KF Math Thread - Discuss Math

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Agent Snigger said:
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
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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.

It was a good analysis problem though: finding for what f,x,k does one have U_k(f)(x) = f(x + k). All I started off with was the premise that f : R ---> R was analytic and you had some x,k real. Each of the three lemmas I had to invent and then prove following a general strategy. And then I noticed the substitution: k = ((x + k) - x), and things just collapsed to simplicity where the three lemmas weren't even necessary and were just their own results.

This is very similar to mathematical research. You start off with some desired result, and in experimenting you both identify and invent lemmas you need to prove to accomplish some grand strategy of accomplishing the desired result. Then, once you've done a few days of work, you give it another look, make a few observations, and then suddenly find a solution that took a few few minutes to complete. I liken it to looking for the switch in a dark room. You're gonna trip over furniture, wander around feeling the wall, and then you find the switch and you saw the geodesic you could've taken but were completely blind to for some reason.
 
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Lurker Child said:
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
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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.
My first time deriving the result, I actually used matrices, because I was less experienced at the time. I don't think I'd even taken a linear algebra class yet, so I have to give myself props for realizing that the polynomials formed a vector space.

I really really enjoy tossing operators into a function with a taylor series and seeing what I get out, it's like a newspaper game for math spergs
 
Does anyone have a recommendation for a good textbook to (re-)learn differential equations from?
Just "Differential Equations For Undergrads" is fine, I don't need everything derived in its full mathematical rigor.
 
Kosher Dill said:
Does anyone have a recommendation for a good textbook to (re-)learn differential equations from?
Just "Differential Equations For Undergrads" is fine, I don't need everything derived in its full mathematical rigor.
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I don't know if there's a good suitable book for your predicament, but I've found the Math Sorceror's recommendations palatable.

 
Kosher Dill said:
Does anyone have a recommendation for a good textbook to (re-)learn differential equations from?
Just "Differential Equations For Undergrads" is fine, I don't need everything derived in its full mathematical rigor.
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I recall this book mentioned previously in this thread had a good DE section:
y a t s said:
Here is a link to the pdf for those that share in my burning hatred for Pearson. And here is a link to the solution manual pdf.
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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.
I would attach it here, but I don't wanna create more copyright headaches for Jersh.
 
Kosher Dill said:
Does anyone have a recommendation for a good textbook to (re-)learn differential equations from?
Just "Differential Equations For Undergrads" is fine, I don't need everything derived in its full mathematical rigor.
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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.

On the topic of books, I re-read E.T. Jaynes' Probability book recently, which is probably one of the most kick ass math books I have had the fortune to come across during my academic career (I may be biased coming from a physics background though). It's one of the books I think anyone doing stats or physics should absolutely read. Jaynes is arguably one of (if not the) godfathers of Bayesianism, but one of the most strange things reading about the rationalist/EA turbo retards like Yidkowski, I get the impression few or none of them know who he is or have understood his work. I think if you reanimated Jaynes, he would be absolutely laying the slap down on those retards, as his thinking seemed way more cautious and (mostly) reservedly qualified on applications outside of physics and stats.
 
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Aren't some infinities bigger than others?

The number of primes are infinite.

But so are the number of even numbers.

But how about the number of numbers divisible by three?

They're both an infinite set of numbers.

But one's smallest. And primes get increasingly sparse.

Mersenne primes are increasingly difficult to find, as an example. The number of Mersenne primes is also infinite. Or is it? There's no proof one way or the other. There should be. But are they?

Prove it or blow me.

Also if you are a mathematician, kill yourself, but not by normal suicide, just devote your life to the Collatz Conjecture, lmao, you'll wish you were dead.
 
AnOminous said:
Aren't some infinities bigger than others?
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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.
 
AnOminous said:
Aren't some infinities bigger than others?

The number of primes are infinite.

But so are the number of even numbers.

But how about the number of numbers divisible by three?

They're both an infinite set of numbers.

But one's smallest. And primes get increasingly sparse.

Mersenne primes are increasingly difficult to find, as an example. The number of Mersenne primes is also infinite. Or is it? There's no proof one way or the other. There should be. But are they?

Prove it or blow me.

Also if you are a mathematician, kill yourself, but not by normal suicide, just devote your life to the Collatz Conjecture, lmao, you'll wish you were dead.
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Some infinities are indeed bigger than others. If you want to have a mind-blowing afternoon, look up ordinal numbers (ordinals).

Here's an easy example:
  • The set of natural numbers is infinite (since you can always add 1) but is a subset of the set of integers.
  • The set of integers is a subset of the set of real numbers.
  • The set of real numbers is a subset of the set of complex numbers.
The fact the set of natural numbers is infinite and a subset of the other sets I listed is proof enough each set is infinite, by definition of a subset. It is also trivial to show some of the sets contain elements some other sets don't (e.g. -1 is an integer but not a natural number), hence the sub in subset.

Therefore, this clearly demonstrates some infinite sets are "bigger" than others.

QED

If you want a proof of infinite primes, I can do that too.

A quick note on your even numbers example: @N Space is correct about cardinality, though countable infinities can seem a bit abstract, so I'll frame it a different way. If you consider the set of integers and the set of even numbers, you can define a one-to-one (bijective) mapping between the two sets (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.

I'll leave you with something to think about: What can we do with this knowledge that some infinite sets can be mapped to other infinite sets using a typical one-to-one function?
 
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To the best of my knowledge, Bertrand Russell is the only guy to have ever directly killed another guy solely using math (ok, technically it was logic/set theory, which are math adjacent, but still).

Anyone know of anything else like that? And no, making a conjecture that some autist works on until he starves to death doesn't count. Unless there's cool circumstances, I guess.
 
N Space said:
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.
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Yeah, I should have picked better examples. Those were dumb.
 
AnOminous said:
Aren't some infinities bigger than others?

The number of primes are infinite.

But so are the number of even numbers.

But how about the number of numbers divisible by three?

They're both an infinite set of numbers.

But one's smallest. And primes get increasingly sparse.

Mersenne primes are increasingly difficult to find, as an example. The number of Mersenne primes is also infinite. Or is it? There's no proof one way or the other. There should be. But are they?

Prove it or blow me.

Also if you are a mathematician, kill yourself, but not by normal suicide, just devote your life to the Collatz Conjecture, lmao, you'll wish you were dead.
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The trick with the hierarchies of infinity is to imagine them in terms of mappings, or labelling them.
Countable infinities are infinites you could take and enumerate. The set of integers for instance, you could go in order and give them a name. Therefore, all infinite sets of integers have the same cardinality. You can create a scheme to do the same for rational numbers. However, once you get to the reals, you end up in a situation where you can't create a system to enumerate them, and so it is a higher cardinality.
This is better argued with bijections and those autistic proof techniques, but that's how it be.
 
Agent Snigger said:
This is better argued with bijections and those autistic proof techniques
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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.

AnOminous said:
Yeah, I should have picked better examples. Those were dumb.
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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.

That's linear algebra.
 
Agent Snigger said:
However, once you get to the reals, you end up in a situation where you can't create a system to enumerate them, and so it is a higher cardinality.
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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.
 
AnOminous said:
Yeah, I should have picked better examples. Those were dumb.
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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.

*The canonical example being Vitali's Set, if you want to know how it's constructed and why you can't measure it, I recommend you watch this video, it's thorough and quite understandable IMO.
 
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AnOminous said:
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.
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Damn you are a renaissance man, did you study any math in undergrad by chance?
 
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