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location Albany, CA
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visits member for 2 years, 10 months
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If I see further than most, it is because I have stood on the toes of giants and they kicked me high into the air...

However, there is nothing to suggest that I see further than most.

I am (inasmuch as one 'is' an occupation) an engineer. My mathematical skills are rather pedestrian, with a rare insight every now and then. I have been lucky enough to sit in the same room with some famous mathematicians & engineers. My Erdős number is 5, which undoubtedly makes me as unique as an Irishman in a pub. And I have been spotted in The Pub from time to time.

In professional circumstances, my value add has usually been the injection of common sense and completion of grunt work when perspectives and energies are unfocused.

My current age is the smallest number composed of the first two primes that would allow me to have watch on RTÉ the slightly contradictory small step for man and giant leap for mankind.

My real name is Joe Higgins. Apparently, my last name means 'of the Viking' in Irish.

My alma mater is University College, Cork in Ireland, and I had the additional privilege of obtaining my Phd. from the University of California at Berkeley under the enlightened tutelage of Lucien Polak.

I can be reached at joe dot higgins at gmail dot com.


2h
comment Is the Law of Large Numbers empirically proven?
This is the weak law. The Wikipedia describes the strong law.
2h
answered Is the Law of Large Numbers empirically proven?
5h
comment Determine the set of points that satisfy $Re\left(\frac{z-z_1}{z-z_2}\right) =0$ for $z_1,z_2$ are fixed
If you treat $a,b$ in my comment above as pairs then the condition is equivalent to $\operatorname{re}a \operatorname{re}b +\operatorname{im}a \operatorname{im}b =0$ which shows them as orthogonal in $\mathbb{R}^2$. Draw a picture and use Thales' theorem.
6h
comment Is it possible to have simultaneously $\int_I(f(x)-\text{sin} x)^2 dx\leq \frac{4}{9}$ and $\int_I(f(x)-\text{cos} x)^2 dx\leq \frac{1}{9}$?
I am truly curious as to why this was downvoted?
6h
comment How to use matlab for plotting functions that contain summations?
@A.Donda: I am only familiar with using sum for adding the elements of an array, what were you thinking? (Aside: How did you highlight sum?)
12h
comment Closed vector space and a subspace of a vector space
I don't understand the question. How do you define closed, open and $\oplus$?
14h
comment Determine the set of points that satisfy $Re\left(\frac{z-z_1}{z-z_2}\right) =0$ for $z_1,z_2$ are fixed
My $a,b$ are not related to yours. It means that $a,b$ are perpendicular (in $\mathbb{R}^2$). I don't know what you mean by dish. My point here is that a little geometry can help...
14h
comment Determine the set of points that satisfy $Re\left(\frac{z-z_1}{z-z_2}\right) =0$ for $z_1,z_2$ are fixed
Think of $z_2-z_1$ as a diameter of a circle. What does $\operatorname{re} {a \over b} = 0$ mean about the 'directions' of $a,b$?
14h
comment Fourier transform all steps walkthrough for wave vector $k$ and $x$
You might want to give a quick initial summary of what you want as that is a lot to wade through...
15h
comment Why $f(z)=z^2$ is single valued?
It is a function, the same as $z \mapsto z\cdot z$. How can it be multiply valued?
15h
comment Fourier spectrum reflected across origin and Nyquist frequency
You can express the Poisson summation formula using the $\cos,\sin$ formulation, but it will end up giving the same sufficient condition.
15h
comment Fourier spectrum reflected across origin and Nyquist frequency
Note that when we talk about the bandwidth of a signal as you have above, it really refers to half of the spectrum of the signal. Any real signal has $\overline{\hat{f}(\omega)} = \hat{f}(-\omega)$, so the Nyquist criterion gives a sufficient condition to avoid overlap.
15h
comment Fourier spectrum reflected across origin and Nyquist frequency
As an aside, I think this is a good question as it brings up much of the loose thinking that is taught in signal processing courses. The underlying concept is the Poisson summation formula which (loosely) shows that the spectrum of a uniformly sampled signal is the sum of the spectra of the original signal shifted by ${k \over T}$, where $T$ is the sampling interval.
15h
comment Fourier spectrum reflected across origin and Nyquist frequency
I don't have the time to properly address the issues you bring up (and there are many). The main one is that the method of analysis doesn't affect the signal, of course, so you can't magically make the negative frequencies go away. In radio broadcast, one can reduce the bandspread of a signal around the carrier by using a technique called single sideband transmission, but with this technique you are changing the signal.
16h
comment Union and Intersection of sets proofs
Glad to be able to help. You will become familiar with the techniques over time.
16h
comment Fourier spectrum reflected across origin and Nyquist frequency
Well, that was my point above. Either you have some $\omega \mapsto \hat{f}(\omega)$ defined on the reals (that is, including the negative axis) or you have $\omega \mapsto \hat{f}_c(\omega)$, $\omega \mapsto \hat{f}_s(\omega)$ defined on the non negative half line. You have twice the content on half the line. Nothing has disappeared. (I'm talking about real valued 'time domain' signals here.)
16h
comment Union and Intersection of sets proofs
I added an answer below outlining steps. This is one way of doing it.
16h
answered Union and Intersection of sets proofs
16h
comment Union and Intersection of sets proofs
No, but it will help show you what is going on.
16h
comment Union and Intersection of sets proofs
Draw a Venn diagram.