mathematic

It loos like an ordinary (not partial) diff. eq. in r. Which symbols represent functions of r?

Light bulb is ordinary (old fashioned) incandescent. House is private home - no one else to handle circuit breaker. Filament breaking and reconnect seems only possible, even if improbable, explanation.

Late night thunderstorm - lightning appeared to hit less than a few hundred feet from house. Immediate effect - light in hall (ordinary bulb dimmed) went out, although electric power was still on (electric clock still working). Ten minutes later light in hall comes back on. How can an ordinary light bulb behave this way?

Your question needs elaboration.

Try Amazon.

I presume there is something which prevents the weight from sinking. Weight of water in tube equals weight of water in basin + 10 kg. If wight sinks to the bottom, it can be ignored and the weight in the basis = weight in tube.

Energy is used to describe photons, etc. which have no rest mast. Pure is a popular term which means there are no particles with mass, such as electrons, quarks, etc. The fundamental question that is under discussion here is why is there matter now?

Forward time travel needs only special relativity. Go from here to some place, say ten light years away, at near the speed of light, so the journey for you is a matter of days. Now turn around and go back to earth at the same speed. For you the time elapsed is a matter of some number of days, but when you get back to earth it will be twenty years later.

Who knows? You can theoretically assume switching to a parallel universe at some time and switch back to a different time. Current state of physical knowledge is that both ideas are pure speculation.

Angular resolution makes sense, but I can't see how to define "size".

The question is "could be a value of y". All the other choices would force the greatest common factor to be greater than 27. For x=3 (times anything which is not divisible by 3) , 27 is fine for the greatest common factor. This gives a restriction on possible x values only.

It is easy to get the angular resolution. A=2arctan(30/d) where d is the distance in feet. I am not sure how to convert angle to apparent size.

Assuming uniform density for the earth, the maximum will be at the surface. However, the interior density is higher, so a curve as shown may be more accurate.

What you have hit upon is a mathematical construction called analytic continuation, where a function is defined for a small domain, i.e. the power series for |z| < 1, is extended to all z, except |z|=1, by using a different expression which coincides with the first expression over its domain (which has to be an open set).

Try this website. You are more likely to get a response. Advise - they have a lot of nitpickers, so define the question as clearly as possible and avoid linking to somewhere else if you can. https://math.stackexchange.com