# kigen

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bio website math.rutgers.edu location New Jersey age member for 1 year, 4 months seen 2 days ago profile views 301

I'm an undergraduate at Rutgers University. As of now I am interested in analysis, especially parts of analysis with strong interactions with topology, but also analysis as a whole. I also have some interest in algebraic topology, as well as problems arising from physics.

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 Apr15 awarded Necromancer Apr13 comment Borel-Cantelli Lemma “Corollary” in Royden and Fitzpatrick Well, just think about what $E^C$ must be: if $E = \{x\in\mathbb{R}^n: \exists~\text{infinitely many}~E_k~\text{such that}~x\in E_k\}$, then just negate the appropriate quantifiers and you get exactly what I claimed. Apr6 comment Measure Theory - Convergence of functions with bounded integrals Why exactly is that obviously false? $L^p(X)$ is a Banach space, if a sequence converges in norm to $0$ then the sequence converges to $0$... Apr6 comment Measure Theory - Convergence of functions with bounded integrals ...$\sup_K |f_n(x)|^2 \to 0$ means $f_n(x)\to 0$ on $K$, so if $f_n(x) \to 0$ on all $X_j$ then $f_n(x)\to0$ on $\cup_j X_j$. Apr6 comment Measure Theory - Convergence of functions with bounded integrals Note that the property you have shown ($\int_K f~d\mu \leq \mu(K)\sup_K f(x)$) holds for any $K$ of finite measure. So $\sup_K |f_n(x)|^2 \to 0$ on any $K$ of finite measure. If $X$ is a union of sets of finite measure, then... Apr6 comment Measure Theory - Convergence of functions with bounded integrals Now use that $X$ is a countable union of sets of finite measure... Apr6 comment Is this counter-intuitive result actually correct? Interesting corollary: take a sphere and cut it up into 4 pieces by making a cut along the equator, then 2 cuts on the latitudes halfway between the equator and each pole. Which piece has the largest area? Ans: the areas are all equal. (My professor posed this question to introduce toric varieties at an MAA sectional meeting.) Apr6 comment Measure Theory - Convergence of functions with bounded integrals If $X$ has finite measure, can you show $\lim |f_n|^2 = 0$ from what you have already? (Ans: yes.) Now how can you use $\sigma$-finiteness to complete the proof? Apr6 comment Estimation of $\pi$ using dice On a more serious note, you could probably roll enough dice in trials to simulate a randomc variable whose distribution is approximately normal, then use something like the Gaussian integral. Apr6 comment Estimation of $\pi$ using dice @Doop Thanks, corrected. Apr6 comment Measure Theory - Convergence of functions with bounded integrals Hint: you only need $\lim |f_n|^2$ to exist. Note that you haven't used $X$ $\sigma$-finite yet. Apr6 comment Estimation of $\pi$ using dice Arrange 100 dice in a circle, find how many dice are needed to make a line from the center to the edge, then divide 100 by $2$ times this number. Apr6 comment Limit Proof Question This proof is pretty direct, especially considering that division cannot be defined without a prior definition of multiplication (algebraically speaking). What do you have in mind exactly? Apr6 comment For Riemann sums involving square roots, why do we let $c_{i} = \frac{i^{2}}{ n^{2}}$? It doesn't matter what the size of the mesh is, as long as it goes to $0$. Using $i^2/n^2$ simply makes calculating $f(c_i)$ a lot easier when $f(x) = \sqrt{x}$. Same deal for $f(x) = \sqrt[3]{x}$. Apr6 comment Simple Grammar Question @user139388 Uncapitalized. "lemmata" (or "lemmas") in that context is not a proper noun. You need to give a name (typically a number, a label in any case) to your lemmas/lemmata to make reference to distinct ones and to use them as proper nouns. Apr6 comment Possible (trivial) error in Folland's “Real Analysis”? Irrelevant, perhaps, but I'll say it anyway: looking at the proof, the precise constant in front of the $\epsilon$ is unimportant. Apr6 comment Simple Grammar Question @user139388 The convention there should be to capitalize "universities," but for a different reason: uncapitalized, it is ambiguous whether the phrase refers to all universities within Atlantis and Valhalla, or just the pair. Unfortunately having a possessive in the phrase ruins the exact analogy. "Lemmas 1 and 2" or "Lemmata 1 and 2" are not so ambiguous. Apr6 answered General Vector Space: Change of basis Apr6 revised Measure Theory - Convergence of functions with bounded integrals added 263 characters in body Apr6 answered Measure Theory - Convergence of functions with bounded integrals