Variety of fiber orientation tensors

Fiber orientation tensors are established descriptors of fiber orientation states in (thermo-)mechanical material models for fiber-reinforced composites. In this paper, the variety of fourth-order orientation tensors is analyzed and specified by parameterizations and admissible parameter ranges. The combination of parameterizations and admissible parameter ranges allows for studies on the mechanical response of different fiber architectures. Linear invariant decomposition with focus on index symmetry leads to a novel compact hierarchical parameterization, which highlights the central role of the isotropic state. Deviation from the isotropic state is given by a triclinic harmonic tensor with simplified structure in the orientation coordinate system, which is spanned by the second-order orientation tensor. Material symmetries reduce the number of independent parameters. The requirement of positive-semi-definiteness defines admissible ranges of independent parameters. Admissible parameter ranges for transversely isotropic and planar cases are given in a compact closed form and the orthotropic variety is visualized and discussed in detail. Sets of discrete unit vectors, leading to selected orientation states, are given.

Classical Limit of Quantum Mechanics for Damped Driven Oscillatory Systems: Quantum–Classical Correspondence

The investigation of quantum–classical correspondence may lead to gaining a deeper understanding of the classical limit of quantum theory. I have developed a quantum formalism on the basis of a linear invariant theorem, which gives an exact quantum–classical correspondence for damped oscillatory systems perturbed by an arbitrary force. Within my formalism, the quantum trajectory and expectation values of quantum observables precisely coincide with their classical counterparts in the case where the global quantum constant ℏ has been removed from their quantum results. In particular, I have illustrated the correspondence of the quantum energy with the classical one in detail.

A New Nonlinear Active Disturbance Rejection Control for the Cable Car System to Restrain the Vibration

The safety of the cable car system is very important for the lives of the people. But, it is easily affected by the environment such as the wind which causes the cable car system to have strong vibration disturbance, thus degrading the safety of the cable car system. In this paper, a new nonlinear active disturbance rejection control (ADRC) is proposed to restrain the vibration of the cable car. First, a new two-mass-spring system model is utilized to establish the cable car system model. The new translation vibration nonlinear model is derived by a linear-invariant two-mass-spring system. Then, a special nonlinear fal• is designed to restrain the vibration, and a new high-order nonlinear ADRC is presented for the cable car system. Finally, simulation results verify the feasibility and accuracy of the proposed model.

Sharp Bounds on the Spectral Radius of Nonnegative Matrices and Comparison to the Frobenius’ Bounds

In this paper, a new upper bound and a new lower bound for the spectral radius of a nοnnegative matrix are proved by using similarity transformations. These bounds depend only on the elements of the nonnegative matrix and its row sums and are compared to the well-established upper and lower Frobenius’ bounds. The proposed bounds are always sharper or equal to the Frobenius’ bounds. The conditions under which the new bounds are sharper than the Frobenius' ones are determined. Illustrative examples are also provided in order to highlight the sharpness of the proposed bounds in comparison with the Frobenius’ bounds. An application to linear invariant discrete-time nonnegative systems is given and the stability of the systems is investigated. The proposed bounds are computed with complexity O(n2).

A LINEAR INVARIANT OF A MATRIX RING OF SIZE TWO OVER A POLYNOMIAL RING

We consider a matrix ring of order two over a ring of polynomials in two variables with coefficients from a commutative associative integrity domain with unity. A linear mapping of this ring into the polynomial ring is presented, depending on a matrix of a special form, whose square is zero matrix. The value of this map is invariant with respect to conjugation by an invertible matrix of elements of the ring, including the matrix by which the map is constructed. The properties of the mapping thus obtained are described.

Comment on: Quantum theory of a non-Hermitian time-dependent forced harmonic oscillator having 𝒫𝒯 symmetry

Pedrosa et al.1 have recently used a [Formula: see text] symmetric linear invariant to study a unidimensional time-dependent [Formula: see text] symmetric harmonic oscillator with a complex time-dependent [Formula: see text] symmetric external force. We show in this comment that the normalization condition of the eigenfunctions of the invariant is not verified as claimed in Ref. 1. In order to obtain the normalization condition, we introduce a novel concept of the pseudoparity-time (pseudo-[Formula: see text]) symmetry.