower power consumption than conventional electronic processors.

We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least 2 over compact intervals equipped with strictly log-concave warping functions. This generalizes earlier results of Cecchini-Zeidler to all dimensions. Moreover, we show scalar curvature rigidity of round spheres of dimension at least 3 with two antipodal points removed. This resolves a problem in Gromov's ''Four Lectures'' in all dimensions. Our arguments are based on spin geometry., Comment: published version