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 Jul 29 comment Why does this expression equal $\pi$? @Charlie: indeed, that explanation is more straightforward but it kind of requires you to know the answer beforehand (i.e. to know derivatives of all the elementary functions). The second method is much more robust. One can e.g. use it to integrate $1 \over P(x)$ for any polynomial $P(x)$ (assuming it grows sufficiently quickly so that we can discard the boundary term coming the half-circle). This is because the first method gives us much more information than the second one: it allows us to integrate $\int_a^b {1 \over 1+x^2} dx$ for any $a,b \in \mathbb R$ not just the infinite ones. Jul 29 answered Why does this expression equal $\pi$? Jul 28 comment Dyson series and T product (II) @Carlos: you're welcome. As for the derivation, it's essentially nothing else than splitting a rectangle into two right-angled triangles. Does it make sense? Picture would go a long way here :/ Jul 28 answered Surprising Generalizations Jul 27 answered Dyson series and T product (II) Jul 26 comment Is there a way of working with the Zariski topology in terms of convergence/limits? So what would be the best way to think about the general topology? It's obviously a hugely useful tool but its general axioms seem to encompass much much more than what the theory originated from (i.e. geometry of spaces). Jul 26 comment How do people apply the Lebesgue integration theory? @paul: sure, I am aware of foundations of both measure theory and topology (and their interplay in Borel sigma algebras and Bourbaki approach to measure theory). I just thought that OP had only standard real analysis stuff in mind (where we exploit the topology also for Lebesgue integration). Rereading the question though, I am not sure what he meant anymore. Jul 25 comment How do people apply the Lebesgue integration theory? @paul: you mean there are people who call all of measure theory a Lebesgue theory? That would indeed be unfortunate. However, I've never heard anybody use term Lebesgue theory that way (I am mostly familiar with measure theory from probability theory literature though). Jul 25 comment How do people apply the Lebesgue integration theory? I don't understand this at all. Lebesgue theory is built on the Borel sigma algebra and so requires a topology. Or one can view it as a Haar measure on a topological group. Again, topology. In what sense is the Lebesgue theory minimal and Riemann integration isn't? Perhaps you were talking about general measure theory (although there is also an approach to this through continuous functionals) but how is that relevant to the question which is explicitly concerned about the real line then? Jul 25 comment Local Homeomorphism of the $S^2$ sphere to $R^2$ @wckronholm: well, if we are to prove anything we need to have some definition of $S^2$. As this is obviously an elementary question it's clear from the context that $S^2$ is understood as a subset of $\mathbb R^3$ (not to mention that OP talks about the stereographic projection) and not as a CW-complex with 1 0-cell and 1 2-cell (say); which requires some definition of spheres anyway... Jul 25 comment Local Homeomorphism of the $S^2$ sphere to $R^2$ @wckronholm: you can label all points of any subset of $\mathbb R^2$ with two real numbers but this surely doesn't assume anything about local homeomorphisms... Jul 25 comment Reference for Quantum groups @El: I've given you some references, try checking them out. I am not sure how much they'll satisfy your requirement of being for non-physicists. I am not aware of any physics oriented book that would be directed explicitly to mathematicians (say). Jul 25 answered Reference for Quantum groups Jul 25 comment Showing that $\int_{\mathbb{R}^2} \frac{df^2 }{dx^2} \frac{df^2 }{dy^2} dx dy = \int_{\mathbb{R}^2} \left( \frac{df^2 }{dx dy} \right)^2 dx dy$ Well, then just integrate by parts, there's nothing else to it... Of course, you need to assume that the surface terms go to zero and this gives you a condition on (derivatives of) $f$. Jul 25 awarded Enthusiast Jul 23 comment Advanced algebraic topology topics overview Thank you, these are all very useful. Incidently, MCAT's Introduction and references therein address precisely my question as well. Jul 23 comment Advanced algebraic topology topics overview Thank you! This is in a little different direction than the one I was hoping for but since I also have quite some interest in knots, Yang-Baxters, quantum groups and related guys I find this answer really useful. Jul 23 comment Reference for Quantum groups Are you more interested in the approach that builds the general theory or rather some nice applications (there are tons in physics alone) with theory mentioned along the way? Jul 23 asked Advanced algebraic topology topics overview Jul 23 accepted What can be said about the lattice of topologies on a given set?