Recent zbMATH articles in MSC 08https://www.zbmath.org/atom/cc/082021-04-16T16:22:00+00:00WerkzeugProbabilistic logic over equations and domain restrictions.https://www.zbmath.org/1456.030452021-04-16T16:22:00+00:00"Mordido, Andreia"https://www.zbmath.org/authors/?q=ai:mordido.andreia"Caleiro, Carlos"https://www.zbmath.org/authors/?q=ai:caleiro.carlosSummary: We propose and study a probabilistic logic over an algebraic basis, including equations and domain restrictions. The logic combines aspects from classical logic and equational logic with an exogenous approach to quantitative probabilistic reasoning. We present a sound and weakly complete axiomatization for the logic, parameterized by an equational specification of the algebraic basis coupled with the intended domain restrictions. We show that the satisfiability problem for the logic is decidable, under the assumption that its algebraic basis is given by means of a convergent rewriting system, and, additionally, that the axiomatization of domain restrictions enjoys a suitable subterm property. For this purpose, we provide a polynomial reduction to Satisfiability Modulo Theories. As a consequence, we get that validity in the logic is also decidable. Furthermore, under the assumption that the rewriting system that defines the equational basis underlying the logic is also subterm convergent, we show that the resulting satisfiability problem is \textsf{NP}-complete, and thus the validity problem is \textsf{coNP}-complete. We test the logic with meaningful examples in information security, namely by verifying and estimating the probability of the existence of offline guessing attacks to cryptographic protocols.Local smooth conjugations of Frobenius endomorphisms.https://www.zbmath.org/1456.390042021-04-16T16:22:00+00:00"Kalnitsky, V. S."https://www.zbmath.org/authors/?q=ai:kalnitskij.v-s"Petrov, A. N."https://www.zbmath.org/authors/?q=ai:petrov.andrei-n|petrov.a-n|petrov.alexander-nSummary: A generalization of the Böttcher equation is considered. It turned out that the parametrized Poisson integral, as a function of its parameters, satisfies an equation of the type described. The structure theorem for splitting maps of Frobenius endomorphisms in a ring and in an algebra over it is proved. The real field case is considered. The generalized Böttcher equation is solved for classical two-dimensional algebras and for the Poisson algebra.Dirac operator on the quantum fuzzy four-sphere \(S_{q F}^4\).https://www.zbmath.org/1456.811712021-04-16T16:22:00+00:00"Lotfizadeh, M."https://www.zbmath.org/authors/?q=ai:lotfizadeh.mSummary: \(q\)-deformed fuzzy Dirac and chirality operators on quantum fuzzy four-sphere \(S_{q F}^4\) are studied in this article. Using the \(q\)--deformed fuzzy Ginsparg-Wilson algebra, the \(q\)--deformed fuzzy Dirac and chirality operators in an instanton and no-instanton sector are studied. In addition, gauged Dirac and chirality operators in both cases have also been constructed. It has been shown that in each step, our results have a correct commutative limit in the limit case when \(q \rightarrow 1\) and the noncommutative parameter \(l\) tends to infinity.
{\copyright 2021 American Institute of Physics}