In this paper, a combination of algebraic and topological methods are applied to obtain new structural results on harmonic rings. Especially, it is shown that if Gelfand ring A modulo its Jacobson radical zero dimensional ring, then clean ring. It also proved that, for given A, the retraction map $${\text {Spec}}(A)\rightarrow {\text {Max}}(A)$$ flat continuous only Dually, mp-ring {Min}}(A)$$...

The dually flat structure introduced by Amari–Nagaoka is highlighted in information geometry and related fields. In practical applications, however, the underlying pseudo-Riemannian metric may often be degenerate, such an excellent geometric rarely defined on entire space. To fix this trouble, present paper, we propose a novel generalization of for certain class singular models from viewpoint L...

Dually flat manifolds constitute fundamental mathematical objects of information geometry. This note establishes some facts on the global properties and topology of dually flat manifolds which, in particular, provide answers to questions and problems in global information geometry posed by Amari and Amari-Nagaoka.

A vertex with neighbours of degrees d1 ≥ · · · ≥ dr has vertex type (d1, . . . , dr). A graph is vertex-oblique if each vertex has a distinct vertex-type. While no graph can have distinct degrees, Schreyer, Walther and Mel’nikov [Vertex oblique graphs, same proceedings] have constructed infinite classes of super vertex-oblique graphs, where the degree-types of G are distinct even from the degre...

Many works related to dually chordal graphs, their cliques and neighborhoods were published [1, 6]. We will undertake a similar study by considering minimal vertex separators and their properties instead. We find a necessary and sufficient condition for every minimal vertex separator to be contained in the closed neighborhood of a vertex and two major characterizations of dually chordal graphs ...

In our previous work, we have generalized the notion of dually flat or Hessian manifold to quasi-Hessian manifold; it admits metric be degenerate but possesses a particular symmetric cubic tensor (generalized Amari-Centsov tensor). Indeed, naturally appears as singular model in information geometry and related fields. A is locally accompanied with possibly multi-valued potential its dual, whose...