Nonlinear Space Research Articles

Fair Numerical Algorithm of Coset Cardinality Spectrum for Distributed Arithmetic Coding.

As a typical symbol-wise solution of asymmetric Slepian-Wolf coding problem, Distributed Arithmetic Coding (DAC) non-linearly partitions source space into disjoint cosets with unequal sizes. The distribution of DAC coset cardinalities, named the Coset Cardinality Spectrum (CCS), plays an important role in both theoretical understanding and decoder design for DAC. In general, CCS cannot be calculated directly. Instead, a numerical algorithm is usually used to obtain an approximation. This paper first finds that the contemporary numerical algorithm of CCS is theoretically imperfect and does not finally converge to the real CCS. Further, to solve this problem, we refine the original numerical algorithm based on rigorous theoretical analyses. Experimental results verify that the refined numerical algorithm amends the drawbacks of the original version.

A Tensor Method based on Enhanced Tensor Nuclear Norm and Hypergraph Laplacian Regularization for Pan-Cancer Omics Data Analysis.

As a powerful data representation technique, tensor robust principal component analysis (TRPCA) has been widely used for clustering and feature selection tasks. However, it ignores the significant difference in singular values of tensor data and the manifold information contained in different views, thereby causing serious degradation of conventional TRPCA performance. In this paper, a novel tensor method based on enhanced tensor nuclear norm and hypergraph Laplacian regularization (ETHLR) is developed to address the above problem. ETHLR can jointly learn the prior knowledge of singular values and high-order manifold structures in the unified tensor space and the view-specific feature spaces, respectively. Specifically, the enhanced tensor nuclear norm, namely, the weighted tensor Schatten p-norm, is used to shrink the singular values by fully considering the salient difference information of singular values and the complementary information embedded in the tensor space; the hypergraph Laplacian constraint helps encode high-order geometric structures among multiple samples in the nonlinear view-specific feature space. Furthermore, we employ inexact augmented Lagrange multipliers (ALM) to optimize the ETHLR method. Numerous experiments on pan-cancer omics data show that the superiority of ETHLR over several state-of-the-art competitors.

Unified Convergence Criteria of Derivative-Free Iterative Methods for Solving Nonlinear Equations

A local and semi-local convergence is developed of a class of iterative methods without derivatives for solving nonlinear Banach space valued operator equations under the classical Lipschitz conditions for first-order divided differences. Special cases of this method are well-known iterative algorithms, in particular, the Secant, Kurchatov, and Steffensen methods as well as the Newton method. For the semi-local convergence analysis, we use a technique of recurrent functions and majorizing scalar sequences. First, the convergence of the scalar sequence is proved and its limit is determined. It is then shown that the sequence obtained by the proposed method is bounded by this scalar sequence. In the local convergence analysis, a computable radius of convergence is determined. Finally, the results of the numerical experiments are given that confirm obtained theoretical estimates.

Numerical analysis of the high-order scheme of the damped nonlinear space fraction Schrödinger equation

Firstly, we construct a fourth-order numerical differential formula for approximating Riesz derivative. Substituting it into the damped nonlinear space fractional Schrödinger equation, the original equation becomes a matrix form of nonlinear ordinary differential equation system. Then, by means of implicit integrating factor method and Padé approximation, we obtain a new numerical scheme whose convergence order is higher than the existing algorithms. Finally, a numerical example is given to verify the effectiveness of the numerical algorithm and the correctness of the theoretical analysis.

An investigation of a closed-form solution for non-linear variable-order fractional evolution equations via the fractional Caputo derivative

Determining the non-linear traveling or soliton wave solutions for variable-order fractional evolution equations (VO-FEEs) is very challenging and important tasks in recent research fields. This study aims to discuss the non-linear space–time variable-order fractional shallow water wave equation that represents non-linear dispersive waves in the shallow water channel by using the Khater method in the Caputo fractional derivative (CFD) sense. The transformation equation can be used to get the non-linear integer-order ordinary differential equation (ODE) from the proposed equation. Also, new exact solutions as kink- and periodic-type solutions for non-linear space–time variable-order fractional shallow water wave equations were constructed. This confirms that the non-linear fractional variable-order evolution equations are natural and very attractive in mathematical physics.

Unified framework for over-damped string stable adaptive cruise control systems

Many recent experiments have shown that the current commercially available adaptive cruise control (ACC) systems demonstrate string instability phenomena. The lower-level time-lag, the sensor time-delay, and the input disturbance primarily contribute to such instability. This paper presents a unified strategy to design ACC systems compensating the lower-level time-lag, the sensor time-delay, and the input disturbance, by fulfilling the over-damped string stability criterion for homogeneous platoons. The over-damped feature associated with string stability explicitly addresses the safety of a platoon in a stricter sense, considering the transient convergence effects, such as over-shoot, under-shoot, and oscillations, and satisfying all the Lp stability norms, where p∈1,∞. In this article, the derived over-damped string-stable strategy compensating lag, delay, and disturbance (OSSCLDD) is presented in a general unified form that applies to any linear or convex velocity-dependent spacing functions. The derived strategy is demonstrated and validated via simulation using a linear constant time gap spacing function and two nonlinear spacing functions, namely, the variable time gap and the quadratic range spacing function.

A Group-Theoretic Approach to the Bifurcation Analysis of Spatial Cosserat-Rod Frameworks with Symmetry

We present a general approach to the bifurcation analysis of spatial framework structures with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of geometrically exact frameworks is not straightforward. We consider spatial structures comprising assemblages of Cosserat rods; the main difficulty arises from the nonlinear configuration space, due to the presence of (cross-sectional) rotation fields. We avoid this via a single-rod formulation, developed earlier by one of the authors, whereby the governing equations are embedded in a slightly enlarged linear space. The field equations for a framework then comprise the assembly of all such rod equations, supplemented by compatibility and equilibrium conditions at the joints. We demonstrate their equivariance under the natural symmetry group action, and the implementation of group-theoretic methods is now clear within the enlarged linear space. All potential generic, symmetry-breaking bifurcations are predicted a-priori. We then employ an open-source path-following code, which can detect and compute simple, one-dimensional bifurcations; multiple bifurcation points are beyond its capabilities. For the latter, we construct symmetry-reduced problems implemented by appropriate substructures. Multiple bifurcations are rendered simple, and the path-following code is again applicable. We first analyze a simple tripod framework, providing all details of our methodology. We then treat a more complex hexagonal space frame via the same approach. Both structures exhibit simple and double bifurcation points.

4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data.

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of objects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered. Also, different deforming objects, hereinafter referred to as 4D surfaces, evolve at different speeds and thus they need to be temporally aligned. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemannian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields (SRNF) where the [Formula: see text] metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces even in the presence of large elastic deformations and large variations in their execution rates; (2) the computation of geodesics between 4D surfaces; (3) the computation of statistical summaries, such as means and modes of variation, of collections of 4D surfaces; and (4) the synthesis of random 4D surfaces. We demonstrate the performance of the proposed framework using 4D facial surfaces and 4D human body shapes.

A Craft Essay on Trauma

A Craft Essay on Trauma

Design Finite Array with Non-linear Element Spacing

The final matrix presented here consists of identical matrix elements, located at random. The fixed patch antenna serves as the base element for the grille. Arrays are created by randomly placing a base element. Arrays of this type are difficult to model and can be resource-intensive. It will show you how you can construct matrices using the available finite matrix tools. When the array is built in this way, DFM (Domain Green Function Method) acceleration can be applied to reduce the required resources. 20 mV, 20 dB general orientation, 0 dB axis ratio

Adaptive progressive learning stochastic resonance for weak signal detection

Stochastic resonance (SR) can enhance signals by using noise. This has attracted more attention in the field of weak signal detection. In practical applications, owing to the non-adjustability of noisy signals, SR is required to adjust the system parameters adaptively to satisfy the conditions of the SR phenomenon. In this paper, an adaptive progressive learning SR method is proposed to improve the detection ability for weak signal, and the SR phenomenon is quantitatively defined. A theoretical learning framework is established with an improved reinforcement learning model by mapping the nonlinear system parameter space to a progressive learning set. By selecting a proper learning layer within a determined constraint range, the matching system parameters can be quickly and accurately searched to generate a desired optimal output. Numerical simulation results show that the signal energy and the output signal-to-noise ratio (SNR) can be enhanced significantly, which reflects an excellent weak signal detection performance especially for low SNR conditions. Finally, a diagnosis of the outer race fault signals of a rolling bearing confirms that the proposed method can effectively detect fault characteristics.

KVNet: An iterative 3D keypoints voting network for real-time 6-DoF object pose estimation

Accurate and efficient object pose estimation holds the indispensable part of virtual/augmented reality (VR/AR) and many other applications. While previous works focus on directly regressing 6D pose from RGB and depth image and thus suffer from the non-linearity of rotation space, we propose an iterative 3D keypoints voting network, named as KVNet. Specifically, our method decouples the pose into separate translation and rotation branch, both estimated by Hough voting scheme. By treating the uncertainty of keypoints’ vote as the Lipschitz continuous function of seed points’ fused embedding feature, our method is able to adaptively select the optimal keypoints vote. In this way, we argue that KVNet bridges the gap between the non-linear rotation space and linear Euclidean space, which introduces inductive bias for our network to learn the intrinsic pattern and infer 6D pose from RGB and depth images. Furthermore, our model will refine the initial keypoints localization with iterative fashion. Experiments show that across three challenging benchmark datasets (LineMOD, YCB-Video and Occlusion LineMOD), our method exhibits excellent performance.

Widely tunable electron bunch trains for the generation of high-power narrowband 1–10 THz radiation

Terahertz radiation enables the coherent excitation of many fundamental modes, such as molecular rotation, lattice vibration and spin precession. To excite the transient state of matter far out of equilibrium, high-power and tunable narrowband terahertz radiation sources have been in great demand for years. However, the terahertz source available at present cannot meet these tunability and power demands, leaving a large scientific gap that has yet to be fully explored. Here we convert the energy modulation induced by the nonlinear longitudinal space charge force to a density modulation and experimentally demonstrate the generation of electron bunch trains with modulation frequencies that are adjustable between 1 and 10 THz. The electron bunch trains can be directly used to produce tunable high-power narrowband terahertz radiation that fully covers the long-standing ‘terahertz gap’, which will open up many more possibilities in terahertz science. A tunable terahertz radiation pulse is demonstrated based on a linear accelerator. The emission frequency of this terahertz radiation is tunable from 1 to 10 THz by changing the bunching frequency of a 34 MeV electron beam. The pulse energy is at the submillijoule level.

ScGMAAE: Gaussian mixture adversarial autoencoders for diversification analysis of scRNA-seq data.

The progress of single-cell RNA sequencing (scRNA-seq) has led to a large number of scRNA-seq data, which are widely used in biomedical research. The noise in the raw data and tens of thousands of genes pose a challenge to capture the real structure and effective information of scRNA-seq data. Most of the existing single-cell analysis methods assume that the low-dimensional embedding of the raw data belongs to a Gaussian distribution or a low-dimensional nonlinear space without any prior information, which limits the flexibility and controllability of the model to a great extent. In addition, many existing methods need high computational cost, which makes them difficult to be used to deal with large-scale datasets. Here, we design and develop a depth generation model named Gaussian mixture adversarial autoencoders (scGMAAE), assuming that the low-dimensional embedding of different types of cells follows different Gaussian distributions, integrating Bayesian variational inference and adversarial training, as to give the interpretable latent representation of complex data and discover the statistical distribution of different types of cells. The scGMAAE is provided with good controllability, interpretability and scalability. Therefore, it can process large-scale datasets in a short time and give competitive results. scGMAAE outperforms existing methods in several ways, including dimensionality reduction visualization, cell clustering, differential expression analysis and batch effect removal. Importantly, compared with most deep learning methods, scGMAAE requires less iterations to generate the best results.

Geometric Deep Neural Network Using Rigid and Non-Rigid Transformations for Landmark-Based Human Behavior Analysis.

Deep learning architectures, albeit successful in most computer vision tasks, were designed for data with an underlying Euclidean structure, which is not usually fulfilled since pre-processed data may lie on a non-linear space. In this article, we propose a geometric deep learning approach using rigid and non-rigid transformations, named KShapenet, for 2D and 3D landmark-based human motion analysis. Landmark configuration sequences are first modeled as trajectories on Kendall's shape space and then mapped to a linear tangent space. The resulting structured data are then input to a deep learning architecture, which includes a layer that optimizes over rigid and non-rigid transformations of landmark configurations, followed by a CNN-LSTM network. We apply KShapenet to 3D human landmark sequences for action and gait recognition, and 2D facial landmark sequences for expression recognition, and demonstrate the competitiveness of the proposed approach with respect to state-of-the-art.

Correction To "Null controllability of a nonlinear age, space and two-sex structured population dynamics model"

Correction To "Null controllability of a nonlinear age, space and two-sex structured population dynamics model"

Recent modification of decomposition method for solving wave-like equation

In this article exact solution for nonlinear wave-like equations with variable coefficients will be obtain by using reliable manner depend on combined Laplace transform with decomposition technique and the results has shown a high-precision, smooth and the series solution is converge rapidly to exact analytic solution compared with other classic approaches. Suggested approach not needs any discretization by data of domain or presents assumption or neglect for a perturbation parameter in problems and not need to use any assumption to convert the non-linear terms into linear. Two examples of strongly nonlinear 2-dimensional space high order have been presented to show the convergence of solution obtained by suggested method to the exact.

Classification via Structure-Preserved Hypergraph Convolution Network for Hyperspectral Image

Graph convolutional network (GCN) as a combination of deep learning and graph learning has gained increasing attention in hyperspectral image (HSI) classification. However, most GCN methods consider the simple point-to-point structure between two pixels rather than the high-order structure of multiple pixels, which is contradict with the real feature distribution of ground object. And the nonlinear property of HSI also brings challenge for precise structural representation in GCN. To tackle these problems, this work proposes a structure preserved hypergraph convolution network (SPHGCN). It first builds a multiple neighborhood reconstruction (MNR) model to reveal the essential resemblance of multiple pixels in nonlinear spectral feature space. With the high-order structure, SPHGCN designs the hypergraph convolution operation for irregular feature aggregation among similar pixels from different regions, which achieves more discriminative features from multiple pixel nodes. Meanwhile, a structure preservation layer is built to optimize the distribution of convolutional features under the guidance of high-order structure. Moreover, SPHGCN integrates local regular convolution and irregular hypergraph convolution to learn the structured semantic feature of HSI. This strategy breaks the boundary restriction in traditional convolution and aggregates semantic feature across different image patches. Experiments on three HSI data sets indicate that SPHGCN outperforms a few state-of-the-art methods for HSI classification.

Null controllability of a nonlinear age, space and two-sex structured population dynamics model

<p style='text-indent:20px;'>In this paper, we study the null controllability of a nonlinear age, space and two-sex structured population dynamics model. This model is such that the nonlinearity and the couplage are at birth level. We consider a population with males and females and we are dealing with two cases of null controllability problems.</p><p style='text-indent:20px;'>The first problem is related to the total extinction, which means that, we estimate a time <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> to bring the male and female subpopulation density to zero. The second case concerns null controllability of male or female subpopulation. Since the absence of males or females in the population stops births; so, if we have the total extinction of the females at time <inline-formula><tex-math id="M2">\begin{document}$ T, $\end{document}</tex-math></inline-formula> and if <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula> is the life span of the individuals, at time <inline-formula><tex-math id="M4">\begin{document}$ T+A $\end{document}</tex-math></inline-formula> one will get certainly the total extinction of the population. Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system, and after the Schauder's fixed point theorem.</p>

Multiple exp-function method to solve the nonlinear space–time fractional partial differential symmetric regularized long wave (SRLW) equation and the (1+1)-dimensional Benjamin–Ono equation

In this study, we apply relatively analytical techniques, the multiple [Formula: see text]-function method, [Formula: see text]-function method and [Formula: see text]-expansion method to get approximate and analytic solutions of some nonlinear partial differential equations (PDEs), i.e., the nonlinear space–time fractional partial differential symmetric regularized long wave equation, an impressive model to characterize ion-acoustic and space change waves, the nonlinear [Formula: see text]-dimensional Fokas PDE, a meaningful multi-dimensional extension of the Kadomtsev–Petviashvili (KP) and Davey–Stewartson (DS) equations, [Formula: see text]-dimensional Bateman–Burgers equation, a simplification of a more complex and sophisticated model, and the [Formula: see text]-dimensional Benjamin–Ono equation, a model for the propagation of unidirectional internal waves in stratified fluids. Finally, we propose the numerical results in tables and discuss advantages and disadvantages of the mentioned methods.