Linear Tensor Research Articles

Geometric phase of cosmological scalar and tensor perturbations in f(R) gravity

By using the conformal equivalence of f(R) gravity in vacuum and the usual Einstein theory with scalar-field matter, we derive the Hamiltonian of the linear cosmological scalar and tensor perturbations in f(R) gravity in the form of time-dependent harmonic oscillator Hamiltonians. We find the invariant operators of the resulting Hamiltonians and use their eigenstates to calculate the adiabatic Berry phase for sub-horizon modes as a Lewis–Riesenfeld phase.

Domain wall brane in a reduced Born–Infeld-f(T) theory

The Born–Infeld f(T) theory is reduced from the Born–Infeld determinantal gravity in Weitzenböck spacetime. We investigate a braneworld scenario in this theory and obtain an analytic domain wall solution by utilizing the first-order formalism. The model is stable against the linear tensor perturbation. It is shown that the massless graviton is localized on the brane, but the continuous massive gravitons are non-localized and will generate a tiny correction with the behavior of 1/(kr)3 to the Newtonian potential. The four-dimensional teleparallel gravity is recovered as an effective infrared theory on the brane. As a physical application, we consider the (quasi-)localization property of spin-1/2 Dirac fermion in this model.

Berry phase of primordial scalar and tensor perturbations in single-field inflationary models

In the framework of the single-field slow-roll inflation, we derive the Hamiltonian of the linear primordial scalar and tensor perturbations in the form of time-dependent harmonic oscillator Hamiltonians. We find the invariant operators of the resulting Hamiltonians and use their eigenstates to calculate the adiabatic Berry phase for sub-horizon modes in terms of the Lewis–Riesenfeld phase. We conclude by discussing the discrepancy in the results of Pal et al. (2013) [21] for these Berry phases, which is resolved to yield agreement with our results.

Heteroepitaxial growth of ferroelectric films on Si substrates and their applications in waveguides and electro-optics

We introduce a general route to integrate epitaxially ferroelectric Sr1.9Ca0.1NaNb5O15 (SCNN) thin films on Si(001) substrates by pulsed laser deposition. The In:Mg0.8Zn0.2O (In:MZO)/TiN buffer layer is induced to overcome the large lattice mismatch between SCNN films and Si(001) substrates. The In:MZO layer acts as the buffer layer to bridge the Si and ferroelectric, the buffer layer with low refractive index for waveguide structure and the bottom electrode for electro-optic (EO) modulators. Waveguide and EO properties of the epitaxial integrated heterostructure were characterized by a prism coupler technique. The observed sharp and distinguishable TM and TE propagation mode lines revealed that favorable light confinements were achieved within the SCNN waveguide layer. The extraordinary and ordinary refractive indices were measured to be ne = 2.1802 and no = 2.2164 at 633 nm, respectively, implying a birefringence of 0.0362 was obtained. This large birefringence indicates that a large optically anisotropic property existed in the epitaxial SCNN films. An angular shift of guided mode was observed under applied electric field along with c-axis direction. The linear EO coefficient tensor r33 was measured to be about 860 pm/V. All of these results indicate that the integrated SCNN/In:MZO/TiN/Si epitaxial heterostructure has great potential for integrated optical EO and waveguide applications.

Tucker Tensor Regression and Neuroimaging Analysis.

Neuroimaging data often take the form of high dimensional arrays, also known as tensors. Addressing scientific questions arising from such data demands new regression models that take multidimensional arrays as covariates. Simply turning an image array into a vector would both cause extremely high dimensionality and destroy the inherent spatial structure of the array. In a recent work, Zhou et al. (2013) proposed a family of generalized linear tensor regression models based upon the CP (CANDECOMP/PARAFAC) decomposition of regression coefficient array. Low rank approximation brings the ultrahigh dimensionality to a manageable level and leads to efficient estimation. In this article, we propose a tensor regression model based on the more flexible Tucker decomposition. Compared to the CP model, Tucker regression model allows different number of factors along each mode. Such flexibility leads to several advantages that are particularly suited to neuroimaging analysis, including further reduction of the number of free parameters, accommodation of images with skewed dimensions, explicit modeling of interactions, and a principled way of image downsizing. We also compare the Tucker model with CP numerically on both simulated data and a real magnetic resonance imaging data, and demonstrate its effectiveness in finite sample performance.

δ M formalism: a new approach to cosmological perturbation theory in anisotropic inflation

We study the evolution of the metric perturbations in a Bianchi background in the long-wavelength limit. By applying the gradient expansion to the equations of motion we exhibit a generalized “Separate Universe” approach to the cosmological perturbation theory. Having found this consistent separate universe picture, we introduce the δ M formalism for calculating the evolution of the linear tensor perturbations in anisotropic inflation models in almost the same way that the so-called δ N formula is applied to the super-horizon dynamics of the curvature perturbations. Similar to her twin formula, δ N, this new method can substantially reduce the amount of calculations related to the evolution of tensor modes. However, it is not as general as δ N; it is a “perturbative" formula and solves the shear only to linear order. In other words, it is restricted to weak shear limit.

The Drazin inverse of an even-order tensor and its application to singular tensor equations

The notion of the Moore–Penrose inverses of matrices was recently extended from matrix space to even-order tensor space with Einstein product in the literature. In this paper, we further study the properties of even-order tensors with Einstein product. We define the index and characterize the invertibility of an even-order square tensor. We also extend the notion of the Drazin inverse of a square matrix to an even-order square tensor. An expression for the Drazin inverse through the core-nilpotent decomposition for a tensor of even-order is obtained. As an application, the Drazin inverse solution of the singular linear tensor equation A∗X=B will also be included.

Thick branes with inner structure in mimetic gravity

In this paper, thick branes generated by mimetic scalar field are investigated. Three typical thick brane models are constructed and the linear tensor and scalar perturbations are analyzed. These branes have different inner structures, some of which are absent in general relativity. For each brane model, the solution is stable under both tensor and scalar perturbations. The tensor zero modes are localized on the branes, while the scalar perturbations do not propagate and they are not localized on the brane. As the branes split into multi sub-branes for specific parameters, the potentials of the tensor perturbations also split into multi-wells, and this may lead to new phenomenon in the resonance of the tensor perturbation and the localization of matter fields.

An Implicit SPH Formulation for Incompressible Linearly Elastic Solids

AbstractWe propose a novel smoothed particle hydrodynamics (SPH) formulation for deformable solids. Key aspects of our method are implicit elastic forces and an adapted SPH formulation for the deformation gradient that—in contrast to previous work—allows a rotation extraction directly from the SPH deformation gradient. The proposed implicit concept is entirely based on linear formulations. As a linear strain tensor is used, a rotation‐aware computation of the deformation gradient is required. In contrast to existing work, the respective rotation estimation is entirely realized within the SPH concept using a novel formulation with incorporated kernel gradient correction for first‐order consistency. The proposed implicit formulation and the adapted rotation estimation allow for significantly larger time steps and higher stiffness compared to explicit forms. Performance gain factors of up to one hundred are presented. Incompressibility of deformable solids is accounted for with an ISPH pressure solver. This further allows for a pressure‐based boundary handling and a unified processing of deformables interacting with SPH fluids and rigids. Self‐collisions are implicitly handled by the pressure solver.

Comparison of the DeWitt metric in general relativity with the fourth-rank constitutive tensors in electrodynamics and in elasticity theory

We perform a short comparison between the local and linear constitutive tensor $\chi^{\lambda\nu\sigma\kappa}$ in four-dimensional electrodynamics (Sec.2), the elasticity tensor $c^{ijkl}$ in three-dimensional elasticity theory (Sec.3), and the DeWitt metric $G^{abcd}$ in general relativity, with ${a,b,\dots=1,2,3}$ (Sec.4). We find that the DeWitt metric has only six independent components.

Inelastic anisotropic constitutive models based on evolutionary linear transformations on stress tensors with application to masonry

The formulation of constitutive models for anisotropic materials such as masonry is a problem of large complexity. One possible way is to define linear transformations on the stress tensors using fourth-order transformation tensors that carry all the anisotropic information of the material. In the present paper, a new type of evolutionary linear transformation tensor is defined, which can change the values of its components along with the evolution of internal variables. This means the transformation laws are defined according to the current plastic and damage levels, and allows the constitutive model to describe totally different hardening and softening behaviours of the material along different directions. First, a general procedure of formulation of anisotropic constitutive models is given. Second, as a specific example, an orthotropic plastic–damage constitutive model for masonry is presented. Finally, the proposed constitutive model is validated by comparing finite element results with experimental ones pertaining to simple masonry structures under static and cyclic loading.

Photomagnetoelectric and magnetophotovoltaic effects in multiferroic BiFeO3

A polarized light beam modifies the equilibrium tensors associated with any measurable property of a crystal. The light-induced linear magnetoelectric susceptibility and electric current density of bismuth ferrite (BFO) are investigated theoretically using their expansion in Wigner spherical functions. Under illumination, the interplay between magnetoelectric and photovoltaic properties yields the emergence in the paramagnetic and multiferroic phases of BFO of linear photomagnetoelectric tensor components changing sign in domains with opposite electric polarization or magnetization, and to magnetophotocurrents changing sense in opposite ferroelectric and magnetic domains.

Anisotropic yield function based on stress invariants for BCC and FCC metals and its extension to ductile fracture criterion

It is essential to accurately model the anisotropic plastic deformation and ductile fracture of metals in order to guarantee the reliable numerical analysis and optimization of metal forming. For this purpose, the Drucker function is revisited. Effect of the third stress invariant in the Drucker function is analyzed and calibrated for metals with body-centered cubic (BCC) and face-centered cubic (FCC) crystal systems based on the yielding and plastic flow of both crystal plasticity and biaxial tensile experiments. The calibrated Drucker function is extended into an anisotropic form using a fourth order linear transformation tensor. The anisotropic flexibility is enhanced by two approaches: non-associate flow rule (non-AFR) and the sum of n-components of the anisotropic Drucker function. The proposed anisotropic Drucker function is applied to model the anisotropic behavior of both BCC and FCC metals. The predicted anisotropic behavior is compared with experimental results. The comparison demonstrates that the anisotropy is accurately modeled for both BCC and FCC metals by the anisotropic Drucker function. The anisotropic Drucker function is also implemented into numerical analysis of tension of specimens with a central hole to investigate its computation efficiency under spatial loading compared with the Yld2000-18p function. It is found that the proposed anisotropic Drucker function can reduce about 60% of computation time in case that the Yld2000-18p function is substituted by the anisotropic Drucker function in numerical computation due to its simplicity compared to the Yld2000-18p function. A ductile fracture criterion is also developed by coupling the Drucker function with the first stress invariant. The modified Drucker function is reformulated to investigate the effect of the stress triaxiality and the normalized third invariant on ductile fracture. Comparison of the modified Drucker fracture locus with the experimental results of AA2024-T351 demonstrates that the modified Drucker criterion accurately illustrates the fracture stress of the alloy in wide stress states with the stress triaxiality ranging from −0.5 in plane strain compression to 0.6 in tension of notched specimens. The modified Drucker fracture criterion is expected to be less sensitive to the change of strain path considering that the criterion describes fracture in the stress space. Accordingly, the anisotropic Drucker yield function and the pressure-coupled Drucker fracture criterion are suggested to model anisotropic plastic deformation and to predict the onset of failure for both BCC and FCC metals due to simple implementation in numerical analysis under spatial loading and computation efficiency with brick elements.

Linear Cherenkov radiation in ferroelectric domain walls.

Nonlinear Cherenkov radiation plays an important role in detecting the internal structure of the ferroelectric crystal. However, linear Cherenkov radiation in the ferroelectric crystal was not observed before. Based on mirror symmetry reduction at the domain walls, we theoretically predicted that the linear Cherenkov radiation could be generated in periodically poled ferroelectric crystals according to the coupled wave equation. Then, we experimentally demonstrated the generation of such Cherenkov radiation in domain walls. Compared with the domain region, new nonzero elements χ13, χ23, χ31 and χ32 appear in the linear susceptibility tensor. The experimental results are consistent with our theoretical prediction.

Pre-inflationary universe in loop quantum cosmology

The evolutions of the flat FLRW universe and its linear perturbations are studied systematically in the dressed metric approach of LQC. When it is dominated by the kinetic energy of the inflaton at the quantum bounce, the evolution of the background can be divided into three different phases prior to the preheating, {\em bouncing, transition and slow-roll inflation}. During the bouncing phase, the evolution is independent of not only the initial conditions, but also the inflationary potentials. In particular, the expansion factor can be well described by the same exact solution in all the cases considered. In contrast, in the potential dominated case such a universality is lost. It is because of this universality that the linear perturbations are also independent of the inflationary models and obtained exactly. During the transition phase, the evolutions of the background and its linear perturbations are found explicitly, and then matched to the ones given in the other two phases. Hence, once the initial conditions are imposed, the linear scalar and tensor perturbations will be uniquely determined. Considering two different sets of initial conditions, one imposed during the contracting phase and the other at the bounce, we calculate the Bogoliubov coefficients and find that the two sets yield the same results and all lead to particle creations at the onset of the inflation. Due to the pre-inflationary dynamics, the scalar and tensor power spectra become scale-dependent. Comparing with the Planck 2015 data, we find constraints on the total e-folds that the universe must have expanded since the bounce, in order to be consistent with current observations.

A New Restriction for Initially Stressed Elastic Solids

We introduce a fundamental restriction on the strain energy function and stress tensor for initially stressed elastic solids. The restriction applies to strain energy functions W that are explicit functions of the elastic deformation gradient F and initial stress τ⁠, that is W:=W(F,τ)⁠. The restriction is a consequence of energy conservation and ensures that the predicted stress and strain energy do not depend upon an arbitrary choice of reference configuration. We call this restriction initial stress reference independence (ISRI). It transpires that most strain energy functions found in the literature do not satisfy ISRI, and may therefore lead to unphysical behaviour, which we illustrate through a simple example. To remedy this shortcoming, we derive three strain energy functions that do satisfy the restriction. We also show that using initial strain (often from a virtual configuration) to model initial stress leads to strain energy functions that automatically satisfy ISRI. Finally, we reach the following important result: ISRI reduces the number of unknowns in the linear stress tensor for initially stressed solids. This new way of reducing the linear stress may open new pathways for the non-destructive determination of initial stresses through ultrasonic experiments, among others.

Analysis of DTI-Derived Tensor Metrics in Differential Diagnosis between Low-grade and High-grade Gliomas.

Purpose: It is critical and difficult to accurately discriminate between high- and low-grade gliomas preoperatively. This study aimed to ascertain the role of several scalar measures in distinguishing high-grade from low-grade gliomas, especially the axial diffusivity (AD), radial diffusivity (RD), planar tensor (Cp), spherical tensor (Cs), and linear tensor (Cl) derived from diffusion tensor imaging (DTI).Materials and Methods: Fifty-three patients with pathologically confirmed brain gliomas (21 low-grade and 32 high-grade) were included. Contrast-enhanced T1-weighted images and DTI were performed in all patients. The AD, RD, Cp, Cs, and Cl values in the tumor zone, peritumoral edema zone, white matter (WM) adjacent to edema and contralateral normal-appearing white matter (NAWM) were calculated. The DTI parameters and tumor grades were statistically analyzed, and receiver operating characteristic (ROC) curve analysis was also performed.Results: The DTI metrics in the affected hemisphere showed significant differences from those in the NAWM, except for the AD values in the tumor zone and the RD values in WM adjacent to edema in the low-grade groups, as well as the Cp values in WM adjacent to edema in the high-grade groups. AD in the tumor zone as well as Cs and Cl in WM adjacent to edema revealed significant differences between the low- and high-grade gliomas. The areas under the curve (Az) of all three metrics were greater than 0.5 in distinguishing low-grade from high-grade gliomas by ROC curve analysis, and the best DTI metric was Cs in WM adjacent to edema (Az: 0.692).Conclusion: AD in the tumor zone as well as Cs and Cl in WM adjacent to edema will provide additional information to better classify gliomas and can be used as non-invasive reliable biomarkers in glioma grading.

Optical Anisotropy of Topologically Distorted Semiconductor Nanocrystals.

Engineering nanostructured optical materials via the purposeful distortion of their constituent nanocrystals requires the knowledge of how various distortions affect the nanocrystals' electronic subsystem and its interaction with light. We use the geometric theory of defects in solids to calculate the linear permittivity tensor of semiconductor nanocrystals whose crystal lattice is arbitrarily distorted by imperfections or strains. The result is then employed to systematically analyze the optical properties of nanocrystals with spatial dispersion caused by screw dislocations and Eshelby twists. We demonstrate that Eshelby twists create gyrotropy in nanocrystals made of isotropic semiconductors whereas screw dislocations can produce it only if the nanocrystal material itself is inherently anisotropic. We also show that the dependence of circular dichroism spectrum on the aspect ratio of dislocation-distorted semiconductor nanorods allows resonant enhancing their optical activity (at least by a factor of 2) and creating highly optically active nanomaterials.

The details of structural disconnectivity in psychotic disorder: A family-based study of non-FA diffusion weighted imaging measures

BackgroundDiffusion tensor imaging (DTI) studies in psychotic disorder have shown reduced FA, often interpreted as disturbed white matter integrity. The observed ‘dysintegrity’ may be of multifactorial origin, as changes in FA are thought to reflect a combination of changes in myelination, fiber organization and number of axons. Examining the structural substrate of the diffusion tensor in individuals with (risk for) psychotic disorder may provide better understanding of the underlying structural changes. MethodsDTI scans were acquired from 85 patients with psychotic disorder, 93 siblings of patients with psychotic disorder and 80 controls. Cross-sectional group comparisons were performed using Tract-Based Spatial Statistics (TBSS) on six DTI measures: axial diffusivity (AXD), radial diffusivity (RD), mean diffusivity (MD), and the case linear (CL), case planar (CP) and case spherical (CS) tensor shape measures. ResultsAXD did not differ between the groups. RD and CS values were significantly increased in patients compared to controls and siblings, with no significant differences between the latter two groups. MD was higher in patients compared to controls (but not siblings), with no difference between siblings and controls. CL was smaller in patients than in siblings and controls, and CP was smaller in both patients and siblings as compared to controls. ConclusionThe differences between individuals with psychotic disorder and healthy controls, derived from detailed diffusion data analyses, suggest less fiber orientation and increased free water movement in the patients. There was some evidence for association with familial risk expressed by decreased fiber orientation.

Excitons in gyrotropic quantum-dot supercrystals

We use quantum theory of molecular crystals to study collective excitations (excitons) of gyrotropic quantum-dot (QD) supercrystals with complex lattices consisting of two or more sublattices of semiconductor QDs. We illustrate the potentials of our approach by applying it to analytically calculate the linear permittivity tensor of supercrystals with two QDs per unit cell. The spatial dispersions of exciton energy bands and permittivity tensor components are examined in detail for two-dimensional supercrystals with a square lattice, which are relatively easy to fabricate in practice. Our results provide a systematic and versatile framework for the engineering of dispersion properties of gyrotropic QD supercrystals and for the analysis of their absorption and circular dichroism spectra.