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In this paper, we explore and compare four criteria for continuous-variable entanglement, which serve as sufficient conditions for determining the separability of Gaussian two-mode states. Our system comprises two nano-mechanical resonators coupled to a hybrid doubly resonant optomechanical cavity system integrated with a non-degenerate optical parametric amplifier. The entanglement between the mechanical oscillators is primarily driven by non-classical state transitions of injected photons from the squeezed vacuum reservoir and intracavity squeezed radiation induced by radiation pressure in which the system operates in the weak coupling regime within good cavity limit. Our findings indicate that while the applied inseparability criteria show similar entanglement patterns within specific parameter ranges, the degree of entanglement varies depending on the chosen criteria. Additionally, the combined effects of injected squeezing and the parametric amplifier significantly enhance the entanglement when optimal parameters are selected. We also observe that the strength of the entanglement is mainly influenced by optomechanical cooperativity and thermal noise from the mechanical baths. The entanglement levels can be controlled by carefully adjusting these parameters, suggesting potential applications in quantum metrology and quantum information processing., Comment: 13 pages, 7 figures

An integer-valued polynomial $P(x,y,z)$ is said to be universal (over $\mathbb Z$) if each nonnegative integer can be written as $P(x,y,z)$ with $x,y,z\in\mathbb Z$. In this paper, we mainly introduce a new technique to determine the universality of some sums in the form $x(a_1x+a_2)/2+y(b_1y+b_2)/2+z(c_1z+c_2)/2$ (with $a_1-a_2,b_1-b_2,c_1-c_2$ all even) conjectured by Sun, using various identities of Ramanujan's theta functions. For example, we prove that $x(3x+1)+y(3y+2)+2z(3z+2)$ and $x(4x+r)+y(3y+1)/2+z(7z+1)/2\ (r=1,3)$ are universal., Comment: 22 pages, refined version with some remarks added

We investigate new properties of the fractional Dirichlet Laplacian on smooth bounded domains and establish fractional product estimates and nonlinear Poincar\'e inequalities. We also use these tools to study the long-time dynamics of the surface quasi-geostrophic equation forced by some given time-independent body forces in the presence of physical boundaries and prove the existence of a finite-dimensional global attractor., Comment: 24 pages

We consider an electrodiffusion model describing the evolution of $N$ ionic species in a three-dimensional fluid flowing through a porous medium and forced by added body charges. We address the global well-posedness and long-time dynamics of the model. In the absence of added charges, we prove that the ionic concentrations decay exponentially fast in time to their initial spatial averages in all Sobolev norms. When the fluid undergoes the influence of given time-independent charges, we obtain the existence of a finite-dimensional global attractor.

Motivated by the recent successful application of physics-informed neural networks (PINNs) to solve Boltzmann-type equations [S. Jin, Z. Ma, and K. Wu, J. Sci. Comput., 94 (2023), pp. 57], we provide a rigorous error analysis for PINNs in approximating the solution of the Boltzmann equation near a global Maxwellian. The challenge arises from the nonlocal quadratic interaction term defined in the unbounded domain of velocity space. Analyzing this term on an unbounded domain requires the inclusion of a truncation function, which demands delicate analysis techniques. As a generalization of this analysis, we also provide proof of the asymptotic preserving property when using micro-macro decomposition-based neural networks.

We consider the \emph{exact} error correction of a noisy Euclidean distance matrix, EDM, where the elements are the squared distances between $n$ points in $R^d$. For our problem we are given two facts: (i) the embedding dimension, $d$, (ii) \emph{exactly one} distance in the data is corrupted by \emph{nonzero noise}. But we do \underline{not} know the magnitude nor position of the noise. Thus there is a combinatorial element to the problem. We present three solution techniques. These use three divide and conquer strategies in combination with three versions of facial reduction that use: exposing vectors, facial vectors, and Gale transforms. This sheds light on the connections between the various forms of facial reduction related to Gale transforms. Our highly successful empirics confirm the success of these approaches as we can solve huge problems of the order of $100,000$ nodes in approximately one minute to machine precision. \\Our algorithm depends on identifying whether a principal submatrix of the \EDM contains the corrupted element. We provide a theorem for doing this that is related to the existing results for identifying \emph{yielding} elements, i.e.,~we provide a characterization for guaranteeing the perturbed EDM remains an EDM with embedding dimension $d$. The characterization is particularly simple in the $d=2$ case. \\In addition, we characterize when the intuitive approach of the nearest EDM problem, solves our problem. In fact, we show that this happens if, and only if, the original distance element is $0$, degenerate, and the perturbation is negative.

Current research in breast cancer Magnetic Resonance Imaging (MRI), especially with Artificial Intelligence (AI), faces challenges due to the lack of expert segmentations. To address this, we introduce the MAMA-MIA dataset, comprising 1506 multi-center dynamic contrast-enhanced MRI cases with expert segmentations of primary tumors and non-mass enhancement areas. These cases were sourced from four publicly available collections in The Cancer Imaging Archive (TCIA). Initially, we trained a deep learning model to automatically segment the cases, generating preliminary segmentations that significantly reduced expert segmentation time. Sixteen experts, averaging 9 years of experience in breast cancer, then corrected these segmentations, resulting in the final expert segmentations. Additionally, two radiologists conducted a visual inspection of the automatic segmentations to support future quality control studies. Alongside the expert segmentations, we provide 49 harmonized demographic and clinical variables and the pretrained weights of the well-known nnUNet architecture trained using the DCE-MRI full-images and expert segmentations. This dataset aims to accelerate the development and benchmarking of deep learning models and foster innovation in breast cancer diagnostics and treatment planning., Comment: 15 paes, 7 figures, 3 tables

In this paper, it was proposed a new concept of the inexact higher degree $(\delta, L, q)$-model of a function that is a generalization of the inexact $(\delta, L)$-model, $(\delta, L)$-oracle and $(\delta, L)$-oracle of degree $q \in [0,2)$. Some examples were provided to illustrate the proposed new model. Adaptive inexact gradient and fast gradient methods for convex and strongly convex functions were constructed and analyzed using the new proposed inexact model. A universal fast gradient method that allows solving optimization problems with a weaker level of smoothness, among them non-smooth problems was proposed. For convex optimization problems it was proved that the proposed gradient and fast gradient methods could be converged with rates $O\left(\frac{1}{k} + \frac{\delta}{k^{q/2}}\right)$ and $O\left(\frac{1}{k^2} + \frac{\delta}{k^{(3q-2)/2}}\right)$, respectively. For the gradient method, the coefficient of $\delta$ diminishes with $k$, and for the fast gradient method, there is no error accumulation for $q \geq 2/3$. It proposed a definition of an inexact higher degree oracle for strongly convex functions and a projected gradient method using this inexact oracle. For variational inequalities and saddle point problems, a higher degree inexact model and an adaptive method called Generalized Mirror Prox to solve such class of problems using the proposed inexact model were proposed. Some numerical experiments were conducted to demonstrate the effectiveness of the proposed inexact model, we test the universal fast gradient method to solve some non-smooth problems with a geometrical nature.

In this paper, we analyze a three-dimensional Nernst-Planck-Boussinesq (NPB) system that describes ionic electrodiffusion in an incompressible viscous fluid. This new model incorporates variational temperature and is forced by buoyancy force stemming from temperature and salinity fluctuations, enhancing its generality and realism. The electromigration term in the NPB system displays a complex nonlinear structure influenced by the reciprocal of the temperature that distinguishes its mathematical aspects from other electrodiffusion models studied in the literature. We address the global existence of weak solutions to the NPB system on the three-dimensional torus for large initial data. In addition, we study the long-time dynamics of these weak solutions and the associated relative entropies and establish their exponential decay in time to steady states., Comment: 28 pages

We prove logarithmic Sobolev inequalities on higher-dimensional bounded smooth domains based on novel Gagliardo-Nirenberg type interpolation inequalities. Moreover, we use them to address the long-time dynamics of some nonlinear nonlocal drift-diffusion models and prove the exponential decay of their solutions to constant steady states.

In this paper, we study the coupled Keller-Segel-Navier-Stokes system, which models chemotaxis occuring in ambient viscous fluid. We consider this nonlinear, nonlocal system on a periodic strip, equipped with homogeneous Neumann boundary conditions for the Keller-Segel part and no-slip boundary condition for the fluid part. We prove the simultaneous space-time analyticity of the solution up to the boundary based on energy methods., Comment: 21 pages

We systematically analyze the accuracy of Physics-Informed Neural Networks (PINNs) in approximating solutions to the critical Surface Quasi-Geostrophic (SQG) equation on two-dimensional periodic boxes. The critical SQG equation involves advection and diffusion described by nonlocal periodic operators, posing challenges for neural network-based methods that do not commonly exhibit periodic boundary conditions. In this paper, we present a novel approximation of these operators using their nonperiodic analogs based on singular integral representation formulas and use it to perform error estimates. This idea can be generalized to a larger class of nonlocal partial differential equations whose solutions satisfy prescribed boundary conditions, thereby initiating a new PINNs theory for equations with nonlocalities., Comment: 21 pages

In this paper, the optimal convergence rate $O\left(N^{-1/2}\right)$ (where $N$ is the total number of iterations performed by the algorithm), without the presence of a logarithmic factor, is proved for mirror descent algorithms with special time-varying step sizes, for solving classical constrained non-smooth problems, problems with the composite model and problems with non-smooth functional (inequality types) constraints. The proven result is an improvement on the well-known rate $O\left(\log (N) N^{-1/2}\right)$ for the mirror descent algorithms with the time-varying step sizes under consideration. It was studied a new weighting scheme assigns smaller weights to the initial points and larger weights to the most recent points. This scheme improves the convergence rate of the considered mirror descent methods, which in the conducted numerical experiments outperform the other methods providing a better solution in all the considered test problems., Comment: Preprint under update

The \emph{simplified} Wasserstein barycenter problem, also known as the cheapest hub problem, consists in selecting one point from each of $k$ given sets, each set consisting of $n$ points, with the aim of minimizing the sum of distances to the barycenter of the $k$ chosen points. This problem is also known as the cheapest hub problem. This problem is known to be NP-hard. We compute the Wasserstein barycenter by exploiting the Euclidean distance matrix structure to obtain a facially reduced doubly nonnegative, \DNNp, relaxation. The facial reduction provides a natural splitting for applying the symmetric alternating directions method of multipliers (\sADMM) to the \DNN relaxation. The \sADMM method exploits structure in the subproblems to find strong upper and lower bounds. In addition, we extend the problem to allow varying $n_j$ points for the $j$-th set. The purpose of this paper is twofold. First we want to illustrate the strength of this \DNN relaxation with a splitting approach. Our numerical tests then illustrate the surprising success on random problems, as we generally, efficiently, find the provable exact solution of this NP-hard problem. Comparisons with current commercial software illustrate this surprising efficiency. However, we demonstrate and prove that there is a duality gap for problems with \emph{enough} multiple optimal solutions, and that this arises from problems with highly symmetrized structure.

We consider an electrodiffusion model that describes the intricate interplay of multiple ionic species with a two-dimensional, incompressible, viscous fluid subjected to stochastic additive noise. This system involves nonlocal nonlinear drift-diffusion Nernst-Planck equations for ionic species and stochastic Navier-Stokes equations for fluid motion under the influence of electric and time-independent forces. Under the selective boundary conditions imposed on the concentrations, we establish the existence and uniqueness of global pathwise solutions to this system on smooth bounded domains. Our study also investigates long-time ionic concentration dynamics and explores Feller properties of the associated Markovian semigroup. In the context of equal diffusive species and under appropriate conditions, we demonstrate the existence of invariant ergodic measures supported on $H^2$. We then enhance the ergodicity results on periodic tori and obtain smooth invariant measures under a constraint on the initial spatial averages of the concentrations. The uniqueness of the invariant measures on periodic boxes and smooth bounded domains is further established when the noise forces sufficient modes, and the diffusivities of the species are large. Finally, in the case of two ionic species with equal diffusivities and valences of $1$ and $-1$, we study the rate of convergence of the Markov transition kernels to the invariant measure and obtain unconditional, unique exponential ergodicity for the model., Comment: 48 pages

The 3rd Generation Partnership Project (3GPP) is currently studying machine learning (ML) for the fifth generation (5G)-Advanced New Radio (NR) air interface, where spatial and temporal-domain beam prediction are important use cases. With this background, this letter presents a low-complexity ML design that expedites the spatial-domain beam prediction to reduce the power consumption and the reference signaling overhead, which are currently imperative for frequent beam measurements. Complexity analysis and evaluation results showcase that the proposed model achieves state-of-the-art accuracy with lower computational complexity, resulting in reduced power consumption and faster beam prediction. Furthermore, important observations on the generalization of the proposed model are presented in this letter.

We consider a nonlocal nonlinear model with fractional diffusion motivated by studies of electroconvection phenomena in incompressible viscous fluids. We address the global well-posedness, global regularity and long time dynamics of the model in bounded smooth domains with Dirichlet boundary conditions. We prove the existence and uniqueness of exponentially decaying in time solutions for $H^1$ initial data regardless of the fractional dissipative regularity. In the presence of time independent body forces in the fluid, we prove the existence of a compact finite dimensional global attractor. In the case of periodic boundary conditions, we prove that the unique smooth solution is globally analytic in time, and belongs to a Gevrey class of functions that depends on the dissipative regularity of the model.