Linear Tensor Research Articles

Growing avascular tumours as elasto-plastic bodies by the theory of evolving natural configurations

The aim of this article is to propose a simple way of describing a tumour as a linear elastic material from a reference configuration that is continuously evolving in time due to growth and remodelling. The main assumption allowing this simplification is that the tumour mass is a very ductile material, so that it can only sustain moderate stresses while the deformation induced by growth, that can actually be quite big, mainly induces a plastic reorganisation of malignant cells. In mathematical terms this means that the deformation gradient can be split into a volumetric growth term, a term describing the reorganisation of cells, and a term that can be approximated by means of the linear strain tensor. A dimensional analysis of the importance of the different terms also allows to introduce a second simplification consisting of decoupling the equations describing the growth of the tumour mass from those describing the flow of the interstitial fluid.

Massive Born-Infeld and other dual pairs

We consider massive dual pairs of p-forms and (D-p-1)-forms described by non-linear Lagrangians, where non-linear curvature terms in one theory translate into non-linear mass-like terms in the dual theory. In particular, for D=2p and p even the two non-linear structures coincide when the non-linear massless theory is self-dual. This state of affairs finds a natural realization in the four-dimensional massive N=1 supersymmetric Born-Infeld action, which describes either a massive vector multiplet or a massive linear (tensor) multiplet with a Born-Infeld mass-like term. These systems should play a role for the massive gravitino multiplet obtained from a partial super-Higgs in N=2 Supergravity.

F-zips with additional structure

An $F$-zip over a scheme $S$ over a finite field is a certain object of semi-linear algebra consisting of a locally free module with a descending filtration and an ascending filtration and a $\Frob_q$-twisted isomorphism between the respective graded sheaves. In this article we define and systematically investigate what might be called "$F$-zips with a $G$-structure", for an arbitrary reductive linear algebraic group $G$. These objects come in two incarnations. One incarnation is an exact linear tensor functor from the category of finite dimensional representations of $G$ to the category of $F$-zips over $S$. Locally any such functor has a type $\chi$, which is a cocharacter of $G$. The other incarnation is a certain $G$-torsor analogue of the notion of $F$-zips. We prove that both incarnations define stacks that are naturally equivalent to a quotient stack of the form $[E_{G,\chi}\backslash G_k]$ that was studied in an earlier paper. By the results obtained there they are therefore smooth algebraic stacks of dimension 0 over $k$. Using our earlier results we can also classify the isomorphism classes of such objects over an algebraically closed field, describe their automorphism groups, and determine which isomorphism classes can degenerate into which others. For classical groups we can deduce the corresponding results for twisted or untwisted symplectic, orthogonal, or unitary $F$-zips. The results can be applied to the algebraic de Rham cohomology of smooth projective varieties (or generalizations thereof such as smooth proper Deligne-Mumford stacks) and to truncated Barsotti-Tate groups of level 1. In addition, we hope that our systematic group theoretical approach will help to understand the analogue of the Ekedahl-Oort stratification of the special fibers of arbitrary Shimura varieties.

Anisotropic linear forcing for synthetic turbulence generation in large eddy simulation and hybrid RANS/LES modeling

A general forcing method for Large Eddy Simulation (LES) is proposed for the purpose of providing the flow with fluctuations that satisfy a desired statistical state. This method, the Anisotropic Linear Forcing (ALF) introduces an unsteady linear tensor function of the resolved velocity which acts as a restoring force in the mean velocity and resolved stress budgets. The ALF generalizes and extends several forcing previously proposed in the literature. In order to make it possible to impose the integral length scale of the turbulence generated by the forcing term, an alternative formulation of the ALF, using a differential spatial filter, is proposed and analyzed. The anisotropic forcing of the Reynolds stresses is particularly attractive, since unsteady turbulent fluctuations can be locally enhanced or damped, depending on the target stresses. As such, it is shown that the ALF is an effective method to promote turbulent fluctuations downstream of the LES inlet or at the interface between RANS and LES in zonal hybrid RANS/LES modeling. The detailed analysis of the influence of the ALF parameters in spatially developing channel flows and hybrid computations where the ALF target statistics are given by a RANS second-moment closure show that this original approach performs as well as the synthetic eddy method. However, since the ALF method is more flexible and significant computational savings are obtained, the method appears a promising all-in-one solution for general embedded LES simulations.

Personalized Tag Recommendation through Nonlinear Tensor Factorization Using Gaussian Kernel

Personalized tag recommendation systems recommend a list of tags to a user when he is about to annotate an item. It exploits the individual preference and the characteristic of the items. Tensor factorization tech- niques have been applied to many applications, such as tag recommendation. Models based on Tucker Decomposition can achieve good performance but require a lot of computation power. On the other hand, mod- els based on Canonical Decomposition can run in linear time and are more feasible for online recommendation. In this paper, we propose a novel method for personalized tag recommendation, which can be considered as a nonlinear extension of Canonical Decomposition. Different from linear tensor factorization, we exploit Gaussian radial basis function to increase the model’s capacity. The experimental results show that our proposed method outperforms the state-of-the-art methods for tag recommendation on real datasets and perform well even with a small number of features, which verifies that our models can make better use of features.

Saliency Mapping Enhanced by Structure Tensor.

We propose a novel efficient algorithm for computing visual saliency, which is based on the computation architecture of Itti model. As one of well-known bottom-up visual saliency models, Itti method evaluates three low-level features, color, intensity, and orientation, and then generates multiscale activation maps. Finally, a saliency map is aggregated with multiscale fusion. In our method, the orientation feature is replaced by edge and corner features extracted by a linear structure tensor. Following it, these features are used to generate contour activation map, and then all activation maps are directly combined into a saliency map. Compared to Itti method, our method is more computationally efficient because structure tensor is more computationally efficient than Gabor filter that is used to compute the orientation feature and our aggregation is a direct method instead of the multiscale operator. Experiments on Bruce's dataset show that our method is a strong contender for the state of the art.

Revisiting the Link between Lipid Membrane Elasticity and Microscopic Continuum Models

Fluid lipid membranes can be described by a continuum-elastic Hamiltonian that features two central parameters mean and Gaussian curvature modulus. Of those two, the Gaussian modulus is much less understood, since it affects the membrane energetics only through topology or boundaries and is thus difficult to measure experimentally. Moreover, recent work by Hu et al. [Biophys. J. 102, 1403 (2012)] to determine this modulus from computational studies revealed discrepancies with more microscopic expectations, according to which this modulus should be given by the second moment of the lateral trans-membrane stress profile. Our goal in the present study is to revisit the arguments linking the Gaussian modulus to properties of thin sheets, using thin plate theory as a starting point, but allowing for a number of generalizations. For instance, membranes are more generally described as anisotropic continua with in-plane fluidity and a nontrivial distribution of pre-stresses. It is easy to see that in this case linear elasticity, combined with the additional but common approximation of a linear strain tensor, will predict that the Gaussian curvature modulus vanishes. We therefore need to include the possibility of nonlinear strains (while still using linear constitutive equations), but this in turn leads to some subtle inconsistencies within Monge gauge. Another difficulty is that virtually all existing theories treat Young's modulus and the Poisson ratio as constant throughout the width of the membrane. Starting from fundamental linear elasticity theory, amended by a nonlinear strain tensor, and using consistent geometrical constructions, we then investigate the elastic properties of membranes, especially the Gaussian curvature modulus.

Partial correlation analyses of global diffusion tensor imaging-derived metrics in glioblastoma multiforme: Pilot study.

To determine existing correlates among diffusion tensor imaging (DTI)-derived metrics in healthy brains and brains with glioblastoma multiforme (GBM). Case-control study using DTI data from brain magnetic resonance imaging of 34 controls (mean, 41.47; SD, ± 21.94 years; range, 21-80 years) and 27 patients with GBM (mean, SD; 48.41 ± 15.18 years; range, 18-78 years). Image postprocessing using FSL software calculated eleven tensor metrics: fractional (FA) and relative anisotropy; pure isotropic (p) and anisotropic diffusions (q), total magnitude of diffusion (L); linear (Cl), planar (Cp) and spherical tensors (Cs); mean (MD), axial (AD) and radial diffusivities (RD). Partial correlation analyses (controlling the effect of age and gender) and multivariate Mancova were performed. There was a normal distribution for all metrics. Comparing healthy brains vs brains with GBM, there were significant very strong bivariate correlations only depicted in GBM: [FA↔Cl (+)], [FA↔q (+)], [p↔AD (+)], [AD↔MD (+)], and [MD↔RD (+)]. Among 56 pairs of bivariate correlations, only seven were significantly different. The diagnosis variable depicted a main effect [F-value (11, 23) = 11.842, P ≤ 0.001], with partial eta squared = 0.850, meaning a large effect size; age showed a similar result. The age also had a significant influence as a covariate [F (11, 23) = 10.523, P < 0.001], with a large effect size (partial eta squared = 0.834). DTI-derived metrics depict significant differences between healthy brains and brains with GBM, with specific magnitudes and correlations. This study provides reference data and makes a contribution to decrease the underlying empiricism in the use of DTI parameters in brain imaging.

Hamiltonian Operators of Dubrovin-Novikov Type in 2D

First order Hamiltonian operators of differential-geometric type were introduced by Dubrovin and Novikov in 1983, and thoroughly investigated by Mokhov. In 2D, they are generated by a pair of compatible flat metrics \({g}\) and \({\tilde g}\) which satisfy a set of additional constraints coming from the skew-symmetry condition and the Jacobi identity. We demonstrate that these constraints are equivalent to the requirement that \({\tilde g}\) is a linear Killing tensor of g with zero Nijenhuis torsion. This allowed us to obtain a complete classification of n-component operators with n ≤ 4 (for n = 1, 2 this was done before). For 2D operators the Darboux theorem does not hold: the operator may not be reducible to constant coefficient form. All interesting (non-constant) examples correspond to the case when the flat pencil \({g, \tilde g}\) is not semisimple, that is, the affinor \({\tilde g g^{-1}}\) has non-trivial Jordan block structure. In the case of a direct sum of Jordan blocks with distinct eigenvalues, we obtain a complete classification of Hamiltonian operators for any number of components n, revealing a remarkable correspondence with the class of trivial Frobenius manifolds modelled on H *(CP n-1).

High field studies on BiFeO3 single crystals grown by the laser-diode heating floating zone method

Magnetic field dependence of magnetization and electric polarization in mono-domain crystals of BiFeO3 were studied in pulsed magnetic fields up to 56T. The observed anisotropic transition fields were reasonably explained by Ginzburg–Landau theory at low temperatures, whereas their temperature dependence could not be in the simple model. The electric polarization measurements provide matrix elements of the linear magnetoelectric tensor |α31|∼|α32|=1.5±0.2ps/m.

The linear elasticity tensor of incompressible materials

With a universally accepted abuse of terminology, materials having much larger stiffness for volumetric than for shear deformations are called incompressible. This work proposes two approaches for the evaluation of the correct form of the linear elasticity tensor of so-called incompressible materials, both stemming from non-linear theory. In the approach of strict incompressibility, one imposes the kinematical constraint of isochoric deformation. In the approach of quasi-incompressibility, which is often employed to enforce incompressibility in numerical applications such as the Finite Element Method, one instead assumes a decoupled form of the elastic potential (or strain energy), which is written as the sum of a function of the volumetric deformation only and a function of the distortional deformation only, and then imposes that the bulk modulus be much larger than all other moduli. The conditions which the elasticity tensor has to obey for both strict incompressibility and quasi-incompressibility have been derived, regardless of the material symmetry. The representation of the linear elasticity tensor for the quasi-incompressible case differs from that of the strictly incompressible case by one parameter, which can be conveniently chosen to be the bulk modulus. Some important symmetries have been studied in detail, showing that the linear elasticity tensors for the cases of isotropy, transverse isotropy and orthotropy are characterised by one, three and six independent parameters, respectively, for the case of strict incompressibility, and two, four and seven independent parameters, respectively, for the case of quasi-incompressibility, as opposed to the two, five and nine parameters, respectively, of the general compressible case.

Reduced-density-matrix description for pump-probe optical phenomena in moving atomic systems

Linear and nonlinear (especially coherent) electromagnetic interactions of moving many-electron atoms are investigated using a reduced-density-matrix description, which is applied to electromagnetically induced transparency and related resonant pump-probe optical phenomena. External magnetic fields are included on an equal footing with the electromagnetic fields and spin-Zeeman interactions are taken into account. Complimentary time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations of the reduced-density-matrix description are self-consistently developed. The general nonperturbative and non-Markovian formulations provide a fundamental framework for systematic evaluations of corrections to the standard Born (lowest-order-perturbation) and Markov (short-memory-time) approximations. The macroscopic electromagnetic response is described semiclassically, employing a perturbation expansion of the reduced-density operator in powers of the classical electromagnetic field. Our primary results are compact Liouville-space operator expressions for the linear and general ($n\text{th}$-order) nonlinear macroscopic electromagnetic-response tensors, which can be evaluated for nonlocal and nonstationary optical media described by multilevel atomic-system representations. Interactions among atoms and with environmental photons are treated as line-broadening effects by means of a general Liouville-space self-energy operator, for which the tetradic-matrix elements are explicitly evaluated in the diagonal, lowest-order, and Markov approximations. The compact Liouville-space operator expressions that are derived for the macroscopic electromagnetic-response tensors are introduced into the dynamical description of the electromagnetic-field propagation. It is pointed out that a quantized-electromagnetic-field approach will be required for a fully self-consistent quantum-mechanical treatment of local-field effects and radiative corrections.

Modeling the optical Kerr effect in periodic structures by the linear Fourier modal method

We present a numerically feasible method for rigorous modeling of crossed diffraction gratings made of isotropic materials with third-order nonlinearity. The approach is based on an iterative solution of the crossed-grating problem with linear but anisotropic materials. We also discuss how the symmetries of the effective linear permittivity tensor can be exploited to speed up the computation when one considers normal incidence of light. Several numerical experiments are performed to demonstrate the accuracy as well as the versatility of this numerical technique.

Coherent [formula omitted]-photoproduction on the deuteron near the [formula omitted]-production threshold including polarization observables

Coherent π0-photoproduction on the deuteron including polarization observables is studied in the energy region near the η-production threshold at backward center-of-mass angles of the outgoing pion. This work is motivated by the measurements of the CLAS Collaboration at Jefferson Lab, where a cusp-like structure in the energy dependence of the differential cross section has been observed at extremely backward pion angles. The present approach is based on the impulse approximation and first-order rescattering diagrams with intermediate production of both π- and η-mesons. Numerical results for unpolarized cross sections, the linear photon asymmetry (Σ), the vector (T11) and tensor (T2M, M=0, 1, 2) deuteron target asymmetries, and the double polarization E-asymmetry are predicted and compared with available experimental data and other theoretical models. The effect of first-order rescattering is found to be much larger in spin asymmetries than in the unpolarized cross sections. It reaches on average about 40% in the tensor target and E asymmetries. Compared to the experimental data from CLAS Collaboration, sizable discrepancies are found. This is not the case for the linear photon asymmetry, for which a better comparison with the data from YerPhI Collaboration is obtained.

Modeling of the optical properties of molecular crystals

In this contribution we present our current findings in the calculations of the linear and second-order nonlinear electric susceptibility tensor components of organic crystals. The methodology used for this purpose is based on a combination of the electrostatic interaction scheme developed by Hurst and Munn (Hurst &amp; Munn, 1986) with electronic structure calculations for the isolated molecules. Our modification of the method consists in i) running periodic boundary condition (PBC) calculations for an adequate chromophore geometry (either experimental or optimized) to obtain atomic charges and in ii) performing the calculations of the molecular properties within a non-uniform embedding field generated by point charges located spherically around the reference molecule. Using this approach good accuracy is achieved on the electric susceptibility tensor components in comparison with the uniform dipole electric field (Seidler et al., 2013). We extend here the application of this method to other molecular crystals as well as we present the first attempt to predict the chi(1) and chi(2) components of two-component organic crystals (Gryl et al., 2014).

Quantization of perturbations in an inflating elastic solid

A sufficiently rigid relativistic elastic solid can be stable for negative pressure values and thus is capable of driving a stage of accelerated expansion. If a relativistic elastic solid drove an inflationary stage in the early Universe, quantum mechanically excited perturbations would arise in the medium. We quantize the linear scalar and tensor perturbations and investigate the observational consequences of having such an inflationary period. We find that slowly varying sound speeds of the perturbations and a slowing varying equation of state of the solid can produce a slightly red-tilted scalar power spectrum that agrees with current observational data. Even in the absence of nonadiabatic pressures, perturbations evolve on superhorizon scales, due to the shear stresses within the solid. As such, the spectra of perturbations are in general sensitive to the details of the end of inflation and we characterize this dependence. Interestingly, we uncover here accelerating solutions for elastic solids with (1 + P/\rho) significantly greater than 0 that nevertheless have nearly scale-invariant scalar and tensor spectra. Beyond theoretical interest, this may allow for the possibility of viable inflationary phenomenology relatively far from the de Sitter regime.

WE-A-17A-11: Implanted Brachytherapy Seed Movement Due to Transrectal Ultrasound Probe-Induced Prostate Deformation

Purpose: To characterize the movement of implanted brachytherapy seeds due to transrectal ultrasound probe-induced prostate deformation and to estimate the effects on prostate dosimetry. Methods: Implanted probe-in and probe-removed seed distributions were reconstructed for 10 patients using C-arm fluoroscopy imaging. The prostate was delineated on ultrasound and registered to the fluoroscopy seeds using a visible subset of seeds and residual needle tracks. A linear tensor and shearing model correlated the seed movement with position. The seed movement model was used to infer the underlying prostate deformation and to simulate the prostate contour without probe compression. Changes in prostate and surrogate urethra dosimetry were calculated. Results: Seed movement patterns reflecting elastic decompression, lateral shearing, and rectal bending were observed. Elastic decompression was characterized by anterior-posterior expansion and superior-inferior and lateral contractions. For lateral shearing, anterior movement up to 6 mm was observed for extraprostatic seeds in the lateral peripheral region. The average intra-prostatic seed movement was 1.3 mm, and the residual after linear modeling was 0.6 mm. Prostate D90 increased by 4 Gy on average (8 Gy max) and was correlated with elastic decompression. For selected patients, lateral shearing resulted in differential change in D90 of 7 Gy between anterior andmore » posterior quadrants, and increase in whole prostate D90 of 4 Gy. Urethra D10 increased by 4 Gy. Conclusion: Seed movement upon probe removal was characterized. The proposed model captured the linear correlation between seed movement and position. Whole prostate dose coverage increased slightly, due to the small but systematic seed movement associated with elastic decompression. Lateral shearing movement increased dose coverage in the anterior-lateral region, at the expense of the posterior-lateral region. The effect on whole prostate D90 was smaller due to the subset of peripheral seeds involved, but lateral shearing movement can have greater consequences for local dose coverage.« less

A GA-based feature selection and parameter optimization for linear support higher-order tensor machine

In the fields of pattern recognition, computer vision, and image processing, many real-world image and video data are more naturally represented as tensors. Recently, based on the supervised tensor learning (STL) framework, a linear support higher-order tensor machine (SHTM) has been proposed. Considering that there are much redundancy information in the tensor data and the model parameter largely affects the performance of SHTM, in this study, we present a genetic algorithm (GA) based feature selection and parameter optimization algorithm for the linear SHTM. The proposed algorithm can remove the redundancy information in tensor data and obtain a better generalized accuracy by searching for the optimal model parameter and feature subset simultaneously. A set of experiments is conducted on nine second-order face recognition datasets and three third-order gait recognition datasets to illustrate the performance of the proposed algorithm. The statistic test shows that compared with the original linear SHTM, the proposed algorithm can provide a significant performance gain in terms of generalized accuracy for tensor classification.

Identification of Anisotropic Yield Criterion Parameters from a Single Biaxial Tensile Test

The present work deals with the calibration strategy of yield functions used to describe the plastic anisotropic behavior of metallic sheets. In this paper, Bron and Besson yield criterion is used to model the plastic anisotropic behavior of AA5086 sheets. This yield model is flexible enough since the anisotropy is represented by 12 parameters (4 isotropic parameters and 8 anisotropic parameters in plane stress condition) in the form of two linear fourth order transformation tensors. The parameters of this anisotropic yield model have been identified from a single dedicated cross biaxial tensile test. It is shown, from finite element simulations, that the strain distribution in the center of the cruciform specimen is significantly dependent on the yield criterion. Moreover, this cross biaxial test involves a large range of strain paths in the center of the specimen. The calibration stage is performed by means of an optimization procedure minimizing the gap between experimental and numerical values of the principal strains along a specified path in the gauge area of the cruciform specimen. It is shown that the material parameters of Bron and Besson anisotropic yield model can be determined accurately by a unique biaxial tensile test.

Linear and non-linear optical properties of GaAs nanowires

The linear and non-linear optical properties of different geometrical structures of gallium arsenide (GaAs) nanowires have been studied by employing ab initio method. We have calculated the optical response of four different GaAs nanowires, viz., two-atom linear wire, two-atom zigzag wire, four-atom square wire and six-atom hexagonal wire. We have investigated imaginary part of the zz component of the linear dielectric tensor and second-order susceptibility for different structures along with bulk material. We revealed that the strongest absorption occurs for four-atom square nanowire configuration.