Planar Tensor Research Articles

Ray Transforms of the Moments of Planar Tensor Fields

Ray Transforms of the Moments of Planar Tensor Fields

Symbol alphabets from tensor diagrams

We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of n-particle amplitudes in planar mathcal{N} = 4 Yang-Mills theory and certain polytopes associated to the Grassmannian Gr(4, n). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of n ≤ 8, and find that the polytope {mathcal{C}}^{dagger}left(4,9right) encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.

Optical beam diffraction tensor in birefringent crystals

We demonstrate that anisotropy of Fresnel diffraction in a birefringent medium can be quantitatively characterized by a planar tensor. Eigenvectors of the tensor correspond to directions of minimum and maximum beam divergence. Zero eigenvalues exist for the slow eigenmode near the optic axis of biaxial crystals and correspond to autocollimated beam propagation along the cone of external conical refraction. Applications of the beam diffraction tensor are demonstrated for analysis of noncritical phase matching in acousto-optics and nonlinear three-wave mixing.

Neural Fiber Integrity in High- Versus Low-Grade Glioma using Probabilistic Fiber Tracking

Gliomatous tumors are known to affect neural fiber integrity, either by displacement or destruction. The aim of this study is to investigate the integrity and distribution of the white matter tracts within and around the glioma regions using probabilistic fiber tracking. Forty-two glioma patients were subjected to MRI using a standard tumor protocol with diffusion tensor imaging (DTI). The tumor and peritumor regions were delineated using snake model with reference to structural and diffusion MRI. A preprocessing pipeline of the structural MRI image, DTI data, and tumor regions was implemented. Tractography was performed to delineate the white matter (WM) tracts in the selected tumor regions via probabilistic fiber tracking. DTI indices were investigated through comparative mapping of WM tracts and tumor regions in low-grade gliomas (LGG) and high-grade gliomas (HGG). Significant differences were seen in the planar tensor (Cp) in peritumor regions; mean diffusivity, axial diffusivity and pure isotropic diffusion in solid-enhancing tumor regions; and fractional anisotropy, axial diffusivity, pure anisotropic diffusion (q), total magnitude of diffusion tensor (L), relative anisotropy, Cp and spherical tensor (Cs) in solid nonenhancing tumor regions for affected WM tracts. In most cases of HGG, the WM tracts were not completely destroyed, but found intact inside the tumor. Probabilistic fiber tracking revealed the existence and distribution of WM tracts inside tumor core for both LGG and HGG groups. There were more DTI indices in the solid nonenhancing tumor region, which showed significant differences between LGG and HGG.

Correlations between DTI-derived metrics and MRS metabolites in tumour regions of glioblastoma: a pilot study.

IntroductionSpecific correlations among diffusion tensor imaging (DTI)-derived metrics and magnetic resonance spectroscopy (MRS) metabolite ratios in brains with glioblastoma are still not completely understood.Patients and methodsWe made retrospective cohort study. MRS ratios (choline-to-N-acetyl aspartate [Cho/NAA], lipids and lactate to creatine [LL/Cr], and myo-inositol/creatine [mI/Cr]) were correlated with eleven DTI biomarkers: mean diffusivity (MD), fractional anisotropy (FA), pure isotropic diffusion (p), pure anisotropic diffusion (q), the total magnitude of the diffusion tensor (L), linear tensor (Cl), planar tensor (Cp), spherical tensor (Cs), relative anisotropy (RA), axial diffusivity (AD) and radial diffusivity (RD) at the same regions: enhanced rim, peritumoral oedema and normal-appearing white matter. Correlational analyses of 546 MRS and DTI measurements used Spearman coefficient.ResultsAt the enhancing rim we found four significant correlations: FA ⇔ LL/Cr, Rs = -.364, p = .034; Cp ⇔ LL/Cr, Rs = .362, p = .035; q ⇔ LL/Cr, Rs = -.349, p = .035; RA ⇔ LL/Cr, Rs = -.357, p = .038. Another ten pairs of significant correlations were found in the peritumoral edema: AD ⇔ LL/Cr, AD ⇔ mI/Cr, MD ⇔ LL/Cr, MD ⇔ mI/Cr, p ⇔ LL/Cr, p ⇔ mI/ Cr, RD ⇔ mI/Cr, RD ⇔ mI/Cr, L ⇔ LL/Cr, L ⇔ mI/Cr.ConclusionsDTI and MRS biomarkers answer different questions; peritumoral oedema represents the biggest challenge with at least ten significant correlations between DTI and MRS that need additional studies. The fact that DTI and MRS measures are not specific of one histologic type of tumour broadens their application to a wider variety of intracranial pathologies.

Direction-averaged diffusion-weighted MRI signal using different axisymmetric B-tensor encoding schemes.

PurposeIt has been shown, theoretically and in vivo, that using the Stejskal‐Tanner pulsed‐gradient, or linear tensor encoding (LTE), and in tissue exhibiting a “stick‐like” diffusion geometry, the direction‐averaged diffusion‐weighted MRI signal at high b‐values ( 7000<b<10000s/mm2) follows a power‐law, decaying as 1/b. It has also been shown, theoretically, that for planar tensor encoding (PTE), the direction‐averaged diffusion‐weighted MRI signal decays as 1/b. We aimed to confirm this theoretical prediction in vivo. We then considered the direction‐averaged signal for arbitrary b‐tensor shapes and different tissue substrates to look for other conditions under which a power‐law exists.MethodsWe considered the signal decay for high b‐values for encoding geometries ranging from 2‐dimensional PTE, through isotropic or spherical tensor encoding to LTE. When a power‐law behavior was suggested, this was tested using in silico simulations and, when appropriate, in vivo using ultra‐strong (300 mT/m) gradients.ResultsOur in vivo results confirmed the predicted 1/b power law for PTE. Moreover, our analysis showed that using an axisymmetric b‐tensor a power‐law only exists under very specific conditions: (a) “stick‐like” tissue geometry and purely LTE or purely PTE waveforms; and (b) "pancake‐like" tissue geometry and a purely LTE waveform.ConclusionsA complete analysis of the power‐law dependencies of the diffusion‐weighted signal at high b‐values has been performed. Only three specific forms of encoding result in a power‐law dependency, pure linear and pure PTE when the tissue geometry is “stick‐like” and pure LTE when the tissue geometry is "pancake‐like". The different exponents of these encodings could be used to provide independent validation of the presence of different tissue geometries in vivo.

A 4D Basis and Sampling Scheme for the Tensor Encoded Multi-Dimensional Diffusion MRI Signal

We propose a 4-dimensional (4D) basis and sampling scheme, along with a corresponding reconstruction algorithm, for the measurement and reconstruction of the $\mathbf {b}$ -tensor encoded diffusion signal in diffusion magnetic resonance imaging (MRI). This is only the second basis proposed for representing the $\mathbf {b}$ -tensor encoded diffusion signal and the first to allow for planar tensor measurements. We design a sampling scheme that attains an efficient number of samples, equal to the degrees of freedom required to represent the diffusion signal in the proposed 4D basis. The properties of the diffusion signal are studied to provide recommendations on how many $\mathbf {b}$ -tensor measurements to use. Evaluation of the proposed scheme using Monte Carlo simulations of the diffusion signal is done to show that the proposed scheme gives accurate interpolation of the signal.

Tensor chain and constraints in tensor networks

We develop our recent work on quantum error correction (QEC) and entanglement spectrum (ES) in tensor networks (arXiv:1806.05007). We propose a general framework for planar tensor network state with tensor constraints as a model for AdS3/CFT2 correspondence, which could be viewed as a generalization of hyperinvariant tensor networks recently proposed by Evenbly. We elaborate our proposal on tensor chains in a tensor network by tiling H2 space and provide a diagrammatical description for general multi-tensor constraints in terms of tensor chains, which forms a generalized greedy algorithm. The behavior of tensor chains under the action of greedy algorithm is investigated in detail. In particular, for a given set of tensor constraints, a critically protected (CP) tensor chain can be figured out and evaluated by its average reduced interior angle. We classify tensor networks according to their ability of QEC and the flatness of ES. The corresponding geometric description of critical protection over the hyperbolic space is also given.

Quantum error correction and entanglement spectrum in tensor networks

A sort of planar tensor networks with tensor constraints is investigated as a model for holography. We study the greedy algorithm generated by tensor constraints and propose the notion of critical protection (CP) against the action of greedy algorithm. For given tensor constraints, a CP tensor chain can be defined. We further find that the ability of quantum error correction (QEC), the non-flatness of entanglement spectrum (ES) and the correlation function can be quantitatively evaluated by the geometric structure of CP tensor chain. Four classes of tensor networks with different properties of entanglement is discussed. Thanks to tensor constraints and CP, the correlation function is reduced into a bracket of Matrix Production State and the result agrees with the one in conformal field theory.

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.

A note on the computation of the extrema of Young’s modulus for hexagonal materials: An approach by planar tensor invariants

A simple method to obtain the highest and lowest Young’s modulus for a material of the hexagonal class is presented. It is based upon the use of tensor invariants of plane anisotropic elasticity; in fact, the cylindrical symmetry of the elastic tensor allows for transforming the 3D original problem into a planar one, with a considerable simplification.

Global diffusion tensor imaging derived metrics differentiate glioblastoma multiforme vs. normal brains by using discriminant analysis: introduction of a novel whole-brain approach.

BackgroundHistological behavior of glioblastoma multiforme suggests it would benefit more from a global rather than regional evaluation. A global (whole-brain) calculation of diffusion tensor imaging (DTI) derived tensor metrics offers a valid method to detect the integrity of white matter structures without missing infiltrated brain areas not seen in conventional sequences. In this study we calculated a predictive model of brain infiltration in patients with glioblastoma using global tensor metrics.MethodsRetrospective, case and control study; 11 global DTI-derived tensor metrics were calculated in 27 patients with glioblastoma multiforme and 34 controls: mean diffusivity, fractional anisotropy, pure isotropic diffusion, pure anisotropic diffusion, the total magnitude of the diffusion tensor, linear tensor, planar tensor, spherical tensor, relative anisotropy, axial diffusivity and radial diffusivity. The multivariate discriminant analysis of these variables (including age) with a diagnostic test evaluation was performed.ResultsThe simultaneous analysis of 732 measures from 12 continuous variables in 61 subjects revealed one discriminant model that significantly differentiated normal brains and brains with glioblastoma: Wilks’ λ = 0.324, χ2 (3) = 38.907, p < .001. The overall predictive accuracy was 92.7%.ConclusionsWe present a phase II study introducing a novel global approach using DTI-derived biomarkers of brain impairment. The final predictive model selected only three metrics: axial diffusivity, spherical tensor and linear tensor. These metrics might be clinically applied for diagnosis, follow-up, and the study of other neurological diseases.

Numerical evaluation of tensor Feynman integrals in Euclidean kinematics

For the investigation of higher order Feynman integrals, potentially with tensor structure, it is highly desirable to have numerical methods and automated tools for dedicated, but sufficiently 'simple' numerical approaches. We elaborate two algorithms for this purpose which may be applied in the Euclidean kinematical region and in d=4-2*epsilon dimensions. One method uses Mellin-Barnes representations for the Feynman parameter representation of multi-loop Feynman integrals with arbitrary tensor rank. Our Mathematica package AMBRE has been extended for that purpose, and together with the packages MB (M. Czakon) or MBresolve (A. V. Smirnov and V. A. Smirnov) one may perform automatically a numerical evaluation of planar tensor Feynman integrals. Alternatively, one may apply sector decomposition to planar and non-planar multi-loop epsilon-expanded Feynman integrals with arbitrary tensor rank. We automatized the preparations of Feynman integrals for an immediate application of the package sector_decomposition (C. Bogner and S. Weinzierl) so that one has to give only a proper definition of propagators and numerators. The efficiency of the two implementations, based on Mellin-Barnes representations and sector decompositions, is compared. The computational packages are publicly available.

1D electro-elastic fields in piezoelectrics excited by intrinsic strains

The electro-elastic fields induced by the transversely 1D distribution of intrinsic strains in an arbitrary infinite piezoelectric plate with metallized surfaces are determined. The results obtained generalize the well-known theory of internal stresses in anisotropic purely elastic plates developed by Indenbom, Sil’vestrova, and Sirotin [1]. The coupled fields found in the piezoelectric are expressed in terms of the 4D formalism and the corresponding generalized planar tensor of electro-elastic moduli. It is shown that the limiting transitions from a piezoelectric plate to an unbounded medium and half-space lead to identical formulas.

The polar analysis of a third order piezoelectricity-like plane tensor

The derivation of the invariants for a tensor of the third order in a two dimensional space is presented in the paper. The mathematical technique to obtain the expression of the invariants is based upon the so-called polar method, proposed as early as 1979 by G. Verchery as an alternative method to represent a plane tensor by its invariants. The polar components of a third order planar tensor are also introduced. When considered in the framework of piezoelectricity, an energetic interpretation of these components is given.

Guided waves on a planar tensor impedance surface

Guided wave mode propagation constants are determined for an infinite planar surface characterized by an anisotropic impedance tensor. Restrictions on the tensor elements implied by the imposition of a passivity requirement are presented. The conditions under which E and H modes can exist are determined. The special case of a lossless surface is considered.

A unified formulation for plasticity models of granular and other materials

A general physically based rule for the kinematics of planar deformation of granular materials is presented and a pair of kinematic equations governing the velocity field is obtained. The method results in a unified derivation of the equations for the double sliding, free rotating model, the double shearing model and the plastic potential model for granular materials. The formulation of the double sliding, free rotating model for the kinematics of granular materials presented here is consistent with the continuum theory of constitutive equations and demonstrates the relationship of this model both to the double shearing model and to the plastic potential models. Expressions for the rate of working of the stresses are presented. A constitutive equation, expressed in terms of planar tensor quantities, is also presented that has the three models as special cases.