Abstract We consider the stochastic damped nonlinear wave equation $$\partial _t^{2}u+\partial _t u+u-\Delta u +u^{3} = \sqrt{2} {\langle {\nabla }\rangle ^{-s}} \xi $$ ∂ t 2 u + ∂ t u + u - Δ u + u 3 = 2 ⟨ ∇ ⟩ - s ξ on the two-dimensional torus $$\mathbb T^2$$ T 2 , where $$\xi $$ ξ denotes a space-time white noise and $$s>0$$ s > 0 . We show that the measure $$\vec {\mu }_s$$ μ → s corresponding to the unique invariant measure for the flow of the associated linear equation is quasi-invariant under the nonlinear stochastic flow.

In this paper, we obtain the sharp bounds of probabilistic Gel’fand width of the multivariate Sobolev space [Formula: see text] with common smoothness equipped with the Gaussian measure in the [Formula: see text] norm by discretization method, where [Formula: see text] is a finite set. Then we obtain the sharp bounds of average Gel’fand width from the results of probabilistic width.

Abstract We study isoperimetric inequalities on “slabs”, namely weighted Riemannian manifolds obtained as the product of the uniform measure on a finite length interval with a codimension‐one base. As our two main applications, we consider the case when the base is the flat torus and the standard Gaussian measure on . The isoperimetric conjecture on the three‐dimensional cube predicts that minimizers are enclosed by spheres about a corner, cylinders about an edge and coordinate planes. This has only been established for relative volumes close to 0, and 1 by compactness arguments. Our analysis confirms the isoperimetric conjecture on the three‐dimensional cube with side lengths in a new range of relative volumes . In particular, we confirm the conjecture for the standard cube () for all , when for the entire range where spheres are conjectured to be minimizing, and also for all . When we reduce the validity of the full conjecture to establishing that the half‐plane is an isoperimetric minimizer. We also show that the analogous conjecture on a high‐dimensional cube is false for . In the case of a slab with a Gaussian base of width , we identify a phase transition when and when . In particular, while products of half‐planes with are always minimizing when , when they are never minimizing, being beaten by Gaussian unduloids. In the range , a potential trichotomy occurs.

ABSTRACT Accurate degradation state estimation is critical for predictive maintenance, yet it is often compromised by measurement outliers and parameter uncertainty. Existing methods either assume Gaussian measurement errors, which are sensitive to outliers, or overlook parameter uncertainty, leading to overconfident predictions. To address these challenges, we propose a Bayesian online degradation state estimation framework that integrates robust error modeling with parameter uncertainty quantification. Specifically, we model measurement errors using a Student's‐ distribution to handle outliers and employ variational Bayes with Laplace and Gamma approximations to estimate posterior distributions of degradation states and parameters efficiently. This framework enables real‐time updates, ensuring adaptability to dynamic operating conditions. Furthermore, based on the estimated degradation states, we derive real‐time remaining useful life predictions and dynamic maintenance strategies under a cost function model. Numerical experiments and case studies demonstrate the framework's robustness, computational efficiency, and practical applicability.

To investigate interocular differences in posterior retinal curvature in anisometropic children and their association with interocular differences in myopia. Forty-two non-highly myopic children underwent ultra-widefield swept-source optical coherence tomography for 24-mm × 20-mm macular-centered retinal imaging. Gaussian curvature measurements were derived from corrected three-dimensional retinal reconstructions, based on Bruch's membrane segmentation. The retina was divided into six regions (I-VI) with diameters of 1, 3, 6, 9, 12, and 15 mm according to the Early Treatment of Diabetic Retinopathy Study grid standard. Regions I to III were classified as the macular zone, while regions IV to VI constituted the peripheral zone. The mean difference in spherical equivalent between the more myopic and less myopic eyes was 1.42 ± 0.51 diopters, while the difference in axial length (AL) was 0.58 ± 0.32 mm. The average retinal curvature in the macular zone was significantly steeper in the more myopic eyes (0.55 ± 0.15 × 10⁻² mm⁻²) compared to the less myopic eyes (0.46 ± 0.11 × 10⁻² mm⁻²; P < 0.001). Retinal curvature was significantly higher in regions I to IV of the more myopic eyes but lower in region VI (all P < 0.001). Furthermore, positive correlation was observed between macular retinal curvature disparities and AL differences (R = 0.340, P = 0.028), while peripheral retinal curvature disparities were negatively correlated with AL differences (R = -0.374, P = 0.015). Axial elongation drives region-specific posterior retinal remodeling in anisometropic children, steepening the macula and flattening the periphery. By investigating retinal curvature in anisometropic children, we provide morphologic insights that establish a basis for strategies to prevent and manage pediatric myopia.

Novel method for Gaussian beam size measurement with sub-pixel resolution using laser speckle analysis

This paper introduces a novel integrated guidance and control (IGC) framework for fixed-wing unmanned aerial vehicles (UAVs) designed for high-precision missions in uncertain and dynamic environments. The core innovation is a controller that explicitly models and actively compensates for structured inter-channel coupling (pitch, yaw, and roll) and external disturbances simultaneously. The proposed methodology synergizes an Adaptive Continuous High-Order Sliding Mode Controller (ACHOSMC) with a Lyapunov-based online neural network for real-time gain tuning. This architecture effectively suppresses chattering while preserving finite-time convergence. The embedded neural network autonomously mitigates the impacts of compound uncertainties, including ±20% parametric variations, Gaussian measurement noise (σ = 0.05), and a 0.2 s actuator delay. The framework’s performance is validated using a full six-degree-of-freedom (6-DOF) nonlinear model of a fixed-wing UAV. Rigorous simulations demonstrate superior performance, achieving over a 75% reduction in convergence time (10.8 s vs a 45 s baseline) and a 90% reduction in terminal miss distance (from 8 km to 750 m). Crucially, the system maintains stability during combined target maneuvers and environmental perturbations. These results confirm the proposed controller’s efficacy for time-critical missions that demand robust, high-precision engagement under multifaceted uncertainties. Future work will focus on bridging the simulation-to-reality gap through hardware-in-the-loop (HIL) and flight test validation.

We propose efficient algorithms for classically simulating Gaussian unitaries and measurements applied to non-Gaussian initial states. The constructions are based on decomposing the non-Gaussian states into linear combinations of Gaussian states. We use an extension of the covariance matrix formalism to efficiently track relative phases in the superpositions of Gaussian states. We get an exact simulation algorithm, which costs quadratically with the number of Gaussian states required to represent the initial state, and an approximate simulation algorithm, which costs linearly with the l1 norm of the coefficients associated with the superposition. We define measures of non-Gaussianity quantifying this simulation cost, which we call the Gaussian rank and the Gaussian extent. From the perspective of quantum resource theories, we investigate the properties of this type of non-Gaussianity measure and compute optimal decompositions for states relevant to continuous-variable quantum computing.

Accurately characterizing sub-nanometer gaps in plasmonic nanoparticle dimers is essential for understanding their optical properties, particularly in the transition from classical to quantum plasmonic behavior. While two-dimensional (scanning) transmission electron microscopy imaging provides high spatial resolution, it lacks the three-dimensional (3D) morphological information needed to reliably extract gap sizes. In this work, we combine electron tomography with a robust data analysis workflow to quantify interparticle gaps in gold nanosphere dimers with sub-nanometer precision. We show that gap size estimates are highly sensitive to reconstruction algorithms, segmentation thresholds, and meshing parameters. To overcome this, we introduce a model-fitting approach based on convolving a step function with a Gaussian, enabling consistent and accurate gap measurements even in the absence of a known ground truth. Validation on simulated datasets confirms pixel-level accuracy, and application to experimental data demonstrates the robustness and general applicability of the method. The resulting 3D reconstructions are directly integrated into electromagnetic simulations, allowing reliable interpretation of the optical response of the dimer. This workflow offers a broadly applicable strategy for correlating morphology and optical function in plasmonic systems and provides a crucial step toward resolving quantum effects in nanoscale light-matter interactions.

Abstract We consider the Schrödinger equation on the one dimensional torus with a general odd-power nonlinearity $p \geq 5$ , which is known to be globally well-posed in the Sobolev space $H^\sigma(\mathbb{T})$ , for every $\sigma \geq 1$ , thanks to the conservation and finiteness of the energy. For regularities σ &lt; 1, where this energy is infinite, we explore a globalization argument adapted to random initial data distributed according to the Gaussian measures µs, with covariance operator $(1-\Delta)^s$ , for s in a range $(s_p,\frac{3}{2}]$ . We combine a deterministic local Cauchy theory with the quasi-invariance of Gaussian measures µs, with additional Lq-bounds on the Radon-Nikodym derivatives, to prove that the Gaussian initial data generate almost surely global solutions. These Lq-bounds are obtained with respect to Gaussian measures accompanied by a cutoff on a renormalization of the energy; the main tools to prove them are the Boué-Dupuis variational formula and a Poincaré-Dulac normal form reduction. This approach is similar in spirit to Bourgain’s invariant argument [7] and to arecent work by Forlano-Tolomeo in [18].

Imaging thermal sources naturally yields Gaussian states at the receiver, raising the question of whether Gaussian measurements can perform optimally in quantum imaging. In this Letter, we establish no-go theorems on the performance of Gaussian measurements for imaging thermal sources in the limit of mean photon number per temporal mode ε→0 or source size L→0. We show that non-Gaussian measurements can outperform any Gaussian measurement in the scaling of the estimation variance with ε (or L). We also present several examples to illustrate the no-go results.

In this paper, we investigate the reverse isoperimetric inequality with respect to the Gaussian measure for convex sets in \mathbb{R}^{2} . While the isoperimetric problem for the Gaussian measure is well understood, many relevant aspects of the reverse problem have not yet been investigated. In particular, to the best of our knowledge, there seem to be no results on the shape that the isoperimetric set should take. Here, through a local perturbation analysis, we show that smooth perimeter-maximizing sets have locally flat boundaries. Additionally, we derive sharper perimeter bounds than those previously known, particularly for specific classes of convex sets, such as the convex sets symmetric with respect to the axes. Finally, for quadrilaterals with vertices on the coordinate axes, we prove that the set maximizing the perimeter “degenerates” into the x -axis, traversed twice.

Abstract It is well known in quantum optics that any process involving the preparation of a multimode gaussian state, followed by a gaussian operation and gaussian measurements, can be efficiently simulated by classical computers. Here, we provide evidence that computing transition amplitudes of Gaussian processes with a single-layer of non-linearities is hard for classical computers. To this end, we show how an efficient algorithm to solve this problem could be used to efficiently approximate outcome probabilities of a Gaussian boson sampling experiment. We also extend this complexity result to the problem of computing transition probabilities of Gaussian processes with two layers of non-linearities, by developing a Hadamard test for continuous-variable systems that may be of independent interest. Given recent experimental developments in the implementation of photon-photon interactions, our results may inspire new schemes showing quantum computational advantage or algorithmic applications of non-linear quantum optical systems realizable in the near-term.&amp;#xD;

IntroductionPredicting the relationship between diseases and microbes can significantly enhance disease diagnosis and treatment, while providing crucial scientific support for public health, ecological health, and drug development.MethodsIn this manuscript, we introduce an innovative computational model named BANSMDA, which integrates Bilinear Attention Networks with sparse autoencoder to uncover hidden connections between microbes and diseases. In BANSMDA, we first constructed a heterogeneous microbe-disease network by integrating multiple Gaussian similarity measures for diseases and microbes, along with known microbe-disease associations. And then, we employed a BAN-based autoencoder and a sparse autoencoder module to learn node representations within this newly constructed heterogeneous network. Finally, we evaluated the prediction performance of BANSMDA using a 5-fold cross-validation framework.ConclusionExperiments results showed that BANSMDA achieved superior performance compared to other cutting-edge methods. To further assess its effectiveness, we carried out case studies on two common diseases (including Asthma and Colorectal carcinoma) and two important microbial genera (including Escherichia and Bacteroides), and in the top 20 predicted microbes, there were 19 and 20 having been confirmed by published literature respectively. Besides, in the top 20 predicted diseases, there were 19 and 19 having been confirmed by published literature separately. Therefore, it is easy to conclude that BANSMDA can achieve satisfactory prediction ability.

Random fields on Hilbert spaces with their equivalent Gaussian measures

We provide a graphical method to describe and analyze non-Gaussian quantum states using a hypergraph framework. These states are pivotal resources for quantum computing, communication, and metrology, but their characterization is hindered by their complex high-order correlations. The framework encapsulates transformation rules for a series of typical Gaussian unitary operation and local quadrature measurement, offering a visually intuitive tool for manipulating such states through experimentally feasible pathways. Notably, we develop methods for the generation of complex hypergraph states with more or higher-order hyperedges from simple structures through Gaussian operations only, facilitated by our graphical rules. We present illustrative examples on the preparation of non-Gaussian states rooted in these graph-based formalisms, revealing their potential to advance continuous-variable general quantum computing capabilities.

We introduce Littlewood Paley functions defined in terms of a reparameterization of the Ornstein-Uhlenbeck semigroup obtaining that these operators are bounded in \(L^p\), \(1&lt;p&lt;\infty\), with respect to the unidimensional gaussian measure, by means of singular integrals theory. In addition, we study the Abel summability of the Fourier Hermite expansions considering their pointwise convergence and their convergence in the \(L^p\) sense, obtaining a version of Tauber's theorem.

We present an asymptotic analysis of a stochastic two-compartmental cell division system with regulatory mechanisms inspired by Getto et al. (Math Biosci 245: 258-268, 2013). The hematopoietic system is modeled as a two-compartment system, where the first compartment consists of dividing cells in the bone marrow, referred to as type 0 cells, and the second compartment consists of post-mitotic cells in the blood, referred to as type 1 cells. Division and self-renewal of type 0 cells are regulated by the population density of type 1 cells. By scaling up the initial population, we demonstrate that the scaled dynamics converges in distribution to the solution of a system of ordinary differential equations (ODEs). This system of ODEs exhibits a unique non-trivial equilibrium that is globally stable. Furthermore, we establish that the scaled fluctuations of the density dynamics converge in law to a linear diffusion process with time-dependent coefficients. When the initial data is Gaussian, the limit process is a Gauss-Markov process. We analyze its asymptotic properties to elucidate the joint structure of both compartments over large times. This is achieved by proving exponential convergence in the 2-Wasserstein metric for the associated Gaussian measures on an [Formula: see text] Hilbert space. Finally, we apply our results to compare the effects of regulating division and self-renewal of type 0 cells, providing insights into their respective roles in maintaining hematopoietic system stability.

The study of quantum thermodynamics aims to elucidate the role played by quantum principles in the emergent features of quantum thermodynamic processes. Specifically, it is of fundamental importance to understand how quantum correlation among different parties enables thermodynamic features distinguishable from those arising in classical thermodynamics. In this work, we investigate the relation between extractable work and quantum correlations for two-mode Gaussian states. We examine the change in local energy occurring at one party due to a Gaussian measurement performed on the other in relation to the quantum correlations of two-mode states classified as separable, entangled, and steerable states. Our analysis reveals a clear quantitative difference in the extractable work, depending on the class of states to which the two-mode state belongs.

Millimeter-wave (mmWave) MIMO systems have emerged as a key enabling technology for next-generation wireless networks, addressing the growing demand for ultra-high data rates through the utilization of wide bandwidths and large-scale antenna configurations. Beyond communication capabilities, these systems offer inherent advantages for integrated sensing applications, particularly in scenarios requiring precise object detection and localization. The sparse mmWave channel in the beamspace domain allows fewer radio-frequency (RF) chains by selecting dominant beams, boosting both communication efficiency and sensing resolution. However, existing channel estimation methods, such as learned approximate message passing (LAMP) networks, rely on computationally intensive iterations. This becomes particularly problematic in large-scale system deployments, where estimation inaccuracies can severely degrade sensing performance. To address these limitations, we propose a low-complexity channel estimator using a non-iterative reconstruction network (NIRNet) with a learning-based selection matrix (LSM). NIRNet employs a convolutional layer for efficient, non-iterative beamspace channel reconstruction, significantly reducing computational overhead compared to LAMP-based methods, which is vital for real-time sensing. The LSM generates a signal-aware Gaussian measurement matrix, outperforming traditional Bernoulli matrices, while a denoising network enhances accuracy under low SNR conditions, improving sensing resolution. Simulations show the NIRNet-based algorithm achieves a superior normalized mean squared error (NMSE) and an achievable sum rate (ASR) with lower complexity and reduced training overhead.