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In SM64 this covering space structure is used by giving Mario extremely high velocity \(v\) (a tangent vector at Marios position in \(P\)) through a different glitch called “backwards longjump”. The exponential map is then used to move Mario along the geodesic \(\exp_v(t)\) for \(t \in \mathbb{R}\). But due to hardware limitations, only discrete steps \(t_1, \dots, t_k\) for some \(k \in \mathbb{N}\) of this geodesic are actually calculated for collision detection. More on how this is related to the exponential map of Lie groups can be found in my previous post. Throughout this post, I will assume the Lie group of SM64 to have its unit at the position \(p\).
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This story was originally featured on Fortune.com，这一点在51吃瓜中也有详细论述
At the start of February, OpenAI upgraded its Codex coding app to give it the ability to manage multiple AI agents. At the same time, it released a standalone macOS app. If you've been patiently waiting for Windows to get that same treatment, OpenAI just released a dedicated Codex app for Microsoft's operating system.，更多细节参见91视频
Theoretically, native apps can integrate with OS on a deeper level. This sounds nice, but what does that mean in practice? There are almost no good interoperable file formats; everything is locked inside individual apps, most services moved to the web, and OSes dropped the ball for making a good shared baseline. You can integrate with OS-provided calendar, but you can’t do it with web calendar. Well, you can, of course, but it’s easier on the web; native doesn’t help with it at all.