Redirected walking (RDW) aims to reduce the collisions in the physical space for VR applications. However, most of the previous RDW methods do not consider future possibilities of collisions after imperceptibly redirecting users. In this paper, we combine the subtle RDW methods and reset strategy in our method design and propose a novel solution for RDW that can make better use of physical space and trigger fewer resets. The key idea of our method is to discretize the representation of possible user positions and orientations by a series of standard poses and rate them based on the possibilities of hitting obstacles of their reachable poses. A transfer path algorithm is proposed to measure the accessibility among standard poses and is used to support the calculation of the scores of standard poses. Using our method, the user can be redirected imperceptibly to the optimal pose with the best score among all the reachable poses from the user’s current pose during walking. Experiments demonstrate that our method outperforms state-of-the-art methods in various environment sizes and obstacle layouts.

Theorem. Let $\pi$ be a finite group of order $n$, $R$ be a Dedekind domain satisfying that (i) $\fn{char}R=0$, (ii) every prime divisor of $n$ is not invertible in $R$, and (iii) $p$ is unramified in $R$ for any prime divisor $p$ of $n$. Then all the flabby (resp.\ coflabby) $R\pi$-lattices are invertible if and only if all the Sylow subgroups of $\pi$ are cyclic. The above theorem was proved by Endo and Miyata when $R=\bm{Z}$ \cite[Theorem 1.5]{EM}. As applications of this theorem, we give a short proof and a partial generalization of a result of Torrecillas and Weigel \cite[Theorem A]{TW}, which was proved using cohomological Mackey functors.

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The$H$^{$p$}corona problem is the following: Let$g$_{1}, ...,$g$_{$m$}be bounded holomorphic functions with 0<δ≤Σ‖$g$_{$i$}‖. Can we, for any$H$^{$p$}function ϕ, find$H$^{$p$}functions$u$_{1}, ...,$u$_{$m$}such that Σ$g$_{$i$}$u$_{$i$}=ϕ? It is known that the answer is affirmative in the polydisc, and the aim of this paper is to prove that it is in non-degenerate analytic polyhedra. To prove this, we construct a solution using a certain integral representation formula. The$H$^{$p$}estimate for the solution is then obtained by localization and some harmonic analysis results in the polydisc.

In this paper we present a rigorous derivation of the material parameters for both the cylinder and rectangle cloaking structures. Numerical results using these material parameters are presented to demonstrate the cloaking effect.