Density in $\mathbb R$
Proving that $m+n\sqrt{2}$ is dense in $\mathbb R$
The world I've been playing crashed recently and took the level.dat with it, so I've recreated it, however I have no idea what my seed used to be, and I really don't want to end up with another corrupted game world from exploring with the wrong seed.  How can I recover the seed which was used to generate my world? How to mount read-write an ext4 partition on Windows? I want to use ext4 on my external HDD, but it must be cross-platform. (At least Linux, Win7 &amp; WinXP).  Unfortunately, I haven't found any proper driver allowing Windows to WRITE to ext4. I've found , but it only reads ext4.     9. ext4 extent read-only, no  size truncating and expanding support   More:     Supported Ext3/4 features:         flexible inode size: > 128 bytes, up to block size   dir_index:    htree directory index   filetype:     extra file mode in dentry   large_file:   > 4G files supported   sparse_super: super block backup in group descriptor   uninit_bg:    fast fsck and group checksum   extent:       reading, writing with no extending.   journal:      only support replay for internal journal         Unsupported Ext3/4 features:         journal: log-based operations, external journal   extent: size truncating &amp; expanding, file deletion   flex_bg: first metadata group   EA (extended attributes), ACL support      Also, I have found a patch for Ext4Fsd v0.50 ():     New: Matt Wu has made a new release 0.50 at   ext2fsd.com.  Support for ext4 extents and fix   for BSOD on Windows 7.   I don't think it will be safe. Also I've read some words about writing ext4 in Windows could cause data loss on the disk...  What is the External HDD used for? - First, it will store backups, then family movies, pictures. Some music, and etc. It will be connected to a Raspberry Pi, if I finally receive it. It won't really move, just when it is used to migrate computer's data to a new one. The largest files it will store are Clonezilla disk images (I don't know, if they will be large files or not).  Can someone provide a solution for this problem? Or is there NO important enough difference between ext4 and ext3, and I should use ext3? - (If yes, then can I use Ext2Fsd for that ext3 mount?)  Thank you! A book on quantum mechanics supported by the high-level mathematics I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators etc, certainly the modern mathematics). If there isn't something similar please give me a reference to the book that is strictly supported by mathematics (given a set of mathematically descripted axioms author develops the theory using mathematics as a main tool). Is there a way to show that $\sqrt{p_{n}} < n$? Is there a way to show that $\sqrt{p_{n}} &lt; n$?  In , I show that $f_{2}(x)=\frac{x}{ln(x)} - \sqrt{x}$ is ascending, for $\forall x\geq e^{2}$. As a result, $\forall n \geq 3$ $$\frac{p_{n}}{ln(p_{n})} - \sqrt{p_{n}}\leq \frac{p_{n+1}}{ln(p_{n+1})} - \sqrt{p_{n+1}}$$ Also (and as a result), $\forall n \geq 3$ $$ \frac{p_{n}}{ln(p_{n})} - \sqrt{p_{n}} &gt; 0$$ Or $$ \frac{\pi (p_{n})}{p_{n}/ln(p_{n})} &lt; \frac{\pi (p_{n})}{\sqrt{p_{n}}}$$  According to  $$\displaystyle\smash{\lim_{n \to \infty }}\frac{\pi (p_{n})}{p_{n}/ln(p_{n})}=1$$ Or, $\forall \varepsilon &gt;0$, $\exists N(\varepsilon )$: $\forall n&gt;N(\varepsilon )$ $$1- \varepsilon &lt; \frac{\pi (p_{n})}{p_{n}/ln(p_{n})} &lt; 1+ \varepsilon$$ Or $$1- \varepsilon &lt; \frac{\pi (p_{n})}{p_{n}/ln(p_{n})} &lt; \frac{\pi (p_{n})}{\sqrt{p_{n}}}$$ As a result $\forall \varepsilon &gt;0$, $\exists N(\varepsilon )$: $\forall n&gt;N(\varepsilon )$ $$(1 - \varepsilon ) \cdot  \sqrt{p_{n}} &lt; \pi (p_{n}) = n$$  But this is not enough.  Interestingly,  is true iff function $f_{4}(x)=\pi (x) - \sqrt{x}$ is strictly ascending ($x &lt; y \Rightarrow f(x) &lt; f(y)$) for prime arguments.  If $f_{4}(p_{n}) &lt; f_{4}(p_{n+1})$ then $$\pi (p_{n}) - \sqrt{p_{n}} &lt; \pi (p_{n+1}) - \sqrt{p_{n+1}}$$ Or $$\sqrt{p_{n+1}} - \sqrt{p_{n}} &lt; \pi (p_{n+1}) - \pi (p_{n}) =1$$  And vice-versa, if $$\sqrt{p_{n+1}} - \sqrt{p_{n}} &lt; 1$$ Then $$-\sqrt{p_{n}} &lt; -\sqrt{p_{n+1}} + 1$$ Or $$\pi (p_{n})-\sqrt{p_{n}} &lt; \pi (p_{n}) + 1 -\sqrt{p_{n+1}} = \pi (p_{n+1}) -\sqrt{p_{n+1}}$$  So, if Andrica's conjecture is true then $\forall n \geq 3$ $$\pi (p_{n})-\sqrt{p_{n}} &gt; 0$$ Or $$\sqrt{p_{n}} &lt; \pi (p_{n})= n$$ Set normal orientation of origin I want to change the orientation of my object to a specified direction.  Say I have an axle of a train whose normal orientation is set off by 90° from the global orientation:    The green arrow (Y-axis) points to the back of the train (along the global X-axis). But I want it to point to the sides of the train, so all I want is to rotate the orientation by 90° without rotating the object itself.  In other words, I want the normal and global orientation of the axle to be the identical:    Is this possible and if so, how?  Background: The game engine to use this model will move the bogies and the axles according to their orientations. After importing this train into the game the axles rotated 90° off. How can I make a tool drop blocks with CanPlaceOn tag In Adventure mode, I'm trying to let a player get blocks that he can place later, is there a way to do it directly or do I have to change them in his inventory?  Situation: The player gets a tool that will allow him to clear a boulder blocking the road. Further down the path, he encounters a broken bridge, and I want him to use the blocks he gathered before to rebuild it.  At the moment, if I build the boulder with blocks having the CanPlaceOn tag, the tag disappear when the player destroys them and they go in its inventory. Firefox running with rare arguments I got this output from ps aux|grep firefox command:  $ ps aux|grep firefox auraham   5443 17.3 11.0 2600628 426484 ?      Sl   18:10   0:54 /usr/lib/firefox/firefox auraham   5496 24.0 17.9 2558108 691700 ?      Sl   18:10   1:15 /usr/lib/firefox/firefox -contentproc -childID 1 -isForBrowser -intPrefs 5:50|6:-1|18:0|28:1000|33:20|34:10|43:128|44:10000|49:0|51:400|52:1|53:0|54:0|59:0|60:120|61:120|91:2|92:1|106:5000|117:0|119:0|130:10000|155:24|156:32768|158:0|159:0|167:5|171:1048576|172:100|173:5000|175:600|176:4|177:1|186:2|200:60000| -boolPrefs 1:0|2:0|4:0|26:1|27:1|30:0|35:1|36:0|37:0|38:0|41:1|42:1|45:0|46:0|47:0|48:0|50:0|55:1|56:1|57:0|58:1|62:1|63:1|64:0|65:1|66:1|67:0|68:1|71:0|72:0|75:1|76:1|80:1|81:1|82:1|83:0|85:0|86:0|87:1|88:0|93:1|94:0|100:0|105:0|108:1|109:0|111:1|112:1|114:1|118:0|120:0|122:0|124:1|125:1|131:0|132:0|133:1|135:0|146:0|153:0|154:0|157:1|160:0|162:1|164:1|165:0|170:0|174:1|179:0|180:0|181:0|182:1|183:0|184:0|185:1|188:1|192:0|193:0|194:1|195:1|196:0|197:1|198:1|199:1|201:0|202:0|204:0|212:1|213:1|214:0|215:0|216:0| -stringPrefs 3:7;release|134:3;1.0|151:332;  ¼½¾ǃː??։֊׃״؉؊٪۔܁܂܃܄ᅟᅠ᜵           ???‐’․‧??????? ‹›⁁⁄⁒ ⅓⅔⅕⅖⅗⅘⅙⅚?⅜⅝⅞⅟∕∶⎮╱⧶⧸⫻⫽⿰⿱⿲⿳⿴⿵⿶⿷⿸⿹⿺⿻　。〔〕〳゠ㅤ㈝㈞㎮㎯㏆㏟꞉︔︕︿﹝﹞?．／｡ﾠ???￼�|152:8;moderate| -greomni /usr/lib/firefox/omni.ja -appomni /usr/lib/firefox/browser/omni.ja -appdir /usr/lib/firefox/browser 5443 true tab auraham   5694  0.0  0.0  15444  1092 pts/2    S+   18:15   0:00 grep --color=auto firefox   According to , this error occurs after firefox crashes. However, every time I run Firefox again, that large line of arguments appears too.  I already try these things:   Run Firefox with addons disabled Reinstall Firefox (sudo apt-get purge firefox; sudo apt-get install firefox)   However, it did not work.  Is it possible to remove such a line of arguments? Update: Failed to lock /var/cache/apt/archives/lock I am running the upgrade(12.10 to 13.04) through terminal and it keeps getting stuck at this:  Failed to lock /var/cache/apt/archives/lock   What do I do? I have root access. I have no idea what else to try other than a clean install.  Checking package manager Reading package lists... Done     Building dependency tree           Reading state information... Done Building data structures... Done   Calculating the changes Calculating the changes  Do you want to start the upgrade?   3 packages are going to be removed. 155 new packages are going to be  installed. 1204 packages are going to be upgraded.   You have to download a total of 595 M. This download will take about  1 hour 15 minutes with a 1Mbit DSL connection and about 23 hours with  a 56k modem.   Fetching and installing the upgrade can take several hours. Once the  download has finished, the process cannot be canceled.    Continue [yN]  Details [d]y  Fetching  Could not download the upgrades   The upgrade has aborted. Please check your Internet connection or  installation media and try again. All files downloaded so far have  been kept.   Failed to lock /var/cache/apt/archives/lock    Restoring original system state  Aborting Reading package lists... Done     Building dependency tree           Reading state information... Done Building data structures... Done  root@chaitanya-XPS-Ubuntu:~# example of a non differentiable manifold Take the manifold: The graph of $|x|$ on $(-1,1)$, with the induced topology from $\mathbb{R}^2$. This is a topological manifold, which is homeomorphic to $(-1,1)$ by projection. Is it a differentiable manifold? I believe it is, because we can take an atlas with only the projection, and then we will have only one transition map which is the identity, and therefore differentiable. But I'm not sure I get the definition of a differentiable manifold right...  Thanks! Why would the First Order have lightsaber-resistant batons? In The Force Awakens there is a new Stormtroopr who is equipped with a lightsaber-resistant baton.    Except for one thing:   In Revenge of the Sith, most of the Jedi are killed. During the original Star Wars trilogy the last few remaining Jedi — except for Luke Skywalker — die.     And before The Force Awakens Kylo Ren kills off everyone at Luke’s school and drives Luke into exile.    There are practically no lightsabers in existence, so why are these weapons deployed at all if their only use is in counter-lightsaber combat? Did Darth Sidious and Vader ever discover how to cheat death? In Episode 3 Revenge of the Sith Palpatine/Sidious strings Anakin along to the dark side with the promise that together the two of them will be able to re-discover Darth Plagueis' secret to cheating death.      "To cheat death is a power only one has achieved, but if we work together, I know we can discover the secret."   From the end of Revenge of the Sith to their deaths in Return of the Jedi did Darth Sidious and Darth Vader ever manage to re-discover this secret? Topological Circuit Simulator Does something like  exist for topological (eg. braided) circuits?   Alternatively, any ideas on how @ is getting  (or something similar)? How can I find the difference between two angles? If neutrinos are disfavoured as DM candidates why aren't axions? Can someone please give me the prerequisites and mathematics required for string theory? Are there some good references to study it, both online and in a book? Please consider I am a newbie in string theory, and only about up to the level explained in Hawking's Brief History of Time and Penrose's Road to Reality. I would prefer something which does not omit rigor, but builds it up.Also, are there good online lectures on this?  (I have covered most of GR, but I am not really comfortable with advanced Quantum mechanics. I have studied QM only from Griffith's book, and a few McGraw hill lecture notes.) I use ubuntu to deal with daily job. Now I buy a new laptop, and I want to migrate my whole system(programs, packages, data, settings, etc) to the new machine.  I know I can copy my home folder to new machine and I will get (almost) all data and settings. But is there a safe way to copy the whole system? Or it's a bad idea? Can anyone recommend a good exposition of the theory behind partial least squares regression (available online) for someone who understands SVD and PCA?  I have looked at many sources online and have not found anything that had the right combination of rigor and accessibility.    I have looked into The Elements of Statistical Learning, which was suggested in a comment on a question asked on Cross Validated, , but I don't think that this reference does the topic justice (it's too brief to do so, and doesn't provide much theory on the subject).  From what I've read, PLS exploits linear combinations of the predictor variables, $z_i=X \varphi_i$ that maximize the covariance $ y^Tz_i $ subject to the constraints $\|\varphi_i\|=1$ and $z_i^Tz_j=0$ if $i \neq j$, where the $\varphi_i$ are chosen iteratively, in the order in which they maximize the covariance.  But even after all I've read, I'm still uncertain whether that is true, and if so, how the method is executed. Role of dependencies while installing software?? What is the role of dependencies while installing any software?I was installing vlc player and it were closed by giving me error something like check dependency .  What is to solution of this and how i'll get these dependencies??  Thanks in Advance. What do we lose passing from the reals to the complex numbers? As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we lose moving from the reals to the complex numbers? Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem. In terms of purely set theory, the axiom of choice says that for any set $A$, its power set (with empty set removed) has a choice function, i.e. there exists a function $f\colon \mathcal{P}^*(A)\rightarrow A$ such that for any subset $S$ of $A$, $f(S)\in S.$ Is this correct?  My question then is about proving this fact, so that we do not need to put it as an axiom. Now as per the research done on this single object- Axiom of Choice, I believe here that there should be some falsity in my argument. I do not find the mistake.  For any $S\in \mathcal{P}^*(A)$, since $S\neq \emptyset$, $\exists s\in S$. Define $f(S)=s$. Then $f$ is a choice function.  This was showing me that the axiom of choice is proved, but then why it had been put as an axiom? For example, in  book, the author asserts that      It is a metatheorem of mathematical logic that it is impossible to specify the function that assigns to each non-empty subset of $\mathbb{R}$, an element of itself.     There are several notes and books on axiom of choice, but here I am trying to understand through doing some argument for some problem, where problem actually arises.