Eigenvalues Of Tensor Research Articles

Structural transitions of a semi-flexible polyampholyte.

Polyampholytes (PAs) are charged polymers composed of positively and negatively charged monomers along their backbone. The sequence of the charged monomers and the bending of the chain significantly influence the conformation and dynamical behavior of the PA. Using coarse-grained molecular dynamics simulations, we comprehensively study the structural and dynamical properties of flexible and semi-flexible PAs. The simulation results demonstrate a flexible PA chain, displaying a transition from a coil to a globule in the parameter space of the charge sequence. In addition, the behavior of the mean-square displacement (MSD), denoted as ⟨(Δr(t))2⟩, reveals distinct dynamics, specifically for the alternating and charge-segregated sequences. The MSD follows a power-law behavior, where ⟨(Δr(t))2⟩ ∼ tβ, with β ≈ 3/5 and β ≈ 1/2 for the alternating sequence and the charge-segregated sequence in the absence of hydrodynamic interactions, respectively. However, when hydrodynamic interactions are incorporated, the exponent β shifts to ∼3/5 for the charge-segregated sequence and 2/3 for the well-mixed alternating sequence. For a semi-flexible PA chain, varying the bending rigidity and electrostatic interaction strength (Γe) leads to distinct, fascinating conformational states, including globule, bundle, and torus-like conformations. We show that PAs acquire circular and hairpin-like conformations in the intermediate bending regime. The transition between various conformations is identified in terms of the shape factor estimated from the ratios of eigenvalues of the gyration tensor.

C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^3$$\\end{document} Matching Conditions for Anisotropic Fluids

The C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^3$$\\end{document} approach is an invariant formalism that utilizes the eigenvalues of the Riemann curvature tensor to match spacetimes across a specific matching surface. We apply this approach to match an anisotropic fluid with an exterior vacuum solution, including the case in which discontinuities appear on the matching surface. As a particular example, a class of analytic solutions, which describe the gravitational field of realistic neutron stars, is matched to the exterior Schwarzschild spacetime.

Evaluation of a decomposition-based interpolation method for fourth-order fiber-orientation tensors: An eigensystem approach

We propose and assess a new decomposition-based interpolation method on fourth-order fiber-orientation tensors. This method can be used to change the resolution of discretized fields of fiber-orientation tensors, e.g., obtained from flow simulations or computer tomography, which are common in the context of short- and long-fiber–reinforced composites. The proposed interpolation method separates information on structure and orientation using a parametrization which is based on tensor components and a unique eigensystem. To identify this unique eigensystem of a given fourth-order fiber-orientation tensor in the absence of material symmetry, we propose a sign convention on tensor coefficients. We explicitly discuss challenges associated with material symmetries, e.g., non-distinct eigenvalues of the second-order fiber-orientation tensor and propose algorithms to obtain a unique set of parameters combined with a minimal number of eigensystems of a given fourth-order fiber-orientation tensor. As a side product, we specify for the first time, parametrizations and admissible parameter ranges of cubic, tetragonal, and trigonal fiber-orientation tensors. Visualizations in terms of truncated Fourier series, quartic plots, and tensor glyphs are compared.

Relative equilibria of mechanical systems with rotational symmetry

We consider the task of classifying relative equilibria for mechanical systems with rotational symmetry. We divide relative equilibria into two natural groups: a generic class which we call normal, and a non-generic abnormal class. The eigenvalues of the locked inertia tensor descend to shape-space and endow it with the geometric structure of a three-web with the property that any normal relative equilibrium occurs as a critical point of the potential restricted to a leaf from the web. To demonstrate the utility of this web structure we show how the spherical three-body problem gives rise to a web of Cayley cubics on the three-sphere, and use this to fully classify the relative equilibria for the case of equal masses.

Solving a class of infinite‐dimensional tensor eigenvalue problems by translational invariant tensor ring approximations

AbstractWe examine a method for solving an infinite‐dimensional tensor eigenvalue problem , where the infinite‐dimensional symmetric matrix exhibits a translational invariant structure. We provide a formulation of this type of problem from a numerical linear algebra point of view and describe how a power method applied to is used to obtain an approximation to the desired eigenvector. This infinite‐dimensional eigenvector is represented in a compact way by a translational invariant infinite Tensor Ring (iTR). Low rank approximation is used to keep the cost of subsequent power iterations bounded while preserving the iTR structure of the approximate eigenvector. We show how the averaged Rayleigh quotient of an iTR eigenvector approximation can be efficiently computed and introduce a projected residual to monitor its convergence. In the numerical examples, we illustrate that the norm of this projected iTR residual can also be used to automatically modify the time step to ensure accurate and rapid convergence of the power method.

The hierarchical cosmic web and assembly bias

Accurate modeling of galaxy distributions is paramount for cosmological analysis using galaxy redshift surveys. However, this endeavor is often hindered by the computational complexity of resolving the dark matter halos that host these galaxies. To address this challenge, we propose the development of effective assembly bias models down to small scales, i.e., going beyond the local density dependence capturing non-local cosmic evolution. We introduce a hierarchical cosmic web classification that indirectly captures up to third-order long- and short-range non-local bias terms. This classification system also enables us to maintain positive definite parametric bias expansions. Specifically, we subdivide the traditional cosmic web classification, which is based on the eigenvalues of the tidal field tensor, with an additional classification based on the Hessian matrix of the negative density contrast. We obtain the large-scale dark matter field on a mesh with ~3.9 h -1 Mpc cell side resolution through Augmented Lagrangian Perturbation Theory. To assess the effectiveness of our model, we conduct tests using a reference halo catalogue extracted from the UNIT project simulation, which was run within a cubical volume of 1 h -1 Gpc side. The resulting mock halo catalogs, generated through our approach, exhibit a high level of accuracy in terms of the one-, two- and three-point statistics. They reproduce the reference power-spectrum within better than 2 percent accuracy up to wavenumbers k ~ 0.8 h Mpc-1 and provide accurate bispectra within the scales that are crucial for cosmological analysis. This effective bias approach provides a forward model appropriate for field-level cosmological inference and holds significant potential for facilitating cosmological analysis of galaxy redshift surveys, particularly in the context of projects such as DESI, EUCLID, and LSST.

Jacobi Stability Analysis of Liu System: Detecting Chaos

By utilizing the Kosambi–Cartan–Chern (KCC) geometric theory, this paper is dedicated to providing novel insights into the Liu dynamical system, which stands out as one of the most distinctive and noteworthy nonlinear dynamical systems. Firstly, five important geometrical invariants of the system are obtained by associating the nonlinear connection with the Berwald connection. Secondly, in terms of the eigenvalues of the deviation curvature tensor, the Jacobi stability of the Liu dynamical system at fixed points is investigated, which indicates that three fixed points are Jacobi unstable. The Jacobi stability of the system is analyzed and compared with that of Lyapunov stability. Lastly, the dynamical behavior of components of the deviation vector is studied, which serves to geometrically delineate the chaotic behavior of the system near the origin. The onset of chaos for the Liu dynamical system is obtained. This work provides an analysis of the Jacobi stability of the Liu dynamical system, serving as a useful reference for future chaotic system research.

Repulsive gravity in regular black holes

We evaluate the effects of repulsive gravity using first order geometric invariants, i.e. the Ricci scalar and the eigenvalues of the Riemann curvature tensor, for three regular black holes, namely the Bardeen, Hayward, and Dymnikova spacetimes. To examine the repulsive effects, we calculate their respective onsets and regions of repulsive gravity. Afterwards, we compare the repulsive regions obtained from these metrics among themselves and then with the predictions got from the Reissner–Nordström and Schwarzschild–de Sitter. A notable characteristic, observed in all these metrics, is that the repulsive regions appear to be unaffected by the mass that generates the regular black hole. This property emerges due to the invariants employed in our analysis, which do not change sign through linear combinations of the mass and the free coefficients of the metrics. As a result, gravity can change sign independently of the specific values acquired by the mass. This conclusion suggests a potential incompleteness of regular solutions, particularly in terms of their repulsive effects. To further highlight this finding, we numerically compute, for the Reissner–Nordström and Schwarzschild–de Sitter solutions, the values of mass, M, that emulate the repulsive effects found in the Bardeen and Hayward spacetimes. These selected values of M provide evidence that regular black holes do not incorporate repulsive effects by means of the masses used to generate the solutions themselves. Implications and physical consequences of these results are then discussed in detail.

Anisotropic generalized polytropic spheres: Regular 3D black holes

We model gravitating relativistic 3D spheres composed of an anisotropic fluid in which the radial and transverse components of the pressure correspond to the vacuum energy and a generalized polytropic equation-of-state, respectively. By using the generalized Tolman–Oppenheimer–Volkoff (TOV) equation, and solving the complete system of equations for these anisotropic generalized polytropic spheres, for a given range of model parameters, we find three novel classes of asymptotically AdS black hole solutions with regular core. We show the regularity of the solutions using curvature scalars and the formalism of geodesic completeness. Then, using the eigenvalues of the Riemann curvature tensor, we consider the effects of repulsive gravity in the three static 3D regular black holes, concluding that their regular behavior can be explained as due to the presence of repulsive gravity near the center of the objects. Finally, we study the stability of the regular black holes under the flow of the energy through the Cauchy horizon.

Likelihood-free posterior estimation and uncertainty quantification for diffusion MRI models

Abstract Diffusion magnetic resonance imaging (dMRI) allows to estimate brain tissue microstructure as well as the connectivity of the white matter (known as tractography). Accurate estimation of the model parameters (by solving the inverse problem) is thus very important to infer the underlying biophysical tissue properties and fiber orientations. Although there has been extensive research on this topic with a myriad of dMRI models, most models use standard nonlinear optimization techniques and only provide an estimate of the model parameters without any information (quantification) about uncertainty in their estimation. Further, the effect of this uncertainty on the estimation of the derived dMRI microstructural measures downstream (e.g., fractional anisotropy) is often unknown and is rarely estimated. To address this issue, we first design a new deep-learning algorithm to identify the number of crossing fibers in each voxel. Then, at each voxel, we propose a robust likelihood-free deep learning method to estimate not only the mean estimate of the parameters of a multi-fiber dMRI model (e.g., the biexponential model), but also its full posterior distribution. The posterior distribution is then used to estimate the uncertainty in the model parameters as well as the derived measures. We perform several synthetic and in-vivo quantitative experiments to demonstrate the robustness of our approach for different noise levels and out-of-distribution test samples. Besides, our approach is computationally fast and requires an order of magnitude less time than standard nonlinear fitting techniques. The proposed method demonstrates much lower error (compared to existing methods) in estimating several metrics, including number of fibers in a voxel, fiber orientation, and tensor eigenvalues. The proposed methodology is quite general and can be used for the estimation of the parameters from any other dMRI model.

Linear to multi-linear algebra and systems using tensors

In the past few decades, multi-linear algebra also known as tensor algebra has been adapted and employed as a tool for various engineering applications. Recent developments in tensor algebra have indicated that several well-known concepts from linear algebra can be extended to a multi-linear setting with the help of a special form of tensor contracted product, known as the Einstein product. Thus, the tensor contracted product and its properties can be harnessed to define the notions of multi-linear system theory where the input, output signals, and the system are inherently multi-domain or multi-modal. This study provides an overview of tensor algebra tools which can be seen as an extension of linear algebra, at the same time highlighting the differences and advantages that the multi-linear setting brings forth. In particular, the notions of tensor inversion, tensor singular value, and tensor eigenvalue decomposition using the Einstein product are explained. In addition, this study also introduces the notion of contracted convolution for both discrete and continuous multi-linear system tensors. Tensor network representation of various tensor operations is also presented. In addition, application of tensor tools in developing transceiver schemes for multi-domain communication systems, with an example of MIMO CDMA system, is presented. This study provides a foundation for professionals whose research involves multi-domain or multi-modal signals and systems.

The correlations between galaxy properties in different environments of the cosmic web

Abstract We study the correlations between (u-r) colour, stellar mass, specific star formation rate (sSFR) and metallicity of galaxies in different geometric environments of the cosmic web using a volume limited sample from the SDSS. The geometric environment at the location of each galaxy is determined using the eigenvalues of the tidal tensor in three dimensions. We use the Pearson correlation coefficient (PCC) and the normalized mutual information (NMI) to quantify the correlations between these galaxy properties in sheets, filaments and clusters after matching the stellar mass distributions of the galaxies in these environments. A two-tailed t-test assesses the statistical significance of the observed differences between these relations in different geometric environments. The null hypothesis can be rejected at >99.99% significance level in most of the cases, suggesting that the scaling relations between the observable galaxy properties are susceptible to the geometric environments of the cosmic web.

Normalized Newton method to solve generalized tensor eigenvalue problems

AbstractThe problem of generalized tensor eigenvalue is the focus of this paper. To solve the problem, we suggest using the normalized Newton generalized eigenproblem approach (NNGEM). Since the rate of convergence of the spectral gradient projection method (SGP), the generalized eigenproblem adaptive power (GEAP), and other approaches is only linear, they are significantly improved by our proposed method, which is demonstrated to be locally and cubically convergent. Additionally, the modified normalized Newton method (MNNM), which converges to symmetric tensors Z‐eigenpairs under the same ‐Newton stability requirement, is extended by the NNGEM technique. Using a Gröbner basis, a polynomial system solver (NSolve) generates all of the real eigenvalues for us. To illustrate the efficacy of our methodology, we present a few numerical findings.

Spectrum Situation Awareness for Space-Air-Ground Integrated Networks Based on Tensor Computing.

The spectrum situation awareness problem in space-air-ground integrated networks (SAGINs) is studied from a tensor-computing perspective. Tensor and tensor computing, including tensor decomposition, tensor completion and tensor eigenvalues, can satisfy the application requirements of SAGINs. Tensors can effectively handle multidimensional heterogeneous big data generated by SAGINs. Tensor computing is used to process the big data, with tensor decomposition being used for dimensionality reduction to reduce storage space, and tensor completion utilized for numeric supplementation to overcome the missing data problem. Notably, tensor eigenvalues are used to indicate the intrinsic correlations within the big data. A tensor data model is designed for space-air-ground integrated networks from multiple dimensions. Based on the multidimensional tensor data model, a novel tensor-computing-based spectrum situation awareness scheme is proposed. Two tensor eigenvalue calculation algorithms are studied to generate tensor eigenvalues. The distribution characteristics of tensor eigenvalues are used to design spectrum sensing schemes with hypothesis tests. The main advantage of this algorithm based on tensor eigenvalue distributions is that the statistics of spectrum situation awareness can be completely characterized by tensor eigenvalues. The feasibility of spectrum situation awareness based on tensor eigenvalues is evaluated by simulation results. The new application paradigm of tensor eigenvalue provides a novel direction for practical applications of tensor theory.

Phenomenological emergent dark energy versus the ΛCDM: ellipticity of cosmic voids

ABSTRACT We disclose the influences of the phenomenological emergent dark energy (PEDE) and its generalization (GEDE) on the distribution of tidal ellipticity of the cosmic voids. We also compare our results with that of the standard ΛCDM cosmology. The underlying models consist with recent observations (specially in favour of H0 tension) and may impact the cosmic voids geometry. We employ the analytic approach based on statistics of the tidal tensor eigenvalues. We confirm a significant sensitivity of the ellipticity distribution function for PEDE and GEDE cosmology. We observe the largest deviation from the standard cosmology for GEDE in the range ($z\gt 0, R_{\mathrm{L}}\ge 4\, h^{-1}\, \mathrm{Mpc}$). Working on these distribution functions we reveal that the mean ellipticity (〈ε〉) profile of GEDE versus redshift lies below than that of PEDE and ΛCDM indicating that GEDE leads to a less elongated society of voids with respect to other models. We also identify a tight correlation between σ8 and the mean ellipticity evolution. Finally, we explore the mean ellipticity versus RL and estimate that in this case GEDE leads to a different ellipticity curve (up to 11 ${{\ \rm per\ cent}}$ smaller than that of ΛCDM).

Defect Wormholes Are Defective

The various “defect wormholes” developed by Klinkhamer have recently attracted considerable attention—especially in view of the fact that the simplest example, the so-called “vacuum defect wormhole”, was claimed to be an everywhere-vacuum everywhere-Ricci-flat exact solution to the Einstein equations. This claim has been conclusively refuted by Feng, and in the current article, we take a deeper look at exactly what goes wrong. The central issue is this: Although Klinkhamer’s specific representation of the metric gab is smooth (C∞), his inverse metric gab is not even everywhere continuous (C0), being undefined at the wormhole throat. This situation implies that one should very carefully investigate curvature tensors at the throat using the Israel–Lanczos–Sen thin-shell formalism. Doing so reveals the presence of a delta-function energy-condition-violating thin shell of matter at the wormhole throat. The “defect wormholes” are thus revealed to be quite ordinary “cut-and-paste” thin-shell wormholes, but represented in a coordinate system that is unfortunately pathological at exactly the same place that all the interesting physics occurs. To help clarify the situation, we shall focus on the behavior of suitable coordinate invariants—the Ricci scalar, the eigenvalues of the mixed Rab Ricci tensor, and the eigenvalues of the mixed Rabcd Riemann tensor.

Tides in clouds: control of star formation by long-range gravitational force

ABSTRACT Gravity drives the collapse of molecular clouds through which stars form, yet the exact role of gravity in cloud collapse remains a complex issue. Studies point to a picture where star formation occurs in clusters. In a typical, pc-sized cluster-forming region, the collapse is hierarchical, and the stars should be born from regions of even smaller sizes (${\approx} 0.1\,\rm pc$). The origin of this spatial arrangement remains under investigation. Based on a high-quality surface density map towards the Perseus region, we construct a 3D density structure, compute the gravitational potential, and derive eigenvalues of the tidal tensor (λmin, λmid, λmax, λmin < λmid < λmax), analyse the behaviour of gravity at every location, and reveal its multiple roles in cloud evolution. We find that fragmentation is limited to several isolated, high-density ‘islands’. Surrounding them, is a vast amount of the gas ($75~{{ \rm per\ cent}}$ of the mass, $95~{{ \rm per\ cent}}$ of the volume) that stays under the influence of extensive tides where fragmentation is suppressed. This gas will be transported towards these regions to fuel star formation. The spatial arrangement of regions under different tides explains the hierarchical and localized pattern of star formation inferred from the observations. Tides were first recognized by Newton, yet this is the first time its dominance in cloud evolution has been revealed. We expect this link between cloud density structure and role gravity to be strengthened by future studies, resulting in a clear view of the star formation process.

Wave Polarization Control in Anisotropic Locally Resonant Materials

Elastic wave propagation in solids can be controlled and manipulated by properly designed metamaterials. In particular, polarization conversion can be obtained by using anisotropic materials. In this paper, we propose a three-component locally resonant material with non-symmetrically coated inclusions, and we study the effect of the anisotropic equivalent mass on band gap formation and the polarization conversion of elastic waves. The equivalent frequency-dependent mass tensor is obtained through the two-scale homogenization approach. The study of the eigenvalues of the mass tensor enables to predict band gaps and polarization bands, as well as identifying a priori the effect of different geometric and material parameters, thus opening the way to metamaterial optimization.

Turbulence anisotropy in a wall-wake flow downstream of two horizontal cylinders

In this study, experimental results of flow past two horizontal cylinders — one above the other are discussed. The research was conducted using experimental data collected on a rough-bed with double cylinders of three different sizes. First, fundamental features such as streamwise velocity, Reynolds shear stress (RSS), and turbulence intensity profiles have been explained at various downstream positions with respect to the cylinders; next, some advanced analysis such as correlation coefficient, stress anisotropy, and dissipation anisotropy have been presented and compared location-wise. To demonstrate anisotropy, stress, dissipation tensors along with anisotropic invariant map and function are plotted and thoroughly analyzed. Lastly, A different diagrammatic representation of the level of anisotropy of turbulence is obtained by incorporating the eigenvalues of the Reynolds stress tensor, namely barycentric coordinates.© 2017 ElsevierInc.Allrightsreserved.

Spherically Symmetric C3 Matching in General Relativity

We study the problem of matching interior and exterior solutions to Einstein’s equations along a particular hypersurface. We present the main aspects of the C3 matching approach that involve third-order derivatives of the corresponding metric tensors in contrast to the standard C2 matching procedures known in general relativity, which impose conditions on the second-order derivatives only. The C3 alternative approach does not depend on coordinates and allows us to determine the matching surface by using the invariant properties of the eigenvalues of the Riemann curvature tensor. As a particular example, we apply the C3 procedure to match the exterior Schwarzschild metric with a general spherically symmetric interior spacetime with a perfect fluid source and obtain that on the matching hypersurface, the density and pressure should vanish, which is in accordance with the intuitive physical expectation.