# Paul Trow

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1. ## Mathematics of Traffic

Take a look at http://www.math.toronto.edu/mathnet/carcompet/carcompet.html
2. ## Kepler's 1st Law

On the contrary, the ellipse does have a second focus - there's just nothing there, as Ophiolite pointed out.
3. ## Maximal and maximum subgroups of semigroups

There is a difference between maximal and maximum. A subgroup is maximal if there is no proper subgroup that contains it. I've rarely heard the term maximum applied to a subgroup, but what it should mean is that every other subgroup is contained in it. For example, in Z8, the subgroup {0, 2, 4, 6} is a "maximum subgroup." This is the same as the meaning of maximum applied to sets of real numbers: the maximum of the set [0, 3] is 3, because every other element of the set is less than it. Note that a group might not have a maximum subgroup - for example, Z2 X Z2 - just as the set (0,3) does not have a maximum. (However, any group does have maximal subgroups.) http://home.comcast.net/~paultrow/writing.htm
4. ## Gravitational force

By analogy, think of the intensity of light, which varies inversely with the square of the distance from the source. If you're one meter from the source and a little patch of light hits your eye, the intensity of the light is basically the number of photons in the little patch divided by its area. But if you're two meters from the source, the area of the same little patch is four times as great, because the surface area of a sphere is proportional to the square of the radius. Since the number of photons in the patch is the same, the density of light at two meters is one fourth what it is at one meter. Gravity works the same way - its "intensity" is inversely proportional to the distince from the source. http://home.comcast.net/~paultrow/writing.htm
5. ## Approaching c

Actually, mass doesn't "increase" - it's just measured differently depending on your frame of reference. If observer A sees you travelling at a constant velocity, he will measure your mass as being greater than you will measure your own mass. And since, according to relativity, you can just as well regard yourself as being at rest, and A as moving at a constant velocity in the opposite direction, you will measure his mass as greater than he will measure his own mass. In other words, mass - like length and time - is a relativistic quantity.
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