Chase’s lemma and its context

Chase’s lemma provides a powerful tool for translating properties of (co)products in abelian categories into chain conditions. This note discusses the context in which the lemma is used, making explicit what is often neglected in the literature because of its technical nature.     

On the complexity of extending the convergence ball of Wang’s method for finding a zero of a derivative

Ball convergence results are very important, since they demonstrate the complexity in choosing initial points for iterative methods. One of the most important problems in the study of iterative methods is to determine the convergence ball. This ball is small in general restricting the choice of initial points. We address this problem in the case of Wang’s method utilized to determine a zero of a derivative. Finding such a zero has many applications in computational fields, especially in function optimization. In particular, we find the convergence ball of Wang’s method using hypotheses up to the second derivative in contrast to earlier studies using hypotheses up to the fourth derivative. This way, we also extend the applicability of Wang’s method. Numerical experiments used to test the convergence criteria complete this study.     

Some unsolvable conjectures in finite p-groups

We survey some unsolvable conjectures in finite p-groups and their research progress.     

Boundedness of semilinear Duffing equations at resonance in a critical situation

In this paper, we propose a sufficient and necessary condition for the boundedness of all the solutions for the equation $\ddot{x}+n^{2}x+g(x)=p(t)$ with the critical situation that $|{\int }_0^{2\pi }p(t)e^{?int}dt|=2|g(+\infty )?g(?\infty )|$ on g and p, where $n\in {\mathbb{N}}^{+}$, $p(t)$ is periodic and $g(x)$ is bounded.     

Convergence of solutions for the fractional Cahn–Hilliard system

This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called ?ojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.     

Heinz type inequalities for mappings satisfying Poisson’s equation

In this paper, we study the Heinz type inequalities for mappings satisfying Poisson’s equation. Some results generalize the ones obtained by Partyka and Sakan.     

Delay of terminal current in systems with abruptly changed Hamiltonian

The current response for the parameter change of a mesoscopic system is a practical issue for future's circuit design. Nowadays most considered cases are various types of bias modulation, while the effect of change of conductor Hamiltonian is seldom addressed. In this paper, we investigate the response of ballistic transport induced by a sudden change of the conductor Hamiltonian. We formulize the terminal current in language of non-equilibrium Green's function. Our method is applied to one-dimensional tight-binding chains and we find that the terminal current has a delay to the Hamiltonian change. The amount of delay is not determined by the velocity of incident electrons in the bias window, but depends on the tight-binding hopping energy γ. The delay of current response at the detecting point away from where the Hamiltonian changes is $C{\gamma }^{?1}$, where C is a constant independent of the system.