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In this thesis, we consider some aspects of$noncommutative classical invariant theory$, i.e., noncommutative invariants of$the classical group SL(2, k)$. We develop a$symbolic method$for invariants and covariants, and we use the method to compute some invariant algebras. The subspace$Ĩ$_{d}^{m}of the noncommutative invariant algebra$Ĩ$_{$d$}consisting of homogeneous elements of degree$m$has the structure of a module over the$symmetric group S$_{$m$}. We find the explicit decomposition into irreducible modules. As a consequence, we obtain the$Hilbert series$of the commutative classical invariant algebras. The$Cayley—Sylvester theorem$and the$Hermite reciprocity law$are studied in some detail. We consider a new power series H($Ĩ$_{d},$t$) whose coefficients are the number of irreducible$S$_{$m$}-modules in the decomposition of$Ĩ$_{d}^{m}, and show that it is rational. Finally, we develop some analogues of all this for covariants.

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.

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We introduce two exponential-type integrators for the “good” Boussinesq equation. They are of orders one and two, respectively, and they require lower spatial regularity of the solution compared to classical exponential integrators. For the first integrator, we prove first-order convergence in H^r for solutions in H^{r+1} with r>1/2. This new integrator even converges (with lower order) in H^r for solutions in H^r, i.e., without any additional smoothness assumptions. For the second integrator, we prove second-order convergence in H^r for solutions in H^{r+3} with r>1/2 and convergence in L^2 for solutions in H^3. Numerical results are reported to illustrate the established error estimates. The experiments clearly demonstrate that the new exponential-type integrators are favorable over classical exponential integrators for initial data with low regularity.