Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

Let $0<\alpha<n$ and $M_{\alpha}$ be the fractional maximal function. The nonlinear commutator of $M_{\alpha}$ and a locally integrable function $b$ is given by $[b,M_{\alpha}](f)=bM_{\alpha}(f)-M_{\alpha}(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_{\alpha}]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or BMO(\rn) spaces, by which some new characterizations for certain subclasses of Lipschitz and BMO(\rn) spaces are obtained.Comment: 20 page

Hot spin spots in the laser-induced demagnetization

Laser-induced femtosecond magnetism or femtomagnetism simultaneously relies on two distinctive contributions: (a) the optical dipole interaction (ODI) between a laser field and a magnetic system and (b) the spin expectation value change (SEC) between two transition states. Surprisingly, up to now, no study has taken both contributions into account simultaneously. Here we do so by introducing a new concept of the optical spin generator, a product of SEC and ODI between transition states. In ferromagnetic nickel, our first-principles calculation demonstrates that the larger the value of optical spin generator is, the larger the dynamic spin moment change is. This simple generator directly links the time-dependent spin moment change {\Delta}Mk z (t) at every crystal- momentum k point to its intrinsic electronic structure and magnetic properties. Those hot spin spots are a direct manifestation of the optical spin generator, and should be the focus of future research.Comment: 10 pages, 2 figures, [email protected]

Numerical cognition in bees and other insects

The ability to perceive the number of objects has been known to exist in vertebrates for a few decades, but recent behavioral investigations have demonstrated that several invertebrate species can also be placed on the continuum of numerical abilities shared with birds, mammals, and reptiles. In this review article, we present the main experimental studies that have examined the ability of insects to use numerical information. These studies have made use of a wide range of methodologies, and for this reason it is striking that a common finding is the inability of the tested animals to discriminate numerical quantities greater than four. Furthermore, the finding that bees can not only transfer learnt numerical discrimination to novel objects, but also to novel numerosities, is strongly suggestive of a true, albeit limited, ability to count. Later in the review, we evaluate the available evidence to narrow down the possible mechanisms that the animals might be using to solve the number-based experimental tasks presented to them. We conclude by suggesting avenues of further research that take into account variables such as the animals' age and experience, as well as complementary cognitive systems such as attention and the time sense.This publication was funded by the German Research Foundation (DFG) and the University of Wuerzburg in the funding program Open Access Publishing. Shaowu Zhang was supported by the ARC-CoE in Vision Science