Linear Tensor Research Articles

On the tractability of linear tensor product problems in the worst case

It has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem to be weakly tractable in the worst case. The complexity of linear tensor product problems in the worst case depends on the eigenvalues { λ i } i ∈ N of a certain operator. It is known that if λ 1 = 1 and λ 2 ∈ ( 0 , 1 ) then λ n = o ( ( ln n ) − 2 ) , as n → ∞ , is a necessary condition for a problem to be weakly tractable. We show that this is a sufficient condition as well.

Generalized Tractability for Multivariate Problems Part II: Linear Tensor Product Problems, Linear Information, and Unrestricted Tractability

We continue the study of generalized tractability initiated in our previous paper “Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information”, J. Complex. 23:262–295, 2007. We study linear tensor product problems for which we can compute linear information which is given by arbitrary continuous linear functionals. We want to approximate an operator S d given as the d-fold tensor product of a compact linear operator S 1 for d=1,2,…, with ‖S 1‖=1 and S 1 having at least two positive singular values. Let n(e,S d ) be the minimal number of information evaluations needed to approximate S d to within e∈[0,1]. We study generalized tractability by verifying when n(e,S d ) can be bounded by a multiple of a power of T(e −1,d) for all (e −1,d)∈Ω⊆[1,∞)×ℕ. Here, T is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity. We study the exponent of tractability which is the smallest power of T(e −1,d) whose multiple bounds n(e,S d ). We also study weak tractability, i.e., when $\lim_{\varepsilon^{-1}+d\to\infty,(\varepsilon^{-1},d)\in \varOmega}\ln n(\varepsilon,S_{d})/(\varepsilon^{-1}+d)=0$. In our previous paper, we studied generalized tractability for proper subsets Ω of [1,∞)×ℕ, whereas in this paper we take the unrestricted domain Ω unr=[1,∞)×ℕ. We consider the three cases for which we have only finitely many positive singular values of S 1, or they decay exponentially or polynomially fast. Weak tractability holds for these three cases, and for all linear tensor product problems for which the singular values of S 1 decay slightly faster than logarithmically. We provide necessary and sufficient conditions on the function T such that generalized tractability holds. These conditions are obtained in terms of the singular values of S 1 and mostly asymptotic properties of T. The tractability conditions tell us how fast T must go to infinity. It is known that T must go to infinity faster than polynomially. We show that generalized tractability is obtained for T(x,y)=x 1+ln y . We also study tractability functions T of product form, T(x,y)=f 1(x)f 2(x). Assume that a i =lim inf x→∞(ln ln f i (x))/(ln ln x) is finite for i=1,2. Then generalized tractability takes place iff $$a_{i}>1\quad\mbox{and}\quad(a_{1}-1)(a_{2}-1)\ge1$$, and if (a 1−1)(a 2−1)=1 then we need to assume one more condition given in the paper. If (a 1−1)(a 2−1)>1 then the exponent of tractability is zero, and if (a 1−1)(a 2−1)=1 then the exponent of tractability is finite. It is interesting to add that for T being of the product form, the tractability conditions as well as the exponent of tractability depend only on the second singular eigenvalue of S 1 and they do not depend on the rate of their decay. Finally, we compare the results obtained in this paper for the unrestricted domain Ω unr with the results from our previous paper obtained for the restricted domain Ω res=[1,∞)×{1,2,…,d *}∪[1,e 0−1)×ℕ with d *≥1 and e 0∈(0,1). In general, the tractability results are quite different. We may have generalized tractability for the restricted domain and no generalized tractability for the unrestricted domain which is the case, for instance, for polynomial tractability T(x,y)=xy. We may also have generalized tractability for both domains with different or with the same exponents of tractability.

Structural and thermal properties of the monoclinic Lu2SiO5single crystal: evaluation as a new laser matrix

The crystal structure of monoclinic Lu2SiO5(LSO) crystals, grown by the Czochralski method, was determined at room temperature by X-ray diffraction. The unit-cell parameters area= 10.2550 (2),b= 6.6465 (2),c= 12.3626 (4) Å, β = 102.422 (1)° in space groupI2/a. The linear thermal expansion tensor was determined along thea,b,candc* directions over the temperature range from 303.15 to 768.15 K, and the principal coefficients of the thermal expansion tensor are found to be αI= −1.0235 × 10−6 K, αII= 4.9119 × 10−6 K and αIII= 10.1105 × 10−6 K. The temperature dependence of the cell volume and monoclinic angle were also evaluated. In addition, the specific heat and the thermal diffusivity were measured over the temperature ranges from 293.15 to 673.15 K and from 303.15 to 572.45 K, respectively. As a result, the anisotropic thermal conductivity could be calculated and is reported for the first time, to the best of the authors' knowledge. The specific heat capacity of LSO is 139.54 J mol−1 K−1, and the principal components of the thermal conductivity arekI= 2.26 W m−1 K−1,kII= 3.14 W m−1 K−1andkII= 3.67 W m−1 K−1at 303.15 K. A new structure model was proposed to better understand the relationships between the crystal structure and anisotropic thermal properties. In comparison with other laser matrix crystals, it is found that LSO possesses relatively large anisotropic thermal properties, and owing to its small heat capacity it has a moderate thermal conductivity, which is similar to those of the tungstates but lower than those of the vanadates.

Propagation Effects in Magnetized Transrelativistic Plasmas

The transfer of polarized radiation in magnetized and nonmagnetized relativistic plasmas is an area of research with numerous flaws and gaps. The present paper is aimed at filling some gaps and eliminating the flaws. Starting from a Trubnikov's linear response tensor for a vacuum wave with |k| = ω/c in thermal plasma, the analytic expression for the dielectric tensor is found in the limit of high frequencies. The Faraday rotation and Faraday conversion measures are computed in their first orders in the ratio of the cyclotron frequency Ω0 to the observed frequency ω. The computed temperature dependencies of propagation effects bridge the known nonrelativistic and ultrarelativistic limiting formulae. The fitting expressions are found for high temperatures, where the higher orders in Ω0/ω cannot be neglected. The plasma eigenmodes are found to become linearly polarized at much larger temperatures than thought before. The results are applied to the diagnostics of the hot interstellar medium, hot accretion flows, and jets.

Dispersive properties of a magnetized degenerate electron gas

Accurate dispersion relations are fundamental in the evaluation of many astrophysical processes such as, for example, heat loss by neutrino emission. In this paper, the parallel propagating wave modes of a magnetized, degenerate, electron gas are evaluated exactly using the S-matrix form of the linear response tensor (for a relativistic quantum magnetized electron gas) and the vertex formalism. The modes are then discussed in terms of the dissipative regions in which they occur. In addition to analogous modes to the unmagnetized case, there also exist doublet modes due to the logarithmic singularites that occur in the dissipative regimes of both the longitudinal and transverse dispersion relations and modes above the gyromagnetic thresholds for one of the transverse components. These additional modes only appear over a restricted range of wave numbers and are unique to the magnetized case.

Ac-driven vortices and the Hall effect in a superconductor with a tilted washboard pinning potential

The Langevin equation for a two-dimensional (2D) nonlinear guided vortex motion in a tilted cosine pinning potential in the presence of an ac is exactly solved in terms of a matrix continued fraction at arbitrary value of the Hall effect. The influence of an ac of arbitrary amplitude and frequency on the dc and ac magnetoresistivity tensors is analyzed. The ac density and frequency dependence of the overall shape and the number and position of the Shapiro-like steps on the anisotropic current-voltage characteristics are considered. The influence of a subcritical or overcritical dc on the time-dependent stationary ac longitudinal and transverse resistive vortex responses (on the frequency of an ac drive W) in terms of the nonlinear impedance tensor Z and the nonlinear ac response at W-harmonics are studied. Analytical formulas for 2D temperature-dependent linear impedance tensor Zl in the presence of a dc which depend on the angle a between the current-density vector and the guiding direction of the washboard planar pinning potential are derived and analyzed. Influence of a-anisotropy and the Hall effect on the nonlinear power absorption by vortices is discussed.

Hypersingular formulation for boundary stress evaluation revisited. Part 2: Corners.

This work focuses on the use of the hypersingular boundary integral equation<br />to evaluate the linear elastic stress tensor on the boundary. A companion<br />paper [1] has been devoted to smooth boundaries: in the present work, the<br />traction equation is analyzed at a corner, in the presence of a Lipschitz<br />boundary. Properties of the hypersingular and of the strongly singular operator<br />are discussed together with properties of the whole linear elastic integral<br />operator.

Strongly orthotropic continuum mechanics and finite element treatment

AbstractDespite successful application to orthotropic analysis, any Lagrangian strain tensor that is symmetric can be classified as an isotropic metric, while the infinitely orthotropic case can be accurately dealt with using one‐dimensional elements, structural tensors or kinematic constraints. In this paper, we present a strongly orthotropic continuum mechanics basis that models the exact kinematic behavior of the intermediary class of materials and also show its application to multi‐axial media and treatment using the finite element method. By asserting that mechanistic strain metrics must be material property dependent and satisfy equilibrium, we are able to derive a novel orthotropic linear strain tensor that is asymmetric and thus capable of describing all levels of orthotropy, while maintaining generality to the well‐established isotropic approach. Subsequently formulated are a material principal rotation tensor, extended orthotropic compliance tensor and an extended Mohr's plot for strain relying on an additional metric denoted as aspectual strain. Using the developed finite element formulation, it is shown that identical stress results to conventional theory for an orthotropic linear problem are predicted, while offering a more informative analysis. A second numerical example demonstrates the unique capability of this approach to solve the erroneous response of strongly orthotropic materials under trellis shear as compared with a number of conventional and contemporary approaches and thus its ability to produce kinematically exact results. Copyright © 2008 John Wiley & Sons, Ltd.

Real-time MR diffusion tensor and Q-ball imaging using Kalman filtering

Diffusion magnetic resonance imaging (dMRI) has become an established research tool for the investigation of tissue structure and orientation. In this paper, we present a method for real-time processing of diffusion tensor and Q-ball imaging. The basic idea is to use Kalman filtering framework to fit either the linear tensor or Q-ball model. Because the Kalman filter is designed to be an incremental algorithm, it naturally enables updating the model estimate after the acquisition of any new diffusion-weighted volume. Processing diffusion models and maps during ongoing scans provides a new useful tool for clinicians, especially when it is not possible to predict how long a subject may remain still in the magnet. First, we introduce the general linear models corresponding to the two diffusion tensor and analytical Q-ball models of interest. Then, we present the Kalman filtering framework and we focus on the optimization of the diffusion orientation sets in order to speed up the convergence of the online processing. Last, we give some results on a healthy volunteer for the online tensor and the Q-ball model, and we make some comparisons with the conventional offline techniques used in the literature. We could achieve full real-time for diffusion tensor imaging and deferred time for Q-ball imaging, using a single workstation.

Maxima: An open source computer algebra system

In one way or another, economists, especially econometricians, wind up spending a good part of their day interacting with computers. We may find ourselves writing Monte Carlo simulations designed to study the finite-sample behaviour of statistical procedures, conducting applied data analysis and visualization, or conducting mathematical analysis with the aid of symbolic mathematical software. Most are aware of the renowned open source GNU compiler collection (gcc.gnu.org), which is useful for low-level programming in, say, Fortran or C, while the profession has embraced R, which is an open source high-level language and environment for statistical computing and graphics; see www.R-project.org or Racine and Hyndman (2002) for further information. However, until recently, the availability of a tested and powerful platform for symbolic mathematics was notably absent from the open source arena. Maxima is a computer algebra package that has a variety of potential uses for economists and econometricians alike. It can be used to manipulate symbolic and numerical expressions including differentiation, integration, Taylor series, Laplace transforms, ordinary differential equations, systems of linear equations, and vectors, matrices, and tensors. Maxima produces high-precision results by using exact fractions and arbitrarily long floating point representations, and can plot functions and data in two and three dimensions, while also supporting gnuplot; see www.gnuplot.info and Racine (2006) for more information regarding this powerful portable interactive data and function plotting utility. In this brief review, we describe the Maxima computer algebra system via a set of examples, and also briefly discuss some additional features that make using Maxima a joy for economists and econometricians alike. For what follows, we employ version 5.10, which was released in the fall of 2006.

First-principles calculation of vibrational Raman spectra of tetrahedral amorphous carbon

The nonresonant vibrational Raman spectra of tetrahedral amorphous carbon are calculated from first principles. The structural model was generated using Car–Parinello molecular dynamics, the vibrational modes are determined using the linear response approach and Raman tensors are calculated using the finite electric field method. Our theoretical visible and reduced Raman spectra show an overall good agreement with experimental spectra, and better than previous calculated results. The analysis in terms of atomic vibrations shows that the Raman spectrum mainly comes from sp 2 contribution, G peak is due to the stretching vibration of any pair of sp 2 atoms and only a small sp 3 contribution can be noticed. The differences between peak intensities of reduced theoretical and experimental results mainly come from defects and the high sp 3 content in our simulated structure.

Thermal properties of monoclinic KLu(WO_4)_2 as a promising solid state laser host

Thermal analysis of the monoclinic solid state laser host KLu(WO4)2 is presented. The specific heat was measured by the relaxation method in the temperature range from 1.9 to 385 K: its value at room temperature is 0.324 J/gK. The Debye temperature and the sound velocity amount to 303+/-3 K and 3734 m/s. The linear thermal expansion tensor was measured by X-ray powder diffraction from room temperature up to 773 K. The eigenvalues of this tensor are alpha'(11)=8.98 x 10(-6) K(-1), alpha'(22)=3.35 x 10(-6) K(-1), and alpha'(33)=16.72 x 10(-6) K(-1), with the maximum value in the a-b crystallographic plane, at 31.94 degrees from the N(g) principal optical axis. The thermal diffusivity and its anisotropy in the temperature range between 300 and 500 K were measured by the pyroelectric method to determine the thermal conductivity tensor. The eigenvalues of the thermal conductivity are kappa'(11)=2.95 Wm(-1)K(-1), kappa'(22)=2.36 Wm-1K-1, and kappa'(33)=4.06 Wm(-1)K(-1), with the maximum value along a direction again in the a-b crystallographic plane, at 40.75 degrees from the N(g) principal optical axis. Simulation of the temperature distribution in a bulk sample of KLu(WO4)2 with dimensions 3 x 3 x 3 mm(3) shows that pump and laser beam directions along the N(p) principal optical axis in terms of thermal effects are preferable because the propagation is along a quasi-isothermal path.

Determination of the midsurface of a deformed shell from prescribed surface strains and bendings via the polar decomposition

We show how to determine the midsurface of a deformed thin shell from known geometry of the undeformed midsurface as well as the surface strains and bendings. The latter two fields are assumed to have been found independently and beforehand by solving the so-called intrinsic field equations of the non-linear theory of thin shells. By the polar decomposition theorem the midsurface deformation gradient is represented as composition of the surface stretch and 3D finite rotation fields. Right and left polar decomposition theorems are discussed. For each decomposition the problem is solved in three steps: (a) the stretch field is found by pure algebra, (b) the rotation field is obtained by solving a system of first-order PDEs, and (c) position of the deformed midsurface follows then by quadratures. The integrability conditions for the rotation field are proved to be equivalent to the compatibility conditions of the non-linear theory of thin shells. Along any path on the undeformed shell midsurface the system of PDEs for the rotation field reduces to the system of linear tensor ODEs identical to the one that describes spherical motion of a rigid body about a fixed point. This allows one to use analytical and numerical methods developed in analytical mechanics that in special cases may lead to closed-form solutions.

Single-shot measurement of ultrafast time-varying phase modulation induced by femtosecond laser pulses with arbitrary polarization

We demonstrate a single-shot measurement of the transient phase modulation due to field free molecular alignment at the revival times of a rotational wave packet. The wave packet is excited by an arbitrarily polarized ultrashort laser pulse in CO2 at room temperature. With this technique the time dependence along the eigenpolarization directions of the linear susceptibility tensor, i.e., the time dependence of its principle components, can be directly observed with high sensitivity.

Anisotropic Diffusion with Nonlinear Structure Tensor

A new coupled diffusion is proposed for joint image restoration and discontinuity detection. The system consists of two types of diffusion, an anisotropic diffusion on an image for restoration and an isotropic diffusion on the structure tensor of the image for singularity extraction. The diffusion on the image is guided by the structure tensor, and the diffusion of the structure tensor is supervised by the image. The coupled system keeps the advantages of the anisotropic model given by Weickert for image restoration, while it performs better in orientation discontinuities than the linear structure tensor. The existence and uniqueness of the solution of the model are proved, and a convergent iteration scheme is given to solve the system. Numerical experiments both on synthetic and natural images are presented to demonstrate the performance of the model.

Structural and Thermodiffractometric Analysis of Coordination Polymers. Part I: Tin Derivatives of the Bim ligand [Bim = Bis(1-imidazolyl)methane

New polynuclear coordination species containing the ditopic bis(1-imidazolyl)methane (Bim) ligand have been prepared as microcrystalline powders and structurally characterized by ab initio X-ray powder diffraction methods. [Bim(Me2SnCl2)]n (1), [Bim(nBu2SnCl2)]n (3), [Bim(Ph2SnCl2)]n (4), [Bim(MeSnCl3)]n (5), and [Bim(PhSnCl3)]n (6) all contain 1D chains with octahedral tin atoms with trans N-Sn-N linkages (but 4, which displays a cis N-Sn-N linkage). Their thermodiffractometric analysis allowed the estimation of the linear thermal expansion coefficients and strain tensors derived there from. The potential-energy surface of the free Bim ligand (as defined by two torsional degrees of freedom about the two N-CH2 bonds), eventually controlling the length of the repeating unit (polymer elongation), has been estimated using molecular mechanics and correlated with experimental observations.

Structural and Thermodiffractometric Analysis of Coordination Polymers. Part II:1 Zinc and Cadmium Derivatives of the Bim ligand [Bim = Bis(1-imidazolyl)methane

New polynuclear coordination species containing the ditopic bis(1-imidazolyl)methane (Bim) ligand have been prepared as microcrystalline powders and structurally characterized by ab initio X-ray powder diffraction methods. [Zn(CH3COO)2(Bim)]n contains 1D chains with tetrahedral metal atoms bridged by Bim ligands; [CdBr2(Bim)]n shows a dense packing with hexacoordinated Cd(II) ions and mu-Br and mu-Bim bridges; at variance, the isomorphous [ZnCl2(Bim)]n and [ZnBr2(Bim)]n species contain cyclic dimers based on tetrahedral Zn(II) ions. Thermodiffractometric analysis allowed estimation of the linear thermal expansion coefficients and strain tensors derived there from. Bim-rich phases, with 2:1 ligand-to-metal ratio, were also isolated: ZnBr2(Bim)2(H2O)3 and [Cd(CH3COO)2(Bim)2]n containing cis and trans MN4O2 chromophores, respectively, show 1D polymers built upon M2Bim2 cycles, hinged on the metal ions. In all species the conformation of the Bim ligands is Cs (or nearly so), while in the few sparse reports of similar coordination polymers the alternative C2 one was preferentially observed.

An energetic approach to the analysis of anisotropic hyperelastic materials

We demonstrate that for the class of anisotropic hyperelastic materials with stiffening behaviour (i.e., the stiffness increases for increasing strain), it is possible to find an approximation by means of the linear superposition of an anisotropic quadratic potential, generated by the true linear elasticity tensor of the target material, and a suitable correction potential that is isotropic and hyperelastic. The proposed method can be implemented into commercially available Finite Element software by use of featured options only. This approach is intended to provide the solution to a stress–strain problem, based entirely on energetic considerations, which ensures the convexity of the potentials, and provides a simple material characterisation procedure.

Double Poisson cohomology of path algebras of quivers

In this note, we give a description of the graded Lie algebra of double derivations of a path algebra as a graded version of the necklace Lie algebra equipped with the Kontsevich bracket. Furthermore, we formally introduce the notion of double Poisson–Lichnerowicz cohomology for double Poisson algebras, and give some elementary properties. We introduce the notion of a linear double Poisson tensor on a quiver and show that it induces the structure of a finite-dimensional algebra on the vector spaces V v generated by the loops in the vertex v. We show that the Hochschild cohomology of the associative algebra can be recovered from the double Poisson cohomology. Then, we use the description of the graded necklace Lie algebra to determine the low-dimensional double Poisson–Lichnerowicz cohomology groups for three types of (linear and nonlinear) double Poisson brackets on the free algebra C 〈 x , y 〉 . This allows us to develop some useful techniques for the computation of the double Poisson–Lichnerowicz cohomology.

Linear strain tensor and differential geometry

The components of the linear strain tensor in spherical coordinates are calculated using three methods. The first is a change of basis method and does not require elaborate concepts. The second requires the concept of covariant derivative, and the third requires the concept of a Lie derivative. The calculation provides an opportunity to become familiar with both tensorial methods and associated computational tools in the context of an elementary physical situation. The components of the linear strain tensor in an arbitrary curvilinear orthonormal basis are interpreted geometrically as half the sum of two scalar products.