Relative locations of subwords in free operated semigroups and Motzkin words

Bracketed words are basic structures both in mathematics (such as Rota-Baxter algebras) and mathematical physics (such as rooted trees) where the locations of the substructures are important. In this paper we give the classification of the relative locations of two bracketed subwords of a bracketed word in an operated semigroup into the separated, nested and intersecting cases. We achieve this by establishing a correspondence between relative locations of bracketed words and those of words by applying the concept of Motzkin words which are the algebraic forms of Motzkin paths.Comment: 14 page

Rota-Baxter operators on the polynomial algebras, integration and averaging operators

The concept of a Rota–Baxter operator is an algebraic abstraction of integration. Following this classical connection, we study the relationship between Rota–Baxter operators and integrals in the case of the polynomial algebra k[x] k[x] . We consider two classes of Rota–Baxter operators, monomial ones and injective ones. For the first class, we apply averaging operators to determine monomial Rota–Baxter operators. For the second class, we make use of the double product on Rota–Baxter algebras

Quantum Dynamics of Mesoscopic Driven Duffing Oscillators

We investigate the nonlinear dynamics of a mesoscopic driven Duffing oscillator in a quantum regime. In terms of Wigner function, we identify the nature of state near the bifurcation point, and analyze the transient process which reveals two distinct stages of quenching and escape. The rate process in the escape stage allows us to extract the transition rate, which displays perfect scaling behavior with the driving distance to the bifurcation point. We numerically determine the scaling exponent, compare it with existing result, and propose open questions to be resolved.Comment: 4 pages, 4 figures; revised version accepted for publication in EP