A Class of Quantum Markov Fields on Tree-like Graphs: Ising-type Model on a Husimi Tree

A new class of forward quantum Markov fields (FQMFs) is introduced. The structure of these quantum Markov fields is investigated in the finer structure of a quasi-local algebra of observable over a tree-like graph. We provide an effective construction of a class of FQMCs. Moreover, we show the existence of three FMRFs associated with an Ising type model on a Husimi tree.

A Remark on Quadrics in Projective Klingenberg Spaces over a Certain Local Algebra

This article is devoted to some polar properties of quadrics in the projective Klingenberg spaces over a local ring which is a linear algebra generated by one nilpotent element. In this case, polar subspaces are described; the notion “degree of neighborhood” is used for the geometric description of polar subspaces of quadrics. The polarity induced by a quadric is also studied.

A Remark on Structure of Projective Klingenberg Spaces over a Certain Local Algebra

This article is devoted to the projective Klingenberg spaces over a local ring, which is a linear algebra generated by one nilpotent element. In this case, subspaces of such Klingenberg spaces are described. The notion of the “degree of neighborhood” is introduced. Using this, we present the geometric description of subsets of points of a projective Klingenberg space whose arithmetical representatives need not belong to a free submodule.

The maximal dimension of formal fibers of local rings of an algebraic scheme of finite type

For a scheme [Formula: see text] of finite type over a Noetherian local ring [Formula: see text] with a closed point [Formula: see text] of the special fiber, we show that the maximal dimension of the formal fibers of the local algebra [Formula: see text] equals to [Formula: see text] provided that either [Formula: see text] is complete of dimension one or the dimensions of the formal fibers of [Formula: see text] are less than [Formula: see text]. This extends Matsumura’s theorem for algebraic varieties.