53 research outputs found
Border rank is not multiplicative under the tensor product
It has recently been shown that the tensor rank can be strictly
submultiplicative under the tensor product, where the tensor product of two
tensors is a tensor whose order is the sum of the orders of the two factors.
The necessary upper bounds were obtained with help of border rank. It was left
open whether border rank itself can be strictly submultiplicative. We answer
this question in the affirmative. In order to do so, we construct lines in
projective space along which the border rank drops multiple times and use this
result in conjunction with a previous construction for a tensor rank drop. Our
results also imply strict submultiplicativity for cactus rank and border cactus
rank.Comment: 25 pages, 1 figure - Revised versio
Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties
These are expository notes from the 2008 Srni Winter School. They have two
purposes: (1) to give a quick introduction to exterior differential systems
(EDS), which is a collection of techniques for determining local existence to
systems of partial differential equations, and (2) to give an exposition of
recent work (joint with C. Robles) on the study of the Fubini-Griffiths-Harris
rigidity of rational homogeneous varieties, which also involves an advance in
the EDS technology.Comment: To appear in the proceedings of the 2008 Srni Winter School on
Geometry and Physic
Decomposability of Tensors
Tensor decomposition is a relevant topic, both for theoretical and applied mathematics, due to its interdisciplinary nature, which ranges from multilinear algebra and algebraic geometry to numerical analysis, algebraic statistics, quantum physics, signal processing, artificial intelligence, etc. The starting point behind the study of a decomposition relies on the idea that knowledge of elementary components of a tensor is fundamental to implement procedures that are able to understand and efficiently handle the information that a tensor encodes. Recent advances were obtained with a systematic application of geometric methods: secant varieties, symmetries of special decompositions, and an analysis of the geometry of finite sets. Thanks to new applications of theoretic results, criteria for understanding when a given decomposition is minimal or unique have been introduced or significantly improved. New types of decompositions, whose elementary blocks can be chosen in a range of different possible models (e.g., Chow decompositions or mixed decompositions), are now systematically studied and produce deeper insights into this topic. The aim of this Special Issue is to collect papers that illustrate some directions in which recent researches move, as well as to provide a wide overview of several new approaches to the problem of tensor decomposition
- …