Will an unhindered (un-scattered) photon go to the edge of the universe?
As I understand it, there are several models for what the universe could look like from the “outside”. I don’t know if any of these are outdated, so if they are please tell me if they are.   Expanding bubble. This model says the universe will never stop expanding, and the universe will die because the energy all turns to photons and gets spread out so much. Bubble that will eventually collapse. In this model the universe will stop expanding, and collapse. Everything merges into super-massive black holes, and the universe dies with fire. The universe is a torus. I'm not clear whether the size of the universe in this model is changing, but the central idea is that space is donut-shaped.   Those are the major ones that I have heard of. My question involves what I have heard to be the most popular of the three, #2. In this model, eventually something will "touch" the edge of the universe. Obviously, the first thing to "touch" it would be photons as they travel at, well, the speed of light. This involves all forms of energy in the universe, but I thought it might be easier to explain for something without mass.  What happens if some form of energy "touches" the edge of the universe? Logically, the energy cannot be lost without violating the law of conservation of energy. The only thing I can think of, is that it would "bounce" or be refracted off the edge.  I realize that it sounds silly, however if model #2 is correct then this situation will arise some number of billions of years from now.  I can only assume that there is either a new model that accounts for this, there is some complex explanation, or that model #2 is incorrect.
Algebraic extension of perfect field in which every polynomial has a root is algebraically closed Let $F$ be a perfect field, i.e. every irreducible polynomial over $F$ has distinct roots in the algebraic closure of $F$. Suppose that $K$ is an algebraic extension of $F$ with the property that every non-constant $p(X) \in F[X]$ has a root in $K$.   I want to show that $K$ is algebraically closed, i.e. that $K = \overline{F}$, the algebraic closure of $F$.  If $K$ were a normal extension of $F$, this would follow immediately: Let $\alpha \in \overline{F}$ and $m(X)\in F[X]$ its minimal polynomial. By our assumption, $m(X)$ has at least one zero in $K$. Since $K$ is a normal extension of $F$, it thus contains all other roots of $m(X)$ including $\alpha$.  I don't have an idea as to how to show that $K$ is a normal extension. How do I survive in the Nether? Whenever I travel to the Nether in Minecraft, I almost instantly get blasted to smithereens and chopped up into little pieces.  How do I survive in the Nether, at least long enough to build some sort of shelter and get situated? Biblatex: submitting to the arXiv I am trying to submit a LaTeX file to the  which uses biblatex.  However, it fails whatever I do.    I first get the message   biblatex.sty is not found   I have tried including (all from TeXLive 2010) biblatex.sty but then I get   logreq.sty is missing then logreq.def is missing   then   ! Package biblatex Error: Outdated 'etoolbox' package.   I then tried adding etoolbox.sty and etoolbox.def and get  ! Package biblatex Error: Style 'alphabetic' not found   So I added alphabetic.bbx then standard.bbx then alphabetic.cbx and finally I get   ! Package biblatex Error: Autocite command 'inline' undefined   I also tried incorporating the bib into the main LaTeX file as in  but it doesn't help.  Has anyone successfully used biblatex with the  and if so, how? Converting transparent gif / png to jpeg using java Help me understand the proof for Bolzano-Weierstrass Theorem! I am trying to find all 1- or 2- dimensional Lie Algebras "a" up to isomorphism. This is what I have so far:  If a is 1-dimensional, then every vector (and therefore every tangent vector field) is of the form $cX$. Then , by anti-symmetry, and bilinearity:  $$[X,cX]=c[X,X]= -c[X,X]==0$$  I think this forces a unique Lie algebra because Lie algebra isomorphisms preserve the bracket.  I also know Reals $\mathbb{R}$ are the only 1-dimensional Lie group, so its Lie algebra ($\mathbb{R}$ also) is also 1-dimensional. How can I show that every other 1-dimensional algebra is isomorphic to this one? Do I use preservation of bracket?  For 2 dimensions, I am trying to use the fact that the dimension of the Lie algebra g of a group $G$ is the same as the dimension of the ambient group/manifold $G$. I know that all surfaces (i.e., groups of dimension 2) can be classified as products of spheres and Tori, and I think the only 2-dimensional Lie group is $S^1\times S^1$, but I am not sure every Lie algebra can be realized as the Lie algebra of a Lie group ( I think this is true in the finite-dimensional case, but I am not sure).  I know there is a result out there that I cannot yet prove that all 1- and  2-dimensional Lie algebras are isomorphic to Lie subalgebras of $GL(2,\mathbb{R})$ (using matrix multiplication, of course); would someone suggest how to show this last? Thanks. Disallow closing of questions with a certain number of question / answer upvotes With reference to this currently open question   At the time of writing, it has 3 close votes yet has 7 question upvotes and the single answerer (11 upvotes) will have been awarded the enlightened badge (silver).  Therefore in the opinion of many (i) the question adds value and (ii) the answer adds value.  As a staring point for discussion, I suggest that 5 net question upvotes (and perhaps 5 answer net upvotes) should be enough to keep a question permanently open. Let's haggle over the numbers though. Thoughts? I have two dates of the form:  Start Date: 2007-03-24  End Date: 2009-06-26   Now I need to find the difference between these two in the following form:  2 years, 3 months and 2 days   How can I do this in PHP? Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges. Let $\{a_n\}$ be defined by $a_1 =1  $ and $a_{n+1} = 1 + {\dfrac{1}{a_n}}$ with $n \in N$. Show that $a_{n+1}= 1 + {\dfrac{1}{a_n}}$ converges.  I know the limit but how can I show that is a Cauchy sequence or that this sequence converges. Where can I find fantasy minis on the cheap? I'm looking to build a collection of fantasy miniatures, preferably pewter - but I don't want to buy them one at a time from my local hobby store... Much too expensive, and they don't often carry anything I'm interested in buying, which makes it a long term venture as well. So I'm looking for ideas on how to obtain many miniatures at once, preferably for cheap. D&amp;D Miniatures aren't a good source for me as they're collectibles, and the prices on those are also prohibitive.  By the way - they don't have to be pewter, that's just my preference. =) How to make excel automatically fill a value in a cell of a sheet when another corresponding cell of another sheet is filled the same value? I have a large file and I would like print from each sequential 50 lines , the 15th and 25th lines.  sed -n '15,25p' inputfile   How to modify this command to print only lines 15 and 25 and to loop over each 50 lines in the file. Why do some adjectives follow the nouns they modify? Right now I can only think of one instance in which this regularly occurs.  The adjective proper is sometimes placed after the noun it modifies, e.g:      A class of cold-blooded oviparous or ovoviviparous vertebrated animals whose skin is covered with scales or scutes; the reptiles proper.   What is this word order called, and why is it ever used?  Edit: I really want to know about proper, and while the word hyperbaton relates to post-positivity when it is meant to be rhetorically impacting, it doesn't seem to fit with ubiquitous post-positivity, such as that of proper.  I've accepted hyperbaton as the correct term/reason for almost all post-positive constructions, but I'd like a complete answer to include proper. SEO in Stack Overflow We've been out of private beta for only two days, and stackoverflow.com is already ranking high in .  Are there any specific techniques applied in the site? I've been researching the subject for some time and haven't found it to be so easy.   even returns site-links.  () Is the Stack Overflow source code available? Please make the multiple comment timer less annoying (non-resetting etc.) As a little feature to prevent comment spam, you can only enter one comment every 15 seconds. That's a very reasonable limitation, since you very rarely have a good reason to write comments that quickly. I've only seen it twice at all, I think, but both times it was extremely annoying.     Only 1 comment allowed per 15 seconds; timer reset.   The reason is that every time you hit the limit, the 15 seconds begin anew. You have no indicator that tells you when you are allowed to comment again, and if you hit enter after just 14 seconds, you're back to zero. Yesterday, I went through four cycles of not quite 15 seconds, counting in my head, before I got a myself a coffee and tried again afterwards.  I suggest that you do one or more of the following (in decreasing order):   Make the counter non-restarting Add a visible countdown Make it clearly visible that you can't comment right now You can actually increase the waiting time, if it is clearly visible and there is no unnerving invisible countdown. Actually, just remove all comment fields after one comment, and tell the user to reload the page. Just please don't have UI elements that look like they would work, but feel like they are trolling you. Do creatures that enter the battlefield at the same time see each other enter? I want to know a proof of an alternative form of Fermat-Euler's theorem   $$a^{\phi (n) +1} \equiv a \pmod n$$   when $a$ and $n$ are not relatively prime.   I searched some number theory books and a cryptography book and internet, but there were only proofs of the original theorem  $a^{\phi (n)} \equiv 1 \pmod n$ or in case n=pq which is used for RSA. So I would be very thankful if someone show me a proof or book I should read. How to remove configuration files completely Recently I uninstalled some software using sudo apt-get --purge autoremove, thinking that this would remove all traces of it including unused dependencies and configuration files. However I discovered that a configuration file was left behind in my home folder.  Is there a more powerful command which would remove even that?   Also, is it safe to delete the hidden files and folders under home, if they are merely configuration files, or are there other kinds of files? Is there a limit of $\cos(n!)$? I encountered a problem today to prove that $(X_n)$ with $X_n = \cos(n!)$ does not have a limit (when $n$ approaches infinity). I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are used) the better. Thanks