If I travel almost side by side with light, and my speed is lets say 99% of speed of light..would I see the light going near me at the speed of light (i mean as same as when i would not be moving, when i would be stationary) or would i be able to actually see it going besides me?
I have just started special theory of relativity. The limiting speed known as speed of light fascinated me most. I asked my teacher:      Consider two massless objects moving in the same direction at the speed of light. What would be their relative velocity w.r.t each other?   My teacher tells me that their relative velocity would be zero. But then the speed of light is universal constant regardless of the motion of its frame of reference, so shouldn't their relative velocity be $c$? What is their relative velocity and how? Do objects moving at the speed of light obey law of addition of velocities?
Why gpedit and the corresponding registry entries are not synchronized? Prove that $\lim_{x \to \infty} \frac{\log(1+e^x)}{x} = 1$ Show that $$\lim_{x \to \infty} \frac{\log(1 + e^x)}{x} = 1$$ How do I prove this? Or how do we get this result? Here $\log$ is the natural logarithm. Do I end up richer paying down a mortgage or contributing to an IRA? I have $10,000 extra cash that I can either use to pay down my 5% mortgage, or contribute to my IRA.  I'm 30 years old.  I hope to pay off the mortgage in about 5 years.  I'm stymied, because:   Paying $10,000 into my mortgage earns me $500 a year (in interest savings) for the life of the mortgage -- hopefully 5 years. Paying $10,000 into my IRA earns me roughly a $3000 lowering of my tax bill now (plus CD ROI of about 1% at retirement time.)   So I can get $2500 ($500 a year over 5 years), after which I will have paid my mortgage off with other money; or I could get $3000 in cash right now plus a bit of extra money when I retire.  Even ignoring the future benefits of the IRA contribution, it feels like I should put the money into my IRA: it's an immediate 30% return on investment, as opposed to a 5% return for 5 years.  But if I always make that decision, I won't be paying off my mortgage early, and that $500 will stretch out over 30 years, costing me $15,000!  Which actually leaves me with more money in the end?  What basic principle of math am I missing?  Update:  The principle of math I was missing is that, assuming I'm taxed at the same rate when I retire as I am taxed now, I end up with exactly as much money as if I had invested the $10k and not received a $3k tax reduction.  (Of course, my income level and tax laws will have changed by then, but let's ignore that.)  So the question simplifies to, "Should I put $10k toward eliminating a 5% debt or toward a 1% investment?" in which case I should clearly put it toward the mortgage.  I was confused by the government incentive into thinking that retirement investment was a sweeter deal than it actually is.  This is probably their goal, and probably a noble one :) It just struck me that Utility Companies world-wide supply domestic users at either 50/60Hz. It could have been selected to be 10 Hz, or 100 Hz ... whatever  Was 50/60 chosen at random? Is there some rationale involved (efficiency, easier to construct/control etc) in restricting domestic supply to either 50/60 Hz?  In other words, if I were designing a power system, why would I pick 50/60 Hz instead of some other frequency? The set of real numbers and power set of the natural numbers I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?  I have also learnt that there exists no set whose cardinality is strictly between the natural numbers and the real numbers. Is there a proof of this or at least some intuitiveness behind it? Why is an object stuck to another? So I created a Cube in Blender and then I created an Empty beside the Cube (in object mode). Now, when I move the Empty around, the Cube follows... Why? On which Stack Exchange site can I ask customer service questions about product X or company Y? On which Stack Exchange site can I ask questions like        When will [some new feature] for [some product] be released?   Where can I download an update for [some product]?   I'm getting errors when accessing the customer portal for [some company] – how can I fix this?   Why was I banned for doing [some action] on [some platform]?   Why is [some company] discontinuing [some product]?   Why is [some company] giving away [some product] for free?   What's the price for [some product]?      or any other product support/customer service questions?  It's not a product related to programming (or I've read  on Meta Stack Overflow) so I know Stack Overflow isn't the right place. Which other site in the network will take these kind of questions? $M= \{ A \in Mat_{2 \times 2}{\mathbb{R}}| \det(A)=1 \}$ is homeomorphic to $S^{1} \times \mathbb{R}^{2}$ How close would you have to be to the merger of two black holes, for the effects of gravitational waves to be detected without instruments? Assume two black holes in the most common size range, spiraling into each other until they merge. The event releases significant amounts of energy via gravitational waves, which warp the space-time.  If the distortion is powerful enough, it would get noticed in daily life, or perhaps even detected by the proprioceptors in the human body.  How close, as an order of magnitude estimate, would you have to be to the merger to perceive it like that, without any instrument? Unicode identifiers in Python? Is there a way to select the children of a parent object? How to find distance from the latitude and longitude of two locations? How can I access cloth properties via python? I want to manipulate options like Disk Cache with python:    Any Ideas? How do you show monotonicity of the $\ell^p$ norms? I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it). Inexact measurement and wavefunction collapse As is usually said, measurement of an observable $q$ leads to collapse of wavefunction to an eigenstate of the corresponding operator $\hat q$. That is, now the wavefunction in $q$ representation is $\psi(q)=\delta(q-q_0)$ where $q_0$ is result of measurement.  Now, in reality measurements are never exact, so the wavefunction should not have that sharp peak. Instead it should be broadened depending on precision of measurement. In this case, can we still introduce an operator, an eigenstate of which would the new wavefunction appear? Is it useful in any way? Or does the new wavefunction depend too much on the way it was measured so that each instrument would have its own operator? How would such opeartor look e.g. for a single-slit experiment? Is it hard to transfer contacts from one iPhone to another iPhone? I just bought a new iPhone 6s, and want to move all the contacts from the old one to the new one. Can iTunes or iCloud help me do this, or are there any other ways? I've recently beat the Ender Dragon. I've heard there is a way to respawn her. How do I do that? Better way of saying "those who are curious"? I am writing a note in which I want to ask (curious) readers to refer another paper. The phrase curious minded comes to mind, but it seems somewhat awkward.  Is there a better way to convey similar import? Active login not recognized on Super User Relationship between Binomial and Beta distributions I'm more of a programmer than a statistician, so I hope this question isn't too naive.  It happens in sampling program executions at random times. If I take N=10 random-time samples of the program's state, I could see function Foo being executed on, for example, I=3 of those samples. I'm interested in what that tells me about the actual fraction of time F that Foo is in execution.  I understand that I is binomially distributed with mean F*N. I also know that, given I and N, F follows a beta distribution. In fact I've verified by program the relationship between those two distributions, which is  cdfBeta(I, N-I+1, F) + cdfBinomial(N, F, I-1) = 1   The problem is I don't have an intuitive feel for the relationship. I can't "picture" why it works.  EDIT: All the answers were challenging, especially @whuber's, which I still need to grok, but bringing in order statistics was very helpful. Nevertheless I've realized I should have asked a more basic question: Given I and N, what is the distribution for F? Everyone has pointed out that it's Beta, which I knew. I finally figured out from Wikipedia () that it appears to be Beta(I+1, N-I+1). After exploring it with a program, it appears to be the right answer. So, I would like to know if I'm wrong. And, I'm still confused about the relationship between the two cdfs shown above, why they sum to 1, and if they even have anything to do with what I really wanted to know.