A real tensor T of rank r is identifiable if it has a unique decomposition with rank 1 tensors. Sometimes identifiability fails over , for general tensors of fixed rank. This behaviour is peculiar when r is sub-generic. In several cases, the failure is due to an elliptic normal curve passing through general points of the corresponding variety of rank 1 tensors (Segre, Veronese, Grassmann). When this happens, we prove the existence of nonempty euclidean open subsets of some varieties of rank r tensors, whose elements have several complex decompositions, but only one of them is real.

We show how methods of algebraic geometry can produce criteria for the identifiability of specific tensors that reach beyond the range of applicability of the celebrated Kruskal criterion. More specifically, we deal with the symmetric identifiability of symmetric tensors in \mathrm{Sym}^4(\mathbb C^{n+1}) , i.e., quartic hypersurfaces in a projective space \mathbb P^n , that have a decomposition in 2n+1 summands of rank 1. This is the first case where the reshaped Kruskal criterion no longer applies. We present an effective algorithm, based on efficient linear algebra computations, that checks if the given decomposition is minimal and unique. The criterion is based on the application of advanced geometric tools, like Castelnuovo's lemma for the existence of rational normal curves passing through a finite set of points, and the Cayley–Bacharach condition on the postulation of finite sets. In order to apply these tools to our situation, we prove a reformulation of these results, hereby extending classical results such as Castelnuovo's lemma and the analysis of Geramita, Kreuzer, and Robbiano, Cayley–Bacharach schemes and their canonical modules, Trans. Amer. Math. Soc. 339:443–452, 1993.

We investigated the Michelangelo effect, i.e. the facilitatory effect of a virtual art therapy in motor rehabilitation (Iosa et al. 2021), with a novel virtual reality paradigm in which users are engaged in motor exercises with 3D sculptures. In particular, thirty young adults were immersed in a virtual environment where they could sculpt, by using the real hands, some famous sculptures in the history of art, such as the David of Michelangelo, the Venus of Milo and the statue of Laocoon and His Sons, and their control stimuli, i.e. statues in very low resolution or cubes. We recorded the kinematics (length, the time to complete each trial, mean normalized jerk) and questionnaire answers (objective and subjective beauty, User Satisfaction Evaluation Questionnaire and Nasa Task Load Index). In general, we found that the perception of subjective and objective beauty was higher when sculpting the statues than control stimuli, the judgment of usability of the system was high. The perceived fatigue was not higher when sculpting the statues despite the longer time spent in completing the task that with respect to the control stimuli. Moreover, we found that the interaction with the experimental statues affected the fluidity and symmetry of hands movements. Finally, we discuss this evidence regarding the art therapy and neuroaesthetics principles for motor rehabilitation in the Metaverse with VR, including the possible role of virtual embodiment (illusory feeling to have a virtual body) for boosting the efficacy of the clinical applications.

The paper deals with the computation of the rank and the identifiability of a specific ternary form. Often, one knows some short Waring decomposition of a given form, and the problem is to determine whether the decomposition is minimal and unique. We show how the analysis of the Hilbert-Burch matrix of the set of points representing the decomposition can solve this problem in the case of ternary forms. Moreover, when the decomposition is not unique, we show how the procedure of liaison can provide alternative, maybe shorter, decompositions. We give an explicit algorithm that tests our criterion of minimality for the case of ternary forms of degree $9$ . This is the first numerical case in which a new phenomenon appears: the span of $18$ general powers of linear forms contains points of (subgeneric) rank $18$ , but it also contains points whose rank is $17$ , due to the existence of a second shorter decomposition which is completely different from the given one.