Would be dope, but Josh has a lot on his place atm
Just an aside, the exponential derivative is the laplace shift operator, and the exp(x*d/dx) operator can actually be written as exp(d/dlnx), and is used in quantum calculus, as the limiting case of the q-analog
I'd give that about 5 minutes before it became yet another way to hack the entire forum through the chat.
Yeah that's where the operator comes from. I personally haven't come across it before but that's because I don't fraternize with mathematical physics all too often. My colleague does though and that's what got us talking about the operator in the first place.
I just got tossed it my intro quantum class, and it bugged me, the prof literally refused to elaborate. I only found out about the formal name of it this year.
I don't know if there's a good suitable book for your predicament, but I've found the Math Sorceror's recommendations palatable.
I recall this book mentioned previously in this thread had a good DE section:
I still have the pdf in my textbook collection, so I can DM it to you if you like; just hmu. It's a bigger file and the non-member download from AA could take a while.
If you're talking about ODEs, Arnold's ODE book is pretty good IMO. Not what I learned out of back in the day, but what I wish they had taught out of. I think when I was an undergraduate it was taught out of Polking which was kind of loosy-goosy, but it I don't remember it being a terrible book. For undergrad PDE's I learned out of Haberman's book, and it was honestly a pretty good book. Stauss' book is also good for PDEs.
Yes, but none of your examples illustrates this because they're all countable infinities, ie, they are as big as the natural numbers. A simple example of a larger infinity is the number of real numbers, another example is the cardinality of the power set of natural numbers.
(((THEY))) are just hoarding them up at the end of the number line.
Some infinities are indeed bigger than others. If you want to have a mind-blowing afternoon, look up ordinal numbers (ordinals).
f(x)=2x). The one-to-one bit is key here; f(-2) can only equal -4 as it is defined, and every value of x maps to a distinct value. Another thing to pay attention to is the fact this mapping can be undone. In math terms, we say these two sets have the same cardinality since they meet this condition, but that means nothing to most people. Instead, consider that you can't create such a one-to-one mapping between the set of real numbers and the set of integers.
The trick with the hierarchies of infinity is to imagine them in terms of mappings, or labelling them.
lol that's where I come in. I could go on for days about the different types of mappings and geometric transformations. It all ties into everything in a really pleasing way.
They're not bad examples; it's clear you've given this some thought. You were just missing part of the huge abstract picture that you start forming once you start asking "what is a number system?", "what is a vector space?", "what happens when we apply this geometrically?", etc. You start getting down to the very core of the rules that define the physical space we interact with and anything potentially behind it.
Imaginaries would also be like that and obviously complex numbers, since even if you presuppose the ability to make an infinite list, you can always come up with a number that isn't in that list, i.e. go down the entire list, take the first digit of the first number, change it, second, change it, until you now have a number that isn't in the original list. And of course you can keep doing that forever. That was Cantor's idea.
To be fair, cardinality isn't the only way to think about the size of sets. In the context of real numbers, you have the Lebesgue Measure λ which is the formalization of the concept of length so for example, the Lebesgue Measure of the interval [0,1] is λ([0,1]) = 1; as you might expect, it's translation invariant so for example, λ([1,2]) = 1 . You can use this to measure pretty much* any set you can think of, for instance the measure of any single number is 0 which makes perfect sense since you're talking about a point on the real line. From this fact it's straightforward to prove that the Lebesgue Measure of the Natural and the Rational Numbers is 0 since they're the countable sum of the lengths of their points which are all 0, in other words, λ(ℚ) = λ(ℕ) = 0. In case you were wondering, there are uncountable sets that still have length 0, the canonical example being Cantor's Set.