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?- love(math) is unrequited. true.


Apr
29
comment Probability and Axiom of Choice
You should read this paper and the Freiling paper mentioned in it.
Apr
25
comment Fourier transform probability distribution
It does seem from Box-Muller that you'd be computing a weighted combination of normal components, and so the result ought to be normal.
Apr
24
comment Undergrad Student Trying to Figure Out What to Study
Of course you should give weight to your preferences. I am merely suggesting that most people give too much weight to what they currently enjoy doing, and not enough weight to the real externalities involved. Obviously, do not do something you don't enjoy. But surely most people enjoy more than one thing. I mostly think that when someone says, "I do what I love", they haven't thought about it very much. Surely you don't love just one vocation. It's a fallacy that is unique to high standard-of-living countries that we teach our young to focus almost exclusively on imagined preferences.
Apr
23
answered Undergrad Student Trying to Figure Out What to Study
Apr
22
comment Efficient principal pivots
You definitely want to ask at scicomp.stackexchange.com. Since this is mostly an implementational / speed issue, they can probably help a lot more.
Apr
22
comment Weird conclusion about variance/covariance from differentiating
In the sense that it minimized portfolio volatility.
Apr
22
revised Cluster Analysis Terminology question
edited body
Apr
21
comment Cluster Analysis Terminology question
In the applied machine learning literature, 'hypergraph' is the appropriate term for most applications, and it is meant in the full mathematical sense. You see that in everything from stats papers to computer vision to social networks. See my answer below.
Apr
21
answered Cluster Analysis Terminology question
Apr
21
answered Solving $Ax = B$ when $A$ has a large condition number.
Apr
21
answered Weird conclusion about variance/covariance from differentiating
Apr
21
comment Weird conclusion about variance/covariance from differentiating
I think something is being lost in translation in the OP. The question should be asking: suppose $a$ is chosen to optimize portfolio risk between $X$ and $Y$, and is chosen such that $\text{COV}(X,Y) = a\text{Var}(X)$. Then show that $a$ must be 1.
Apr
18
answered Books for Understanding Bayesian probability from the Beginning
Apr
18
comment Can't see how $e^{\operatorname{Log}(z)} = z$ in these notes
What is $Re[z]$ and $Im[z]$ in terms of $|z|$ and the angle $z$ makes with the positive real axis? It's just polar coordinates, basically.
Apr
16
comment How can Radon-Nikodym and Borel-Cantelli be used to calculate Probability distribution?
The paragraph on the Radon-Nikodym connection makes sense to me. After all, that's the precise way in which the Cantor function (which is a valid CDF) is shown to have no PDF.
Apr
15
comment How to expand undifferential function as power series?
Check out the basics linked here. Your question was something that spurred decades of math research and resulted in radically changing the notion of a function. Fourier series is one method to get some kinds of convergence properties for series expansions of non-differentiable functions, when the points causing non-differentiability are sufficiently well-behaved.
Apr
15
comment How to expand undifferential function as power series?
Re-derive Fourier series?
Apr
15
comment Why bother with Mathematics, if Gödel's Incompleteness Theorem is true?
@CarlMummert I totally agree. If you follow my link to the answer to the other question, you'll see a long passage that discusses a rebuttal to Penrose's ideas. Actually, Penrose took that idea from Lucas, who proposed it much earlier, I think in the 60s, without all of the quantum gravity hoopla.
Apr
15
comment Why bother with Mathematics, if Gödel's Incompleteness Theorem is true?
@Didier, I agree with you 100%. That component of this question should be isolated from question that the title of the post asks. Godel himself asked the question that is in the title and the only answered that seemed to satisfy him was "well, let's hope $P\neq{NP}$". But the stuff about approximating absolute truth, etc., is not appropriate for Godel's theorems. That's a whole different philosophical beast and the analogy with Newtonian-to-Quantum physics isn't a good one given the state of the art in the philosophy of truth.
Apr
14
comment What philosophical consequence of Goedel's incompleteness theorems?
Be closed-minded about it if you want to, but the bulk of the uses of Godel's theorems in philosophy come from mathematicians, especially decision theorists. It's a non-trivial part of the philosophy of mathematics. You are certainly free to disagree with it if you want, but it's nothing but disingenuous to claim that there are virtually no philosophical ramifications of Godel's theorem.