Construction of orthonormal bases in higher symmetry classes of tensors

We present a method for constructing an orthonormal basis for a symmetry class of tensors from an orthonormal basis of the underlying vector space. The basis so obtained is not composed of decomposable symmetrized tensors. Indeed, we show that, for symmetry classes of tensors whose associated character has degree higher than one, it is impossible to construct an orthogonal basis of decomposable symmetrized tensors from any basis of the underlying vector space. We end with an open problem on the possibility of a symmetry class having an orthonormal basis of decomposable symmetrized tensors.     

On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group

A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.     

可合对称张量相等的条件

本文讨论了由一般对称化算子诱导的两非零可合对称张量相等的必要充分条件与可合对称张量为零的必要充分条件。     

On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group

A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.     

A high symmetry class of tensors with an orthogonal basis of decomposable symmetrized tensors

We present a high symmetry class of tensors with an orthogonal basis of decomposable symmetrized tensors, and this is a counter-example of the claim presented in [1].     

Orthogonality of cosets relative to irreducible characters of finite groups

Studied is an assumption on a group that ensures that no matter how the group is embedded in a symmetric group, the corresponding symmetrized tensor space has an orthogonal basis of standard (decomposable) symmetrized tensors.     

Lower bounds for the norms of decomposable symmetrized tensors

Lower bounds are given for the difference of two decomposable symmetrized tensors. The first bound uses a norm which makes the component vectors in a decomposable symmetrized tensor part of an orthonormal basis. The second bound holds only for decomposable elements of symmetry classes whose associated characters are linear.     

Orthogonal decomposable symmetrized tensors

In this article we present a sufficient condition for orthogonality of decompassable symmetrized tensors.     

Orthogonal bases of symmetrized tensor spaces

It is shown that a symmetrized tensor space does not have an orthogonal basis consisting of standard symmetrized tensors if the associated permutation group is 2-transitive. In particular, no such basis exists if the group is the symmetric group or the algernating group as conjectured by T.-Y. Tam and the author.     

Symmetric tensor fields of bounded deformation

We introduce and study spaces of symmetric tensor fields of bounded deformation for tensors of arbitrary order, i.e., where the symmetrized derivative is still a Radon measure. A Sobolev–Korn type estimate, a boundary trace theorem and continuous as well as compact embedding properties into Lebesgue spaces are obtained, showing that these spaces can be regarded as a natural generalization of the spaces of bounded deformation to higher-order symmetric tensors.     

Choice of strategies for extrapolation with symmetrization in the constant stepsize setting

Symmetrization has been shown to be efficient in solving stiff problems. In the constant stepsize setting, we study four ways of applying extrapolation with symmetrization. We observe that for stiff linear problems the symmetrized Gauss methods are more efficient than the symmetrized Lobatto IIIA methods of the same order. However, for two-dimensional nonlinear problems, the symmetrized 4-stage Lobatto IIIA method is more efficient. In all cases, we observe numerically that passive symmetrization with passive extrapolation is more efficient than active symmetrization with active extrapolation.     

Estimates of the Carathéodory metric on the symmetrized polydisc

Estimates for the Carathéodory metric on the symmetrized polydisc are obtained. It is also shown that the Carathéodory and Kobayashi distances of the symmetrized three-disc do not coincide.     

Inequalities between rank moments of overpartitions

F.G. Garvan proved an inequality between crank moments and rank moments of partitions which utilizes an inequality for symmetrized rank and crank moments. In this paper, we study two symmetrized rank moments of overpartitions and prove an inequality between two rank moments of overpartitions.     

Matkowski–Sutô Type Equation on Symmetrized Weighted Quasi-Arithmetic Means

We solve a functional equation involving symmetrized weighted quasi-arithmetic means. More precisely we investigate the invariance of the arithmetic mean in the class of symmetrized weighted quasi-arithmetic means. Some regularity on the unknown generator functions is assumed.     

On some properties of the determinants of tensors

We use the Hilbert?s Nullstellensatz (Hilbert?s Zero Point Theorem) to give a direct proof of the formula for the determinants of the products of tensors. By using this determinant formula and using tensor product to represent the transformations of the slices of tensors, we prove some basic properties of the determinants of tensors which are the generalizations of the corresponding properties of the determinants for matrices. We also study the determinants of tensors after two types of transposes. We use the permutational similarity of tensors to discuss the relation between weakly reducible tensors and the triangular block tensors, and give a canonical form of the weakly reducible tensors.     

Higher order spt-functions

Andrews? spt-function can be written as the difference between the second symmetrized crank and rank moment functions. Using the machinery of Bailey pairs a combinatorial interpretation is given for the difference between higher order symmetrized crank and rank moment functions. This implies an inequality between crank and rank moments that was only known previously for sufficiently large n and fixed order. This combinatorial interpretation is in terms of a weighted sum of partitions. A number of congruences for higher order spt-functions are derived.     

Generalized inverses of tensors via a general product of tensors

We define the {i}-inverse (i = 1, 2, 5) and group inverse of tensors based on a general product of tensors. We explore properties of the generalized inverses of tensors on solving tensor equations and computing formulas of block tensors. We use the {1}-inverse of tensors to give the solutions of a multilinear system represented by tensors. The representations for the {1}-inverse and group inverse of some block tensors are established.     

Definition of the boundary in the local Charzynski-Tammi conjecture

According to the Charzynski-Tammi conjecture, the symmetrized Pick function is extremal in the problem on the estimate for the nth Taylor coefficient in the class of holomorphic univalent functions close to the identical one. In this paper we find the exact value of M ₄ such that the symmetrized Pick function is locally extremal in the problem on the estimate for the 4th Taylor coefficient in the class of holomorphic normalized univalent functions, whose module is bounded byM ₄.     

A fast algorithm for the spectral radii of weakly reducible nonnegative tensors

In this paper, we propose a fast algorithm for computing the spectral radii of symmetric nonnegative tensors. In particular, by this proposed algorithm, we are able to obtain the spectral radii of weakly reducible symmetric nonnegative tensors without requiring the partition of the tensors. As we know, it is very costly to determine the partition for large‐sized weakly reducible tensors. Numerical results are reported to show that the proposed algorithm is efficient and also able to compute the spectral radii of large‐sized tensors. As an application, we present an algorithm for testing the positive definiteness of Z‐tensors. By this algorithm, it is guaranteed to determine the positive definiteness for any Z‐tensor.     

Standard tensor and its applications in problem of singular values of tensors

In this paper, first we give the definition of standard tensor. Then we clarify the relationship between weakly irreducible tensors and weakly irreducible polynomial maps by the definition of standard tensor. And we prove that the singular values of rectangular tensors are the special cases of the eigen-values of standard tensors related to rectangular tensors. Based on standard tensor, we present a generalized version of the weak Perron-Frobenius Theorem of nonnegative rectangular tensors under weaker conditions. Furthermore, by studying standard tensors, we get some new results of rectangular tensors. Besides, by using the special structure of standard tensors corresponding to nonnegative rectangular tensors, we show that the largest singular value is really geometrically simple under some weaker conditions.