Pair Of Tensors Research Articles

Quantization of the canonical tensor model and an exact wave function

Tensor models in various forms are being studied as models of quantum gravity. Among them the canonical tensor model has a canonical pair of rank-three tensors as dynamical variables, and is a pure constraint system with first-class constraints. The Poisson algebra of the first-class constraints provides an algebraically consistent way of discretizing the Dirac algebra for general relativity. This paper successfully formulates the Wheeler-DeWitt quantization of the canonical tensor model. Formally one can obtain wave functions of the "universe" by solving the partial differential equations representing the constraints. For the simplest non-trivial case, the unique wave function is exactly and globally obtained. Although this case is far from being realistic, the wave function is physically interesting; locality is favored, and there exists a locus of configurations with features of the beginning of the universe.

QUANTUM CANONICAL TENSOR MODEL AND AN EXACT WAVE FUNCTION

Tensor models in various forms are being studied as models of quantum gravity. Among them the canonical tensor model has a canonical pair of rank-three tensors as dynamical variables, and is a pure constraint system with first-class constraints. The Poisson algebra of the first-class constraints has structure functions, and provides an algebraically consistent way of discretizing the Dirac first-class constraint algebra for general relativity. This paper successfully formulates the Wheeler–DeWitt scheme of quantization of the canonical tensor model; the ordering of operators in the constraints is determined without ambiguity by imposing Hermiticity and covariance on the constraints, and the commutation algebra of constraints takes essentially the same form as the classical Poisson algebra, i.e. is first-class. Thus one could consistently obtain, at least locally in the configuration space, wave functions of "universe" by solving the partial differential equations representing the constraints, i.e. the Wheeler–DeWitt equations for the quantum canonical tensor model. The unique wave function for the simplest nontrivial case is exactly and globally obtained. Although this case is far from being realistic, the wave function has a few physically interesting features; it shows that locality is favored, and that there exists a locus of configurations with features of beginning of universe.

RETRACTED ARTICLE: Sparse tensor CCA for color face recognition

This paper proposes a subspace learning method, named as sparse tensor canonical correlation analysis (ST-CCA), for color face recognition. A sample image is formalized as high-order tensors to preserve the inherent structure of the color face images. We utilize sparse canonical correlation analysis (SCCA) to choose gene. For each pair of tensors, SCCA generates the sparse loadings alternately, which is helpful for choosing significant variables to reduce dimensions and eliminate the redundancies of tensors. We use the elastic net as constraint condition to attack the collinearity problem by decorrelating and selecting the sufficient variables irrespective of the limited dimensions. Furthermore, ST-CCA gains stable recognition rates because the alternating least square solution converges. ST-CCA is convex with different initials of the projection matrices. Experimental results on AR face database and LFW face database show the superior performance of our method over the state-of-the-art ones.

Nonequilibrium thermodynamical inequivalence of quantum stress-energy and spin tensors

It is shown that different pairs of stress-energy and spin tensors of quantum relativistic fields related by a pseudo-gauge transformation, i.e. differing by a divergence, imply different mean values of physical quantities in thermodynamical nonequilibrium situations. Most notably, transport coefficients and the total entropy production rate are affected by the choice of the spin tensor of the relativistic quantum field theory under consideration. Therefore, at least in principle, it should be possible to disprove a fundamental stress-energy tensor and/or to show that a fundamental spin tensor exists by means of a dissipative thermodynamical experiment.

Nijenhuis and compatible tensors on Lie and Courant algebroids

We show that well known structures on Lie algebroids can be viewed as Nijenhuis tensors or pairs of compatible tensors on Courant algebroids. We study compatibility and construct hierarchies of these structures.

The existence and uniqueness of eigenvalues for monotone homogeneous mapping pairs

In this paper, the concept of eigenvalue is introduced to the monotone homogeneous mapping pairs (f,g), and its existence and uniqueness are established successfully under the boundedness of some orbits of f,g in the Hilbert semi-norm. Also, the Collatz–Wielandt min–max type property is obtained for such a class of mapping pairs. In particular, the nonlinear Perron–Frobenius property for nonnegative tensor pairs (A,B) is obtained without involving the calculation of the tensor inversion. By particularizing the mapping g or f, several results can emerge as corollaries. They lead to further existence results and open problems.

Extreme diffusion values for non-Gaussian diffusions

A new magnetic resonance imaging (MRI) model, called diffusion kurtosis imaging (DKI), was recently proposed, to characterize the non-Gaussian diffusion behaviour in tissues. DKI involves a fourth-order three-dimensional tensor and a second-order three-dimensional tensor. Similar to those in the diffusion tensor imaging (DTI) model, the extreme diffusion values and extreme directions associated to this tensor pair play important roles in DKI. In this paper, we study the properties of the extreme values and directions associated to such tensor pairs. We also present a numerical method and its preliminary computational results.

PARAFAC-Based Blind Estimation Of Possibly Underdetermined Convolutive MIMO Systems

In this paper, we consider the problem of blind identification of a convolutive multiple-input-multiple-output (MIMO) system with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">No</i> outputs and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ni</i> inputs. While many methods have been proposed to blindly identify convolutive MIMO systems with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">No</i> ges <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ni</i> (overdetermined), very scarce results exist for the case of (underdetermined), all of which refer to systems that either have some special structure or special and values. In this paper, we show that, as long as , independent of whether the system is overdetermined or underdetermined, we can always find the appropriate order of statistics that guarantees identifiability of the system response within trivial ambiguities. We also propose an algorithm to reach the solution, that consists of parallel factorization (PARAFAC) of a -way tensor containing th-order statistics of the system outputs, followed by an iterative scheme. For a certain order of statistics , we provide the description of the class of identifiable MIMO systems. We also show that this class can be expanded by applying PARAFAC decomposition to a pair of tensors instead of one tensor. The proposed approach constitutes a novel scheme for estimation of underdetermined systems, and improves over existing approaches for overdetermined systems.

Chemical-Shift Anisotropy Measurements of Amide and Carbonyl Resonances in a Microcrystalline Protein with Slow Magic-Angle Spinning NMR Spectroscopy

Chemical shifts are fundamental to interpretation of NMR spectra and provide constraints for macromolecular structure determination, refinement, and validation, as well as details of active-site chemistry in enzymes. Insights into the origins of the chemical shift can be leveraged to improve chemical analysis, including conformation, bonding, and dynamics. To exploit this information fully, it is desirable to measure not only isotropic chemical shifts but also the full chemical-shift anisotropy (CSA) tensor. Until recently, most efforts to measure backbone amide and carbonyl CSAs have relied upon cross-correlated relaxation and residual anisotropic shifts in solution NMR. In solid-state NMR, analysis of sideband patterns has typically been reserved for site-specifically labeled samples or small peptides. Here we demonstrate that the Herzfeld−Berger method (Herzfeld, J.; Berger, A. E. J. Chem. Phys. 1980, 73, 6021−6030) can be applied to highly 13C,15N-enriched solid proteins, using 2D heteronuclear correlation in combination with high magnetic fields (750 MHz 1H frequency) and pattern labeling of 13C sites. The experiments report on 42 pairs of amide and carbonyl tensors in the microcrystalline protein GB1.

An orbit analysis approach to the study of superintegrable systems in the Euclidean plane

We classify the superintegrable potentials in the Euclidean plane by means of an orbit analysis of the space of valence two Killing tensors under the action of the group of rigid motions. Our approach generalizes the classical approach of Winternitz and collaborators by considering pairs of Killing tensors that are not both in canonical form.

Irreducible correlation functions of the Ŝ matrix in the coordinate representation: Application in calculating Lorentzian half-widths and shifts

By introducing the coordinate representation, the derivation of the perturbation expansion of the Liouville S matrix is formulated in terms of classically behaved autocorrelation functions. Because these functions are characterized by a pair of irreducible tensors, their number is limited to a few. They represent how the overlaps of the potential components change with a time displacement, and under normal conditions, their magnitudes decrease by several orders of magnitude when the displacement reaches several picoseconds. The correlation functions contain all dynamical information of the collision processes necessary in calculating half-widths and shifts and can be easily derived with high accuracy. Their well-behaved profiles, especially the rapid decrease of the magnitude, enables one to transform easily the dynamical information contained in them from the time domain to the frequency domain. More specifically, because these correlation functions are well time limited, their continuous Fourier transforms should be band limited. Then, the latter can be accurately replaced by discrete Fourier transforms and calculated with a standard fast Fourier transform method. Besides, one can easily calculate their Cauchy principal integrations and derive all functions necessary in calculating half-widths and shifts. A great advantage resulting from introducing the coordinate representation and choosing the correlation functions as the starting point is that one is able to calculate the half-widths and shifts with high accuracy, no matter how complicated the potential models are and no matter what kind of trajectories are chosen. In any case, the convergence of the calculated results is always guaranteed. As a result, with this new method, one can remove some uncertainties incorporated in the current width and shift studies. As a test, we present calculated Raman Q linewidths for the N2-N2 pair based on several trajectories, including the more accurate "exact" ones. Finally, by using this new method as a benchmark, we have carried out convergence checks for calculated values based on usual methods and have found that some results in the literature are not converged.

Super-energy and Killing-Yano tensors

Abstract In this paper we investigate a class of basic super-energy tensors, namely those constructed from Killing-Yano tensors, and give a generalization of super-energy tensors for cases when we start not with a single tensor, but with a pair of tensors.

A rigorous framework for diffusion tensor calculus

In biological tissue, all eigenvalues of the diffusion tensor are assumed to be positive. Calculations in diffusion tensor MRI generally do not take into account this positive definiteness property of the tensor. Here, the space of positive definite tensors is used to construct a framework for diffusion tensor analysis. The method defines a distance function between a pair of tensors and the associated shortest path (geodesic) joining them. From this distance a method for computing tensor means, a new measure of anisotropy, and a method for tensor interpolation are derived. The method is illustrated using simulated and in vivo data.

Force-Free States, Relative Strain, and Soft Elasticity in Nematic Elastomers

The remarkable ability of nematic elastomers to exhibit large deformations under small applied forces is known as soft elasticity. The recently proposed neo-classical free-energy density for nematic elastomers, derived by molecular-statistical arguments, has been used to model soft elasticity. In particular, the neo-classical free-energy density allows for a continuous spectrum of equilibria, which implies that deformations may occur in the complete absence of force and energy cost. Here we study the notion of force-free states in the context of a continuum theory of nematic elastomers that allows for isotropy, uniaxiality, and biaxiality of the polymer microstructure. Within that theory, the neo-classical free-energy density is an example of a free-energy density function that depends on the deformation gradient only through a nonlinear strain measure associated with the deformation of the polymer microstructure relative to the macroscopic continuum. Among the force-free states for a nematic elastomer described by the neo-classical free energy density, there is, in particular, a continuous spectrum of states parameterized by a pair of tensors that allows for soft deformations. In these force-free states the polymer microstructure is material in the sense that it stretches and rotates with the macroscopic continuum. Limitations of and possible improvements upon the neo-classical model are also discussed.

The forms of the relation between the stress and strain tensors in a non-linearly elastic material

New relations for the stress and strain tensors, which comprise energy pairs, are obtained for a non-linearly elastic material using a similar method to that employed by Novozhilov, based on a trigonometric representation of the tensors. Shear strain and stress tensors, not used previously, are introduced in a natural way. It is established that the unit tensor, the deviator and the shear tensor form an orthogonal tensor basis. The stress tensor can be expanded in a strain-tensor basis and vice versa. By using this expansion, the non-linear law of elasticity can be written in a compact and physically clear form. It is shown that in the frame of the principal axes the stresses are expressed in terms of the strains and vice versa using linear relations, while the non-linearity is contained in the coefficients, which are functions of mixed invariants of the tensors, introduced by Novozhilov, the generalized moduli of bulk compression and shear and the phase of similitude of the deviators. Relations for different energy pairs of tensors are considered, including for tensors of the true stresses and strains, where the generalized moduli of elasticity have a physical meaning for large strains.

A three-dimensional Fe nonlinear analysis of membranes

This paper presents a three-dimensional finite element method for computing stress and deformation of membrane structures. A conjugate pair of strain and stress tensors, the Green strain tensor and the Second Piola-Kirchhoff stress tensor are adopted to give the constitutive equation of Mooney material. By means of exploiting the plane stress condition of membrane, the hydrostatic pressure is directly expressed as functions of strains. Complex membrane structures, like those initially in box shape with abrupt changes in geometry, can also be analyzed by the proposed method.

Finite elastoplastic deformation of membrane shells

Under restriction of an isotropic elastic response of deformed lattice, develops a covariant theory of finite elastoplasticity in principal axes of a pair of deformation tensors. In material description, the tensor pair consists of the plastic deformation tensor and the total deformation Cauchy‐Green tensor. Applies the proposed theory to elastoplastic membrane shells, whose references and current configurations can be arbitrary space‐curved surfaces. Pressure‐insensitive von Mises yield criterion with isotropic hardening and a quadratic form of the strain energy function given in terms of elastic principal stretches are considered as a model problem. Through an explicit enforcement of the plane stress condition we arrive at a reduced two‐dimensional problem representation, which is set in the membrane tangent plane. Numerical implementation of the presented theory relies crucially on the operator split methodology to simplify the state update computation. Presents a set of numerical examples in order to illustrate the performance of the presented methodology and indicate possible applications in the area of sheet metal forming.

Finite elastoplastic deformations of space-curved membranes

Under restriction of an isotropic elastic response of deformed lattice, a covariant theory of finite elastoplasticity is developed in principal axes of a pair of deformation tensors. In material description, the tensor pair consists of the plastic deformation tensor and the total deformation Cauchy-Green tensor. The proposed theory is applied to elastoplastic membranes, whose reference and current configurations can be arbitrary space-curved surfaces. Pressure-insensitive von Mises yield criterion and isotropic hardening are considered as a model problem. With a particular form of the strain energy function, given in terms of elastic principal stretches, through an explicit enforcement of the plane stress condition we arrive at a reduced two-dimensional problem representation, which is set in the membrane tangent plane. Numerical implementation details are given to show an important role of the operator split methodology in simplifying the state update computation and the computation of the consistent tangent modulus. A set of numerical examples illustrates the performance of the presented theory and indicates some of the possible areas of application.

Mathematical Description of Anisotropic Damage State in Continuum Damage Mechanics.

A fictitious equivalent undamaged configuration B*t of an element of a damaged material is first introduced into a three-dimensional Riemannian manifold by excluding the irrelevant rigid rotation of the element. Then a new symmetric damage tensor for three-dimensional anisotropic damage is developed in terms of Finger's tensor of a fictitious deformation gradient from the current damaged configuration Bt to the equivalent undamaged configuration B*t. The damage tensor components are expressed in terms of the metric tensor components of the convective coordinate systems of Bt and B*t. Furthermore, the effective stress tensor of damage mechanics is defined by the equality of stress vectors acting on the damaged and equivalent undamaged plane elements, and the mechanical interpretation of the effective stress tensor and its symmetry condition are discussed. The symmetry condition is also formulated by means of a commutator tensor formed by the tensor pairs : damage tensor-Cauchy stress tensor and damage tensor-reversible gradient. Finally, the symmetrized effective stress tensor is introduced and interpreted.

How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle

The bidomain model, which describes the behavior of many electrically active tissues, is equivalent to a multi-dimensional cable model and can be represented by a network of resistors and capacitors. For a two-dimensional sheet of tissue, the intracellular and extracellular conductivity tensors can be visualized as two ellipses. For any pair of conductivity tensors, a coordinate transformation can be found that reduces the extracellular ellipse to a circle and aligns the intracellular ellipse with the coordinate axes. The eccentricity of the intracellular ellipse in this new coordinate system is an important parameter. It can have two special values: zero (in which case the tissue has equal anisotropy ratios) or one (in which case the tissue is comprised of one-dimensional fibers coupled through the two-dimensional extracellular space). Thus the bidomain model provides a unifying framework within which the electrical behavior of a wide variety of nerve and muscle tissues can be studied. When the anisotropy ratios in the intracellular and extracellular domains are not equal, stimulation with an anode always causes depolarization of some region of tissue. An analogous effect occurs in models that describe one-dimensional fibers, in which an "activating function" determines the site of stimulation. Experiments indicate that cardiac muscle does not have equal anisotropy ratios. Therefore, models developed to describe stimulation of axons may also help in understanding stimulation of two- or three-dimensional cardiac tissue, and may explain the concept of anodal stimulation of cardiac tissue through a "virtual cathode".