where ni = nβ 2 i (i = 1, 2); β1 and β2 denote the main angles of μ1 and μ2, respectively. Further, let G be any connected or disconnected graph (not necessarily with two main eigenvalues). Let S be any subset of the vertex set V (G) and let GS be the graph obtained from the graph G by adding a new vertex x which is adjacent exactly to the vertices from S. If σ(GS1) = σ(GS2) then we prove that ...

A Deep Neural Network (DNN) is used to estimate the Advanced Scatterometer (ASCAT) C-band microwave normalized backscatter (σ40o), slope (σ′) and curvature (σ″) over France. The Interactions between Soil, Biosphere Atmosphere (ISBA) land surface model (LSM) produce variables (LSVs) that are input DNN. DNN trained simulate σ40o, σ′ σ″ from 2007 2016. predictive skill of evaluated during an indep...

Introducing the notions of (inner) σ-derivation, (inner) σ-endomorphism and oneparameter group of σ-endomorphisms (σ-dynamics) on a Banach algebra, we correspond to each σ-dynamics a σ-derivation named as its σ-infinitesimal generator. We show that the σ-infinitesimal generator of a σ-dynamics of inner σ-endomorphisms is an inner σ-derivation and deal with the converse. We also establish a nice...

Let A be an Artin group with standard generating set Σ. Tits conjectured that the only relations in A amongst the squares of the generators are the obvious ones, namely that σ2 and τ2 commute whenever σ and τ commute, for σ, τ ∈ Σ. In this paper we prove Tits’ conjecture for all Artin groups. In fact, given a number m(σ) ≥ 2 for each σ ∈ Σ, we show that the elements {T (σ) = σm(σ) : σ ∈ Σ} gene...

Throughout this paper, all groups are finite. A group class closed under taking homomorphic images and finite subdirect products is called a formation . The symbol σ denotes some partition of the set primes. V. G. Safonov, I. N. Safonova, A. Skiba (Commun. Algebra. 2020. Vol. 48, № 9. P. 4002–4012) defined generalized σ-function. Any function f form : È {Ø} → {formations groups}, where (Ø) ≠ ∅,...