Symmetric tensor decomposition is of great importance in applications. Several studies have employed a greedy approach, where the main idea is to first find a best rank-one approximation of a given tensor, and then repeat the process to the residual tensor by subtracting the rank-one component. In this paper, we focus on finding a best rank-one approximation of a given orthogonally order-3 symmetric tensor. We give a geometric landscape analysis of a nonconvex optimization for the best rank-one approximation of orthogonally symmetric tensors. We show that any local minimizer must be a factor in this orthogonally symmetric tensor decomposition, and any other critical points are linear combinations of the factors. Then, we propose a gradient descent algorithm with a carefully designed initialization to solve this nonconvex optimization problem, and we prove that the algorithm converges to the global minimum with high probability for orthogonal decomposable tensors. This result, combined with the landscape analysis, reveals that the greedy algorithm will get the tensor CP low-rank decomposition. Numerical results are provided to verify our theoretical results.

In this paper, we are concerned with a class of impulsive neutral stochastic functional different equations driven by tempered fractional Brownian motion in the Hilbert space. We obtain the global attracting and quasi-invariant sets of the considered equations driven by tempered fractional Brownian motion $B^{α,λ}(t)$ with $0<α<1/2$ and $λ>0.$ In particular, we give some sufficient conditions which ensure the exponential decay in the $p$-th moment of the mild solution of the considered equations. Finally, an example is given to illustrate the feasibility and effectiveness of the results obtained.

Objectivity is the name given to covariance with respect to changes in observers by the action of the Euclidean group and is a central concept for mathematical modelling in Physics and Continuum mechanics. However, this notion is delicate and therefore hard to understand by both mechanicians and mathematicians, precisely because it resides at the junction of the two disciplines and hence appeals to knowledge and notions from both fields to become clear. In a first part ("behind objectivity") some classical notions of objectivity and frame-dependence are presented, discussed and critically revisited with the purpose to introduce in a second part ("beyond objectivity") a novel modelling approach in incompatible elasticity, based on intrinsic and geometric concepts, related but distinct from conventional objectivity. This contribution is primarily conceived to assist students, academics and researchers, mathematicians in particular, to find their way through some difficult and often ill-understood concepts of Physics and Continuum mechanics, since objectivity still remains nowadays a profound concept that deserves our attention.