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Quick Questions: June 01, 2022

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ELI maths enthusiast: I'm a bit hazy about the link between provability/decidability of a proposition, and existence of models. Does independence imply the existence of models going "either way"?

* The parallel postulate is independent from the other axioms of Euclidean Geometry. That means we cannot formally deduce the parallel postulate nor its negation from the other axioms, correct? And now crossing the river to the "existence of models" side, there are indeed models of Euclidean geometry (i.e. structures satisfying or "implementing" the axioms) which satisfy the other axioms plus the PP, and models which satisfy the other axioms plus the negation of PP.

* The axiom of choice is independent from ZF. That means that neither it nor its negation can be proven from ZF. Is the same about models true here? Are there models of set theory satisfying ZFC and other models satisfying ZF and the negation of C?

* Goodstein's theorem is independent from PA. Does that mean there are models of PA where Goodstein's theorem is true and others where it's false?

* If yes to the last question, what on earth do those models of PA where Goodstein's theorem fails look like? (Is this where these "nonstandard models of arithmetic" come into play?) If so, what does a sequence violating Goodstein's theorem actually look like? How can such a sequence fail to end up at 0?
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Recommendations for a good book on dynamical systems?
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Literally just bored. Not super mathematical and dont have to use it often anymore. I took up to multivariable Calc and Difeq in college but most of it is gone from my mind.

So if you had a function e\^x could you get the exact same curve with a polynomial, or is that impossible? I remember doing taylor series but can a Taylor series actually ever match e\^x exactly, like most of the examples I looked at even out to 45 is obviously different than E\^x.

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Thanks!
by
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(hope this comment show up this time)

Can someone explain the concept of de Rham-Witt complex to me? The only source with details I know is in French, and English sources lack too much details that I'm confused.
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I'm studying electrical-electronics eng., aside from generic engineering calculus and differential eqs. i also would like to learn more about theoric math like graph theory (i also have a basic knowledge about this due to circuit theory), network theory or if there is a advanced linear algebra.

But tbh, i don't really know where to start or which fields i should try to learn in math or which fields exists or how are they interconnected in general. So how do i start?
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What's a good introduction to measures on manifolds that doesn't require too much differential geometry? Doesn't have to be much, one or two chapters of some textbook will probably be enough.
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What are prerequisites for stochastic calculus(ie shreve) and what are some book recommendations. Also is the book Basic Stochastic Processes by brzezniak a book on stochastic calc? Thnx in advance.
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Is there a notion of holder conjugates of Lp spaces when 0<p<1?
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Somehow struggling to see whether the following is true: Let A and B be symmetric and positive definite with x^T A x >= x^T B x for all x. Is it necessarily true that x^T A^-1 x <= x^T B^-1 x ?

There are a few special cases in which it is easy to see that this is true, for example if A and B are diagonalizable with respect to the same eigenbasis or if the smallest eigenvalue of A is larger than the largest eigenvalue of B, but these two conditions don't need to be fulfilled.

Are there any conditions on A and B equivalent to x^T A^-1 x <= x^T B^-1 x ?
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Will Khan Academy teach me all the multi calc (Calc 3, say) I need to know as a maths graduate? My uni did not teach me enough, and now that I've embarked on the long journey (through Tu's book) to learning generalised Stokes, I've decided enough is enough (I also need it for electromagnetism), and I wondered if Khan Academy was considered sufficiently comprehensive.

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