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In this paper, we prove the Hyers-Ulam-Rassias stability of isometric homomorphisms in proper CQ*-algebras for the following Cauchy-Jensen additive mapping: The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias’ stability theorem that appeared in the paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. This is applied to investigate isometric isomorphisms between proper CQ*-algebras. [ABSTRACT FROM AUTHOR]

Let $${\fancyscript G}$$ be a generalized Kac–Moody algebra with an integral Borcherds–Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac–Moody algebra $$ wU^{d}_{q} {\left( {\fancyscript G} \right)} $$ , which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra $$ wU^{d}_{q} {\left( {\fancyscript G} \right)} $$ and weak A-forms of $$ wU^{d}_{q} {\left( {\fancyscript G} \right)} $$ . [ABSTRACT FROM AUTHOR]

In this paper the following theorem is proved: Every group L3( q) for q = 3(2 m–1)( m ≥ 2) is characterized by its set of element orders. [ABSTRACT FROM AUTHOR]

Let ( X, $${\mathfrak{S}}$$ ( X), $${\mathfrak{m}})$$ be a probability space with σ-algebra , $${\mathfrak{S}}$$ ( X) and probability measure $${\mathfrak{m}}.$$ The set V in $${\mathfrak{S}}$$ is called P-admissible, provided that for any positive integer n and positive-measure set V n ∈ $${\mathfrak{S}}$$ contained in V , there exists a Z n ∈ $${\mathfrak{S}}$$ such that Z n ⊂ V n and 0 < $${\mathfrak{m}}$$ ( Z n ) < 1/ n. Let T be an ergodic automorphism of ( X, $${\mathfrak{S}})$$ preserving $${\mathfrak{m}},$$ and A belong to the space of linear measurable symplectic cocycles $${\fancyscript G}$$ IC ( X, SL(2,ℝ)) := { A : X → SL(2,ℝ)| log ∥ A ±1( x)∥ ∈ $${\Bbb L}$$ 1( X, $${\mathfrak{m}}).$$ We prove that for any P–admissible set V and ε > 0, there exists a B ∈ $${\fancyscript G}$$ IC ( X, SL(2,ℝ)) such that max{∥ A( x) − B( x)∥, ∥ A −1( x) − B −1( x)∥} < ε for all x ∈ V, A( x) = B( x) for all x ∈ X \ V, and B has the simple Lyapunov spectrum over the system ( X, $${\mathfrak{m}},$$ T). [ABSTRACT FROM AUTHOR]