Kronecker powers of tensors and Strassen’s laser method

We answer a question, posed implicitly in [18, §11], [11, Rem. 15.44] and explicitly in [9, Problem 9.8], showing the border rank of the Kronecker square of the little Coppersmith-Winograd tensor is the square of the border rank of the tensor for all q > 2, a negative result for complexity theory. We further show that when q > 4, the analogous result holds for the Kronecker cube. In the positive direction, we enlarge the list of explicit tensors potentially useful for the laser method. We observe that a well-known tensor, the 3×3 determinant polynomial regarded as a tensor, det3 ∈ C9 C9 C9, could potentially be used in the laser method to prove the exponent of matrix multiplication is two. Because of this, we prove new upper bounds on its Waring rank and rank (both 18), border rank and Waring border rank (both 17), which, in addition to being promising for the laser method, are of interest in their own right. We discuss “skew” cousins of the little Coppersmith-Winograd tensor and indicate why they may be useful for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in C3 C3 C

Rank and border rank of Kronecker powers of tensors and Strassen's laser method

We prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor Tcw,q is the square of its border rank for q> 2 and that the border rank of its Kronecker cube is the cube of its border rank for q> 4. This answers questions raised implicitly by Coppersmith & Winograd (1990, §11)and explicitly by Bläser (2013, Problem 9.8) and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith–Winograd tensor in this range. In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen's laser method, introducing a skew-symmetric version of the Coppersmith–Winograd tensor, Tskewcw,q. For q= 2 , the Kronecker square of this tensor coincides with the 3 × 3 determinant polynomial, det 3∈ C9⊗ C9⊗ C9, regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two. We determine new upper bounds for the (Waring) rank and the (Waring) border rank of det 3, exhibiting a strict submultiplicative behaviour for Tskewcw,2 which is promising for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in C3⊗ C3⊗ C3

Factors associated with functional decline in hand and hip/knee osteoarthritis after one year: data from a population‐based study

Public Health and primary careGeriatrics in primary car

Minimal clinically important decline in physical function over one year: EPOSA study

Analysis and Stochastic

Streamlining bioactive molecular discovery through integration and automation

The discovery of bioactive small molecules is generally driven via iterative design–make–purify–test cycles. Automation is routinely harnessed at individual stages of these cycles to increase the productivity of drug discovery. Here, we describe recent progress to automate and integrate two or more adjacent stages within discovery workflows. Examples of such technologies include microfluidics, liquid-handling robotics and affinity-selection mass spectrometry. The value of integrated technologies is illustrated in the context of specific case studies in which modulators of targets, such as protein kinases, nuclear hormone receptors and protein–protein interactions, were discovered. We note that to maximize impact on the productivity of discovery, each of the integrated stages would need to have both high and matched throughput. We also consider the longer-term goal of realizing the fully autonomous discovery of bioactive small molecules through the integration and automation of all stages of discovery

Partially symmetric variants of Comon's problem via simultaneous rank

A symmetric tensor may be regarded as a partially symmetric tensor in several different ways. These produce different notions of rank for the symmetric tensor which are related by chains of inequalities. By exploiting algebraic tools such as apolarity theory, we show how the study of the simultaneous symmetric rank of partial derivatives of the homogeneous polynomial associated to the symmetric tensor can be used to prove equalities among different partially symmetric ranks. This approach aims to understand to what extent the symmetries of a tensor affect its rank. We apply this to the special cases of binary forms, ternary and quaternary cubics, monomials, and elementary symmetric polynomials