The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling. We show for random rank-2 tensors that the expected value of the condition number is infinite for a wide range of choices of the density. Under a mild additional assumption, we show that the same is true for most higher ranks \(r\ge 3\) as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 tensors have finite expected angular condition number. Based on numerical experiments, we conjecture that this could also be true for higher ranks. Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms computing tensor rank decompositions.

     

On the probability of Magnus Carlsen reaching 2900

How likely is it that Magnus Carlsen will achieve his goal of a 2900 Elo rating? At what level of play does Magnus have a reasonable chance of reaching the 2900 goal? These two questions are of great current interest to Magnus and the chess community. The probabilistic properties of Elo's rating system are well known, and together with a Brownian motion model of rating evolution, we use simulation-based methods to address these questions. Our model assesses that Magnus has a 4.5% chance of reaching 2900 if he continues his 2020–2022 level of play. However, this increases dramatically to $80$% chance if he can repeat his hot streak performance of 2019 which is not an easy undertaking. The probabilities are intimately related to Elo's choice $K$-factor used for grandmaster chess play. Finally, we conclude with a discussion of the policy issues involved with the choice of $K$-factor.     

Cooperative Lewis Acid-1,2,3-Triazolium-Aryloxide Catalysis: Pyrazolone Addition to Nitroolefins as Entry to Diaminoamides

Pyrazolones represent an important structural motif in active pharmaceutical ingredients. Their asymmetric synthesis is thus widely studied. Still, a generally highly enantio- and diastereoselective 1,4-addition to nitroolefins providing products with adjacent stereocenters is elusive. In this article, a new polyfunctional Cu^II-1,2,3-triazolium-aryloxide catalyst is presented which enables this reaction type with high stereocontrol. DFT studies revealed that the triazolium stabilizes the transition state by hydrogen bonding between C(5)−H and the nitroolefin and verify a cooperative mode of activation. Moreover, they show that the catalyst adopts a rigid chiral cage/pore structure by intramolecular hydrogen bonding, by which stereocontrol is achieved. Control catalyst systems confirm the crucial role of the triazolium, aryloxide and Cu^II, requiring a sophisticated structural orchestration for high efficiency. The addition products were used to form pyrazolidinones by chemoselective C=N reduction. These heterocycles are shown to be valuable precursors toward β,γ’-diaminoamides by chemoselective nitro and N−N bond reductions. Morphological profiling using the Cell painting assay identified biological activities for the pyrazolidinones and suggest modulation of DNA synthesis as a potential mode of action. One product showed biological similarity to Camptothecin, a lead structure for cancer therapy.     

Effect of a π-donor on the radical bulk polymerization of methyl methacrylate

Bulk polymerizations of methyl methacrylate (MMA) at 60°C initiated with 2,2′-azoisobutyronitrile are influenced by the presence of an organic π-donor such as tetrathiafulvalene (TTF). Upon addition of TTF, the ratio of weight- to number-average molecular weights M̄_w/M̄_n are significantly reduced and the thermal stability of the poly(methyl methacrylate) samples is increased. Kinetic investigations indicate that TTF acts as a retarder on the polymerization mechanism.     

Numerical simulation of interfacial debonding in particulate composites considering material inhomogeneities

A comparative study in the framework of large deformations has been conducted to get insight into micro mechanical response of particulate polymeric composites. The propound model enables finite element predictions of displacement response and resulting debonding of a three-phase material subjected to incremental loading conditions. Predictions employ a unit cell of certain shape under specified loading and constraints. The proposed model involves nonlinear material properties and incorporates a strong or weak non-homogeneous interphase region. The interphase toughness is expressed in terms of the adhesion efficiency between the filler and the matrix, though contact conditions are preserved utilizing special contact elements. Interfacial nonlinear spring joint elements regulate debonding initialization, in relation to the imposed failure criterion. Numerical results are presented and discussed for axial tensile and compressive loadings for a variety of values of the imposed parameters.