The Representation of D-Invariant Polynomial Subspaces Based on Symmetric Cartesian Tensors

Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the D-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a D-invariant polynomial subspace Pn in terms of Cartesian tensors, where Pn is a subspace with a maximal total degree equal to n,n≥1. For an arbitrary homogeneous polynomial p(k) of total degree k in Pn, p(k) can be rewritten as the inner products of a kth order symmetric Cartesian tensor and k column vectors of indeterminates. We show that p(k) can be determined by all polynomials of a total degree one in Pn. Namely, if we treat all linear polynomials on the basis of Pn as a column vector, then this vector can be written as a product of a coefficient matrix A(1) and a column vector of indeterminates; our main result shows that the kth order symmetric Cartesian tensor corresponds to p(k) is a product of some so-called relational matrices and A(1).

Identifying the Positive Definiteness of Even-Order Weakly Symmetric Tensors via Z-Eigenvalue Inclusion Sets

The positive definiteness of even-order weakly symmetric tensors plays important roles in asymptotic stability of time-invariant polynomial systems. In this paper, we establish two Brauer-type Z-eigenvalue inclusion sets with parameters by Z-identity tensors, and show that these inclusion sets are sharper than existing results. Based on the new Z-eigenvalue inclusion sets, we propose some sufficient conditions for testing the positive definiteness of even-order weakly symmetric tensors, as well as the asymptotic stability of time-invariant polynomial systems. The given numerical experiments are reported to show the efficiency of our results.

An accurate full-dimensional potential energy surface for the reaction OH + SO → H + SO2

An accurate full-dimensional PES for the OH + SO ↔ H + SO2 reaction is developed by the permutation invariant polynomial-neural network approach.

An optimal $ Z $-eigenvalue inclusion interval for a sixth-order tensor and its an application

<abstract><p>An optimal $ Z $-eigenvalue inclusion interval for a sixth-order tensor is presented. As an application, a sufficient condition for the positive definiteness of a sixth-order real symmetric tensor (also a homogeneous polynomial form) is obtained, which is used to judge the asymptotically stability of time-invariant polynomial systems.</p></abstract>

A new global potential energy surface of the SH2+(X4A′′) system and quantum calculations for the S+ + H2(v = 0–3, j = 0) reaction

A new global potential energy surface (PES) for the ground state of the SH2+(X4A′′) system is constructed using a permutation invariant polynomial neural network method.