ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅱ)──CRYSTAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS RESTRICTED BY VARIOUS MATERIAL SYMMETRIES

The explicit representations for tensorial Fourier expansion of 3_D crystal orientation distribution functions (CODFs) are established. In comparison with that the coefficients in the mth term of the Fourier expansion of a 3_D ODF make up just a single irreducible mth_order tensor, the coefficients in the mth term of the Fourier expansion of a 3_D CODF constitute generally so many as 2m+1 irreducible mth_order tensors. Therefore, the restricted forms of tensorial Fourier expansions of 3_D CODFs imposed by various micro_ and macro_scopic symmetries are further established, and it is shown that in most cases of symmetry the restricted forms of tensorial Fourier expansions of 3_D CODFs contain remarkably reduced numbers of mth_order irreducible tensors than the number 2m+1 . These results are based on the restricted forms of irreducible tensors imposed by various point_group symmetries, which are also thoroughly investigated in the present part in both 2_ and 3_D spaces.

ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅰ) ──DIRECTIONAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS

In this two_part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically_based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (Ⅰ), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N_dimensional (N_D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2_ and 3_D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point_group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work -Part (Ⅱ), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point_group symmetries are derived.

Geometric simplicity of spectral radius of nonnegative irreducible tensors

We study irreducible tensors. the real and complex geometric simplicity of nonnegative First, we prove some basic conclusions. Based on the conclusions, the real geometric simplicity of the spectral radius of an even- order nonnegative irreducible tensor is proved. For an odd-order nonnegative irreducible tensor, sufficient conditions are investigated to ensure the spectral radius to be real geometrically simple. Furthermore, the complex geometric simplicity of nonnegative irreducible tensors is also studied.

An Inequality for the Perron Pair of an Irreducible and Symmetric Nonnegative Tensor with Application

The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular M-tensor.We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors.We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors.We finally consider the perturbation of the smallest eigenvalue of nonsingular M-tensors and design a strategy to compute its smallest eigenvalue.We verify our results via random numerical examples.

Quantum State Transfer in Engineered Spin Chain under Influence of Spatially Distributed Environment

It has been shown that a quantum state could be perfectly transferred via a spin chain with engineered'always-on interaction'.In this paper,we study a more realistic problem for such a quantum state transfer (QST)protocol,how the efficacy of QST is reduced by the quantum decoherence induced by a spatially distributed environment.Here,the environment is universally modeled as a bath of fermions located in different positions.By making use of theirreducible tensor method in angular momentum theory,we investigate the effect of environment on the efficiency of QSTfor both cases at zero and finite temperatures.We not only show the generic exponential decay of QST efficiency as thenumber of sites increase,but also find some counterintuitive effect,the QST can be enhanced as temperature increasesin some cases.

On the Construction of n-Electron States with Symplectic Symmetry

Under quasispin scheme, a complete group theoretical classification of fermion states with symplectlc symmetry is proposed. Furthermore, the first and second order irreducible tensor operators are investigated in detail to approach the fermion states with explicit forms.

A new definition of geometric multiplicity of eigenvalues of tensors and some results based on it

We give a new definition of geometric multiplicity of eigenvalues of tensors, and based on this, we study the geometric and algebraic multiplicity of irreducible tensors＇ eigenvalues. We get the result that the eigenvalues with modulus p have the same geometric multiplicity. We also prove that two- dimensional nonnegative tensors＇ geometric multiplicity of eigenvalues is equal to algebraic multiplicity of eigenvalues.

LINEAR CONVERGENCE OF THE LZI ALGORITHM FOR WEAKLY POSITIVE TENSORS

We define weakly positive tensors and study the relations among essentially positive tensors, weakly positive tensors, and primitive tensors. In particular, an explicit linear convergence rate of the Liu-Zhou-Ibrahim（LZI） algorithm for finding the largest eigenvalue of an irreducible nonnegative tensor, is established for weakly positive tensors. Numerical results are given to demonstrate linear convergence of the LZI algorithm for weakly positive tensors.