It is shown that the existential theory of G with rational constraints, over an HNN-extension G = 〈H, t; tat = φ(a)(a ∈ A)〉 is decidable, provided that the same problem is decidable in the base group H and that A is a finite group. The positive theory of G is decidable, provided that the existential positive theory of G is decidable and that A and φ(A) are proper subgroups of the base group H w...

We prove that the amalgamated free product of two free groups of rank two over a common cyclic subgroup, admits an amenable, faithful, transitive action on an infinite countable set. We also show that any finite index subgroup admits such an action, which applies for example to surface groups and fundamental groups of surface bundles over S.

Let $R$ be a commutative ring with $1 \ne 0$ and let $m$ and
 $n$ integers $1\leq n < m$. A proper ideal $I$ of is
 called an $(m, n)$-closed if whenever $a^m \in I$ for
 some $a\in R$ implies $a^n I$. $ f:A\rightarrow B$ a
 homomorphism $J$ $B.$ This paper
 investigates the concept $(m,n)$-closed ideals in the
 amalgamation $A$ $B$ along respect $f$ denoted by&...

A generalized-statistics variational principle for source separation is formulated by recourse to Tsallis’ entropy subjected to the additive duality and employing constraints described by normal averages. The variational principle is amalgamated with Hopfield-like learning rules resulting in an unsupervised learning model. The update rules are formulated with the aid of q-deformed calculus. Num...