Caesium carbonate supported on hydroxyapatite‐encapsulated Ni0.5Zn0.5Fe2O4 nanocrystallites as a novel magnetically basic catalyst for the one‐pot synthesis of pyrazolo[1,2‐b]phthalazine‐5,10‐diones

A novel nanomagnetic basic catalyst of caesium carbonate supported on hydroxyapatite‐coated Ni_0.5Zn_0.5Fe₂O₄ magnetic nanoparticles (Ni_0.5Zn_0.5Fe₂O₄@HAP‐Cs₂CO₃) was prepared. This new catalyst was fully characterized using Fourier transform infrared spectroscopy, transmission and scanning electron microscopy, X‐ray diffraction and vibrating sample magnetometry techniques, and then the catalytic activity of this catalyst was investigated in the synthesis of 1H‐pyrazolo[1,2‐b]phthalazine‐5,10‐dione derivatives. Also, Ni_0.5Zn_0.5Fe₂O₄@HAP‐Cs₂CO₃ could be reused at least five times without significant loss of activity and could be recovered easily by applying an external magnet. Thus, the developed nanomagnetic catalyst is potentially useful for the green and economic production of organic compounds. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Enlargement of the Wiener filtration by an absolutely continuous random variable via Malliavin’s calculus

Summary. The analytic treatment of problems related to the asymptotic behaviour of random dynamical systems generated by stochastic differential equations suffers from the presence of non-adapted random invariant measures. Semimartingale theory becomes accessible if the underlying Wiener filtration is enlarged by the information carried by the orthogonal projectors on the Oseledets spaces of the (linearized) system. We study the corresponding problem of preservation of the semimartingale property and the validity of a priori inequalities between the norms of stochastic integrals in the enlarged filtration and norms of their quadratic variations in case the random element F enlarging the filtration is real valued and possesses an absolutely continuous law. Applying the tools of Malliavin’s calculus, we give smoothness conditions on F under which the semimartingale property is preserved and a priori martingale inequalities are valid. Received: 12 April 1995 / In revised form: 7 March 1996