In this paper, we consider the cutoff Boltzmann equation near Maxwellian, we proved the global existence and uniqueness for the cutoff Boltzmann equation in polynomial weighted space for all $\gamma \in (-3, 1]$. We also proved initially polynomial decay for the large velocity in $L^2$ space will induce polynomial decay rate, while initially exponential decay will induce exponential rate for the convergence. Our proof is based on newly established inequalities for the cutoff Boltzmann equation and semigroup techniques. Moreover, by generalizing the $L_x^\infty L^1_v \cap L^\infty_{x, v}$ approach, we prove the global existence and uniqueness of a mild solution to the Boltzmann equation with bounded polynomial weighted $L^\infty_{x, v}$ norm under some small condition on the initial $L^1_x L^\infty_v$ norm and entropy so that this initial data allows large amplitude oscillations.

We study an asymptotic estimate on the number of negative eigenvalues of the Schr\"odinger operators on unbounded fractal spaces which admit a cellular decomposition. We first give some sufficient conditions for Weyl-type asymptotic formula to hold. Second, we verify these conditions for the infinite blowup of Sierpi\'nski gasket and unbounded generalized Sierpi\'nski carpets. Final, we demonstrate how the result can be applied to the infinite blowup of certain fractals associated with iterated function systems with overlaps, including those defining the classical infinite Bernoulli convolution with golden ratio.

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With the recent rise of Metaverse, online multiplayer VR applications are becoming increasingly prevalent worldwide. Allowing users to move easily in virtual environments is crucial for high-quality experiences in such collaborative VR applications. This paper focuses on redirected walking technology (RDW) to allow users to move beyond the confines of the limited physical environments (PE). The existing RDW methods lack the scheme to coordinate multiple users in different PEs, and thus have the issue of triggering too many resets for all the users. We propose a novel multi-user RDW method that is able to significantly reduce the overall reset number and give users a better immersive experience by providing a more continuous exploration. Our key idea is to first find out the ”bottleneck” user that may cause all users to be reset and estimate the time to reset, and then redirect all the users to favorable poses during that maximized bottleneck time to ensure the subsequent resets can be postponed as much as possible. More particularly, we develop methods to estimate the time of possibly encountering obstacles and the reachable area for a specific pose to enable the prediction of the next reset caused by any user. Our experiments and user study found that our method outperforms existing RDW methods in online VR applications.