I am just starting familiarizing myself with special relativity, and want to understand if the following is a correct thought process: Consider a star that is 100 light years away from Earth, from my perspective, where I am a stationary observer on earth at some moment in time. Consider another observer that is passing me heading towards that star with a speed of 0.8c. I see it as both myself and that other observer to be at position x=0 at time t=0, from both of our perspectives, and the star to be at position x=100 at t=0 from my perspective. However, it appears to me that at time t=0 from perspective of the moving observer the star is at position 60. The way I arrive to this conclusion is the following: say at time t=80 from my perspective there's an explosion on the star. The coordinates of the event for me are t=80, x=100. Applying the Lorenz transformation I get the coordinates of the event for the moving observer as t'=0, x'=60, thus I conclude that at time t'=0 (when the moving observer passes me on Earth) for them the star is 60 light years away. So, from perspective of the moving observer it will take them 75 years to reach the star after the moment they passed me by. The next step in my thinking is the following: say at time t=0 I start accelerating towards the star with acceleration of 1g (such that from my perspective Earth moves 9.8 meters / s faster away from me every second). In approximately 300 days I will be moving with the speed of 0.8c relative to Earth, and thus be in the frame of reference of the moving observer who recently passed me. Given I was moving away from Earth, the Earth in my frame of reference is behind me, thus the star must be closer than 60 light years, and it should take me less than 75 years to reach it. Correspondingly, the total time to reach the star will be bounded above a 75 + 2 years to accelerate and decelerate. Thus, I reached a star that was 100 light years away from me in less than 100 years. If my understanding is correct, does it mean that if we find a way to reach speeds close to the speed of light, interstellar travel won't require multiple generations to reach distant stars?
Suppose I wanted to travel to one of the recently discovered potentially Earth-like planets such as Kepler 186f that is 490 light years away. Assuming I had a powerful rocket and enough fuel, how long would it take me?
Why is my starting gravity low? In the fall of 2011 I made the leap from extract/partial kits to all-grain kits. I'm enjoying AG, but my SG is continuously below the expected. I have brewed 5 AG kits and 4 have had a low SG.   My assumption is my sugar extraction during mashing (or sparging) isn't optimal. I heat my mash water to 165 and mash the grains for 60 minutes. Mashing levels out at 152. For Lautering I heat the water to 175-180. After I drain the initial mash I add the lautering water and let that soak for 10 minutes. the temperature levels out around 170-172. Then I drain into the kettle and being the boil.  I aim for a gentle boil just enough to break the surface tension. I boil for 60 minutes, chill, rack, take a reading and pitch yeast.  Here are some of the numbers  type            expected     actual  kit st paul porter  1.052        1.039   http://www.northernbrewer.com/shop/st-paul-porter-all-grain-kit.html belgium strong  1.067        1.055   http://www.midwestsupplies.com/superior-strong-ale-all-grain-kit.html troegs hop back 1.063        1.049   http://www.byo.com/stories/recipeindex/article/recipes/90-american-amber-a-pale-ale/2196-troeegs-hopback-amber-ale-clone   The fuorth was the Phat Tyre kit from Northern Brewer. Can't remember the SG though. It was low. They taste great, but the numbers weren't "right". What can I adjust so the numbers are more inline with the recipe?   Update Thank you for all the feedback. There are a number of factors for me to review and adjust. Given my current setup I will follow @mdma advice and recirculate the wort once and extend lautering to 30 minutes.  It will be a week or two before I can attempt this, so it will be awhile before I report back again.  Update #2 I now have some awesome local contacts who are helping me out with technique and trouble shooting. Another (major) issue I could be experiencing is draining the wort way too fast. According to my friend. The drain should be low and slow after cycling a gallon or 2 of wort. sparging should take ~2 hours. for each gallon drained plato/brix should drop 3 points and go no lower than 3-5 points at the end.  the second recommendation is to heat up 3-5 gallons of water to 180 and poor it in prior to/while sparging.  my first attempt using this technique proved successful with the gravity being a lot closer to the anticipated OG. I had some temp issues with the batch, but I was so focused on sparging that I ignored the temp. My guess is I would have been right on if my temp didn't fall so low.  Update #3 I had the chance to brew with my mentor this weekend and he pointed out 2 glaring problems with my mashing technique.   Starting mash temp is too low. I was heating my water to 10-12 degrees above target temp. He recommends 17 degrees. between grain absorption &amp; the tun itself most of the heat dissipates within the first few minutes. This does go against what I originally said. That I was hitting my temp marks. Looking back I was hasty with my readings. I should not dump my sparge water (or lauter?) into the tun while draining the wort. The idea is to compact the liquid into the bottom of the tun thus compacting the sugars. I am stirring up the mash when I dump all sparge water in. We kept our sparge water 1-2 inches above the grain bed and our final results we excellent.   A 3rd recommendation he made was to lower the pH of the water. tap water is ~7 and the ideal mash pH is 5.2. He added a solvent, which I cannot remember the name of, which did this for us. I'll be picking up some of that this weekend.  Thanks to all who helped me out with this problem. I want to better understand why the construction something + [adjective] makes grammatical sense. Background: I was recently reading a Washington Post article and came across the following sentence.  So if you fall in that category, [the movie] feels more like magic than repetitive.  (Note: I've copied the relevant spoiler-free sentence above; following the link to the article will open you up to movie spoilers! ) The phrase more like magic than repetitive struck me as odd, presumably because it was non-parallel (more like [noun] than [adjective]). I would have expected &quot;...more like magic than something repetitive.&quot; (Actually, that needs to be rewritten to flow better, but the point is that at least it appears to be grammatical.) Thus, one would have the noun phrase &quot;something repetitive,&quot; of the form &quot;[noun] [adjective].&quot; This got me to thinking about the use of adjectives after nouns. There is a great Stack Exchange article  that involves the characteristics of the adjective that indicate it should come after the noun. But I haven't been able to find examples of how the word &quot;something&quot; requires a postponed adjective. Consider the following sentences:  I want to buy something red. I am looking for something sweet to eat.  You cannot say:  *I want to buy red something. *I am looking for sweet something to eat.  But you can say:  I want to buy something (that is) red. I am looking for something (that is) sweet to eat.  So, my question:  Do the above examples mean that the &quot;something [adjective]&quot; construction is an example of the ellipsis of &quot;that is&quot;?  Is there something special about the word &quot;something&quot; that requires the postponed adjective? Note that I am NOT looking for details about adjectives that are postponed; I'm interested in the word that precedes the adjective (either &quot;something&quot; or similar words).   By the way, I am not interested in the following constructions, which are different:  I want to paint the house red. I tried to build the brick house taller.  Thanks in advance! I'm reading the proof of the multiplicative property of $$s_k(n)=\sum_{d|(n,k)}f(d)g\bigg( \frac kd\bigg)$$ The book wrote that in order to understand the proof, we need to know if $a,b,k,m$ are integers such that $(a,k)=(b,m)=1$, then $$(ab,km)=(a,m)(b,k).$$  I have written proof, but I don't know is it consistent, and this seems used more tools than it necessarily has to.   My proof: Let $$S_1=\{d: d|(a,m), d&gt;0\}\\S_2=\{d: d|(b,k), d&gt;0\}\\S=\{d:d|(ab,km), d&gt;0\}.$$ I will prove the statement by showing that the function $\phi:S_1\times S_2\to S$ defined as $$\phi(d_1,d_2)=d_1d_2$$ is bijective, and complete the proof by connecting the maximal elements from these partial ordering sets.  Obviously, for $d_1\in S_1, d_2\in S_2$, this tells us $d_1 $ is both divisor of $a$ and $m$, and $d_2$ is both divisor of $b$ and $k$, hence $d_1d_2$ is both divisor of $ab$ and $km$, so we can tell $\phi(S_1\times S_2)\subseteq S$.  To show one-to-one, if we have $c_1,d_1\in S_1$ and $c_2,d_2\in S_2$ such that $$\phi(c_1,c_2)=\phi(d_1,d_2)$$ then $$c_1c_2=d_1d_2$$  In particular, $c_1|d_1d_2$, we need to show $c_1|d_1$. Indeed, if $(c_1,d_2)&gt;1$, this implies $a,k$ have common factor larger than $1$, contradicts the assumption. Therefore $c_1|d_1$. A similar argument implies $d_1|c_1$, therefore $c_1=d_1$. Again a similar argument implies $c_2=d_2$, which shows one-to-one.   To show onto, let $d\in S$, since $d|ab$, we can split $d$ into $d_1d_2$ such that $d_1|a$ and $d_2|b$. Since $(a,k)=1$, we see $(d_1,k)=1$, by property of $S$ we see also $d_1|m$, therefore $d_1\in S_1$, a similar argument implies $d_2\in S_2$. So every number in $S$ has a preimage in $S_1\times S_2$  This completes the proof. Is there any mistake in the proof?     Edit: A proof about the statement: If $d|ab$ then there are $d_1,d_2$ such that $d=d_1d_2$ and $d_1|a, d_2|b$.  Let $ab=p_1^{e_1}\cdots p_r^{e_r}$ for distinct prime factors $p_1,\cdots,p_r$, then we can write $$a=p_1^{x_1}\cdots p_r^{x_r}\\b=p_1^{y_1}\cdots p_r^{y_r}$$ where some of $x_i, y_i$ can be zero, and $x_i+y_i=e_i$ for all $i=1,\cdots,r$.  Since $d|ab$, we can write $$d=p_1^{f_1}\cdots p_r^{f_r}$$ with $f_i\leq e_i$ for all $i=1,\cdots,r$. Then since $$f_i\leq x_i+y_i$$ Split into two cases:   Case 1: $f_i&gt; x_i$, then we can choose $g_i=x_i$ and $h_i=f_i-x_i$.  Case 2: $f_i\leq x_i$, then we can choose $g_i=f_i$ and $h_i=0$.  Then the choice $$d_1=\prod_{i=1}^r p_i^{g_i}, \quad d_2=\prod_{i=1}^r p_i^{h_i}$$ will prove the statement. AES plaintext is smaller than 128 bits - how to expand? We are currently developing a little AES implementation in a crypto-course at university.  As far as I know, AES uses a 128 bit block length, so all data that will be encrypted gets split up into blocks of 128 bits (16 bytes) into a 2d-array (or 1d in implementation) with 4 &times; 4 fields.  I was wondering and didn't find out what to do if the entered string to encrypt is not 128 bits long? The question is going further by thinking that there will never be exactly n &times; 128 bits of data to encrypt.  For a smaller key given by the user (e.g. "test") I know there is a key expansion algorithm defined.  But what should be done for data that is less than 128 bits? What are the transactional costs when buying and selling a house? Why does pdfTeX hang on this file? Stack Exchange doesn't allow me to answer protected questions because I gave away earned rep in bounties I noticed that I can't answer protected questions. I don't want to actually, but it says:        It was , but I also didn't have an ability to answer other protected questions.  I earned 270 reputation points + 100 reputation point bonus. Then I  350 reputation points as a bounty. Then I earned 59 reputation points. Today I started another bounty (50 reputation points), etc – I offered 4 bounties with total 500 rep.  Despite the fact I started from the association bonus, I must have an ability to answer protected questions, as I earned reputation points here. But offered them as bounties.    I now have enough reputation points, but I would like it to be fixed, because I like offering bounties very much.  I would like system to check for my bounties when it checks if I can answer protected question. Does a Grapple or Shove count as an attack for the Mobile feat? The  feat enables you to not get opportunity attacks from creatures you try to attack.  Does it still apply if the attack you made was a Grapple or Shove ? How to remove app icons in system tray on Ubuntu 18.04 with GNOME How can I remove the apps in right top corner (Google Chrome, flux, TeamViewer)?  Before my update from Ubuntu GNOME 16 these were even more annoying in the left down corner. How do I tell if matrices are similar? I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not.  I solved this by using a matrix called $S$: $$\left(\begin{array}{cc} a&amp; b\\ c&amp; d \end{array}\right)$$ and its inverse in terms of $a$, $b$, $c$, and $d$, then showing that there was no solution to $A = SBS^{-1}$. That worked fine, but what will I do if I have $3\times 3$ or $9\times 9$ matrices? I can't possibly make system that complex and solve it.  How can I know if any two matrices represent the "same" linear transformation with different bases?  That is, how can I find $S$ that change of basis matrix?  I tried making $A$ and $B$ into linear transformations... but without the bases for the linear transformations I had no way of comparing them.  (I have read that similar matrices will have the same eigenvalues... and the same "trace" --but my class has not studied these yet. Also, it may be the case that some matrices with the same trace and eigenvalues are not similar so this will not solve my problem.)  I have one idea. Maybe if I look at the reduced col. and row echelon forms that will tell me something about the basis for the linear transformation? I'm not really certain how this would work though? Please help. True random generation in Java I was reading the Math.random() javadoc and saw that random is only psuedorandom.    Is there a library (specifically java) that generates random numbers according to random variables like environmental temperature, CPU temperature/voltage, or anything like that? Why is there a temperature gradient on Earth? I was thinking about why the poles are colder and I came out with three possible explanations.  1. The atmosphere blocks light beams coming to poles    The Sun is seen only slightly over the horizon at the poles, so the light beams have longer way to the poles. If only this case were true, then the total difference between the poles and the equator would be the same as the energy that was absorbed by the atmosphere.  2. Sun is only slightly over the horizon at the poles.    Poles are often in shadow of the parts of the Earth which are closer to the equator.  3. Poles are farther from the Sun    The longer is the way to Sun the lower is the chance to catch a light from the Sun.  Please, correct me wherever I'm wrong. I'd like to know   Which of these three is most correct?  How significant are the described effects of the other explanations? I can't get the coordinates on my computer because it keeps changing the brightness.  I have a windows 8 Samsung, and I'm playing Minecraft version 1.6.4. AudioSessionSetActive and AudioSessionSetProperty is deprecated in iOS 7 Image data URI and SEO Do you think the use of image data URI is SEO friendly?   On the one hand saves you a HTTP request. On the other hand it really messes up your code. Especially for image files larger than 20KB or moving GIF image files of up to 200KB, the quantity of code inserted is enormous. As Googlebot reads usually from top to bottom, all data after the image data URI encoding will be harder to rank well in Google.   Consider that image data URI is not in the CSS files as they should normally be and for the time being I can use only TinyMCE editor which provides me with just two options, either insert image with the image data URI method or the traditional image link.   I would like to know which is more search engine friendly. PS4 Controller not connecting to PS4 and won't reset? How to run Visual Studio post-build events for debug build only How can I limit my post-build events to running only for one type of build?  I'm using the events to copy DLL files to a local IIS virtual directory, but I don't want this happening on the build server in release mode. I have two different machines (home and work) running Ubuntu 18.04. Last night vim froze at home. I was in insert mode and typing and went to save (esc :w) and nothing happened. The status bar still reads -- INSERT --, the cursor is still blinking where it was. I was stuck. I couldn't find a way out. I couldn't type (nothing happened when I type), I couldn't move around (the up and down arrows did nothing). It was stuck in insert mode with the cursor blinking where it was.   I was definitely multitasking and probably hit some other keys in there, but I don't know what keys. It was late, though, so I closed the terminal window and tried again (I was entering a git commit message). It happened again partway through my typing so I switched to git commit -m "don't need an editor for this" instead. And then I shut down my computer and stopped working.  I figured I was just tired, but then it happened to me today at work on a different laptop altogether. Again I was multitasking and can't swear I didn't type any bizarro key sequence but if I did it was accidental. And other tabs in the same terminal aren't frozen.   I'm used to getting trapped in visual mode in vim. That's a trick I've learned. But stuck in insert mode? Any ideas on what I might've done and how to get out of it?   Per a comment suggestion I tried looking at .viminfo but the only .viminfo I see is owned exclusively by root and only appears to show things I would have edited with sudo:   # Input Line History (newest to oldest):  # Debug Line History (newest to oldest):  # Registers:  # File marks: '0  1  0  /etc/neomuttrc |4,48,1,0,1531789956,"/etc/neomuttrc" '1  1  66  /etc/apt/sources.list.d/signal-bionic.list |4,49,1,66,1530816565,"/etc/apt/sources.list.d/signal-bionic.list" '2  51  0  /etc/apt/sources.list |4,50,51,0,1530816531,"/etc/apt/sources.list"  # Jumplist (newest first): -'  1  0  /etc/neomuttrc |4,39,1,0,1531789956,"/etc/neomuttrc" -'  1  66  /etc/apt/sources.list.d/signal-bionic.list |4,39,1,66,1530816565,"/etc/apt/sources.list.d/signal-bionic.list" -'  1  66  /etc/apt/sources.list.d/signal-bionic.list |4,39,1,66,1530816565,"/etc/apt/sources.list.d/signal-bionic.list" -'  51  0  /etc/apt/sources.list |4,39,51,0,1530816531,"/etc/apt/sources.list" -'  51  0  /etc/apt/sources.list |4,39,51,0,1530816531,"/etc/apt/sources.list" -'  51  0  /etc/apt/sources.list |4,39,51,0,1530816531,"/etc/apt/sources.list" -'  51  0  /etc/apt/sources.list |4,39,51,0,1530816531,"/etc/apt/sources.list" -'  1  0  /etc/apt/sources.list |4,39,1,0,1530816447,"/etc/apt/sources.list" -'  1  0  /etc/apt/sources.list |4,39,1,0,1530816447,"/etc/apt/sources.list" -'  1  0  /etc/apt/sources.list |4,39,1,0,1530816447,"/etc/apt/sources.list" -'  1  0  /etc/apt/sources.list |4,39,1,0,1530816447,"/etc/apt/sources.list"  # History of marks within files (newest to oldest):  &gt; /etc/neomuttrc     *   1531789952  0     "   1   0  &gt; /etc/apt/sources.list.d/signal-bionic.list     *   1530816564  0     "   1   66     ^   1   67     .   1   66     +   1   66  &gt; /etc/apt/sources.list     *   1530816454  0     "   51  0   It seems odd that I wouldn't have an unprivileged .viminfo but I did sudo udpatedb and locate .viminfo and still didn't surface more than the one root-owned file. Applying Arzela-Ascoli to show pointwise convergence on $\mathbb{R}$. The statement I am trying to prove:  Let $\{ f_n \}$ be a sequence of equicontinuous, real valued, uniformly bounded   continuous functions on $\mathbb{R}$. Show that $\{ f_n \}$ has a convergent subsequence   which converges uniformly on any bounded subset of $\mathbb{R}$ and pointwise on all   of $\mathbb{R}$ to a continuous function. (Royden pg 210, problem 9).  I believe the first portion follows directly by Arzela Ascoli, once it is noted that if we are given some bounded subset $E$, we can look at $\overline{E}$ a compact set on which the family is equicontinuous and uniformly bounded.  For pointwise convergence on all of $\mathbb{R}$, the question seems to require some sort of diagonalization argument: Take some dense sequence of $\{x_i\}$ (the reals are separable) and write columns of $f_n$ applied to each individual point. There will be a convergent subsequence by Bolzano-Weirstrass and since the outputs are real and bounded by the uniform boundedness of the family. I am a bit fuzzy on this part: Repeating this literally pointwise and moving far enough diagonally, we can correct for different convergence "speeds" pluck out a single convergent subsequence for any given epsilon and have $\sup|f_n(x)-f(x)|&lt;\epsilon$ for any given epsilon. Finally, continuity should follow from an $\epsilon/3$ argument and equicontinuity of the family.  However, in formalizing the above, I am getting caught up in the notation of that sort of argument and am hoping for a more elegant solution. Is there a better way? Am I just generally confused and that's why I'm having trouble writing things down formally?   Thanks! Command "find -name" must be enclosed in quotes or it doesn't work. Why is that?