@TedShifrin and what if we consider any other measure? let say $\mu$ a generic measure, if instead of $\int u d\mu$ I write $\int u \mu(x) dx$ what does $\mu(x)$ represents inside the integral?
I am actually doing this for personal pleasure (?) and hence I don't have a "required textbook" though my main reference is "Measures, Integrals and Martingales"
@RScrlli May I suggest Folland's Real Analysis for a nice exposition on Measure Theory as well as Stein and Shakarchi's Book 3 Real Analysis. Both have served me very well
@NicholasRoberts I will definitely check them out, I started reading the last chapters (on measurement theory) from Baby Rudin, but I was advised not to
@NicholasRoberts yes actually I am Ms. in Economics
@RScrlli The second book I mentioned, Stein and Shakarchi, is more forgiving than Folland. I see it as a nice bridge between Undergrad and Graduate analysis. Folland is definitely a graduate text.
very nice @RScrlli ! Depending on your background in analysis, Stein may be good for you to look at.
@TedShifrin random question, but do you know any good Functional Analysis books? My only background in the subject is Chapter 5 in Folland (which is very brief) and I'm interested in going deeper in it.
@TedShifrin It's possible to take multivariable calculus course in high school itself. I'll be taking this November, after the single variable analysis ends at the school
@TedShifrin Right now, I need some good stuff on tensor analysis, can you recommend one?
@TedShifrin um? uh? I needed to learn about GR in physics, needed to learn tensors... Then, I'll attempt for QFT, if everything goes alright, right before december.
@TedShifrin I have got a solid foundation of Multivariable calculus, at least required for high school, just don't have a rigorous idea of multivariable analysis for it. (not rigorous, just that much which is required for physics)
Tensors are generalizations of matrices, so you actually need to understand some linear algebra and, in particular, the Jacobian matrix in multivariable calculus.
I am trying to show that the center of $GL_n(\Bbb{R})$ is $\Bbb{R}^\times I_n$. If $A$ is in the center, then $AB = BA$ or $B^{-1}AB = A$ for every $B \in GL_n(\Bbb{R})$. My thought was to take $B$ to be some combination of elementary matrices such that $B^{-1}AB$ is a constant diagonal matrix, but I'm not sure how to make that more precise.
A family of positive convex functions $C_{\lambda}:\mathbb{R}^N\to\mathbb{R}$, with parameter $\lambda \in (0,\infty)$. Let $x_{\lambda}$ denote the unique minimizer of $C_{\lambda}(x)$ over $x \in \mathbb{R}^N$. Another positive convex function $ D : \mathbb{R}^N \to \mathbb{R}$ and a real value...
Give an irrational number, we can choose an interval small enough around it such that every point in that interval has the same decimal expansion up to some $n$th place right? There's a theorem (the name of which I can't recall right now), which says that each term of the continued fraction gives roughly ~1 decimal precision, and the worst case is generally with the golden ratio which for which we need something ~2.3 terms of the expansion to get an extra decimal digit of precision.
This would suggest what I said above is true, but we'd probably need uniform lower bounds on the number of terms required to get extra precision.
Hi there, I'm trying to sketch the boundary $\delta M$ where $M$ is a subset in $\mathbb{R}^2$ consisting of all points $(x,y)$ such that $\frac{y^2-x^4}{x^2+y^2-1}>0$. How does one evaluate the inequality?
Just a regular continued fraction, representing some irrational number in $[0,1]$. Check out this website: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfCALC.html
Type in 0.61 in the decimal section and look at the continued fraction, then type in 0.618 and check it again
the idea stills seems more or less correct, the number of $1$'s will increase as we get closer to the $0.618033 \ldots$, but it's not completely correct that adding extra decimal digits only extends the continued fraction
@Semiclassical Could you show where the equation of the circle is hidden in the inequality? I kind of see it in the denominator, but as a reciprocal and with -1 attached to it.
take a continued fraction and remove the last number ==> you will likely stay in this 1 decimal of difference
the way the continued fraction are build is tricky and some early number will change in the continued fraction representation when you add decimal to the usual representation
I can't rely on a probabilistic/average result though, I need to guarantee that at some decimal precision, we obtain some number of terms of the continued fraction that agree. I'm willing to require huge decimal precision for even a few terms of the continued fraction to agree.
Maybe what I'm saying turns out to be false but I certainly hope not
Given $n$ and irrational $x \in [0,1]$, does there exist an $\epsilon$ such that for all $|x-y| < \epsilon$, the first $n$ terms of the continued fraction expansions of $x$ and $y$ coincide?
from wikipedia : Moreover, every irrational number α {\displaystyle \alpha } \alpha is the value of a unique infinite continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values α {\displaystyle \alpha } \alpha and 1
@rapasite I don't think this follows that directly from uniqueness, but I'll spend some time right now thinking about it and I'll come back with observations/comments.
Ah sorry you emphasized the word UNIQUE I thought you were implying it followed from that. I see now that the algorithm for computing them can give a lot of insight, but it's still not that easy I think. At each step of the algorithm we want to get the integer part of performing some division, so the question translates to how these division are affected by increasing # of decimal digits
but yes thank you for pointing that out, the algorithm seems helpful
