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 Mar 17 answered Find m for which $f(x)=\sqrt[3]{2x^2-mx+1}$ is differentiable on $\mathbb{R}$ Mar 15 comment $f = g$ a.e., $f$ is measurable, $g$ is not You might take the following proposition from this exercise: "If there exist $f$, $g$ that are equal almost everywhere and one of them is measurable while the other is not then the measure space is necessarily incomplete." Mar 15 comment $f = g$ a.e., $f$ is measurable, $g$ is not @Jake: I am not sure I understand your question. This doesn't affect the outcome, it is the outcome. The point of the exercise is to conclude that the functions are the same almost everywhere and that this is compatible with one being measurable and the other not and thus give you some intuition about incomplete measure spaces. Mar 14 accepted On surjectivity of exponential map for Lie groups Mar 14 answered $f = g$ a.e., $f$ is measurable, $g$ is not Mar 14 comment Difference between crossing and touching curves Regarding the rest of the OP, I think your construction works (of course, assuming we are working in some nice category of spaces and with reasonable notion of curves in those spaces), but am not at all sure what one gains by this viewpoint. In any case, what is your question? Do you just want to know how to rigorously prove this starting from some hypotheses on the space and the curves? Mar 14 comment Difference between crossing and touching curves "...loops small enough to split the space in two. (Right?)" You are thinking about a class of spaces that is locally contractible or some such (e.g. manifolds). It certainly need not hold for pathological spaces. Altogether, for the purposes of intersections, it seems to me it would be better to specialize this question to the category of manifolds where these notions are traditionally studied. Mar 14 answered Complex Variables…contour integral Mar 14 comment Complex Variables…contour integral Is $C$ supposed to be a circle around origin? Are $m, n$ integral? Mar 13 comment intersection form of $CP^2$ @kave: I see. I added additional explanation. Does it help or do you need still more details? Mar 13 revised intersection form of $CP^2$ Added a concrete explanation Mar 13 answered intersection form of $CP^2$ Mar 13 answered Summation using residues Mar 12 comment Summation using residues @joriki: if I understand the situation correctly, the standard residue method doesn't help with this problem. Is that a satisfactory answer? I was hoping someone would come up with a method that actually works and solve the problem. Mar 8 comment What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) @Dylan: many arguments involving $\pi$ are circular :) [couldn't resist bad pun, sorry..] Feb 27 comment staircase functions with compact support are dense in $L^1(\mathbb{R}^n)$ You can use essentially the same argument, just cover the support by small hypercubes. Feb 27 comment condition for roots of quartic equation to be purely imaginary When the roots are purely imaginary then $a_1$ is also purely imaginary and so cannot be "real constant different from zero". Something's wrong here... Feb 27 comment staircase functions with compact support are dense in $L^1(\mathbb{R}^n)$ I just realized that your question perhaps isn't about $L^1(\mathbb R)$ but general $L^1$ spaces. Nevertheless, one can use the compactness in much the same way to cover the support with finite number of sets on which $f$ is nearly constant. Feb 27 answered staircase functions with compact support are dense in $L^1(\mathbb{R}^n)$ Feb 18 comment Continuous surjective functions from the unit disk to itself that agree nowhere Please be more specific. Beyond superficial connection with isomorphisms of the unit disk I don't see any relevance of your answer to the OP.