Abstract
AbstractSome properties of tensor ranks and the non-closeness issue of sets with given restrictions on the rank of tensors entering those sets are studied. It is proved that the rank of the d-dimensional Laplacian equals d. The following conjecture is formulated: for any tensor of non-maximal rank there exists a nonzero decomposable tensor (tensor of rank 1) such that the rank increases by one after adding this tensor. In the general case, it is proved that this property holds algebraically almost everywhere for complex tensors of fixed size whose rank is strictly less than the generic rank.
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