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The aim of this paper is the Ore extension of group-cograded Hopf coquasigroups. This paper first shows a categorical interpretation and some examples of group-cograded Hopf coquasigroups, and then gives a necessary and sufficient conditions for the Ore extensions of group-cograded Hopf coquasigroups to be group-cograded Hopf coquasigroups. Finally, a certain isomorphism between Ore extensions are considered., Comment: 15pages

In this paper, we generalizing the main result in Liu[10] to weak Hopf coquasigroups case. We first define and study group-cograded weak Hopf quasigroups, which generalize both group-cograded Hopf quasigroups and weak Hopf group-coalgebras. Then we introduce the notion of p-Yetter-Drinfeld weak quasimodule over group-cograded weak Hopf quasigroups H. If the antipode of H is bijective, we show that the category YDWQ(H) of Yetter-Drinfeld weak quasimodules over H is a crossed category, and the subcategory YD(H) of Yetter-Drinfeld modules is a braided crossed category., Comment: 19pages

We present studies on crystal structure and ionic diffusion behaviors of superionic Cu$_{2-\delta}$Se ($\delta$=0, 0.04, and 0.2) by utilizing neutron powder diffraction and quasi-elastic neutron scattering. In the superionic phase, the structural model with Cu ions occupying the Wyckoff sites of 8$c$ and 32$f$ provides the best description of the structure. As the content of Cu increasing in Cu$_{2-\delta}$Se, the Cu occupancy increases on the 32$f$ site, but decreases on the 8$c$ site . Fitting to the quasi-elastic neutron scattering spectra reveals two diffusion modes, the localized diffusion between the 8$c$ and 32$f$ sites and the long-range diffusion between the adjacent 8$c$ sites using the 32$f$ site as a bypass, respectively. Between 430 and 650 K, we measured that the compound with more Cu content exhibits a larger long-range diffusion coefficient. Temperature in this range does not affect the long-range diffusion process obviously. Our results suggest the two diffusion modes cooperative and thus provide a microscopic understanding of the ionic diffusion of the Cu ions in superionic Cu$_{2-\delta}$Se., Comment: 6 pages, 3 figures

For a class of $\Lambda$-Fleming-Viot processes with Brownian spatial motion in $\mathbb{R}^d$ whose associated $\Lambda$-coalescents come down from infinity, we obtain sharp global and local modulus of continuities for the ancestry processes recovered from the lookdown representations. As applications, we prove both global and local modulus of continuities for the $\Lambda$-Fleming-Viot support processes. In particular, if the $\Lambda$-coalescent is the Beta$(2-\beta,\beta)$ coalescent for $\beta\in(1,2]$ with $\beta=2$ corresponding to Kingman's coalescent, then for $h(t)=\sqrt{t\log (1/t)}$, the global modulus of continuity holds for the support process with modulus function $\sqrt{2\beta/(\beta-1)}h(t)$, and both the left and right local modulus of continuities hold for the support process with modulus function $\sqrt{2/(\beta-1)}h(t)$., Comment: 27 pages

Let $H$ be a crossed group-cograded Hopf quasigroup. We first introduce the notion of $p$-Yetter-Drinfeld quasimodule over $H$. If the antipode of $H$ is bijective, we show that the category $\mathscr Y\mathscr D\mathscr Q(H)$ of Yetter-Drinfeld quasimodules over $H$ is a crossed category, and the subcategory $\mathscr Y\mathscr D(H)$ of Yetter-Drinfeld modules is a braided crossed category., Comment: 19pages. Add a mirror structure and modify some typos. Comments are welcomed

A system of mutually interacting superprocesses with migration is constructed as the limit of a sequence of branching particle systems arising from population models. The uniqueness in law of the superprocesses is established using the pathwise uniqueness of a system of stochastic partial differential equations with non-Lipschitz coefficients, which is satisfied by the corresponding system of distribution-function-valued processes., Comment: 19 pages

A generalized stepping stone model with $\Xi$-resampling mechanism is a two dimensional probability-measure-valued stochastic process whose moment dual is similar to that of the classical stepping stone model except that Kingman's coalescent is replaced by $\Xi$-coalescent. We prove the existence of such a process by specifying its moments using the dual function-valued $\Xi$-coalescent process with geographical labels and migration, and then verifying a multidimensional Hausdorff moment problem. We also characterize the stationary distribution of the generalized stepping stone model and show that it is not reversible if the mutation operator is of uniform jump-type., Comment: 19 pages