Ah. Based on what I found earlier, a double factorial sign is used in modern notation to indicate both.

Did you mean "!!" like that?
Or is it a different notation?
Maybe they have something now I did not know about or was not around yet back then when I needed that.

Oh I see it now:
5!! would be 1*3*5
and
6!! would be 2*4*6

Funny I do not remember seeing that. I do not like that notation, but it's a lot easier to write out. But what happens when we have a disenfranchised (haha) function:
"n!!"
how do we know the author meant odd or even.
They would have to specify that separately, right?
For a constant no problem, we can have 7!!, 8!!, 103!!, 42!!, but for variable 'n' it's not as clear.

Also, I think there is a second definition for n!! which could make it even more unclear.

 

Did you mean "!!" like that?
Or is it a different notation?
Maybe they have something now I did not know about or was not around yet back then when I needed that.

Oh I see it now:
5!! would be 1*3*5
and
6!! would be 2*4*6

Funny I do not remember seeing that. I do not like that notation, but it's a lot easier to write out. But what happens when we have a disenfranchised (haha) function:
"n!!"
how do we know the author meant odd or even.
They would have to specify that separately, right?
For a constant no problem, we can have 7!!, 8!!, 103!!, 42!!, but for variable 'n' it's not as clear.

Also, I think there is a second definition for n!! which could make it even more unclear.

Not a huge fan of it myself, but that is correct. I've not seen another definition for it, though.

 

Not a huge fan of it myself, but that is correct. I've not seen another definition for it, though.

Yes but I have a problem in that:
n!!
is basically undefined even though this is:
7!!
and so is this:
8!!

That's what I really don't like.
What this means is we have to add other expressions to explain which 'n' we are using, even or odd, which negates the whole purpose of having a different way of expressing odd or even factorials.
How about
n!o! (ha ha)
or:
n!e! (ha ha)

or maybe:
n!!o
or:
n!!e

This looks like of sloppy though maybe that is why I always used the overtyped symbols.

Allowing the use of n!! means we can't state a formula like:
y=x/n!!
because we did not specify which 'n' we are using, odd or even. We'd have to state it separately:
y=x/n!!, for n odd

It's too bad that:
n!!!
is already taken for something else, or else we could have used:
n!!
for even 'n' and:
n!!!
for odd 'n'.

Maybe a subsript:
y=x/n!_e
for even and:
y=x/n!_o
for odd, where the underscore just indicates a subscript here.

I guess another variation would be just n! for both odd and even, with the modification that instead of a dot below the vertical line for the exclamation point we use either a small 'o' or a small 'e' to replace the dot.

 

 

Two teenagers have once again proved an ancient math rule


​

But in 1927, mathematician Elisha Loomis asserted that the feat could not be done using rules from trigonometry, a subset of geometry that deals with the relationships between angles and side lengths of triangles. He believed that Pythagoras’ theorem is so fundamental to trigonometry, any trigonometry-based attempt to prove the theorem would have to first assume it was true, thereby resorting to circular logic.

 

 

https://forum.allaboutcircuits.com/threads/why-does-the-universe-exist.178775/post-1735756
https://forum.allaboutcircuits.com/...r-than-the-speed-of-light.186645/post-1738184

So when you hear (simplifying) 'hamiltonian' (Find The Hamiltonian) in a physics paper, usually it means the Hamiltonian Mechanics framework (total mechanical energy) of, with 'least action' of, some process, in some sort of space geometry.
https://profoundphysics.com/hamiltonian-mechanics-for-dummies/

 

 

 

 

 

 

 

My head is spinning. Does that make it a boson, a fermion, or a paraparticle?

 

Paranormal paraparticles. As she says, there are valid mathematical solutions that are non-physical.

Paranormal Paraparticles on a Plate.

 

My head is spinning. Does that make it a boson, a fermion, or a paraparticle?

I read it is a sort of cross between two types.

 

Mathematicians Solve Infamous ‘Moving Sofa Problem’


​

A quick tour through the problem’s history reveals the extensive effort that mathematicians have poured into it—they were no couch potatoes. Faced with an empty hallway, what is the largest shape you could squeeze through it?

 

Mathematicians Solve Infamous ‘Moving Sofa Problem’
View attachment 342009
​

Alternatively, use a time machine.

 

 

Mathematicians Solve Infamous ‘Moving Sofa Problem’
View attachment 342009
​

This is one of the nuttiest "problems" I've ever seen "solved". I also think there are problems with the solution.

Real life objects are not so rigid that they can not bend slightly. There is a chance that the shape they came up with is not the shape that can more around a corner. That's because most large objects can bend slightly, and even if it's just a tiny amount, that reduces increases the size of the object that can move around a corner.

A good example is a mattress because these things have to be moved when someone moves from one place of residence to another. Most mattresses bend, and some bend more than others.
I happen to know someone who move recently and wanted to bring their huge mattress with them. This was a special type though because they had back problems, so it was much stiffer than others. It would not fit up the steps because the ceiling as lower and in order to get it up the stairs it would have to bend around that corner the ceiling created. The mattress had to be discarded. However, if it was more flexible, it would have made it up the stairs, and they did buy one that was actually compressed and that easily fit up the stairs.
So that's another issue. Can the object be compressed.

I suppose theory is good for itself sometimes though because it answers academic questions that we like to see solved. So ok, I am off now to see if I can figure out how many clowns can fit into a Volks Wagon automobile. This has been an open question since the dawn of time