Recent content by chogg

Hah! Good point.

FWIW, one easy way to generate a random point on the sphere is to take a random draw from a standard 3D Gaussian, and normalize it. The 3D Gaussian is isotropic; hence, all directions are equally likely. The only time it doesn't work is if you get the origin. Technically, that's the single...

How many datapoints?

In geometric calculus, div(f) is perfectly well-defined for a scalar function f: it is zero everywhere. See Macdonald's excellent text for more details. Of course, your main point (that div(grad(f)) is not the same as grad(div(f))) is spot-on.

That helps me understand. I was confused because Gaussians have continuous output and no boundaries, but your system is discrete and bounded. I thought the reverse-biased zener diode had Poisson noise, not Gaussian? But presumably in the range you're measuring a Gaussian would be a good...

Ah, but since time passes more slowly for the spaceship's occupants, they can indeed reach distant systems in a reasonable time -- for them! From Earth's point of view, they are traveling near the speed of light for the amount of Earth-time which passes -- say, 50 years, or whatever you want...

The idea that mass increases at high velocities is outmoded. Certainly the passengers on the ship would not feel any heavier! There are no "nasty problems of increasing mass". For the plot point you want, warp drives don't matter. Just a regular, fast ship gets you the time difference...

Well, your scenario violates physics, so I can't answer your question directly. :-) You can't go 10x the speed of light. You can't even go 1x the speed of light, but any number less than 1 is possible in principle. Still, I can explain a scenario where time passes more slowly. There are...

Thinking more about it: do you by any chance have a binomial distribution? If so, then forget my question about standard deviation; your distribution is already determined: (n, p) = (1024, 0.5). If your data don't look like this, I guess you could fit them to a beta-binomial, or some other...

You can't have a normal distribution between 0 and 1024 (not to mention a normal distribution which gives integers!). It could be approximately normal, though. What's the standard deviation?

I disagree that vector division does not give a unique answer. It does when that division is with respect to the geometric product. And this division works in arbitrarily many dimensions. A vector divided by another vector is a complex number, even in arbitrarily many dimensions. This...

I guess that's one way to look at it, but I don't think it's the most natural. You get quaternions from an orthonormal basis in 3D space, just as you get complex numbers from an orthonormal basis in 2D space. The "extra dimension" is really just a plain vanilla scalar. (The three basis...

Yes, there is a division for 2-d vectors (indeed, arbitrary-D vectors) which does have geometric meaning. It requires the geometric product, which unifies the dot product and the wedge product (the latter is somewhat like the cross product, but more general). tl;dr: Pictures would help, I...

Yes there are; they are usually called "quaternions". Reading about geometric algebra will clarify the relationships among vector algebra, complex numbers, quaternions, and many other mathematical systems. I'd recommend the first two chapters of Geometric Algebra for Physicists for some...

All you need for a basis is linear independence. If you also have orthogonality -- i.e., if e_i \cdot e_j = 0 when i \ne j -- that can be convenient, but it's not necessary. The dual basis is the tool we use to get this same convenience (zero dot products) with an arbitrary basis. The main...