Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex matrices and $\|\cdot \|_{(p,k)}$ the $(p,k)$ norm on $M_{mn}$ with a positive integer $k\leq mn$ and a real number $p>2$. We show that a linear map $\phi :M_{mn}\rightarrow M_{mn}$ satisfies

$$ \begin{align*}\|\phi(A\otimes B)\|_{(p,k)}=\|A\otimes B\|_{(p,k)} \mathrm{\quad for~ all\quad}A\in M_m\ \mathrm{and}\ B\in M_n\end{align*} $$

if and only if there exist unitary matrices $U,V\in M_{mn}$ such that

$$ \begin{align*}\phi(A\otimes B)=U(\varphi_1(A)\otimes \varphi_2(B))V \mathrm{\quad for~ all\quad}A\in M_m\ \mathrm{and}\ B\in M_n,\end{align*} $$

where $\varphi _s$ is the identity map or the transposition map $X\mapsto X^T$ for $s=1,2$. The result is also extended to multipartite systems.

Let $M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$ denote the matrix multiplication tensor (and write $M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$ ), and let $\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$ denote the determinant polynomial considered as a tensor. For a tensor T, let $\underline {\mathbf {R}}(T)$ denote its border rank. We (i) give the first hand-checkable algebraic proof that $\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$ , (ii) prove $\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$ and $\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$ , where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was $M_{\langle 2\rangle }$ , (iii) prove $\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$ , (iv) prove $\underline {\mathbf {R}}(\operatorname {det}_3)=17$ , improving the previous lower bound of $12$ , (v) prove $\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$ for all $\mathbf {n}\geq 25$ , where previously only $\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$ was known, as well as lower bounds for $4\leq \mathbf {n}\leq 25$ , and (vi) prove $\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$ for all $\mathbf {n} \ge 18$ , where previously only $\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$ was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors.

The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, called border apolarity developed by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensor T and an integer r, in a finite number of steps, either outputs that there is no border rank r decomposition for T or produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable when T has a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory.

Kruskal’s theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. We prove a ‘splitting theorem’ for sets of product tensors, in which the k-rank condition of Kruskal’s theorem is weakened to the standard notion of rank, and the conclusion of uniqueness is relaxed to the statement that the set of product tensors splits (i.e., is disconnected as a matroid). Our splitting theorem implies a generalization of Kruskal’s theorem. While several extensions of Kruskal’s theorem are already present in the literature, all of these use Kruskal’s original permutation lemma and hence still cannot certify uniqueness when the k-ranks are below a certain threshold. Our generalization uses a completely new proof technique, contains many of these extensions and can certify uniqueness below this threshold. We obtain several other useful results on tensor decompositions as consequences of our splitting theorem. We prove sharp lower bounds on tensor rank and Waring rank, which extend Sylvester’s matrix rank inequality to tensors. We also prove novel uniqueness results for nonrank tensor decompositions.

The polarization constant of a Banach space X is defined as

\[{\text{c}}(X){\text{ }}{\text{ }}\mathop {\lim }\limits_{k \to \infty } {\text{ }}\sup {\text{c}}{(k,X)^{\frac{1}{k}}},\]

where \[{\text{c}}(k,X)\] stands for the best constant \[C > 0\] such that \[\mathop P\limits^ \vee \leqslant CP\] for every k-homogeneous polynomial \[P \in \mathcal{P}{(^k}X)\]. We show that if X is a finite dimensional complex space then \[{\text{c}}(X) = 1\]. We derive some consequences of this fact regarding the convergence of analytic functions on such spaces.

The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak’s complexification procedure.

We also study some other properties connected with polarization. Namely, we provide necessary conditions related to the geometry of X for \[c(2,X) = 1\] to hold. Additionally we link polarization constants with certain estimates of the nuclear norm of the product of polynomials.

Let p be a prime and let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field F of characteristic p. For positive integers r and s with $r \leq s$ , the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes J_{s}$ has the form $J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _{r}}$ . This decomposition determines a partition $\lambda (r,s,p)=(\lambda _1,\lambda _2,\ldots , \lambda _{r})$ of $r s$ . Let $n_1, \ldots , n_k$ be the multiplicities of the distinct parts of the partition and set $c(r,s,p)=(n_1,\ldots ,n_k)$ . Then $c(r,s,p)$ is a composition of r. We present a new bottom-up algorithm for computing $c(r,s,p)$ and $\lambda (r,s,p)$ directly from the base-p expansions for r and s.

For a C*-algebra A, determining the Cuntz semigroup Cu(A ⊗) in terms of Cu(A) is an important problem, which we approach from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups by analysing the passage of significant properties from Cu(A) to Cu(A)⊗Cu Cu(). We describe the effect of the natural map Cu(A) → Cu(A)⊗Cu Cu() in the order of Cu(A), and show that if A has real rank 0 and no elementary subquotients, Cu(A)⊗Cu Cu() enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, non-elementary, real rank 0 and stable rank 1 situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with is inert at the level of the Cuntz semigroup.

We show that the product rank of the $3\,\times \,3$ determinant ${{\det }_{3}}$ is $5$ , and the product rank of the $3\,\times \,3$ permanent $\text{per}{{\text{m}}_{3}}$ is $4$ . As a corollary, we obtain that the tensor rank of ${{\det }_{3}}$ is $5$ and the tensor rank of $\text{per}{{\text{m}}_{3}}$ is $4$ . We show moreover that the border product rank of $\text{per}{{\text{m}}_{3}}$ is larger than $n$ for any $n\,\ge \,3$ .

We consider higher secant varieties to Veronese varieties. Most points on the rth secant variety are represented by a finite scheme of length r contained in the Veronese variety – in fact, for a general point, the scheme is just a union of r distinct points. A modern way to phrase it is: the smoothable rank is equal to the border rank for most polynomials. This property is very useful for studying secant varieties, especially, whenever the smoothable rank is equal to the border rank for all points of the secant variety in question. In this note, we investigate those special points for which the smoothable rank is not equal to the border rank. In particular, we show an explicit example of a cubic in five variables with border rank 5 and smoothable rank 6. We also prove that all cubics in at most four variables have the smoothable rank equal to the border rank.

We study the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n-2/3. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and n is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.

In this short note, we study a relation between the tensor product of matrices and a multilinear map defined by the optimal operator. In this particular case, the linear transform (mediating morphism) hidden in the abstract definition of the general tensor product can be determined explicitly.

We consider the $k$-osculating varieties ${{O}_{k,\,n.d}}$ to the (Veronese) $d$-uple embeddings of ${{\mathbb{P}}^{n}}$. We study the dimension of their higher secant varieties via inverse systems (apolarity). By associating certain 0-dimensional schemes $Y\,\subset \,{{\mathbb{P}}^{n}}$ to $O_{k,n,d}^{s}$ and by studying their Hilbert functions, we are able, in several cases, to determine whether those secant varieties are defective or not.

Let U be a finite dimensional vector space over an infinite field F. Let U(r) denote the r–th symmetric product space over U. Let T: U(r) → U(s) be a linear transformation which sends nonzero decomposable elements to nonzero decomposable elements. Let dim U ≥ s + 1. Then we obtain the structure of T for the following cases: (I) F is algebraically closed, (II) F is the real field, and (III) T is injective.