Tensor Character Research Articles

On advantages of the Kelvin mapping in finite element implementations of deformation processes

Classical continuum mechanical theories operate on three-dimensional Eu-clidian space using scalar, vector, and tensor-valued quantities usually up to the order of four. For their numerical treatment, it is common practice to transform the relations into a matrix-vector format. This transformation is usually performed using the so-called Voigt mapping. This mapping does not preserve tensor character leaving significant room for error as stress and strain quantities follow from different mappings and thus have to be treated differently in certain mathematical operations. Despite its conceptual and notational difficulties having been pointed out, the Voigt mapping remains the foundation of most current finite element programmes. An alternative is the so-called Kelvin mapping which has recently gained recognition in studies of theoretical mechanics. This article is concerned with benefits of the Kelvin mapping in numerical modelling tools such as finite element software. The decisive difference to the Voigt mapping is that Kelvin's method preserves tensor character, and thus the numerical matrix notation directly corresponds to the original tensor notation. Further benefits in numerical implementations are that tensor norms are calculated identically without distinguishing stress or strain-type quantities and tensor equations can be directly transformed into matrix equations without additional considerations. The only implementational changes are related to a scalar factor in certain finite element matrices and hence, harvesting the mentioned benefits comes at very little cost.

On the gradient of the Green tensor in two-dimensional elastodynamic problems, and related integrals: Distributional approach and regularization, with application to nonuniformly moving sources

The two-dimensional elastodynamic Green tensor is the primary building block of solutions of linear elasticity problems dealing with nonuniformly moving rectilinear line sources, such as dislocations. Elastodynamic solutions for these problems involve derivatives of this Green tensor, which stand as hypersingular kernels. These objects, well defined as distributions, prove cumbersome to handle in practice. This paper, restricted to isotropic media, examines some of their representations in the framework of distribution theory. A particularly convenient regularization of the Green tensor is introduced, that amounts to considering line sources of finite width. Technically, it is implemented by an analytic continuation of the Green tensor to complex times. It is applied to the computation of regularized forms of certain integrals of tensor character that involve the gradient of the Green tensor. These integrals are fundamental to the computation of the elastodynamic fields in the problem of nonuniformly moving dislocations. The obtained expressions indifferently cover cases of subsonic, transonic, or supersonic motion. We observe that for faster-than-wave motion, one of the two branches of the Mach cone(s) displayed by the Cartesian components of these tensor integrals is extinguished for some particular orientations of source velocity vector.

Resonant X-ray scattering study of spinel Co2MnO4

X-ray resonant enhancement of the (200) crystallographically forbidden reflection in the inverse cubic spinel Co2MnO4 single crystal is presented. Anisotropic X-ray scattering results near the K absorption edges of both Co and Mn ions observed in the azimuthal angle dependence are interpreted with the anisotropic tensorial character of the X-ray susceptibility. Spectra calculations including the various electric multipole transition channels, such as dipole-dipole (E1-E1), dipole-quadrupole (E1-E2), and quadrupole-quadrupole (E2-E2), in two different polarization states of σ − σ and σ − π are discussed and compared with the experimental data to determine the channel of major contribution in the resonant scattering intensity.

Electromagnetic wave propagation in anisotropic metamaterial waveguides

SummaryIn this paper, we present the formalism of the transverse operator method (TOM). A rigorous study of propagation in anisotropic and multi‐layer medium using the tensor character of the permeability and permittivity is presented. With the application of the Galerkin method, the propagating modes in metallic rectangular waveguides filled with anisotropic metamaterial are exploited. The results are compared with those previously published and show a good agreement. The complex modes have been obtained.The advantages of the techniques used in this paper lie in the proper analytical formulation of the problem studied, on the one hand, and the speed of convergence, on the other hand. TOM offers a fast convergence of the propagation constant. This shows the effectiveness of our numerical model. As such, the formulation of the transverse operator could be a useful tool for microwave engineers. This type of materials known as metamaterial is widely used and needed by industries and information technology, especially in microwave and radiofrequency devices such as patch antennas, antennas waveguides, resonators, circulators, isolators, phase shifters, and filters. Copyright © 2014 John Wiley & Sons, Ltd.

Dimensionality Reduction for Hyperspectral Data Based on Class-Aware Tensor Neighborhood Graph and Patch Alignment.

To take full advantage of hyperspectral information, to avoid data redundancy and to address the curse of dimensionality concern, dimensionality reduction (DR) becomes particularly important to analyze hyperspectral data. Exploring the tensor characteristic of hyperspectral data, a DR algorithm based on class-aware tensor neighborhood graph and patch alignment is proposed here. First, hyperspectral data are represented in the tensor form through a window field to keep the spatial information of each pixel. Second, using a tensor distance criterion, a class-aware tensor neighborhood graph containing discriminating information is obtained. In the third step, employing the patch alignment framework extended to the tensor space, we can obtain global optimal spectral-spatial information. Finally, the solution of the tensor subspace is calculated using an iterative method and low-dimensional projection matrixes for hyperspectral data are obtained accordingly. The proposed method effectively explores the spectral and spatial information in hyperspectral data simultaneously. Experimental results on 3 real hyperspectral datasets show that, compared with some popular vector- and tensor-based DR algorithms, the proposed method can yield better performance with less tensor training samples required.

Hopf algebras, distributive (Laplace) pairings and hash products: a unified approach to tensor product decompositions of group characters

We show for bicommutative graded connected Hopf algebras that a certain distributive (Laplace) subgroup of the convolution monoid of 2-cochains parameterizes certain well behaved Hopf algebra deformations. Using the Laplace group, or its Frobenius subgroup, we define higher derived hash products, and develop a general theory to study their main properties. Applying our results to the (universal) bicommutative graded connected Hopf algebra of symmetric functions, we show that classical tensor product and character decompositions, such as those for the general linear group, mixed co- and contravariant or rational characters, orthogonal and symplectic group characters, Thibon and reduced symmetric group characters, are special cases of higher derived hash products. In the appendix we discuss a relation to formal group laws.

Analytical calculation of four-point correlations for a simple model of cages involving numerous particles

Dynamics of a one-dimensional system of Brownian particles with short-range repulsive interaction (diameter σ) is studied with a liquid-theoretical approach. The mean square displacement, the two-particle displacement correlation, and the overlap-density-based generalized susceptibility are calculated analytically by way of the Lagrangian correlation of the interparticulate space, instead of the Eulerian correlation of density that is commonly used in the standard mode-coupling theory. In regard to the mean square displacement, the linear analysis reproduces the established result on the asymptotic subdiffusive behavior of the system. A finite-time correction is given by incorporating the effect of entropic nonlinearity with a Lagrangian version of mode-coupling theory. The notorious difficulty in derivation of the mode-coupling theory concerning violation of the fluctuation-dissipation theorem is found to disappear by virtue of the Lagrangian description. The Lagrangian description also facilitates analytical calculation of four-point correlations in the space-time, such as the two-particle displacement correlation. The two-particle displacement correlation, which is asymptotically self-similar in the space-time, illustrates how the cage effect confines each particle within a short radius on one hand and creates collective motion of numerous particles on the other hand. As the time elapses, the correlation length grows unlimitedly, and the generalized susceptibility based on the overlap density converges to a finite value which is an increasing function of the density. The distribution function behind these dynamical four-point correlations and its extension to three-dimensional cases, respecting the tensorial character of the two-particle displacement correlation, are also discussed.

Stefan-Boltzmann’S Law under Relativistic Conditions; Generalized Case

An expression was obtained for the energy density of the moving black-body radiation, i.e., the Stefan-Boltzmann law valid in the interval of object velocities from zero to the velocity of light in vacuo. The object temperature is shown to comprise two parts. The first one is a scalar invariant under the Lorentz transformations. The second one is a vector depending on the velocity of system motion. The scalar component of the temperature is a contraction of two tensor components of rank 3. Under normal conditions this mathematical object is a scalar. Taking account of a tensor character of the temperature a new formulation is given for the second thermodynamics law. The results obtained are of the great practical importance, in particular, while designing devices to measure the radiation temperature of moving cosmic objects, e.g., quasars.

Engineering Spin Waves in a High-Spin Ultracold Fermi Gas

We report on the detailed study of multicomponent spin waves in an s=3/2 Fermi gas where the high spin leads to novel tensorial degrees of freedom compared to s=1/2 systems. The excitations of a spin-nematic state are investigated from the linear to the nonlinear regime, where the tensorial character is particularly pronounced. By tuning the initial state we engineer the tensorial spin-wave character, such that the magnitude and the sign of the counterflow spin currents are effectively controlled. A comparison of our data with numerical and analytical results shows good agreement.

典型断层模型的重力全梯度张量特征研究<br>Study on Characteristics of the Full Gravity Gradient Tensors of the Typical Fault Models

以高斯定理为出发点，将计算梯度张量的体积分变成包围该积分域的全表面积分，利用格林公式把对每个面的积分变成对每条边的线积分之和，实现了对5个具有不同断层倾角的三维断层模型的梯度张量数值计算，得到其梯度张量平面分布图。断层模型的Vzz分量能够直接反映断层上下盘的轮廓，并且异常的符号直接反映场源体剩余密度的盈亏；Vxx和Vyy分量以密集的等值线带反映出断层上下盘在南北向和东西向的边缘；而Vzx和Vzy则以极值的形式反映出上下盘的边缘位置，且Vzx分量以极小值的形式揭示出断层面的位置；Vxy分量以极值圈闭的特征揭示上下盘的各个角点位置。对三维断层模型梯度张量分布特征的研究为从重力学方面对断裂构造的识别提供了理论依据。 The gravity gradient tensors of five three-dimensional fault models with different dip angle were calculated and got their plane distribution map by applying the divergence theorem, which changed the volume integral into surface integral, then changed into line integral by using Green’s theorem. The characteristics of gravity gradient tensors of five fault models show that the Vzz component can directly reflect the shapes of the fault footwall, and its abnormal symbols directly reflect the profit or loss of the residual density. Vxx and Vyy components with dense isoline zone reflect the edge of fault footwall in north-south or east-west direction. While Vzx and Vzy reflect the edge of the lower plate in the form of extremum, and Vzx component reveals a fault surface in the form of minimum. Vxy component reflects the cornor of the footwall in the form of extremal. The research on the gradient tensor distribution characteristics of three dimensional fault models could provide a theory basis for finding fault tectonic.

Principal growth directions in development of the lateral root in Arabidopsis thaliana

During lateral root development a new meristem is formed within the mother root body. The main objective of this work was to simulate lateral root formation in Arabidopsis thaliana and to study a potential role of the principal directions in this process. Lateral root growth is anisotropic, so that three principal directions of growth can be distinguished within the organ. This suggests a tensorial character of growth and allows for its description by means of the growth tensor method. First features of the cell pattern of developing lateral roots were analysed in A. thaliana and then a tensorial model for growth and division of cells for this case was specified, assuming an unsteady character of the growth field of the organ. Microscopic observations provide evidence that the principal directions of growth are manifested at various developmental stages by oblique cell walls observed in different regions of the primordium. Other significant features observed are atypically shaped large cells at the flanks of young apices, as well as distinct boundaries between the mother root and the primordium. Simulations were performed using a model for growth. In computer-generated sequences the above-mentioned features could be identified. An attempt was made to reconstruct the virtual lateral root that included a consideration of the formation of particular tissue types based on literature data. In the cell pattern of the developing lateral root the principal directions of growth can be recognized through occurrence of oblique cell divisions. In simulation the role of these directions in cell pattern formation was confirmed, only when cells divide with respect to the principal directions can realistic results be obtained.

On the conformal mappings of quasi-Einstein spaces

We have studied the conformal mappings of quasi-Einstein spaces. We have proved that they are closed with respect to concircular mappings. We have introduced the notion of mobility of pseudo-Riemannian spaces with respect to concircular mappings. The tensor characteristic of maximal mobility for quasi-Einstein spaces has also been found.

Non-linear oscillatory rheological properties of a generic continuum foam model: Comparison with experiments and shear-banding predictions

The occurrence of shear bands in a complex fluid is generally understood as resulting from a structural evolution of the material under shear, which leads (from a theoretical perspective) to a non-monotonic stationary flow curve related to the coexistence of different states of the material under shear. In this paper we present a scenario for shear-banding in a particular class of complex fluids, namely foams and concentrated emulsions, which differs from other scenarios in two important ways. First, the appearance of shear bands is shown to be possible both without any intrinsic physical evolution of the material (e.g. via a parameter coupled to the flow such as concentration or entanglements) and without any finite critical shear rate below which the flow does not remain stationary and homogeneous. Secondly, the appearance of shear bands depends on the initial conditions, i.e. the preparation of the material. In other words, it is history dependent. This behaviour relies on the tensorial character of the underlying model (2D or 3D) and is triggered by an initially inhomogeneous strain distribution in the material. The shear rate displays a discontinuity at the band boundary whose amplitude is history dependent and thus depends on the sample preparation.

Unexpected Geometrical Effects on Paramagnetic Spin–Orbit and Spin–Dipolar 2JFF Couplings

The second-rank tensor character of the paramagnetic spin-orbit and spin-dipolar contributions to nuclear spin-spin coupling constants is usually ignored when NMR measurements are carried out in the isotropic phase. However, in this study it is shown that isotropic (2)J(FF) couplings strongly depend on the relative orientation of the C-F bonds containing the coupling nuclei and the eigenvectors of such tensors. Predictions about such effect are obtained using a qualitative approach based on the polarization propagator formalism at the RPA, and results are corroborated performing high-level ab initio spin-spin coupling calculations at the SOPPA(CCSD)/EPR-III//MP2/EPR-III level in a model system. It is highlighted that no calculations at the RPA level were carried out in this work. The quite promising results reported in this paper suggest that similar properties are expected to hold for the second-rank nuclear magnetic shielding tensor.

Flow and capacity tensor characteristics of anisotropic porous media: Theory and experiment

Tensors determining and describing the main flow and capacity characteristics of anisotropic media are introduced. A system of laboratorymeasurements for the determination of the tensors of areal porosity, permeability, limiting gradients, and characteristic linear dimensions is considered. The pressure dependences of permeability are generalized to take into account the medium anisotropy. The results of an experiment for the determination of the permeability and limiting (initial) gradient tensors in band sandstone with orthotropic flow characteristics are presented. The measurements were carried out on four cores: two along the principal stratification directions, one perpendicular to the stratification, and one inclined at an angle of 45° to the stratification plane. The latter (fourth) specimen was taken for reference: to test the tensor nature of the introduced mathematical objects and formulas. The good agreement of the theoretical and experimental results makes it possible to recommend for engineering calculations both the formulas proposed and the developed method of laboratory investigation of the flow characteristics of anisotropic reservoirs.

Tensor total variation approach to optical coherence tomography reconstruction for improved visualization of retinal microvasculature

A novel optical coherence tomography (OCT) reconstruction approach is introduced for improved visualization of inner-retina capillaries in retinal OCT tomograms. The proposed method utilizes a minimization framework based on a tensor total variation (TTV) energy functional, to enforce capillary structural characteristics in the spatial domain. By accounting for structure tensor characteristics, the TTV reconstruction method allows for contrast enhancement of capillary structural characteristics. The novel TTV method was tested on high resolution OCT images acquired in-vivo from the foveal region of the retina of a healthy human subject. Experimental results demonstrate significant contrast and visibility enhancement of the inner retina capillaries in the retinal OCT tomograms, achieved by use of the TTV reconstruction method. Therefore, the TTV method has a strong potential for improved disease progression analysis based on the study of disease-induced changes in the inner retina vasculature.

Hyperfine characterization and spin coherence lifetime extension in Pr3+:La2(WO4)3

Rare-earth ions in dielectric crystals are interesting candidates for storing quantum states of photons. A limiting factor on the optical density and thus the conversion efficiency is the distortion introduced in the crystal by doping elements of one type into a crystal matrix of another type. Here, we investigate the system Pr3+:La2(WO4)3, where the similarity of the ionic radii of Pr and La minimizes distortions due to doping. We characterize the praseodymium hyperfine interaction of the ground state (3H4) and one excited state (1D2) and determine the spin Hamiltonian parameters by numerical analysis of Raman-heterodyne spectra, which were collected for a range of static external magnetic field strengths and orientations. On the basis of a crystal field analysis, we discuss the physical origin of the experimentally determined quadrupole and Zeeman tensor characteristics. We show the potential for quantum memory applications by measuring the spin coherence lifetime in a magnetic field that is chosen such that additional magnetic fields do not shift the transition frequency in first order. Experimental results demonstrate a spin coherence lifetime of 158 ms - almost three orders of magnitude longer than in zero field.

Simultaneous measurement of 1H–15N and Methyl 1Hm–13Cm residual dipolar couplings in large proteins

A two-dimensional TROSY-based SIM-(13)C(m)-(1)H(m)/(1)H-(15)N NMR experiment for simultaneous measurements of methyl (1) D (CH) and backbone amide (1) D (NH) residual dipolar couplings (RDC) in {U-[(15)N,(2)H]; Ileδ1-[(13)CH(3)]; Leu,Val-[(13)CH(3)/(12)CD(3)]}-labeled samples of large proteins is described. Significant variation in the alignment tensor of the 82-kDa enzyme Malate synthase G is observed as a function of only slight changes in experimental conditions. The SIM-(13)C(m)-(1)H(m)/(1)H-(15)N data sets provide convenient means of establishing the alignment tensor characteristics via the measurement of (1) D (NH) RDCs in the same protein sample.

Response of an atom interacting with an arbitrarily polarized electromagnetic field

We develop the theory of interaction of the electromagnetic field and a single atom being in an arbitrary state and having an arbitrary direction of the angular momentum of the atomic electron with respect to the direction of the field polarization vector. It is shown that the atom response current has a tensor structure and depends on both the direction of the angular momentum of the atom, and the polarization vector of the external field. The tensor character of the response is determined by the externally induced anisotropic distribution of the probability density of spatial localization of the atomic electron. It is shown that the induced-anisotropy effects clarify the harmonic generation mechanism at play during the non-resonance interaction of laser radiation with atomic media. The developed theory is applied to the analysis of the problem about the generation of terahertz waves in a two-color laser field. It is shown that the change in the mutual orientation of wave polarization vectors leads to a significant increase in the efficiency of conversion of high-frequency fields to low-frequency ones. It is shown for the first time that the generation of terahertz waves is possible in the preionization regime, when the generation mechanism is related to atomic nonlinearity.

Nonlinear optical response of an atom in the field of femtosecond laser pulses with near-atomic intensity

A theory of the nonresonant response of a single atom in a state with arbitrary magnitude and direction of the angular momentum of an atomic electron with respect to the polarization vector of the acting electromagnetic field has been developed. It has been shown that the atomic response current has a tensor structure and depends both on the direction of the angular momentum of the atom and on the polarization vector of the external field. The tensor character of the response is due to the effects of the anisotropy of probability density distribution of the atomic electron as compared to the case of the free atom. The selection rules for the axisymmetric problem of the atom in the field have been analyzed. The manifestation of the selection rules in the angular spectra of photoelectrons has been demonstrated. The probability of the ionization of the atom has been analyzed as a function of the amplitude and duration of the pulse. It has been shown that the width of the generation spectrum is a nonlinear function of the field strength and is saturated in the region of nearly atomic fields. Methods for controlling the parameters of the atomic response spectrum have been proposed on the basis of the use of a sequence of laser pulses with various time profiles, carrier frequencies, and polarization states. It has been shown that the generation of terahertz radiation is possible in the preionization regime, where the generation mechanism is attributed to atomic nonlinearity.