Sharp convex bounds on the aggregate sums--An alternative proof

It is well known that a random vector with given marginal distributions is comonotonic if and only if it has the largest sum with respect to the convex order [ Kaas, Dhaene, Vyncke, Goovaerts, Denuit (2002), A simple geometric proof that comonotonic risks have the convex-largest sum, ASTIN Bulletin 32, 71-80. Cheung (2010), Characterizing a comonotonic random vector by the distribution of the sum of its components, Insurance: Mathematics and Economics 47(2), 130-136] and that a random vector with given marginal distributions is mutually exclusive if and only if it has the minimal convex sum [Cheung and Lo (2014), Characterizing mutual exclusivity as the strongest negative multivariate dependence structure, Insurance: Mathematics and Economics 55, 180-190]. In this note, we give a new proof of this two results using the theories of distortion risk measure and expected utility.Comment: 11page

Joint Mixability of Elliptical Distributions and Related Families

In this paper, we further develop the theory of complete mixability and joint mixability for some distribution families. We generalize a result of R\"uschendorf and Uckelmann (2002) related to complete mixability of continuous distribution function having a symmetric and unimodal density. Two different proofs to a result of Wang and Wang (2016) which related to the joint mixability of elliptical distributions with the same characteristic generator are present. We solve the Open Problem 7 in Wang (2015) by constructing a bimodal-symmetric distribution. The joint mixability of slash-elliptical distributions and skew-elliptical distributions is studied and the extension to multivariate distributions is also investigated.Comment: 15page

Convergence to diffusion waves for solutions of Euler equations with time-depending damping on quadrant

This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant $(x,t)\in \mathbb{R}^+\times\mathbb{R}^+$, \begin{equation}\notag \partial_t v - \partial_x u=0, \qquad \partial_t u + \partial_x p(v) =\displaystyle -\frac{\alpha}{(1+t)^\lambda} u, \end{equation} with null-Dirichlet boundary condition or null-Neumann boundary condition on $u$. We show that the corresponding initial-boundary value problem admits a unique global smooth solution which tends time-asymptotically to the nonlinear diffusion wave. Compared with the previous work about Euler equations with constant coefficient damping, studied by Nishihara and Yang (1999, J. Differential Equations, 156, 439-458), and Jiang and Zhu (2009, Discrete Contin. Dyn. Syst., 23, 887-918), we obtain a general result when the initial perturbation belongs to the same space. In addition, our main novelty lies in the facts that the cut-off points of the convergence rates are different from our previous result about the Cauchy problem. Our proof is based on the classical energy method and the analyses of the nonlinear diffusion wave

Skyrmion dynamics in a chiral magnet driven by periodically varying spin currents

In this work, we investigated the spin dynamics in a slab of chiral magnets induced by an alternating (ac) spin current. Periodic trajectories of the skyrmion in real space are discovered under the ac current as a result of the Magnus and viscous forces, which originate from the Gilbert damping, the spin transfer torque, and the $\beta$-nonadiabatic torque effects. The results are obtained by numerically solving the Landau-Lifshitz-Gilbert equation and can be explained by the Thiele equation characterizing the skyrmion core motion