Related papers: Wildness for tensors
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
Let $A$ be a real $n\times n$ matrix and $z,b\in \mathbb R^n$. The piecewise linear equation system $z-A\vert z\vert = b$ is called an absolute value equation. It is equivalent to the general linear complementarity problem, and thus NP hard…
We prove the tame-wild dichotomy conjecture, due to D. Simson, for infinite dimensional algebras and coalgebras. The key part of the approach is proving new representation theoretic characterizations local finiteness. Among other, we show…
Let d be any number greather than or equal to 3. We show that the intersection of the set mdeg(Aut(C^3))\ mdeg(Tame(C3)) with {(d_1,d_2,d_3) : d=d_1 =< d_2 =< d_3} has infinitely many elements, where mdeg h = (deg h_1,...,deg h_n) denotes…
Given an arbitrary (commutative) field K, let V be a linear subspace of M_n(K) consisting of matrices of rank lesser than or equal to some r<n. A theorem of Atkinson and Lloyd states that, if dim V>nr-r+1 and #K>r, then either all the…
We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, it is an invariant of this function with respect to a certain group of transformations of variables; on the other…
We desribe vector bundles over a class of noncommutative curves, namely, over noncommutative nodal curves of string type and of almost string type. We also prove that in other cases the classification of vector bundles over a noncommutative…
We show that determining the rank of a tensor over a field has the same complexity as deciding the existential theory of that field. This implies earlier NP-hardness results by H{\aa}stad~\cite{H90}. The hardness proof also implies an…
In this note, we consider the problem of constructing an enlargement of the category of Betti sheaves that supports an ``exponential local system'' on $\mathbb R$, and a Fourier equivalence defined on all sheaves. We show that there is a…
In this paper, we introduce a class of variational inequalities, where the involved function is the sum of an arbitrary given vector and a homogeneous polynomial defined by a tensor; and we call it the tensor variational inequality (TVI).…
Recent discussion in the public sphere about algorithmic classification has involved tension between competing notions of what it means for a probabilistic classification to be fair to different groups. We formalize three fairness…
There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…
This paper formalizes a latent variable inference problem we call {\em supervised pattern discovery}, the goal of which is to find sets of observations that belong to a single ``pattern.'' We discuss two versions of the problem and prove…
We collect open problems in permutation patterns on four themes: rank-unimodality in the permutation pattern poset, Wilf-equivalence and shape-Wilf-equivalence, the enumeration of derangements in permutation classes, and sorting by stacks…
Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero-dimensional schemes…
Classical results for exchangeable systems of random variables are extended to multi-class systems satisfying a natural partial exchangeability assumption. It is proved that the conditional law of a finite multi-class system, given the…
Consider the Deligne-Simpson problem: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset GL(n,{\bf C})$ (resp. $c_j\subset gl(n,{\bf C})$) so that there exist irreducible $(p+1)$-tuples of…
A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness…
We introduce the notion of interlaced weak ditalgebras and apply reduction procedures to their module categories to prove a tame-wild dichotomy for the category ${\cal F}(\Delta)$ of $\Delta$-filtered modules for an arbitrary finite…
The problem of classifying all unitary R-matrices of arbitrary finite dimension that have precisely two distinct eigenvalues is described, working up to a natural equivalence relation given by the characters of their braid group…