Recent content by CantorSet

Hi everyone, I have a question on the discrete Fourier transform. I already know its a change of basis operator on C^N between the usual orthonormal basis and the "Fourier" basis, which are vectors consisting of powers of the N roots of unity. But if i recall correctly from complex...

Hi folks, The CRT says there's a unique solution to the system of congruences x = a (mod m) x = b (mod n) x = c (mod p) in (mod mnp) when m, n, p are pairwise relatively prime. But what if m, n, p are NOT pairwise relatively prime. Is there a systematic way to solve...

Got it. Thanks to everyone who responded on this thread. I bow before true masters of numbers. :shy:

That's very interesting. But how do we justify the step (2^{a}-1-(2^{r}-1)(2^{b}-1),2^{b}-1)=(2^{r}-1,2^{b}-1)

Thanks for the responses guys, those seem like really helpful hints. But I still could not proceed with the proof. =(

Hi everyone, this is not a homework question just a math puzzle I came across. Let a and b be any two natural numbers. And let (m,n) denote the GCD of m and n as usual. Prove (2^{a}-1,2^{b}-1) = 2^{(a,b)}-1 I'm thinking of double induction on a and b but I'm having trouble with the...

Thanks for the responses, guys.

Hi everyone, This is not a homework question but something I thought of while reading. In the method of maximum likelihood estimation, they're trying to maximize the likelihood function f(\vec{x}| \theta ) with respect to \theta. But shouldn't the likelihood function be defined as...

Hi everyone, This is not a homework question but a clarification on the following proof: Suppose h is an infinitely differentiable real-valued function defined on /Re such that h(1/n)=0 for all n \in N . Then prove h^{(k)}(0)=0 for all k \in . Proof: Since h is infinitely...

Thanks for responding, Stephen. Yea, that was my own confusion for making that assumption. Thanks for clearing that up. By the way, total least squares is just a generalization of linear regression in that the curve you're fitting the data points to can be polynomials with degrees higher...

Hi everyone, This is not a homework question. I just want to understand an aspect of linear regression better. The book "Applied Linear Models" by Kutchner et al, states that a linear regression model is of the form Y_i = B_0 + B_1 X_i + \epsilon_i where Y_i is the value of the...

Thanks for your help, Fleem.

Hi everyone, This is not a homework question but I question I have from reading a signals processing paper on acoustics. Suppose there is a sound source in a room S(t) and two microphones X_1(t) and X_2(t). Then the standard acoustic propagation model has that X_1(t) =...

Oh, I see... So we have \int_{-a}^a e^{2\pi i \lambda(t-d)}d\lambda = \frac{Sin(2a\pi (d-t)}{\pi(d-t)} So the function \frac{Sin(2a\pi (d-t)}{\pi(d-t)} achieves its max of \frac{2a}{\pi} when d = t . Thanks for the help. By the way, was there an easy way to see this...

Hi everyone, this is not a homework question but from my reading of a signals processing paper. This paper says if f(t) is the inverse Fourier transform of a function f(\lambda) = e^{-2i\pi\lambda d} then we can "easily see" that f(t) will have a peak d. Part of the issue here is...