Special relativity different time dilatation?
What if a light clock travels perpendicular to mirrors that make up the clock?
To show that ${x_n}$ is convergent where $|x_{n+1} - x_n|< \frac{1}{n^2}$ Let ${x_n}$ be a sequence in $R$ such that $|x_{n+1} - x_n|&lt; \frac{1}{n^2}$ for all $n \in N$.  Show that the sequence is convergent.  If it were $|x_{n+1} - x_n|= \frac{1}{n^2}$, could take help of the fact $\sum \frac{1}{n^2}$ is convergent and of triangular inequality.  But what to do here? Please help. Thanks in advance. Capitalization of "Theorem 1" in mathematics papers In many mathematics papers, I read sentences like:     With an appeal to Theorem 4.5 we get …   Is the capitalization of theorem in this case correct? If it is correct why do we capitalize the word? Past tense: learned/learnt, dreamed/dreamt? In my very first question, Carlo made an edit stating that learnt is a rare past tense of learn. I am accustomed to using learnt for past tense and learned as an adjective (as in He is a learned scholar.) I checked into a couple of dictionaries and they say both are valid past tenses. Do I stand a risk of not being understood if I use learnt?  In a similar vein, should dreamed be used as a past tense for dream instead of dreamt? I can't think of other such conflicts, but I guess there must be more. Can I ask a postdoc who has closely supervised my research to write me a letter of recommendation? What statistical test can I use to compare two ratios from two independent samples. The ratios are after to before results. I need to compare the after/before ratios for two independent models and show whether they are have significant difference or not. Please help! I created a custom counter 'rtaskno' and a command '\rtask' as following:  \newcounter{rtaskno} \newcommand{\rtask}{%    \stepcounter{rtaskno}%    \thertaskno}   I like to use them as following:  \section{Task \rtask. Blah Blah..}  \section{Task \rtask. Blah Blah..}  \section{Task \rtask. Blah Blah..}  \section{Task \rtask. Blah Blah..}   Later in the text, if I want to cross-reference 'Task 3', how can I do it? Where to put label? Destroy chunked block like Minecraft So, I understand that for a NPN transistor to work the emitter-base junction needs to be forward biased and the collector-base junction needs to be reverse biased.  I understand how current flows from the emitter to the base when forward biased, but I can't seem to wrap my head around how it flows from the emitter, through the base, and into the collector.  From what I understand about the depletion region of a PN junction diode, when forward biased, the electrons in the N-type material are repulsed by the negative charge of the supply (say a battery) and the holes in the P-type material are repulsed by the positive charge from the supply, thus narrowing the depletion region and allowing electrons to flow from the N region to the P region.  I also understand that the holes and electrons are attracted to the source charges when reverse biased, widening the depletion region and blocking any electrons from flowing.  What I don't understand is what happens to the depletion region of the collector-base junction that allows electrons to flow from the emitter to the collector even though the collector-base junction is reverse biased.  Take a simple common emitter circuit for example.  I apparently am not cool enough to post pictures yet so you'll have to use your imagination.  So pretend we have the base connected to the positive terminal on a 1v battery (Vbb).  The collector is connected to the positive terminal on a 10v battery (Vcc).  The emitter and both negative terminals are connected to ground.  Also pretend that there is some kind of current limiting.  How does current get from the emitter through to the collector?  I can see how they are at different potentials, but I don't understand how this overcomes the reverse biasing of the collector-base junction.  Can anyone tell me what actually happens at the junction to allow current flow and how changing Vbb will affect that current? Possible (but not serious) glitch in close/reopen I just had one of my questions closed. (Hey - it happens to everyone) When I clicked on reopen - lo and behold - it worked. I cast a reopen vote!  Is this a glitch? I know we need 4 more votes anyway but still...  And no  tag please :) Remove spaces from std::string in C++ What is the preferred way to remove spaces from a string in C++? I could loop through all the characters and build a new string, but is there a better way? Replace Line Breaks in a String C# How can I replace Line Breaks within a string in C#? How would I remove the border from an iframe embedded in my web app? An example of the iframe is:  &lt;iframe src="myURL" width="300" height="300"&gt;Browser not compatible.&lt;/iframe&gt;   I would like the transition from the content on my page to the contents of the iframe to be seamless, assuming the background colors are consistent. The target browser is IE6 only and unfortunately solutions for others will not help. touchdown held down libgdx White cloudy areas on bottom of new stainless steel skillet I just bought my first few pot and pans "Calaphon" stainless steel trip-ply  I used my new 10" skillet tonight, and cooked pork chops. I heated the pan first, added olive oil, let it heat, then added chops. After I was done, cleaned it immediately under water, and noticed white cloud looking areas on the bottom of the skillet. They won’t wash or rub off  Do I need to return these? I did buy them for the look as well. It's possible to do a array of Matrix in Matlab? How to redirect with "www" URL's to without "www" URL's or vice-versa? I am using ASP.NET 2.0 C#. I want to redirect all request for my web app with "www" to without "www"  www.example.com to example.com  Or  example.com to www.example.com  Stackoverflow.com is already doing this, I know there is a premade mechanism in PHP (.htaccess) file. But how to do it in asp.net ?  Thanks Clever bounty reputation hack I say "hack" in the most favorable light. It's a clever idea and theoretically beneficial.  I noticed that an old question of mine suddenly started getting a lot of renewed attention. The question had been answered, with an answer accepted. But another user added a well reasoned and insightful answer to the list. He also opened a +100 bounty on that question.  The resulting increased attention quickly got his answer quite a lot of up-votes. Well more than the 100 points he had spent on the bounty. Therefore, he gets a net rep profit out of the transaction.  So here's the hack:   Find an interesting question Research and provide a very good answer Open up a bounty on that question to attract attention and votes to your new answer Profit!!!   2 and 3 may be out of order--I didn't see the timing of it. But either way, assuming he gets more than 10 votes, he still gets increased rep out of the deal.  I don't think this is necessarily bad for SO - it does generate some useful discussion and insightful answers. Plus you can increase your own rep very rapidly by making your answers highly visible, and therefore frequently voted upon.   However, it does slightly change the meaning and purpose of bounties away from starting a discussion and toward self-promotion. Still, interesting. Scenario: My friend John has a personality disorder, intensely paranoid for example.  Can I say:     There were three events last year that told me John had a problem.   or can I say:     There were threee events last year that told me John has a problem. A coin is tossed $n$ times. What is the probability of getting odd number of heads? A coin is tossed $n$ times. What is the probability of getting odd number of heads?   I started this chapter sometimes ago and faced in front of a tough problem. At first I started considering cases. Case-I : The probability of getting 1 head.Case-II : The probability of getting 3 head and so on. But there are many cases. So how can I solved this . Please help me. Thank you! Are there any versions of t-SNE for streaming data? My understanding of  and the Barnes-Hut approximation is that all data points are required so that all force interactions can be calculated at the same time and each point can be adjusted in the 2d (or lower dimensional) map.  Are there any versions of t-sne that can efficiently deal with streaming data? So if my observations are arriving one at a time, it will find the best location on the 2d map to place the new observation, or continuously update all points on the 2d map to account for ht new observation.  Would this even make sense or does it go against the setup of t-sne.