Mr. F
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 Apr3 comment Using Homotopy to solve system of nonlinear equations Just as an aside, there's a whole theory of convex optimization to consider. You can use interior point methods, semi-Newton methods, gradient methods, or even basic line search methods to find roots. I'm not suggesting that your interest in homotopy methods is bad, but you'll probably be much better served by considering the half-century old ideas on root-finding with linear algebra before going off to homotopy. Apr3 revised Searching for numerical algorithm realization added 589 characters in body Apr3 revised Searching for numerical algorithm realization added 589 characters in body Apr3 revised Searching for numerical algorithm realization added 589 characters in body Apr3 revised Searching for numerical algorithm realization added 589 characters in body Apr3 comment Show/Prove two conditions when $P$ is an $n \times n$ matrix such that $P^2 = P$ and $P^t = P$. Hint for the hint: What is $P(x-Px)$? Apr2 answered Searching for numerical algorithm realization Apr2 comment Making bounded an unbounded integral Alternatively, it might be possible to pick any strictly-increasing, concave function $g(x)$ that is integrable w.r.t. $p(x)$, and then compose it with $F^{-1}(x)$ to get $f(x) = g(F^{-1}(F(x))) = g(x)$. I'm not sure under what conditions $g(F^{-1}(y))$ will also be strictly-increasing and concave though. Apr2 comment How can we bound $P\{X \ge t\}$ from below? @J.D. Chebyshev's inequality is still an upper bound on probability (for a non-negative variable). It is where Markov's inequality is derived from, so it's likely that the poster already knew of this bound regardless. I suppose one can expand the absolute value usually used in it into two events, then use complements and multiply by -1 to get a statement of lower bounded probability, but it won't be in terms of a useful bound as it will always involve the other probability of the other "half-event" that you split out of the absolute value. Mar30 comment How can I calculate the CDF of this random variable? When I do the computation by hand, I am able to easily perform the integral over the variable $x_{2}$, and then the resulting integral over $x_{3}$ is of the form $\int_{0}^{\infty}\frac{\alpha}{\epsilon + \beta x_{3}}\cdot{}\exp{(\frac{-x_{3}}{\Omega_{3}})}$, for constants $\alpha$, $\epsilon$, and $\beta$. According to Wolfram integrator, an integral of this type requires the Exponential Integral function Ei() to express the solution. Mar27 awarded Tumbleweed Mar26 comment An example of a “pathological” power-spectral density function? I don't like the 'avoid extended discussions in comments' rule, so I abstain from participating in chat-migrated things. I think it's most helpful if everything appears right here on the same page. But I will think much more about this and I'll post any follow-up result as an answer to avoid making the comments thread too long. Mar26 comment An example of a “pathological” power-spectral density function? To clarify, you can construct functions with properties like $x^{2}\sin(1/x^{2})$ that have the symmetry and derivative properties you wanted. But the fact that PSD functions come specifically as Fourier transforms restricts what kinds of functions they can be. So it's a harder analysis problem to prove that you could't concoct a weird trig function with a singularity in its argument such that the singularity blows up for higher powers. Mar26 comment An example of a “pathological” power-spectral density function? This makes the assumption that it is Riemann integrable, but there are surely many pathological functions that should be considered here which aren't. Things like $x^{2}\sin(1/x^{2})$ and the sort. Many of these satisfy the properties of a PSD, but I haven't been able to find one yet where raising it to higher powers makes it less integrable. The fact that we can assume the function has no singularities makes it seem like you are correct, but I can't quite convince myself that it's airtight. Mar26 awarded Scholar Mar26 comment Computing the derivative of a quadratic form and matrix chain rule Thank you for the clarifying remarks. After I had gone back and did some debugging, I found a different reason why the code wasn't converging. In the end, with the derivatives I had calculated, I did get it to converge. Mar26 accepted Computing the derivative of a quadratic form and matrix chain rule Mar26 comment An example of a “pathological” power-spectral density function? As an aside, I used to work as a radar engineer, and once an analyst kept grumbling about why his radar simulation was giving him nonsense $\infty$ values. I looked at his code and it was because he was using a non-integrable function for the antenna gain pattern (hence infinite energy radar). When I tried to explain this, he got angry that I was using the word "integrable" (a concept he apparently thought he could forget about after college). Mar26 comment An example of a “pathological” power-spectral density function? Consider the function $\frac{1}{\sqrt{|f|}}$. The singularity is just barely integrable, but if you square it, then it's no longer integrable. Really, you're looking for function that are in $L^{k}(\mathbb{R})$ for some \$k