Toshio Sumi

Rank of 3-tensors with 2 slices and Kronecker canonical forms

Tensor type data are becoming important recently in various application fields. We determine the rank of a 3-tensor with 2 slices in comparison with its Kronecker canonical form over the complex and real number field.

Polynomial invariants of 2-bridge knots through 22 crossings

We calculate the homfly, Kauffman, Jones, Q, and Conway polynomials of 2-bridge knots through 22 crossings and list all the pairs sharing the same polynomial invariants.

Homology of the universal covering of a co-H-space

The problem 10 posed by Tudor Ganea is known as the Ganea conjecture on a co-IS-space, a space with co-H-structure. Many efforts are devoted to show the Ganea conjecture under additional. assumptions on the given co-H-structure. In this paper, we show a homological property of coH-spaces in a slightly general situation. As a corollary, we get the Ganea conjecture for spaces up to dimension 3.

Algebraic and Computational Aspects of Real Tensor Ranks

Basics of Tensor Rank.- 3-Tensors.- Simple Evaluation Methods of Tensor Rank.- Absolutely Nonsingular Tensors and Determinantal Polynomials.- Maximal Ranks.- Typical Ranks.- Global Theory of Tensor Ranks.- 2 x 2 x * * * x 2 Tensors.

The Laitinen Conjecture for finite solvable Oliver groups

For smooth actions of G on spheres with exactly two fixed points, the Laitinen Conjecture proposed an answer to the Smith question about the G-modules determined on the tangent spaces at the two fixed points. Morimoto obtained the first counterexample to the Laitinen Conjecture for G = Aut(A 6 ). By answering the Smith question for some finite solvable Oliver groups G, we obtain new counterexamples to the Laitinen Conjecture, presented for the first time in the case where G is solvable.

Typical ranks of certain 3-tensors and absolutely full column rank tensors

In this paper, we study typical ranks of 3-tensors and show that there are plural typical ranks for tensors over in the following cases: (1) and , where is the Hurwitz–Radon function, (2) , and , (3) , , and , (4) , , and . (5) , and .

Typical ranks of m × n × (m − 1)n tensors with 3 ≤ m ≤ n over the real number field

Let . We study typical ranks of tensors over the real number field. Let be the Hurwitz–Radon function defined as for nonnegative integers such that and . If , then the set of tensors has two typical ranks . In this paper, we show that the converse is also true: if , then the set of tensors has only one typical rank .

\(2\times 2\times \cdots \times 2\) Tensors

We study an upper bound of the ranks of n-tensors with format \((2,2,\ldots ,2)\) over the complex and real number fields. We consider Cayley’s hyperdeterminant and give the necessary and sufficient condition for a tensor of format (2, 2, 2) to have maximal rank 3. Brylinski showed that the maximal rank of complex tensors with format (2, 2, 2, 2) is 4, and Kong and Jiang showed that the maximal rank of real tensors with format (2, 2, 2, 2) is less than or equal to 5. Catalisano, Geramita, and Gimigliano showed that the generic rank of complex n-tensors with format \((2,2,\ldots ,2)\) is equal to \(\lceil 2^n/(n+1)\rceil \), and then, the maximal rank of real n-tensors with format \((2,2,\ldots ,2)\) is greater than or equal to \(\lceil 2^n/(n+1)\rceil \). The maximal rank of real n-tensors with format \((2,2,\ldots ,2)\) is less than or equal to \(2^{n-1}+1\). Bleckherman and Teitler showed that the maximal rank is less than or equal to twice the generic rank, which gives a more strict upper bound of the maximal rank of n-tensors with format \((2,2,\ldots ,2)\) over the real number field.

Global Theory of Tensor Ranks

The generic rank is considered under the complex number field, and it corresponds with the dimension of the secant variety. The dimension is studied in the area of algebraic geometry. In this chapter, we introduce known results and discuss the typical rank from the point of view of the Jacobian matrix. The generic rank attains the minimal typical rank. The typical rank of the set of tensors with format (m, n, p) for \(3\le m\le n\) is equal to \(\min (p,mn)\) if \(p>(m-1)n\). Note that the typical rank is sometimes not unique and the set of typical ranks consists of integers between some integers a and b. For a positive integer r, we consider the image of the summation map, which gives the set of tensors with rank less than or equal to r. Strassen and Tickteig introduced the idea of computing the dimension of the secant variety via the Jacobi criterion and the splitting technique. The generic rank of the set of tensors with format (m, n, p) is \(\lceil mnp/(m+n+p-2)\rceil \) if m, n, and p are sufficiently large integers, although the generic rank of the set of tensors with format (n, n, 3) is equal to \((3n+1)/2\) which is greater than \(\lceil 3n^2/(2n+1)\rceil \) if n is odd.

Simple Evaluation Methods of Tensor Rank

In this chapter, we illustrate simple evaluations of the rank of 3-mode tensors, which might facilitate readers’ understanding of tensor rank.