Corrigendum to: Free Rota-Baxter systems and a Hopf algebra structure

Abstract

In this corrigendum, we give a correction that a free Rota-Baxter system is a left counital Hopf algebra but not a Hopf algebra.     

Fekete and Szeg problem in one and higher dimensions

In this paper, we establish the Fekete and Szeg inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in C~n.     

Gröbner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras

In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Gröbner-Shirshov bases of free Rota-Baxter algebra, free λ-differential algebra and free λ-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to the recent results obtained by K. Ebrahimi-Fard-L. Guo, and L. Guo-W. Keigher by using other methods.     

The structure and expression of power group

With the application of the topological upgrade, more and more experts are interested in the problems concerning the mathematical structure’s upgrade to its power set. This paper is a further study on the structure and interrelationship of a power group. By revealing the nature of relationship between the power group and a regular one, the present paper proves some theorems of the structure of the power group and constructs some theorems on its homomorphism and isomorphism as well, proposing an expression form and isomorphism replacement construct methods of the power group.     

Asymptotic behavior of the periodic wave solution for the (3+1)-dimensional Kadomtsev-Petviashvili equation

In this paper, the one- and two-periodic wave solutions for the (3+1)-dimensional Kadomtsev-Petviashvili equation are presented by means of the Hirota’s bilinear method and the Riemann theta function. The rigorous proofs on asymptotic behaviors of these two solutions are given that soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure.     

A note on chaos via Furstenberg family couple

The concepts of the first type of distributional chaos in the Tan-Xiong sense (Abbrev. DC1 in the Tan-Xiong sense), the second type of strong-distributional chaos (Abbrev. strong DC2) and the third type of strong-distributional chaos (Abbrev. strong DC3) were introduced by Tan et al. [F. Tan, J. Xiong. Chaos via Furstenberg family couple, Topology Appl. (2008), doi:10.1016/j.topol.2008.08.006] for continuous maps of a metric space. However, it turns out that, for continuous maps of a compact metric space, the three mutually nonequivalent versions of distributional chaos can be discussed. Let X be a compact metric space and f:X→X a continuous map. In this paper, we show that for any integer N>0, f is strong DC2 (resp. strong DC3) if and only if f^N is strong DC2 (resp. strong DC3). We also show that the above three versions of distributional chaos are topological conjugacy invariant. In addition, as an application, we present an example.     

Effects of γ rays irradiation on structure and property of TiO2 thin films

TiO₂ thin films were deposited on a glass substrate by the radio frequency magnetron sputtering method, and annealed for 2 h at temperatures of 550°C. Then, ⁶⁰Co γ rays with different doses were used to irradiate the resulting TiO₂ thin films. The surface features of films before and after irradiation were observed by scanning electron microscope (SEM). Simultaneously, the crystal structure and optical properties of films before and after irradiation were studied by X-ray diffraction (XRD), UV–VIS transmission spectrum and Photoluminescence (PL) spectrum, respectively. The SEM analysis shows that the film is smooth with tiny particles on the film surface, and non-crystallization trend was clear after irradiated with γ rays. The XRD results indicated that the structure of the film at the room temperature mainly exists in the form of amorphous and mixed crystal at a sputtering power of 200 W, and non-crystallinity was more obvious after irradiation. Obvious difference can be found for the transmissibility of the irradiated and pre irradiation TiO₂ films by the UV-VIS spectra. The color becomes light yellow, and the new absorption edge also appeared at about 430 nm. PL spectra and photocatalysis experiments indicate that the photocatalysis degradation rate of the TiO₂ films on methylthionine chloride solution irradiated with the maximum dose can be increased to 90%.     

Notes on Homoclinic Solutions of the Steady Swift-Hohenberg Equation

This paper considers the steady Swift-Hohenberg equation u＇＇＇＋β2u＇＇＋u^3-u=0.Using the dynamic approach, the authors prove that it has a homoclinic solution for each β∈ （4√8-ε0,4 √8）, where ε0 is a small positive constant. This slightly complements Santra and Wei＇s result [Santra, S. and Wei, J., Homoclinic solutions for fourth order traveling wave equations, SIAM J. Math. Anal., 41, 2009, 2038-2056], which stated that it admits a homoclinic solution for each β∈C （0,β0） where β0 = 0.9342 ....