We revisit a geometric lower bound for Waring rank of polynomials (symmetric rank of symmetric tensors) of [LT10] and generalize it to a lower bound for rank with respect to arbitrary varieties, improving the bound given by the “non-Abelian” catalecticants recently introduced by Landsberg and Ottaviani. This is applied to give lower bounds for ranks of multihomogeneous polynomials (partially sy...

A general result on the structure and dimension of the root subspaces of a linear operator under finite rank perturbations is proved: The increase of dimension from the n-th power of the kernel of the perturbed operator to the (n+ 1)-th power differs from the increase of dimension of the corresponding powers of the kernels of the unperturbed operator by at most the rank of the perturbation. Thi...

We consider the conjecture stating that a matrix with rank o(n) and ones on the main diagonal must contain nonzero entries on a 2 2 submatrix with one entry on the main diagonal. We show that a slightly stronger conjecture implies that an explicit linear transformation cannot be computed by linear size and logarithmic depth circuits. We prove some partial results supporting the conjecture.

We investigate the new, Turing-complete class of layered systems, whose lefthand sides of rules can only be overlapped at a multiset of disjoint or equal positions. Layered systems define a natural notion of rank for terms: the maximal number of non-overlapping redexes along a path from the root to a leaf. Overlappings are allowed in finite or infinite trees. Rules may be non-terminating, non-l...

For different models of the electron-phonon interaction, the asymptotic behaviour of the moments of the stationary homogeneous solution of the linear Boltzmann equation is determined in the limit of a high external field. For Hilbert-Schmidt kernels of a finite rank, a result recently proven for kernels of rank one is found generally valid; as a consequence velocity saturation is excluded for t...

impact responses of rectangular composite plates with embedded shape memory alloy (sma) wires are investigated in the present research. the plate is assumed to be placed in a thermal environment; so that in contrast to the available researches in the field, the shape memory and ferroelasticity effects have to be considered also in addition to the superelasticity. the governing equations are der...

We survey two basic techniques for showing that the monadic second-order theory of a structure is decidable. In the first approach, one deals with finite fragments of the theory (given for example by the restriction to formulas of a certain quantifier rank) and – depending on the fragment – reduces the model under consideration to a simpler one. In the second approach, one applies a global tran...