We prove that the property of a free group endomorphism being irreducible is invariant ascending HNN extension it defines. This answers question posed by Dowdall-Kapovich-Leininger. further and atoroidal commensurability invariant. The invariance follows from an algebraic characterization extensions determines exactly when their defining endomorphisms are atoroidal; specifically, we show if onl...

We give an elementary, case-free, Coxeter-theoretic derivation of the formula hnn!/|W| for number maximal chains in noncrossing partition lattice NC(W) a real reflection group W. Our proof proceeds by comparing Deligne-Reading recursion with parabolic characteristic polynomial W-Laplacian matrix considered our previous work. further discuss consequences this geometric theory spherical and affin...

We develop a version of Bass-Serre theory for Lie algebras (over field k) via homological approach. define the notion fundamental algebra graph and show that this construction yields Mayer-Vietoris sequences. extend some well known results in group to N-graded algebras: example, we one relator are iterated HNN extensions with free bases which can be used cohomology computations apply sequence g...

The most useful constructions in Combinatorial Group Theory are amalgamated free products and HNN-extensions, and they are the two basic examples in the theory of graphs of groups due to Bass and Serre (see [9]). We recall that given a group G and a subgroup H 6 G together with monomorphisms (respectively homomorphisms) ψ, φ : H −→ G, the group determined by the presentation 〈 G, t; t−1ψ(h)t = ...

This article presents a comprehensive overview of the hardware realizations of artificial neural network (ANN) models, known as hardware neural networks (HNN), appearing in academic studies as prototypes as well as in commercial use. HNN research has witnessed a steady progress for more than last two decades, though commercial adoption of the technology has been relatively slower. We study the ...

This paper is devoted to studying both the global and local stability of dynamical neural networks. In particular, it has focused on nonlinear neural networks with perturbation. Properties relating to asymptotic and exponential stability and instability have been detailed. This paper also looks at the robustness of neural networks to perturbations and examines if the related properties have bee...