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Simple answer: By default, all state is preserved.

Complicated answer: There is an optional timeout parameter that you can pass to the notebook command when you start the notebook server. If you set it to 360, then start worksheet "running", and close your web browser, then after 6 minutes the corresponding Sage worksheet process will step running. Internally what happens is that the web browser sends a ping every so often, and if it doesn't see one after timeout seconds it kills the worksheet. This does not happen by default. However, on a public server like, it does happen since otherwise there will eventually be too many tty's and files allocated, and the server will grind to a halt.