Kruskal's theorem states that a sum of product tensors constitutes unique tensor rank decomposition if the so-called k-ranks are large. We prove "splitting theorem" for sets tensors, in which k-rank condition is weakened to standard notion rank, and conclusion uniqueness relaxed statement set splits (i.e. disconnected as matroid). Our splitting implies generalization theorem. While several exte...

The Decomposition-Separation Theorem generalizing the classical KolmogorovDoeblin results about the decomposition of finite homogeneous Markov chains to the nonhomogeneous case is presented. The ground-breaking result in this direction was given in the work of David Blackwell in 1945. The relation of this theorem with other problems in probability theory and Markov Decision Processes is discuss...

The question of computing the group complexity of finite semigroups and automata was first posed in K. Krohn and J. Rhodes, Complexity of finite semigroups, Annals of Mathematics (2) 88 (1968), 128–160, motivated by the Prime Decomposition Theorem of K. Krohn and J. Rhodes, Algebraic theory of machines, I: Prime decomposition theorem for finite semigroups and machines, Transactions of the Ameri...

Both the Doob-Meyer and the Graversen-Rao decomposition theorems can be proved following an approach based on predictable compensators of discretizations and weak-L technique, which was developed by K.M. Rao. It is shown that any decomposition obtained by Rao’s method gives predictability of compensators without additional assumptions (like submartingality in the original Doob-Meyer theorem or ...

We describe a new approach to the canonical decompositions of 3–manifolds along tori and annuli due to Jaco–Shalen and Johannson (with ideas from Waldhausen) — the so-called JSJ–decomposition theorem. This approach gives an accessible proof of the decomposition theorem; in particular it does not use the annulus–torus theorems, and the theory of Seifert fibrations does not need to be developed i...

The aim of this paper is to prove ergodic decomposition theorems for probability measures quasi-invariant under Borel actions of inductively compact groups (Theorem 1) as well as for σ-finite invariant measures (Corollary 1). For infinite measures the ergodic decomposition is not unique, but the measure class of the decomposing measure on the space of projective measures is uniquely defined by ...