1: No difference. It is kept around to allow old S-code to continue to function. This is documented a "Note" in ?Math?Arithmetic
2: Yes: But you already know it:
`^`(x,y)
#[1] 1024
In R the mathematical operators are really functions that the parser takes care of rearranging arguments and function names for you to simulate ordinary mathematical infix notation. Also documented at ?Math.
Edit: Let me add that knowing how R handles infix operators (i.e. two argument functions) is very important in understanding the use of the foundational infix "[[" and "["-functions as (functional) second arguments to lapply and sapply:
> sapply( list( list(1,2,3), list(4,3,6) ), "[[", 1)
[1] 1 4
> firsts <- function(lis) sapply(lis, "[[", 1)
> firsts( list( list(1,2,3), list(4,3,6) ) )
[1] 1 4
Answer from IRTFM on Stack OverflowVideos
So, I need to calculate power, or rather, how many samples we need to show a particular size of effect. Unfortunately, this is one of the areas that was left out of my education.
Could you please dear rstats people point me on a good source for this?
Or should I just do a simulation given a particular model and assumption so that I can calculate the required number of samples for different scenarios? (anything from anova, or t-test if you want, populations with different variance, interactions, methods like CART, RF, boosting...)?
So far I feel like one equation from Wikipedia and table with different effect sizes and required populations, and then invest time into simulations might be more sustainable deal.
For context, I am not mathematician, I'm studying engineering. I'm often seeing some power over R (set of real numbers). Every text and note I've found so far just casually starts using it. I do not understand it's meaning. I can infer a little from context but can't confirm since Google search isn't helping. I'd be very thankful if someone could explain it for me.