backprojection

That doesn't sound so bad. I wonder if the process could be made to work with salt water.

his point is that bandwidth sets a lower limit on lag, for a fixed packet size. high bandwidth is a necessary condition, but not sufficient.

I kind of feel like we should be able to use scientific notation for this

300GB hardrive becomes 300 10^9 bytes or 3 10^10. 32GB or memory becomes 32 2^30, or 2^35.

No ambiguity at all, and if you're spending hundreds of dollars for hardware, you should at least be able to grasp exponentiation.

Risk of dying as a function of age, obviously.

Well your vote would be proportional to your investment, $100 out of $2.4M, in this case.

> That would basically mean potential acquisitions would become public knowledge

That would kind of be the point, it would be about fairness. People may not want to invest in a promising project that could change the world, just for it to be bought up by the next FB/Google.

What if, rather than offering equity, crowdfunders got voting rights instead. I feel that most of the outrage from the Oculus deal is that people feel betrayed. They could almost not have picked a worse outfit to have been bought by (rightly or wrongly, the point here is sentiment). If there had been a shareholder-esque vote, I think it's unlikely the deal would have been approved.

So maybe that could be the deal going forward - sure I'll put up $100 to fund your project, but that comes at the cost of you not selling out in the future.

EDIT: Clearly the weight of your vote would be proportional to your investment.

> What are some ways that such an investment could make sense?

My intuition is that patronage systems lead to undue influence. If Coke will fund your video series, they'll want something in return, usually advertisements. That then leads a subtle (or not) influence on the content.

I think the best way to avoid that ultimately is crowd funding on a more massive scale, i.e. 10s-100s of millions of dollar, but that comes with a whole bag of issues.

Are there materials dense enough to be useful for this, and thermally non-conductive/stable enough to withstand the heat of reentry?

> Not a new idea.

Who cares? Why do people always feel compelled to make this kind of comment?

Will there be pirate drones that fly around looting the Amazon drones?

I'm glad I'm not the only one. But how do you handle situation? My family usually takes offense, and I can see that I would seem arrogant for doing so. I just don't know how to communicate how intrusive and strident the TV is to me.

This is a really nice demo, but note that it's not the full Fourier series. This is only the 'sine' contribution. I'm not sure how you could expand the animation to include cosine as well.

I'm confused, why wouldn't they just tax whatever replaces Comcast? Maybe your argument is that the government has an overall interest in maintaining a monopoly, since monopolies raise prices, and under a fixed tax rate, that means more revenue for government.

[EDIT] As others have said below, this comment, while maybe useful, doesn't really apply to the article (or AD), since AD is about differentiating functions, not data.

> I find it funny how you think of differentiation in terms of "frequency domain", "bandlimit" and "filter". Very signal-processy, very engineery. A mathematician would speak about unbounded operators in a Banach space. :-)

This more or less depends on what kind of mathematician you're talking to. A mathematician working on Frames and signals almost certainly uses that language. But someone working in PDEs probably doesn't, even if the theory intersects a lot. It's probably a sign that there's nothing wrong with mathematicians and engineers sharing concepts :).

> It's not meant for differentiating a function known only by its points. It's meant for evaluating nth-order derivatives of functions you can already evaluate at arbitrary values

This is correct. So my comment "I think generally a better approach to AD is to redefine your differentiation", doesn't really apply to AD. Essentially, AD is about differentiating functions, not data.

This is pretty much exactly what I'm saying. So you're convolving with a gaussian kernel. That's the exact same thing as multiplying, in the frequency domain, by another gaussian. (convolution theorem, and the FT of a gaussian is a gaussian). The Gamma function filter goes one step further, it throws in a polynomial to make that gaussian 'flatter', which is more suitable for a low-pass filter.

Try the script at the end of this comment. Here's the output http://imgur.com/5948cW6

Notice that the 0-th one is just a gaussian. If you're interested, these things have traditionally been called HDAFs, I have a library to do this kind of thing https://bitbucket.org/nmaxwell/hdaf it may need more documentation. I don't mind if anyone provides constructive feedback on it.

The functionality you'd be interested in is hdaf.hdaf_periodic_samples, - you can convolve with that to low-pass filter, and use hertz_to_sigma to get the sigma parameter (from cutoff frequency).

  import numpy, scipy.special
  import matplotlib.pyplot as plt

  x=numpy.linspace(-2,5,1000)
  for n in [0,1,4,10]:
      s=1 if n==0 else n
      h=scipy.special.gammaincc(n+1,s*x**2)
      plt.plot(x, h)
  plt.savefig("filters")

EDIT: formatting.

I think it's worth noting that the problem with numerical differentiation, fundamentally, is that differentiation is an unbounded operator. In finite-differences, (the more obvious approach), you assume that your data are samples of some, general, function.

The problem then, is that that general functions have no (essential) bandlimit [1]. Remember that differentiation acts as a multiplication by a monomial, in the frequency domain [2]. Non-constant polynomials always eventually blow up away from 0, so in differentiating, you're multiplying a function by something that blows up, in the frequency domain. This means that, in the result, higher frequencies are going to dominate over lower frequencies, at a polynomial rate.

Let me be clear, the problem with numerical differentiation is not just that rounding errors accumulate, it's that differentiation is fundamentally unstable, and not something you want to apply to real-world data.

It depends very much on what your application is, however, I think generally a better approach to AD is to redefine your differentiation, by composing it with a low-pass filter. If designed properly, your low-pass filter will 'go to zero' faster (in the frequency-domain) than any polynomial, thus making this new operator bounded, and hence numerically more stable. It's not a panacea, but it begins to address the fundamental problem.

One example of such a filter is Gamma(n+1, n x^2)/Factorial[n], where Gamma is the incomplete gamma function [3].

In Python:

scipy.special.gammaincc(n+1,nx2) or mpmath.gammainc(n+1,nx2, regularized=True)

To see why this is a nice choice, notice item 2 in [4]. This filter is simply the product of exp(- x^2) (the Gaussian) multiplied by the first n-terms of the Taylor series of exp(+ x^2), (1/ the-Gaussian). Since this series converges unconditionally everywhere, as n-> +infinity, this filter converges to 1 for a fixed x (as you increase n), however, since it's still a gaussian times a polynomial, it always converges to 0 as you increase x, but fix n.

This is my area of research, so if anyone's interested I can give more details.

[1] https://en.wikipedia.org/wiki/Band-limit [2] https://en.wikipedia.org/wiki/Fourier_transform#Analysis_of_... [3] https://en.wikipedia.org/wiki/Incomplete_gamma_function [4] https://en.wikipedia.org/wiki/Incomplete_gamma_function#Spec...

? lobbyists are exceptionally well payed. Capitol hill is now a farm league for K street...

[1] https://en.wikipedia.org/wiki/Republic,_Lost

Well I guess we won't know until they release part numbers on the CPU, and that part number is available form Newegg/Amazon. It's not clear to me if they're going Haswell or Ivy-Bridge. The I-B E5 Xeons on Newegg now are all 6+ cores and very expensive (into the thousands of $s). The closest I can find are Haswell quad core (the new Mac Pro is 4-core) and are much cheaper: http://www.newegg.com/Product/Product.aspx?Item=N82E16819116...

> I would be interested in seeing the cost of a truly equivalent generic PC build. ("Truly" equivalent meaning no non-Xeon CPUs and other "almost as good" parts, something that's all too commonly done with Mac Pro comparisons.

I agree with that. We'll have to see.

Well, to be fair, it's more a comment on Intel's progress than Apple's. I'm sure you can throw together an ugly generic PC with the same parts for less. Anyway, workstations are supposed to look big and intimidating :)

I'm surprised this isn't even mentioned in the article. I guess there are issues with boundary effects, but those can be overcome in a variety of ways.

Also, if your convolution kernel is short enough, you can beat an FFT. So, in 2D with an NxN image, an FFT filter will have a complexity of about (N^2 + N^2 Log N). If your kernel is KxK, then direct convolution will be (N^2 K^2), so you have to compare Log N with K^2. But keep in mind 1.) these are asymptotic complexities the coefficients matter 2.) K may or may not depend on N 3.) FFTs are may have larger memory requirements, although you may be able to get away with in-place FFTs, and you don't have to explicitly compute the FFT of the gaussian - you can work that out analytically (you should get another gaussian...).

Also, as someone pointed out, the gaussian is separable, so then the complexity of the direct convolution is (K N^2).

Anyway, what are FFT libraries like for iOS? I suppose with Apples policies, you can't compile FFTW for your App?

I don't know if my perspective is just so heavily biased or impractical at this point, but I'm almost embarrassed that my country's Navy would use windows for such a critical system.

> Does the tesla have regenerative braking system, and I wonder if there are rare earth metals there?

As far as I understand it, they use the motors for regenerative breaking, and the motors don't contain permanent magnets. Where permanent magnets would be used, they use more coils, and thus can vary the strength of the magnetic fields, much like your car's alternator. This is also how they forgo the need for a gear box.

This is interesting since I'm sure the feds want cooperation from Amazon, and hence Bezos. If Bezos is behind it, then that might make some sense - NSA forces Bezos to comply with whatever they want, and this is his way of fighting back.

Is this just speculation? It's fine if it is, I'd just like to know.

This is great, thanks!

Someone posted ambient warp-core sound from Star Trek on Reddit earlier today, maybe you could add this.

http://www.reddit.com/r/startrek/comments/1nwzd6/ambient_war...

There's still an incentive to work with a negative income tax. This is an interesting discussion about it:

https://www.youtube.com/watch?v=xtpgkX588nM

In case anyone didn't catch it, Stephen Fry extolling GNU and free software:

https://gnu.org/fry/happy-birthday-to-gnu.html

This man is a wonderful person.