Geometric complexity theory and matrix powering

Valiant's famous determinant versus permanent problem is the flagship problem in algebraic complexity theory. Mulmuley and Sohoni (Siam J Comput 2001, 2008) introduced geometric complexity theory, an approach to study this and related problems via algebraic geometry and representation theory. Their approach works by multiplying the permanent polynomial with a high power of a linear form (a process called padding) and then comparing the orbit closures of the determinant and the padded permanent. This padding was recently used heavily to show no-go results for the method of shifted partial derivatives (Efremenko, Landsberg, Schenck, Weyman, 2016) and for geometric complexity theory (Ikenmeyer Panova, FOCS 2016 and B\"urgisser, Ikenmeyer Panova, FOCS 2016). Following a classical homogenization result of Nisan (STOC 1991) we replace the determinant in geometric complexity theory with the trace of a variable matrix power. This gives an equivalent but much cleaner homogeneous formulation of geometric complexity theory in which the padding is removed. This radically changes the representation theoretic questions involved to prove complexity lower bounds. We prove that in this homogeneous formulation there are no orbit occurrence obstructions that prove even superlinear lower bounds on the complexity of the permanent. This is the first no-go result in geometric complexity theory that rules out superlinear lower bounds in some model. Interestingly---in contrast to the determinant---the trace of a variable matrix power is not uniquely determined by its stabilizer

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

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

Analysis and Stochastic

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

Depletion and voids formation in the substrate during high temperature oxidation of Ni-Cr alloys

A numerical model to treat the kinetics of vacancy annihilation at the metal/oxide interface but also in the bulk metal has been implemented. This was done using EKINOX, which is a mesoscopic scale 1D-code that simulates oxide growth kinetics with explicit calculation of vacancy fluxes. Calculations were performed for high temperature Ni-Cr alloys oxidation forming a single chromia scale. The kinetic parameters used to describe the diffusion in the alloy were directly derived from an atomistic model. Our results showed that the Cr depletion profile can be strongly affected by the cold work state of the alloy. In fact, the oversaturation of vacancies is directly linked to the efficiency of the sinks which is proportional to the density of dislocations. The resulting vacancy profile highlights a supersaturation of vacancy within the metal. Based on the classical nucleation theory, the possibility and the rate of void formation are discussed