As Einstein's theory tell us it takes infinite amount of energy to get something with a mass to reach the speed limit $c$.   How true is that when observed experimentally either in a laboratory or elsewhere in the universe?
I'm in a freshmen level physics class now, so I don't know much, but something I heard today intrigued me. My TA was talking about how at the research facility he worked at, they were able to accelerate some certain particle to "99.99% the speed of light". I said why not 100%, and I didn't quite understand his explanation, but he said it wasn't possible. This confused me. Since the speed of light is a finite number, why can we go so close to its speed but not quite?  Edit: I read all the answers, and I think I'm sort of understanding it. Another silly question though: If we are getting this particle to 99.99% the speed of light by giving it some sort of finite acceleration, and increasing it more and more, why cant we increase it just a little more? Sorry I know this is a silly question. I totally accept the fact we cant reach 100%, but I'm just trying to break it down. If we've gotten so close by giving it larger and larger acceleration every time, why cant we just supply it with more acceleration? And how much of a difference is there between 99.99% the speed of light, and the speed of light? (I'm not really sure if "difference" is a good word to use, but hopefully you get what I'm asking).
I want to prove the assertion:      The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 \pmod{4}$.   My first attempt is this. In $\mathbb{Z}[\zeta_p]$, $1-\zeta_p$ is prime and $$ p = \epsilon^{-1} (1-\zeta_p)^{p-1} $$ where $\epsilon$ is a unit.  Since $p$ is an odd prime, $(p-1)/2$ is an integer and  $$ \sqrt{\epsilon p } = (1-\zeta_p)^{(p-1)/2} $$ makes sense and belongs to $\mathbb{Z}[\zeta_p]$.    How do I deal with the $\epsilon$ under the square root?  I guess the condition on the congruence class of $p$ comes from that.  Is this even the right way to proceed?  Uniqueness is not clear to me either.  I thought about looking at the valuation $v_p$ on $\mathbb{Q}$, extending it to two possible quadratic extensions beneath $\mathbb{Q}(\zeta_p)$, then seeing how those have to extend to common valuations on $\mathbb{Q}(\zeta_p)$, but I didn't see how to make it work.  I would appreciate some help. How does TikZ sort intersections? I used the sort by key for intersections to sort along a line, expecting the intersections to be numbered 1 through 5. Why were they numbered 4, 5, 3, 1, 2? Playing around with it, I could not seem to figure out how the order is determined. The code and image follow.  \documentclass{standalone} \usepackage{tikz} \usetikzlibrary{intersections}  \begin{document} \begin{tikzpicture}     \draw[name path=grid] [xstep=3,ystep=2] (0,0) grid (9,8);     \draw[-&gt;, name path=line] (2,1) -- (7,7);     \draw[name intersections={of=grid and line, sort by=line, name=i, total=\t}]         \foreach \s in {1,...,\t}{(i-\s) node {\s}}; \end{tikzpicture} \end{document} How can I pad a String in Java? Is there some easy way to pad Strings in Java?  Seems like something that should be in some StringUtil-like API, but I can't find anything that does this. GeoReference tiff image into the Google map? I want to Geo-reference a tiff image contain a Map for a specific region, I need to add this region(image) and Geo-reference it on the Google map, SO I can see on the map for the region I select the image which I have inserted.  the image also need information such as the Latitude and Longitude of corners and size, and the pixel per meter measurement.  How to do the correct Geo-reference for that ? Eclipse says: “Workspace in use or cannot be created, chose a different one.” How do I unlock a workspace? Hi I'm traveling to Uzbekistan and have 12 hours layover in Seoul via Korean Airlines. Do I need a visa to go out of the airport for sightseeing? I'm a Philippine passport holder. I am Sri-Lankan staying in Bangladesh working in well established company in Bangladesh. I, my wife and two kids applied for Schengen visa from German embassy in Dhaka. I got only sponser letter from my paternal uncle in Germany. We submit all company letter, bank account, school letters, insurance, booking air tickets, etc.  Our visa refused and the reason given "your intention to leave the territory of the member states before the expiry of the visa could not be ascertained" so I want to know what it mean. And how I apply again fix by this reason? Our annual vacation starts on 15 July for one month. I applied visa for 15 days. We plan to stay 15 days in Germany and travel to Sri Lanka and back to Bangladesh. In swimming, is there a competitive advantage to being in one lane or another? Upon observing swimming competitions in the Olympics, it is not clear to me whether one of the eight lanes is more advantageous versus another. I've seen vague claims but nothing backed up with science.  In competitive swimming, is there an advantage to being in one lane or another? Furthermore, does the stroke (ie: breast, fly, freestyle, etc.) affect this effect? Diminishing returns / non-linear reputation from “great” answers Over time I realized a very annoying property of the Stack Overflow reputation system: Easy answers to popular question are rewarded with a lot more reputation points than actually great answers. In case anybody actually needs examples for this, here are two examples from :   gained me 420 rep, I invested 2 minutes writing it, and it only was a wild guess.   took me about an hour for the research and the math and gained me 40 rep.  Now I know that you are bound to get more rep for answers that are useful to more people, and I'd say that's fine. But the current situation is quite extreme: 30% of my reputation is from 5 of my 330 answers, and they are definitely far from being my best ones.  A final example that is not from my profile, but more extreme: Do you really think  is worth 40000 reputation?  My proposition to improve that situation would be to make the reputation gain from a single answer non-linear, maybe even have it capped at a certain point. I'm not sure on how that non-linear function should look exactly, and would leave good proposals for that to users with more experience than me. Objects not rotating with each other (Rubik's cube) Should I really pay off my entire credit card balance each month or should I maintain some balance? How to select top 10 records from each category Let's get rid of the 10K flag queue are pretty cool... You get a birds-eye view of activity on the site, a &quot;dashboard&quot; view of what's happening. Some of the individual tools haven't scaled particularly well with Stack Overflow's growth, but the concept behind them is still sound: we trust you to enough to be a bee watcher now.  ...and then there's . What a let-down! Once upon a time, this queue contained spam and offensive flags, which one might reasonably assume were important enough to put in front of the more trusted members of the site. Re-flagging them brought them that much closer to deletion, while editing them offered salvation to some hapless author. Nowadays, it's a bunch of Not an Answer flags and a smattering of assorted cruft. 10K users can't even vote to delete these; only 20K users have that privilege. Re-flagging them does nothing but increase their priority in the moderator flag queue, where they frequently outrank more pressing issues;  has . Also, it's , and the behavior has diverged far enough from that of the moderator flag queue on which it is based that it has become an active hindrance to further development of the tooling there. Worst of all, we have  that's available to anyone with the editing privilege. It even has logic built in to prioritize answers likely to be deleted for users with . It's like we gave you a car for your 10th birthday, and then replaced it with a rusty bicycle when you turned 16. It was a nice idea, but it has outlived its usefulness. Proposed changes The /tools/flagged route goes away. Period. No replacement. The rest of the 10K tools stick around as informational pages. Not an Answer flags go into /review/low-quality, . Then we beef up the Low Quality review process to make better use of more experienced reviewers and solve this whole &quot;declined / helpful / disputed&quot; flag debate once and for all:  Effective # of reviews required == ReviewsRequired + # of applicable flags (where ReviewsRequired is 2 on Stack Overflow, 1 everywhere else). So 1 VLQ or NAA flag means EffectiveReviewsRequired=3. LQ tasks are not dequeued until one of the following conditions is met:  Post is edited from within review. Outcome: flags are marked &quot;helpful&quot; (current behavior). Post accumulates 3 Delete votes (can only happen when post scores &lt;= 0 and reviewers have &gt;= 20K rep). Outcome: post is deleted, flags are marked &quot;helpful&quot;. Task accumulates EffectiveReviewsRequired &quot;Looks Good&quot; reviews. Outcome: if the number of (Recommend)Delete reviews is &gt;= the number of Looks Good reviews, mark flags &quot;disputed&quot; and raise DisputedLowQuality mod flag. Otherwise, mark flags &quot;declined&quot;. Task accumulates 6* RecommendDelete + Delete reviews. Outcome: mark flags &quot;helpful&quot;. If the post scores &gt; 0 then raise DisputedLowQuality mod flag, else just delete post (current behavior).   As under the current system, flags on posts that've already completed one full review cycle without being deleted should skip /review and go directly into the mod queue.  End result  Flags get handled faster, more accurately, and with less wasted effort from 10K reviewers. Moderators are free to focus on situations that can't be handled by the community - the ! Developers are free to make enhancements to the moderator tooling without having to work around 10K user restrictions.  Questions Am I forgetting anything here? I haven't really spent much time in the 10K flag queue since back when it was filled with spam flags and therefore useful - is there a use-case that you depend on that would be lost with its removal? Post 'em below. *: on Stack Overflow, only . Change folder permissions and ownership I would like the user to have full rights on this folder (as well as all sub-directories and files in it):  ~/.blabla   currently owned by root.  I have found numerous posts (in this forum and elsewhere) on how to do this for files but I can't find a way to do it for whole folders. Why was my "shopping list" question closed? My question was closed for being a "shopping list question", even though it was very much related to the topic of the site. Why? Interpretation of R's lm() output Deleting external/auxiliary files? $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function. Assume that $\lim_{x \rightarrow \pm \infty} f(x)$ exist and are finite. Prove that $f$ is bounded.  So to show that $f$ is bounded, I must show that $\exists M \in \mathbb{R}$ such that $|f(x)| \leq M$ for all $x \in Dom(f)$. Assuming that $f(x)$'s limit exist, then I know that $\forall \epsilon  &gt; 0$, $\exists N$ such that for $n &gt; N$, $|f(n) - L| &lt; \epsilon$. But I am having trouble with figuring out how to connect the two ideas together. Any Video formats that render with Alpha Channel? Does anyone know if there are any video formats (besides the HUGE uncompressed avis) that you can render in that maintain the alpha channel? (i.e. for after FX etc.)   I used to use quicktime, but that's no longer an option.  I know you can render in png. or other image formats, but video is significantly smaller, and I often work on machines with less space &amp; resources. Error: mysqli_stmt::bind_param(): Number of elements in type definition string doesn't match number of bind variables