We present a method for contracting a square-lattice tensor network in two dimensions based on auxiliary tensors accomplishing successive truncations (renormalization) of eight-index tensors for 2 × 2 plaquettes into four-index tensors. Since all approximations are done on the wave function (which also can be interpreted in terms of different kinds of tensor networks), the scheme is variational...

We show how to generalize the classical electric-magnetic decomposition of the Maxwell or the Weyl tensors to arbitrary fields described by tensors of any rank in general n-dimensional spacetimes of Lorentzian signature. The properties and applications of this decomposition are reviewed. In particular, the definition of tensors quadratic in the original fields and with important positivity prop...

We introduce M -tensors. This concept extends the concept ofM -matrices. We denote Z-tensors as the tensors with nonpositive off-diagonal entries. We show that M -tensors must be Ztensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric M -tensor must be nonnegative. A symmetric M -tensor is copositive. Based on the spectral theory of nonnegative tensors,...

The nonlinear version of the well known PCA called the Prinicipal Geodesic Analysis (PGA) was introduced in the past decade for statistical analysis of shapes as well as diffusion tensors. PGA of diffusion tensor fields or any other manifold-valued fields can be a computationally demanding task due to the dimensionality of the problem and thus establishing motivation for an incremental PGA (iPG...

We review two similar concepts of hierarchical rank of tensors (which extend the matrix rank to higher order tensors): the TT-rank and the H-rank (hierarchical or H-Tucker rank). Based on this notion of rank, one can define a data-sparse representation of tensors involving O(dnk + dk 3) data for order d tensors with mode sizes n and rank k. Simple examples underline the differences and similari...

We look at extensions (i.e., stronger logics in the same language) of the Belnap-Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap-Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterize the reduced algebraic models of these new logics and prove a completeness result for the first ...

We show how to generalize the classical electric-magnetic decomposition of the Maxwell or the Weyl tensors to arbitrary fields described by tensors of any rank in general n-dimensional spacetimes of Lorentzian signature. The properties and applications of this decomposition are reviewed. In particular, the definition of tensors quadratic in the original fields and with important positivity prop...

We present the conditions when a Killing-Yano tensor becomes a Nambu tensor. We have shown that in the flat space case all Killing-Yano tensors are Nambu tensors. In the case of Taub-NUT metric and Kerr-Newmann metric we found that a Killing-Yano tensor of order two generate a Nambu tensor of order three. after 1st September 1999 at Institute of Space Sciences,Bucharest-Magurele, Romania e-mail...

The solution of the Lax tensor equations in the case Lαβγ = −Lβαγ was analyzed. The Lax tensors on the dual metrics were investigated. We classified all two dimensional metrics having the symmetric Lax tensor Lαβγ . The Lax tensors of the flat space, Rindler system and its dual were found.