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Yannan Chen

Publications and source records attributed to Yannan Chen.

33 records · Page 2Linked to original sources
arXiv2020

A Polynomially Irreducible Functional Basis of Elasticity Tensors

Tensor function representation theory is an essential topic in both theoretical and applied mechanics. For the elasticity tensor, Olive, Kolev and Auffray (2017) proposed a minimal integrity basis of 297 isotropic invariants, which is also a functional basis. Inspired by Smith's and Zheng's works, we use a novel method in this article to seek a functional basis of the elasticity tensor, that contains less number of isotropic invariants. We achieve this goal by constructing 22 intermediate tensors consisting of 11 second order symmetrical tensors and 11 scalars via the irreducible decomposition of the elasticity tensor. Based on such intermediate tensors, we further generate 429 isotropic invariants which form a functional basis of the elasticity tensor. After eliminating all the invariants that are zeros or polynomials in the others, we finally obtain a functional basis of 251 isotropic invariants for the elasticity tensor.

math-ph↗
arXiv2020

Triple Decomposition and Tensor Recovery of Third Order Tensors

In this paper, we introduce a new tensor decomposition for third order tensors, which decomposes a third order tensor to three third order low rank tensors in a balanced way. We call such a decomposition the triple decomposition, and the corresponding rank the triple rank. For a third order tensor, its CP decomposition can be regarded as a special case of its triple decomposition. The triple rank of a third order tensor is not greater than the middle value of the Tucker rank, and is strictly less than the middle value of the Tucker rank for an essential class of examples. These indicate that practical data can be approximated by low rank triple decomposition as long as it can be approximated by low rank CP or Tucker decomposition. This theoretical discovery is confirmed numerically. Numerical tests show that third order tensor data from practical applications such as internet traffic and video image are of low triple ranks. A tensor recovery method based on low rank triple decomposition is proposed. Its convergence and convergence rate are established. Numerical experiments confirm the efficiency of this method.

math.NA↗
arXiv2019

Tensor Norm, Cubic Power and Gelfand Limit

We establish two inequalities for the nuclear norm and the spectral norm of tensor products. The first inequality indicates that the nuclear norm of the square matrix is a matrix norm. We extend the concept of matrix norm to tensor norm. We show that the $1$-norm, the Frobenius norm and the nuclear norm of tensors are tensor norms, but the infinity norm and the spectral norm of tensors are not tensor norms. We introduce the cubic power for a general third order tensor, and show that a Gelfand formula holds for a general third order tensor. In that formula, for any norm, a common spectral radius-like limit exists for that third order tensor. We call such a limit the Gelfand limit. The Gelfand limit is zero if the third order tensor is nilpotent, and is one or zero if the third order tensor is idempotent. The Gelfand limit is not greater than any tensor norm of that third order tensor, and the cubic power of that third order tensor tends to zero as the power increases to infinity if and only if the Gelfand limit is less than one. The cubic power and the Gelfand limit can be extended to any higher odd order tensors.

math.NA↗
arXiv2018

An Irreducible Polynomial Functional Basis of Two-dimensional Eshelby Tensors

Representation theorems for both isotropic and anisotropic functions are of prime importance in both theoretical and applied mechanics. The Eshelby inclusion problem is very fundamental, and is of particular importance in the design of advanced functional composite materials. In this paper, we discuss about two-dimensional Eshelby tensors (denoted as $M^{(2)}$). Eshelby tensors satisfy the minor index symmetry $M_{ijkl}^{(2)}=M_{jikl}^{(2)}=M_{ijlk}^{(2)}$ and have wide applications in many fields of mechanics. In view of the representation of two-dimensional irreducible tensors in complex field, we obtain a minimal integrity basis of ten isotropic invariants of $M^{(2)}$. Remarkably, note that an integrity basis is always a functional basis, we further confirm that the minimal integrity basis is also an irreducible function basis of isotropic invariants of $M^{(2)}$.

math-ph↗
arXiv2018

Two Irreducible Functional Bases of Isotropic Invariants of A Fourth Order Three-Dimensional Symmetric and Traceless Tensor

The elasticity tensor is one of the most important fourth order tensors in mechanics. Fourth order three-dimensional symmetric and traceless tensors play a crucial role in the study of the elasticity tensors. In this paper, we present two isotropic irreducible functional bases of a fourth order three-dimensional symmetric and traceless tensor. One of them is the minimal integrity basis introduced by Smith and Bao in 1997. It has nine homogeneous polynomial invariants of degrees two, three, four, five, six, seven, eight, nine and ten, respectively. We prove that it is also an irreducible functional basis. The second irreducible functional basis also has nine homogeneous polynomial invariants. It has no quartic invariant but has two sextic invariants. The other seven invariants are the same as those of the Smith-Bao basis. Hence, the second irreducible functional basis is not contained in any minimal integrity basis.

math-ph↗
arXiv2018

Irreducible Function Bases of Isotropic Invariants of A Third Order Three-Dimensional Symmetric and Traceless Tensor

Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless tensor has four invariants with degrees two, four, six and ten respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no polynomial syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.

math-ph↗
arXiv2017

Octupolar Tensors for Liquid Crystals

A third-order three-dimensional symmetric traceless tensor, called the \emph{octupolar} tensor, has been introduced to study tetrahedratic nematic phases in liquid crystals. The octupolar \emph{potential}, a scalar-valued function generated on the unit sphere by that tensor, should ideally have four maxima capturing the most probable molecular orientations (on the vertices of a tetrahedron), but it was recently found to possess an equally generic variant with \emph{three} maxima instead of four. It was also shown that the irreducible admissible region for the octupolar tensor in a three-dimensional parameter space is bounded by a dome-shaped surface, beneath which is a \emph{separatrix} surface connecting the two generic octupolar states. The latter surface, which was obtained through numerical continuation, may be physically interpreted as marking a possible \emph{intra-octupolar} transition. In this paper, by using the resultant theory of algebraic geometry and the E-characteristic polynomial of spectral theory of tensors, we give a closed-form, algebraic expression for both the dome-shaped surface and the separatrix surface. This turns the envisaged intra-octupolar transition into a quantitative, possibly observable prediction. Some other properties of octupolar tensors are also studied.

math-ph↗
arXiv2017

Spectral Analysis of Piezoelectric Tensors

A third order real tensor is called a piezoelectric-type tensor if it is partially symmetric with respect to its last two indices. The piezoelectric tensor is a piezoelectric-type tensor of dimension three. We introduce C-eigenvalues and C-eigenvectors for piezoelectric-type tensors. Here, "C" names after Curie brothers, who first discovered the piezoelectric effect. We show that C-eigenvalues always exist, they are invariant under orthonormal transformations, and for a piezoelectric-type tensor, the largest C-eigenvalue and its C-eigenvectors form the best rank-one piezoelectric-type approximation of that tensor. This means that for the piezoelectric tensor, its largest C-eigenvalue determines the highest piezoelectric coupling constant. We further show that for the piezoelectric tensor, the largest C-eigenvalue corresponds to the electric displacement vector with the largest $2$-norm in the piezoelectric effect under unit uniaxial stress, and the strain tensor with the largest $2$-norm in the converse piezoelectric effect under unit electric field vector. Thus, C-eigenvalues and C-eigenvectors have concrete physical meanings in piezoelectric effect and converse piezoelectric effect. Finally, we apply C-eigenvalues and associated C-eigenvectors for various piezoelectric crystals with different symmetries.

math-ph↗
arXiv2016

Finding the maximum eigenvalue of a class of tensors with applications in copositivity test and hypergraphs

Finding the maximum eigenvalue of a symmetric tensor is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a semi-definite program algorithm for computing the maximum $H$-eigenvalue of a class of tensors with sign structure called $W$-tensors. The class of $W$-tensors extends the well-studied nonnegative tensors and essentially nonnegative tensors, and covers some important tensors arising naturally from spectral hypergraph theory. Our algorithm is based on a new structured sums-of-squares (SOS) decomposition result for a nonnegative homogeneous polynomial induced by a $W$-tensor. This SOS decomposition enables us to show that computing the maximum $H$-eigenvalue of an even order symmetric $W$-tensor is equivalent to solving a semi-definite program, and hence can be accomplished in polynomial time. Numerical examples are given to illustrate that the proposed algorithm can be used to find maximum $H$-eigenvalue of an even order symmetric $W$-tensor with dimension up to $10,000$. We present two applications for our proposed algorithm: we first provide a polynomial time algorithm for computing the maximum $H$-eigenvalues of large size Laplacian tensors of hyper-stars and hyper-trees; second, we show that the proposed SOS algorithm can be used to test the copositivity of a multivariate form associated with symmetric extended $Z$-tensors, whose order may be even or odd. Numerical experiments illustrate that our structured semi-definite program algorithm is effective and promising.

math.SP↗
arXiv2016

Computing Eigenvalues of Large Scale Sparse Tensors Arising from a Hypergraph

The spectral theory of higher-order symmetric tensors is an important tool to reveal some important properties of a hypergraph via its adjacency tensor, Laplacian tensor, and signless Laplacian tensor. Owing to the sparsity of these tensors, we propose an efficient approach to calculate products of these tensors and any vectors. Using the state-of-the-art L-BFGS approach, we develop a first-order optimization algorithm for computing H- and Z-eigenvalues of these large scale sparse tensors (CEST). With the aid of the Kurdyka-Łojasiewicz property, we prove that the sequence of iterates generated by CEST converges to an eigenvector of the tensor.When CEST is started from multiple randomly initial points, the resulting best eigenvalue could touch the extreme eigenvalue with a high probability. Finally, numerical experiments on small hypergraphs show that CEST is efficient and promising. Moreover, CEST is capable of computing eigenvalues of tensors corresponding to a hypergraph with millions of vertices.

math.CO↗
arXiv2015

Computing Eigenvalues of Large Scale Hankel Tensors

Large scale tensors, including large scale Hankel tensors, have many applications in science and engineering. In this paper, we propose an inexact curvilinear search optimization method to compute Z- and H-eigenvalues of $m$th order $n$ dimensional Hankel tensors, where $n$ is large. Owing to the fast Fourier transform, the computational cost of each iteration of the new method is about $\mathcal{O}(mn\log(mn))$. Using the Cayley transform, we obtain an effective curvilinear search scheme. Then, we show that every limiting point of iterates generated by the new algorithm is an eigen-pair of Hankel tensors. Without the assumption of a second-order sufficient condition, we analyze the linear convergence rate of iterate sequence by the Kurdyka-Łojasiewicz property. Finally, numerical experiments for Hankel tensors, whose dimension may up to one million, are reported to show the efficiency of the proposed curvilinear search method.

math.NA↗
arXiv2015

Three Dimensional Strongly Symmetric Circulant Tensors

In this paper, we give a necessary and sufficient condition for an even order three dimensional strongly symmetric circulant tensor to be positive semi-definite. In some cases, we show that this condition is also sufficient for this tensor to be sum-of-squares. Numerical tests indicate that this is also true in the other cases.

math.SP↗
arXiv2015

Positive Semi-Definiteness and Sum-of-Squares Property of Fourth Order Four Dimensional Hankel Tensors

A positive semi-definite (PSD) tensor which is not a sum-of-squares (SOS) tensor is called a PSD non-SOS (PNS) tensor. Is there a fourth order four dimensional PNS Hankel tensor? Until now, this question is still an open problem. Its answer has both theoretical and practical meanings. We assume that the generating vector $v$ of the Hankel tensor $A$ is symmetric. Under this assumption, we may fix the fifth element $v_4$ of $v$ at $1$. We show that there are two surfaces $M_0$ and $N_0$ with the elements $v_2, v_6, v_1, v_3, v_5$ of $v$ as variables, such that $M_0 \ge N_0$, $A$ is SOS if and only if $v_0 \ge M_0$, and $A$ is PSD if and only if $v_0 \ge N_0$, where $v_0$ is the first element of $v$. If $M_0 = N_0$ for a point $P = (v_2, v_6, v_1, v_3, v_5)^\top$, then there are no fourth order four dimensional PNS Hankel tensors with symmetric generating vectors for such $v_2, v_6, v_1, v_3, v_5$. Then, we call such a point $P$ PNS-free. We show that a $45$-degree planar closed convex cone, a segment, a ray and an additional point are PNS-free. Numerical tests check various grid points, and find that they are also PNS-free.

math.SP↗
arXiv2013

Manual Calibration System for Daya Bay Reactor Neutrino Experiment

The Daya Bay Reactor Neutrino Experiment has measured the last unknown neutrino mixing angle, θ13, to be non-zero at the 7.7σ level. This is the most precise measurement to θ13 to date. To further enhance the understanding of the response of the antineutrino detectors (ADs), a detailed calibration of an AD with the Manual Calibration System (MCS) was undertaken during the summer 2012 shutdown. The MCS is capable of placing a radioactive source with a positional accuracy of 25 mm in R direction, 20 mm in Z axis and 0.5° in Φ direction. A detailed description of the MCS is presented followed by a summary of its performance in the AD calibration run.

physics.ins-det↗