Sharp constants for a class of multilinear integral operators and some applications

We investigate a class of multilinear integral operators with the nonnegative kernels, and prove that the norms of the operators can be obtained by integral of the product of the kernel function and finitely many basic functions. Using the integral, we can easily calculate the sharp constants for the multilinear Hilbert inequality, the generalized Hardy-Littlewood-Sobolev inequality and the multilinear Hardy operator.

Magnetic transformation of Mn from anti-ferromagnetism to ferromagnetism in FeCoNiZMn_(x)(Z=Si,Al,Sn,Ge)high entropy alloys

We design high entropy alloys(HEAs) with different induction elements(Si/Al/Sn).In order to keep the crystal structure invariant and to investigate how the increment in saturation magnetization(Ms)is caused only by the change of electron spin state,each set of HEAs contains a different amount of Mn.Synergistic effects among induction elements that induce the magnetic transformation of Mn from anti-ferromagnetism to ferromagnetism are found.Ms of added Mn reduces when a particular induction element(Si_(0.4)/Al_(0.4)/Sn_(0.4)) exists,while a larger increment of Ms appears when two induction elements coexist,Si_(0.4)Al_(0.4)(25.79 emu/g) and Sn_(0.4)Al_(0.4)(15.43 emu/g).This is reflected in the microcosmic magnetic structure for the emergence of closed domains due to large demagnetization energy,which is confirmed by the Lorentz transmission electron microscope(LTEM) data.The calculated magnetic moments and the exchange integral constants from density functional theory based on the Exact Muffin-Tin Orbits fo rmalism reveal that the magnetic state and the strength of fe rromagnetic and anti-ferromagnetic coupling determine the variation of Ms in different chemical environments.The difference in energy levels of coexisting multiple induction elements also leads to a larger increment of Ms,Si_(0.4)Al_(0.4)Sn_(0.4)(29.78 emu/g),and Si_(0.4)Al_(0.4)Ge_(0.4)Sn_(0.4)(31.00 emu/g).