Tensor Form Research Articles

Low-rank tensor recovery for 2-D DOA estimation using cylindrical nested conformal array

In recent years, nested conformal arrays have received considerable attention because they are capable of providing extended array aperture and increased degrees of freedom compared to uniform conformal arrays. In this article, we suggest a low-rank tensor recovery algorithm for two-dimensional (2-D) direction-of-arrival (DOA) estimation with cylindrical nested conformal array (CNCA). First of all, we derive the tensor form coarray output of the CNCA and use the coarray output to construct a third-order augmented tensor. Subsequently, we reconstruct a noise-free third-order augmented tensor by formulating a tensor nuclear norm (TNN) minimization problem. With the reconstructed tensor, we finally derive the closed-form expressions of 2-D DOA estimates. Compared with the existing approaches, the proposed approach achieves better performance by exploiting the multidimensional structure inherent in the coarray output signal. Numerical results illustrate the advantage of the proposed method over several existing techniques.

The terahertz isolator and switch based on the nonreciprocal feature of magnetized InSb layered photonic structure

In this paper, a multifunctional device based on a one-dimensional (1-D) magnetized InSb layered photonic structure (LPS) that can achieve the function of a magnetic-controlled isolator and switch is proposed, with the advantages of multi-function. The external conditions for realizing the LPS function are temperature 303 K, magnetic field B1 = 0–0.3 T, B2 = 0 T, which are easier to obtain through an applied magnetic field device. B1 and B2 differ only in intensity. Both B1 and B2 are applied magnetic fields parallel to the positive y-axis. The change of magnetic field can influence the the cyclotron frequency of the electron which affect the dielectric constant of the tensor form, and finally lead to the change of refractive index (RI), resulting in interesting electromagnetic phenomena. Temporal inversion symmetry and spatial inversion symmetry could be easily broken to actualize nonreciprocal phenomena because of the magneto-optic effect of Indium antimonide (InSb) and an asymmetric arrangement of quasi-periodic media. For the isolator, the transmittance of forwarding propagation arrives at 0.994 and the backward scale is much closer to 0. The isolation and insertion loss achieves 34 dB and 0.0269 dB, and the range of the angle stabilization is from 17° to 31°. For the switch operating, just changing the value of B1 from 0.3 T to 0 T, can realize the function of the switch operating instead of an isolator. This shift is easier to reach by turning off the magnetic field. The extinction ratio attains 99.8 % and angle stabilization is between 9° and 31°. This magnetic-structure multifunctional LPS can play a vital constructive role in medicine, computing, and other fields.

Goldstone Mode of Skyrmion Crystal

We discuss the Goldstone mode of skyrmion crystal in a model of two-dimensional ferromagnet with the Dzyaloshinskii–Moriya interaction in magnetic field. We use stereographic projection approach to construct skyrmion crystal and consider skyrmion displacement field. The small overlap of the individual skyrmion images restricts the potential energy to the interaction of nearest neighboring displacements. The closed form of the Goldstone mode dispersion is found and its dependence on the magnetic field is studied. We use semiclassical quantization to define the Green’s function and show that the propagation of displacements through the crystal changes its tensorial form from anisotropic to isotropic one at large times.

Phonon-Phonon Interactions in the Polarization Dependence of Raman Scattering.

We have found that the polarization dependence of Raman scattering in organic crystals at finite temperatures can only be described by a fourth-rank tensor formalism. This generalization of the second-rank Raman tensor stems from the effect of off-diagonal components in the crystal self-energy on the light scattering mechanism. We thus establish a novel manifestation of phonon-phonon interaction in inelastic light scattering, markedly separate from the better-known phonon lifetime.

Probing nonperturbative third and fifth harmonic generation on silicon without and with thermal oxide layer

We examine Si with and without additional SiO2 thin film coating as a candidate for producing powerful 3rd and 5th harmonics of Ti:sapphire laser pulses for future spectroscopic application. Polarization rotation experiments have been performed at different incident angles to determine the origin of the generated harmonics and a strong polarization-dependency of the harmonic signals was observed. A simplified tensor formalism is introduced to reproduce the measurements with high accuracy. Comparing the measurements with the Oh symmetry of the bulk crystal, the C2v structural symmetry for the uncoated Si sample and a C4v symmetry for the SiO2 coated sample, we conclude that the polarization anisotropies are determined by the surface/interface symmetries.

Physics-to-Geometry Transformation to Construct Identities between Reynolds Stresses

Modeling has become firmly established as a methodology to close the Reynolds-averaged Navier–Stokes (RANS) equations, owing to theoretical and empirical efforts towards a complete formulation of the Reynolds stress tensor and, recently, breakthroughs in data-processing technology. However, mathematical exactness is not generally ensured by modeling, which is an intrinsic reason why the reliability of RANS closure models is not supposed to be consistent for all kinds of turbulent flow. Rather than straightforwardly overcoming this inherent limitation, most of the studies to date were reasonably directed towards broadening the range of turbulent flows, where reliable prediction accuracy can be obtained via modeling. In this paper, we present three identities between components of the Reynolds stress tensor, constructed via spatial mapping on the basis of the differential version of the Gauss–Bonnet formula. Further, we present a constraint condition that gives a set of equations as numerous as the parameters within a RANS model.

A novel method for depolarization tensor and average form of an arbitrarily shaped inclusion: Extension to different physical fields and their effective transport properties of composites

Depolarization tensor (DT), exclusively controlled by the inclusion shape, is crucial in assessing composite transport behavior. However, the solution of DT for a non-ellipsoidal inclusion exhibits nonuniformity and lacks a closed-form expression. Average DT is then introduced to ensure uniformity and seamlessly integrate the evaluation of effective transport properties of composites. In this work, we devise a novel method to obtain DT and average DT of an arbitrarily shaped inclusion by leveraging mathematical similarity from Greenʼs function to degenerate the Eshelby tensor and average Eshelby tensor in an elastic field. The method circumvents singularity and complexity in direct integration and successfully extracts all tensor components. As natural extensions, we also derive the Eshelby conduction tensor, polarizability tensor, and their averages. The accuracy of the method is validated by comparing the average polarizabilities of Platonic solids and stem cells with existing literature data. Furthermore, we incorporate average DT into a developed homogenization model to predict effective transport properties of two-phase composites comprising randomly distributed congruent irregular inclusions. Comparisons with finite element simulations demonstrate accurate predictions of the effective transport properties like effective permeability, diffusivity, conductivity, and permittivity. The proposed framework offers valuable guidance for tailoring inclusion shapes to design particulate/fibrous/porous/biological composites.

Generalized quantum geometric tensor for excited states using the path integral approach

The quantum geometric tensor, composed of the quantum metric tensor and Berry curvature, fully encodes the parameter space geometry of a physical system. We first provide a formulation of the quantum geometrical tensor in the path integral formalism that can handle both the ground and excited states, making it useful to characterize excited state quantum phase transitions (ESQPT). In this setting, we also generalize the quantum geometric tensor to incorporate variations of the system parameters and the phase-space coordinates. This gives rise to an alternative approach to the quantum covariance matrix, from which we can get information about the quantum entanglement of Gaussian states through tools such as purity and von Neumann entropy. Second, we demonstrate the equivalence between the formulation of the quantum geometric tensor in the path integral formalism and other existing methods. Furthermore, we explore the geometric properties of the generalized quantum metric tensor in depth by calculating the Ricci tensor and scalar curvature for several quantum systems, providing insight into this geometric information.

Geometrically consistent nonlinear plane strain and stress constitutive models: Application to soft-material oscillations

Position-gradient constraints are used in this study to develop new geometrically consistent plane strain and stress constitutive models for the nonlinear large-displacement analysis of soft materials. The position gradients are used to define strain and stress constraints that enter into the formulation of the deformation tensor, strain-energy density function, and new constitutive models. The new approach is applied to three nonlinear hyperelastic constitutive models for incompressible materials: Neo-Hookean, Mooney Rivlin, and Yeoh Models. The underlying geometric plane-stress and plane-strain assumptions are preserved by using the finite elements (FE) of the absolute nodal coordinate formulation (ANCF). Limitations of widely used incompressibility penalty functions that produce stresses at initial stress-free configurations are discussed. Several numerical examples developed using planar ANCF shear-deformable beam and rectangular elements are used to assess the proposed formulation and verify its results using spatial solid elements implemented in commercial FE software. It is demonstrated that the ANCF plane-stress and plane-strain formulations for the incompressible materials predict accurately soft-material oscillations, demonstrating that stiffer behavior can be attributed to material-model inconsistencies and cannot always be attributed to locking.

Seq-SVF: An unsupervised data-driven method for automatically identifying hidden governing equations

In this work, an unsupervised data-driven method based on sequential singular value filtering (Seq-SVF) is proposed to simultaneously identify multiple partial differential equations from observed data considering potential noises. This method is aimed to extend the Sparse Identification of Nonlinear Dynamics (SINDy) to the identification of general nonlinear partial differential equations by transforming the paradigm based on regression to an unsupervised paradigm. To discover the complex coupled equations of vector or tensor forms without prior knowledge, the techniques of singular value decomposition (SVD) and strong rank-revealing QR factorization (sRRQR) are applied to the data matrix, which ensures that the method can automatically identify the number and the corresponding linearly independent terms as the left-hand terms of governing equations. To balance the complexity and the precision of modeling, a strategy for filtering singular values is designed to determine the sparse structure of governing equations from a large number of nonlinear basis functions. We show the success of the method to extract explicit and succinct models from many complex linear, nonlinear, and multiphysics mechanical systems, and the examples show more accuracy compared with using traditional sparse learning methods.

Tensor based low rank representation of hyperspectral images for wheat seeds varieties identification

Hyperspectral image (HSI) based methods are widely used in identifying seeds varieties with high accuracy. However, the excessive spectral bands in HSI may contain redundant information and degrade the model performance. To address this challenge, we propose a novel feature extraction method called low rank tensor approximation (LRTA). Unlike traditional methods, LRTA extracts joint discriminative information of all wheat seeds from hyperspectral images in a 3-order tensor form, preserving intrinsic information. Our model has three key steps: extracting the region of interest from hyperspectral images and presenting average spectral information in tensor form, using LRTA to extract jointly discriminative information, and feeding this information into a classifier to identify seeds varieties. Experiments on our proposed dataset show that the proposed method has 4% improved to the conventional methods on average. Our method shows promise for improving the accuracy of seeds identification while reducing the dimensionality of HSI.

Disformal symmetry in the Universe: mimetic gravity and beyond

Symmetries play an important role in fundamental physics. In gravity and field theories, particular attention has been paid to Weyl (or conformal) symmetry. However, once the theory contains a scalar field, conformal transformations of the metric can be considered a subclass of a more general type of transformation, so-called disformal transformation. Here, we investigate the implications of pure disformal symmetry in the Universe. We derive the form of general disformal invariant tensors from which we build the most general disformal invariant action. We argue that, in cosmology, disformal symmetry amounts to require that the lapse function is fully replaced by a (time-like) scalar field at the level of the action. We then show that disformal symmetry is in general an exactly equivalent formulation of general mimetic gravity. Lastly, we go beyond mimetic gravity and find that a particular class of invariance leads to seemingly Ostrogradski-like (with higher derivatives) Lagrangians, which are nevertheless absent of Ostrogradski ghosts in a cosmological background, despite having an additional degree of freedom. We also propose an application of our formalism to find new invertible disformal transformations, where the coefficient involves higher derivatives and curvature, further expanding the theory space of scalar-tensor theories.

Tensor Robust Principal Component Analysis With Side Information: Models and Applications

As a domain-dependent prior knowledge, side information has been introduced into Robust Principal Component Analysis (RPCA) to alleviate its degenerate or suboptimal performance in some real applications. It has recently realized that the natural structural information can be better retained if the observed data is kept in the original tensor form rather than matricizing it or other order reduction means. Hence, studies on RPCA of tensor version have attracted more and more attentions. To share the merits from both direct tensor modeling and side information, we propose three models to deal with the problem of Tensor RPCA with side information based on tensor Singular Value Decomposition (t-SVD). To solve these models, we develop an efficient algorithm with convergence guarantee using the well-known alternating direction method of multiplier. Extensive experimental studies on both synthetic and real-world tensor data have been carried out to demonstrate the superiority of the proposed models over several other state-of-the-arts. Our code is released at https://github.com/zsj9509/TPCPSF.

On the magneto-mechanical response of piezomagnetic microbeam with size effects

Classical piezomagnetic elasticity theory fails to explain the size-dependent magneto-mechanical behaviour. The extended piezomagnetic elasticity theory incorporating the strain gradient elasticity and flexomagneticity is developed to describe the size effects phenomenon. Compared with the flexomagneticity whose contribution is explicit, the influence mechanism of strain gradient elasticity on the magneto-mechanical response is not clarified. In this paper, we clarify the relation and difference among the extended piezomagnetic elasticity theory considering different strain gradient elasticity. For the relation analysis, according to the orthogonal decomposition of the general strain gradient elasticity, we express the extended piezomagnetic elasticity theory in form of strain gradient tensor and its components of symmetric/anti-symmetric splitting, respectively. By ignoring some strain gradients, the extended piezomagnetic elasticity theory with the general strain gradient elasticity can reduce to that with the reduced strain gradient elasticity. For the difference analysis, we perform the magneto-mechanical analysis of the Terfenol-D/Silicon bilayer microbeam. The flexomagnetic coefficient of Terfenol-D is obtained by fitting deflection solution with experiment results. Then, the size-dependent direct/inverse magneto-mechanical behaviour is revealed. Results reveal that the strain gradient effects and flexomagnetic effects significantly affect the magnetic potential and bending defection when the beam thickness is comparable to the material length-scale parameters. Moreover, compared with the extended piezomagnetic elasticity theory with the reduced strain gradient elasticity, the extended piezomagnetic elasticity theory with the general strain gradient elasticity predicts smaller magnetic potential, bending deflection, and thus can describe the magneto-mechanical behaviour more appropriately.

Nonequilibrium thermodynamic modeling of case II diffusion in glassy polymers

In this article, we formulate a nonequilibrium thermodynamic theory for case II diffusion in a glassy polymer. In the model, we integrate the three-dimensional elasto-visco-plasticity of a glassy polymer, the Flory-Rehner theory for polymer network swelling, and the kinetics law for solvent migration with the concentration-dependent diffusivity. Our theory is given in a general 3D tensorial form, which, therefore, can be used to model case II diffusion in glassy polymers with arbitrary geometries and under different mechano-chemical loading conditions. An intrinsic length determined by the combination of the polymer viscosity and solvent diffusivity arises from the theory, which dictates the size-dependent case II diffusion in glassy polymers. The theory developed in the article may provide important guidance for designing semi-permeable membranes, waterproof coating, and drug-delivery systems, where case II diffusion can commonly occur.

UNMF: a unified nonnegative matrix factorization for multi-dimensional omics data.

Factor analysis, ranging from principal component analysis to nonnegative matrix factorization, represents a foremost approach in analyzing multi-dimensional data to extract valuable patterns, and is increasingly being applied in the context of multi-dimensional omics datasets represented in tensor form. However, traditional analytical methods are heavily dependent on the format and structure of the data itself, and if these change even slightly, the analyst must change their data analysis strategy and techniques and spend a considerable amount of time on data preprocessing. Additionally, many traditional methods cannot be applied as-is in the presence of missing values in the data. We present a new statistical framework, unified nonnegative matrix factorization (UNMF), for finding informative patterns in messy biological data sets. UNMF is designed for tidy data format and structure, making data analysis easier and simplifying the development of data analysis tools. UNMF can handle a wide range of data structures and formats, and works seamlessly with tensor data including missing observations and repeated measurements. The usefulness of UNMF is demonstrated through its application to several multi-dimensional omics data, offering user-friendly and unified features for analysis and integration. Its application holds great potential for the life science community. UNMF is implemented with R and is available from GitHub (https://github.com/abikoushi/moltenNMF).

A tensor formalism for multilayer network centrality measures using the Einstein product

Complex systems that consist of diverse kinds of entities that interact in different ways can be modeled by multilayer networks. This paper uses the tensor formalism with the Einstein product to model this type of networks. Several centrality measures, that are well known for single-layer networks, are extended to multilayer networks using tensors and their properties are investigated. In particular, subgraph centrality based on the exponential and resolvent of a tensor are considered. Krylov subspace methods based on the tensor format are introduced for computing approximations of different measures for large multilayer networks.

Comparative analysis of dynamic stability of cylindrical and conical shells under periodic axial compression

A comparative analysis of the dynamic stability of cylindrical and conical shells with the same geometric and mechanical characteristics under periodic uniformly distributed axial compression was presented. The study of the stability of steady periodic vibrations of thin elastic shells was based on the joint use of the method of curvilinear grids, the projection method and the parameter continuation method combined with the Newton–Kantorovich method. Geometrically nonlinear relations of the thin elastic shells theory are formulated on the basis of the vector approximation of the displacements function in the general curvilinear coordinate system in tensor form and satisfy the Kirchhoff-Love hypothesis. The discretization of the differential equations of the steady forced vibrations in the direction of the generating shells using the method of curvilinear grids was carried out. The components of the elements displacement vectors of the shells middle surface in the circular directionare approximated by trigonometric series. Reduction of the number of generalized coordinates of the discrete dynamic model of shells steady forced vibrations was performed by the Bubnov-Galerkin basis reduction method. A transition from vector ordinary differential equations to a nonlinear system of algebraic equations was made. The construction of a mathematical model of the dynamic stability of steady forced nonlinear vibrations of thin elastic shells was performed according to Floquet's theory using the projection method. The criterion for the loss of stability was the equality to zero of the determinant of the matrix of linearized equations of steady forced nonlinear vibrations of shells according to the Lyapunov theorem. A comparative analysis of frequencies and modes of natural vibrations of cylindrical and conical shells with the same geometric and mechanical characteristics and boundary conditions was performed. Nonlinear steady vibrations of the shells due to periodic axial compression were studied. The critical values of the dynamic load and the corresponding forms of loss of shell stability in the range of lower frequencies of their natural vibrations were obtained.

Parity-odd 3-point functions from CFT in momentum space and the chiral anomaly

We illustrate how the Conformal Ward Identities (CWI) in momentum space for parity-odd correlators determine the structure of a chiral anomaly interaction, taking the example of the VVA (vector/vector/axial-vector) and AAA correlators in momentum space. Only the conservation and the anomalous WIs, together with the Bose symmetry, are imposed from the outset for the determination of the correlators. We use a longitudinal/transverse decomposition of tensor structures and form factors. The longitudinal (L) component is fixed by the anomaly content and the anomaly pole, while in the transverse (T) sector we define a new parameterization. We relate the latter both to the Rosenberg original representation of the VVA and to the longitudinal/transverse (L/T) one, first introduced in the analysis of g-2 of the muon in the investigation of the diagram in the chiral limit of QCD. The correlators are completely identified by the conformal constraints whose solutions are fixed only by the anomaly coefficient, the residue of the anomaly pole. In both cases, our CFT result matches the one-loop perturbative expression, as expected. The CWIs for correlators of mixed chirality J_L J J_R generate solutions in agreement with the all-orders nonrenormalization theorems of perturbative QCD and in the chiral limit.

Computation of minimal covariants bases for 2D coupled constitutive laws

We produce minimal integrity bases for both isotropic and hemitropic invariant algebras (and more generally covariant algebras) of most common bidimensional constitutive tensors and — possibly coupled — laws, including piezoelectricity law, photoelasticity, Eshelby and elasticity tensors, complex viscoelasticity tensor, Hill elasto-plasticity, and (totally symmetric) fabric tensors up to twelfth-order. The concept of covariant, which extends that of invariant is explained and motivated. It appears to be much more useful for applications. All the tools required to obtain these results are explained in detail and a cleaning algorithm is formulated to achieve minimality in the isotropic case. The invariants and covariants are first expressed in complex forms and then in tensorial forms, thanks to explicit translation formulas which are provided. The proposed approach also applies to any n-uplet of bidimensional constitutive tensors.