On warped product manifolds satisfying some curvature conditions

We determine curvature properties of pseudosymmetric type of certain warped product manifolds, and in particular of generalized Robertson–Walker spacetimes, with Einsteinian or quasi-Einsteinian fibre.     

An accurate gradient and Hessian reconstruction method for cell‐centered finite volume discretizations on general unstructured grids

In this paper, a novel reconstruction of the gradient and Hessian tensors on an arbitrary unstructured grid, developed for implementation in a cell‐centered finite volume framework, is presented. The reconstruction, based on the application of Gauss' theorem, provides a fully second‐order accurate estimate of the gradient, along with a first‐order estimate of the Hessian tensor. The reconstruction is implemented through the construction of coefficient matrices for the gradient components and independent components of the Hessian tensor, resulting in a linear system for the gradient and Hessian fields, which may be solved to an arbitrary precision by employing one of the many methods available for the efficient inversion of large sparse matrices. Numerical experiments are conducted to demonstrate the accuracy, robustness, and computational efficiency of the reconstruction by comparison with other common methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Tensor Products of Non-Archimedean Weighted Spaces of Continuous Functions

It is shown that the completion of the tensor product of two non-Archimedean weighted spaces of continuous functions is topologically isomorphic to another weighted space. Several applications of this result are given.     

Classification of hypersurfaces with constant Laguerre eigenvalues in R~n

Let x:M → Rn be an umbilical free hypersurface with non-zero principal curvatures.Then x is associated with a Laguerre metric g,a Laguerre tensor L,a Laguerre form C,and a Laguerre second fundamental form B,which are invariants of x under Laguerre transformation group.An eigenvalue of Laguerre tensor L of x is called a Laguerre eigenvalue of x.In this paper,we classify all oriented hypersurfaces with constant Laguerre eigenvalues and vanishing Laguerre form.     

A characterization of the Ejiri torus in <Emphasis Type="Italic">S</Emphasis><Superscript>5</Superscript>

We conjecture that a Willmore torus having Willmore functional between 2π ² and 2π ² \(\sqrt 3 \) is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri’s torus in S ⁵ is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S ⁿ by reducing them into elastic curves in S ³, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S ⁵ attains the minimum 2π ² \(\sqrt 3 \), which indicates our conjecture holds true for Willmore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S ⁵. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in S ³. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough.     

Anisotropic Reynolds stress tensor representation in shear flows using DNS and experimental data

Tensorial decompositions and projections are used to study the performance of algebraic non-linear models and predict the anisotropy of the Reynolds stresses. Direct numerical simulation (DNS) data for plane channel flows at friction Reynolds number (Re_τ = 180, 395, 590, 1000), and for the boundary layer using both DNS (Re_τ = 359, 830, 1271) and experimental data (Re_τ = 2680, 3891, 4941, 7164) are used to build and evaluate the models. These data are projected into tensorial basis formed from the symmetric part of mean velocity gradient and non-persistence-of-straining tensor. Six models are proposed and their performances are investigated. The scalar coefficients for these six different levels of approximations of the Reynolds stress tensor are derived, and made dimensionless using the classical turbulent scales, the kinetic turbulent energy (κ) and its dissipation rate (ε). The dimensionless coefficients are then coupled with classical wall functions. One model is selected by comparing the predicted Reynolds stress components with experimental and DNS data, presenting a good prediction for the shear and normal Reynolds stresses.     

Fast tensor method for summation of long‐range potentials on 3D lattices with defects

In this paper, we present a method for fast summation of long‐range potentials on 3D lattices with multiple defects and having non‐rectangular geometries, based on rank‐structured tensor representations. This is a significant generalization of our recent technique for the grid‐based summation of electrostatic potentials on the rectangular L × L × L lattices by using the canonical tensor decompositions and yielding the O(L) computational complexity instead of O(L³) by traditional approaches. The resulting lattice sum is calculated as a Tucker or canonical representation whose directional vectors are assembled by the 1D summation of the generating vectors for the shifted reference tensor, once precomputed on large N × N × N representation grid in a 3D bounding box. The tensor numerical treatment of defects is performed in an algebraic way by simple summation of tensors in the canonical or Tucker formats. To diminish the considerable increase in the tensor rank of the resulting potential sum, the ?‐rank reduction procedure is applied based on the generalized reduced higher‐order SVD scheme. For the reduced higher‐order SVD approximation to a sum of canonical/Tucker tensors, we prove the stable error bounds in the relative norm in terms of discarded singular values of the side matrices. The required storage scales linearly in the 1D grid‐size, O(N), while the numerical cost is estimated by O(NL). The approach applies to a general class of kernel functions including those for the Newton, Slater, Yukawa, Lennard‐Jones, and dipole‐dipole interactions. Numerical tests confirm the efficiency of the presented tensor summation method; we demonstrate that a sum of millions of Newton kernels on a 3D lattice with defects/impurities can be computed in seconds in Matlab implementation. The tensor approach is advantageous in further functional calculus with the lattice potential sums represented on a 3D grid, like integration or differentiation, using tensor arithmetics of 1D complexity. Copyright © 2015 John Wiley & Sons, Ltd.     

Tensor logarithmic norm and its applications

Matrix logarithmic norm is an important quantity, which characterize the stability of linear dynamical systems. We propose the logarithmic norms for tensors and tensor pairs, and extend some classical results from the matrix case. Moreover, the explicit forms of several tensor logarithmic norms and semi‐norms are also derived. Employing the tensor logarithmic norms, we bound the real parts of all the eigenvalues of a complex tensor and study the stability of a class of nonlinear dynamical systems. Copyright © 2016 John Wiley & Sons, Ltd.