Ambiguous grammar to equivalent unambiguous grammar

I stumbled on this ambiguous grammar and I’ve been trying to make it unambiguous but it’s still ambiguous.

Given the ambiguous CFG :
$ S \to A\mid B$
$ A \to aAb\mid ab$
$ B \to abB\mid \epsilon$

My closet try was:

Given the ambiguous CFG :
$ S \to A$
$ A \to aAb\mid C$
$ B \to b$
$ C \to abC\mid\epsilon$

But the string “$ ab$ ” is ambiguous

Mac Numbers equivalent to Microsoft Excel’s Control+Arrow Keyboard Shortcut

Is there a feature in the MacOS Numbers application that allows me to jump to non-empty cells? In Excel if I hold the control key and press any of the keys or the cursor will navigate to the next non empty cell in that direction from the currently selected cell or hit the outer boundaries of the spreadsheet if all the remaining cells in that direction are empty.

Is there some analogous operation I can do in Numbers that’s equivalent?

“Mathematical Physics”‘s equivalent for Computer Science

Mathematical Physics is a well defined scientific fields that deals with the application of mathematics , mathematical tools , and mathematical methods in the theories and the problems of physics.

Is there a well defined scientific field that deals with the application of mathematics , mathematical tools , and mathematical methods in the theories and the problems of computer science ?

Quillen equivalent module categories

Let $ f:A \rightarrow B$ be a weak equivalence of simplicial commutative rings. There is a Quillen pair $ (-\otimes_{A}B, f_{\ast})$ which is an equivalence. In this situation, $ (-\otimes_{A}B, f_{\ast})$ is an equivalence precisely if the counit map

$ $ \eta_{M} : M \rightarrow f_{\ast}(M \otimes_{A}B)$ $

is a weak equivalence for all cofibrant $ A$ -modules $ M$ . I am having trouble seeing why this map is a weak equivalence. My guess is use the sequence of $ A$ -modules $ \ker(f) \rightarrow A \rightarrow B$ to compute the homotopy groups of the sequence $ \ker(\eta_{M}) \rightarrow M \rightarrow f_{\ast}(M\otimes_{A}B)$ .

Is this the right idea, or is there some easier way to demonstrate the pair is a Quillen equivalence?

Two equivalent python functions (

Our first year students keep struggling with coding conventions. Although everyone knows the theory why those are useful, I currently don’t have material that conclusively illustrates this in a minimalistic way (thinking powerpoint slide).

Therefore, I am looking for a way to illustrate the benefits of following coding conventions by comparing two python functions that perform the same action – function A should hurt as many conventions as possible, while function B should follow those.

My initial attempt was two functions containing print('hello world') & The Most complex “Hello world” program you can justify. However, this is probably a better example to demonstrate the benefits of code readability.


  • Both functions should not exceed 15 SLOC (excluding imports and shebang)
  • The action both snippets perform doesn’t matter, I just want to print identical output
  • The exact number of disregarded coding conventions also does not matter (the more, the better) as long as the point gets across: conventions are there for a reason and not subject of debate.

I would be grateful for any answers, comments or hints towards existing answers in case I missed them.

Mac equivalent of MobXterm

I know you can use Terminal for SSH connections, but what I love about MobaXterm is that you can store different SSH connections very easily and just choose the one you want to use by double-clicking on one of the preconfigured sessions in the sidebar. Furthermore, once you’ve made a connection MobaXterm presents side-by-side windows; one with a GUI-based SFTP that follows the path based in the Terminal, and the terminal window itself. You don’t have to use a combination of different applications for either, e.g. FileZilla and Terminal.

It also has tabbed connections, so you can switch between multiple SSH connections very easily.

Does such an equivalent exist?

How to make upcasting equivalent to using a higher level spell?

In general, damage-dealing spells do not seem worth upcasting; the damage just doesn’t scale very well. This is probably good design, because no one would bother casting higher level spells if an upcast lower one was better. Regardless, I’d like to homebrew a magic item that increases the damage output of spell upcasting to be roughly the same as using an equivalent higher level spell. This is hard for me to figure out, however, because there are so many qualitative differences between spells – it’s hard to sort out how much I need to buff in order to quantitatively make things balance out. I know upcasting damage spells usually isn’t effective, but I don’t know how much I need to add to make it worth it.

I am open to making the item have a limited number of daily charges.