We prove the ultimate boundedness of solutions of some third order nonlinear ordinary differential equations using the Lyapunov method. The results obtained generalize earlier results of Ezeilo, Tejumola, Reissig, Tunç and others. The Lyapunov function used does not involve the use of signum functions as used by others.
We define an ultra $LI$-ideal of a lattice implication algebra and give equivalent conditions for an $LI$-ideal to be ultra. We show that every subset of a lattice implication algebra which has the finite additive property can be extended to an ultra $LI$-ideal.
A mistake concerning the ultra $LI$-ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an $LI$-ideal to be an ultra $LI$-ideal are given. Moreover, the notion of an $LI$-ideal is extended to $MTL$-algebras, the notions of a (prime, ultra, obstinate, Boolean) $LI$-ideal and an $ILI$-ideal of an $MTL$-algebra are introduced, some important examples are given, and the following notions are proved to be equivalent in $MTL$-algebra: (1) prime proper $LI$-ideal and Boolean $LI$-ideal, (2) prime proper $LI$-ideal and $ILI$-ideal, (3) proper obstinate $LI$-ideal, (4) ultra $LI$-ideal.
Recently, more and more efforts are directed towards developing new imaging and drug-delivery options based on various nanoparticles, exploiting their unique properties. Here, ultra-small gold nanoparticles functionalized with widely used polyethylene glycol and its amine-terminated form were tested in respect of their potential interactions with human immune cells (cell line and primary cells). The results showed that differently terminated ultrasmall gold nanoparticles represent an interesting theranostic platform as they are harmless to immune cells (not inducing cytotoxicity and severe immune response) and on the other hand, they can serve as imaging and/or drug delivery agents using e.g. monocytes/ macrophages as “Trojan horses” to deliver these nanoparticles across the blood-brain barrier and diagnose or treat pathologies of the central nervous system.