The $3x+1$ conjecture is defined on $\mathbb{N}$ as  $$ f(x) = (3x+1)/2 $$ if $x$ is odd and $$f(x)=x/2$$ if $x$ is even.  As far as I know the only periodic orbit is the trivial  $$1\to 2 \to 1\to 2....$$  If we extend the domain to include the integers, then I found nontrivial periodic orbits such as $$-5\to -7\to -10\to -5\to -7\to -10.....$$  and $$-1\to -1\to -1\to -1....$$  and $$ -91\to -136\to -68\to -34\to -17\to -25 \to -37 \to -55 \to -82 \to-41\to -61\to -91 .....$$  My first question is " Are these the only periodic orbits in the extended $3x+1$ conjecture ?"  The other question is "Are all orbits in the extended $3x+1$ conjecture  eventually periodic?"
Does applying the Collatz conjecture algorithm to negative integers always result in sequences ending with $-1$ and then repeating themselves an infinite number of times? It is so for the first several seed values.  After hours of thinking and research, I don't know how to answer the question.
Boot to console (standard method fails) I'm trying to get Ubuntu to boot directly to the console. I don't want to completely remove the GUI because I still kind of need it for training wheels sometimes, but I don't want to deal with it if I don't need it. In the past I have been able to achieve this pretty easily in previous versions by editing /etc/default/grub, but for some reason it's not working on 15.10. Here is my /etc/default/grub file: # If you change this file, run 'update-grub' afterwards to update # /boot/grub/grub.cfg. # For full documentation of the options in this file, see: #   info -f grub -n 'Simple configuration'  GRUB_DEFAULT=0 GRUB_HIDDEN_TIMEOUT=0 GRUB_HIDDEN_TIMEOUT_QUIET=true GRUB_TIMEOUT=10 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` #GRUB_CMDLINE_LINUX_DEFAULT=&quot;quiet splash&quot; GRUB_CMDLINE_LINUX=&quot;text&quot;  # Uncomment to enable BadRAM filtering, modify to suit your needs # This works with Linux (no patch required) and with any kernel that obtains # the memory map information from GRUB (GNU Mach, kernel of FreeBSD ...) #GRUB_BADRAM=&quot;0x01234567,0xfefefefe,0x89abcdef,0xefefefef&quot;  # Uncomment to disable graphical terminal (grub-pc only) GRUB_TERMINAL=console  # The resolution used on graphical terminal # note that you can use only modes which your graphic card supports via VBE # you can see them in real GRUB with the command `vbeinfo' #GRUB_GFXMODE=640x480  # Uncomment if you don't want GRUB to pass &quot;root=UUID=xxx&quot; parameter to Linux #GRUB_DISABLE_LINUX_UUID=true  # Uncomment to disable generation of recovery mode menu entries #GRUB_DISABLE_RECOVERY=&quot;true&quot;  # Uncomment to get a beep at grub start GRUB_INIT_TUNE=&quot;480 440 1&quot;  As you can see, I have commented out quiet splash, changed the value of GRUB_CMDLINE_LINUX to &quot;text&quot;, and uncommented GRUB_INIT_TUNE. However, the only thing that seems to make a difference is the quiet boot. When I add and remove the hash from that line it affects the behavior when the machine boots, so I know grub is updating when I tell it to. But no matter what I do it still loads the GUI immediately. Today I came across an OBVIOUSLY good answer in /review. It had 7 upvotes, contained lists, paragraphs of explanation, and well-applied sample code. It was also moderately long. How does a question like this get flagged for review? I don't really know how the algorithm works.   If this is low-quality then I feel bad about all of my answers.  Here's a  if you care to see it in context. It's not really a huge problem (as Bart's comment emphasizes), but this seems like an extreme outlier in every measurable aspect. Is it possible to have the Corruption and the Crimson Biomes in the same world? I was reading on the wiki how there is a chance that the Corruption will be replaced with the Crimson.  I am wondering if when you get a world where you have the Crimson is it possible to also have the Corruption if you bring over corruption seeds from another world and use them. Or would these seeds just get you more of the Crimson? Prove that if the positve term series $\sum^{\infty}_{n=1}a_n$ is convergent, also $\sum^{\infty}_{n=1}\sqrt{a_na_{n+1}}$ is convergent.  Prove that if the positive term series $\sum^{\infty}_{n=1}a_n$ and $\sum^{\infty}_{n=1}b_n$ are convergent, also $\sum^{\infty}_{n=1}a_nb_n$ is convergent.  I've tried to solve it using comparison test, but no results. Fact  The intersection of a closed set $A$ and a compact set $B$ is closed.  Background  I wanted to prove that the intersection of closed and compact sets is compact. There might be other ways, but I am not interested in them for now and am looking to prove the above statement.  Relevant Facts  I don't have much idea what to use. $A^c$ is open. Every cover $\cup_{i\ge1}U_i$ of $B$ contains a subcover $\cup_{i=1}^kU_i$. I might want to prove $A\cap B^c$ is open. Limit points may be another points to consider but I don't get any idea in that direction.  I also don't have that $B$ is closed otherwise it would be trivial.  Some Clarification  I am aware that compact sets are not closed in arbitrary topological spaces. The reason why I was motivated to believe tha intersection of closed and compact set is compact is true is: See Matt E.'s Answer      So if $K$ is compact in an arbitrary topological space (which is just to say that   it is a compact topological space when given its induced topology) and     $L$ is closed then $K \cap L$ is a closed subset of $K$ in its induced topology   Is it true if $A$ and $B$ are not whole spaces?  I would appreciate any hints. I am trying to solve this exercise from Jech's book on set theory:  Ex. 3.6: There is a well ordering of the class of all finite sequences of ordinals such that for each $\alpha$, the set of all finite sequences in $\omega_{\alpha}$ is an initial segment and its order-type is $\omega_{\alpha}$. Definition: If $W$ is a well ordered set and $u \in W$, then the set $\{ x \in W \mid  x &lt; u \}$ is an initial segment in W  Any help? I have played and tried to GM games since I was in elementary school, beginning with games like Shadowrun, Earthdawn, and Champions. Recently, I've gotten dack into gaming with a variety of other games (newer Shadowrun and Earthdawn, plus TOON! and Outbreak:Undead).  I've found examples of actual play and advice on how to actually GM a game in games to be lacking. I can only help but feel I am not alone as someone who isn't already a LITERAL dungeon MASTER.  I fully realize that there are an unlimited number of GM styles, personalities, etc. I do not think I am a very good GM though. No one else wants to be the GM, and few care enough to learn the complex rules that I often have to assist with and explain AS WE PLAY. Being left with the responsibility is quite overwhelming unless I create dynamic encounters or make it all up on the spot. I'm often left having no idea what to do.  Examples of Why I Think I Need Help  This question isn't about these examples so please don't dig into them in the answers - they're here to explain the basic things that leave me feeling like I don't know how to proceed as a GM.  I've read preparation is important, but last time I tried to make a story, the players immediately killed the primary character (a friendly person.) within about 2 minutes of starting the game. While I realize now that I let the dice take control over the story, it was rather difficult to continue a story that relied on this friendly character, with the players making enemies with the character's faction. It is still discouraging when my only other attempt at a prepared story ended with the players wanting to go in a complete opposite direction of where I planned. Both times, the hours of work I put into it were completely wasted, leaving me as ignorant as I always was before.  My group often wants to play Shadowrun, Earthdawn, or L5R- but I feel as though I am a horrible GM. The last time we played Shadowrun, I was bored out of my mind as we rolled hundreds of dice to do an obvious straight-forward "kick down the door" mission.   If I make security as I believe it should or as the forum threads describe their security or a few example missions I can scrounge up (examples of stories, not actual gameplay), my players literally feel completely hopeless as to how to achieve their mission without total failure.   One time when I was a teenager, I had players blow up at me saying "We have no idea what would work. You're telling us that our character's experience or the ally operative says that would get us killed, but what else can we do?" Since there are no examples of what players can do - I am left with the same feeling as the players. As an adult, I've had a similar occurrence where a player said to me, "I don't know what to do. I don't think we can even begin this mission." I had to make something up and just go with it, creating an NPC group (that fit in with the story) that goes in before the players to do something stupid so they don't have to. Mainly because I ALSO didn't know how to accomplish the mission, so I acted as the players would using my NPC group. Extremely boring and IMO a failed dungeon run.  I've had successful stories, but I've had more boring encounters than successful ones. Seeing as how roleplaying takes endless hours to complete a story, I do not see benefit in an activity that wastes more time than is enjoyable.  How Do You Learn To GM?  Where are the resources? Books, blogs, gameplay examples? Am I missing them? How did YOU learn to be a good GM? Or do you feel as bad at this as I do, even when it's a successful, fun encounter? Mount ext4 partition with >4KiB block size I'm attempting to mount a partition from a NAS and the block size is 65536 bytes, this is above my distribution's precompiled kernel memory paging size of 4KiB, so I cannot mount it normally.  Which kernel compilation parameters would I need to change to get a memory paging size large enough?  Would it be possible under the x86-64 architecture?  EDIT: There doesn't seem to be an option for it in make xconfig unless I'm just missing it.  From page_types.h under arch\x86\include\asm  /* PAGE_SHIFT determines the page size */ #define PAGE_SHIFT  12 #define PAGE_SIZE   (_AC(1,UL) &lt;&lt; PAGE_SHIFT) #define PAGE_MASK   (~(PAGE_SIZE-1))`   It doesn't look to be configurable.  I could change this number directly to 16 and it would probably make pretty explosions and destroy Tokyo.  I shall probably try it later this evening. Package pgfplots Error: CRITICAL: shader=interp: got unsupported pdf shading type '0' For the set of data , the following code can't compile showing the error     Package pgfplots Error: CRITICAL: shader=interp: got unsupported pdf   shading type '0'. This may corrupt your pdf!.   \documentclass[border= 5mm]{standalone} \usepackage{pgfplots} \pgfplotsset{compat=newest} \begin{document}     \begin{tikzpicture}     \begin{axis}[unbounded coords=jump, view={70}{40}]     \addplot3 [surf,shader=interp]      table [x index={0}, y index={1}, col sep=tab] {mySurfaceData.txt};     \end{axis}     \end{tikzpicture} \end{document}    So, what went wrong to cause this?  P.S. the original data without NaN is available , where I would like to skip plotting those points of z=1. How does Java calculate negative numbers? How to know the types of windowing system, window manager and desktop environment of a Unix-like OS Given a pointed category $X$, is $|N(X)|$ contractible ($N(X)$ is the nerve of $X$)? Or are there counter-examples? Script to convert filenames to lowercase depending on extension Count number of increasing functions, nondecreasing functions $f: \{1, 2, 3, \ldots, n\} \to \{1, 2, 3, \ldots, m\}$, with $m \geq n$. I stumbled upon a question given like:  Let $m$ and $n$ be two integers such that $m \geq n \geq 1$.   Count the number of functions $$f: \{1, 2, · · · , n\} \to \{1, 2, · · · , m\}$$ of the following two types:  (a) strictly increasing; i.e., whenever $x &lt; y, f(x) &lt; f(y)$, and  (b) non-decreasing; i.e., whenever $x &lt; y, f(x) \leq f(y)$.  I tried in the following way.  For (a). We can say $f(n)&gt;f(n-1)$ AND $f(n)&gt;f(n-2) \ldots f(n)&gt;f(1)$ (total $n-1$ elements)  Similarly for $f(n-1)&gt;f(n-2), f(n-1)&gt;f(n-3) \ldots f(n-1)&gt;f(1)$ (total $n-2$ elements)...  and $f(2)&gt;1$  Like this $$(n-1) + (n-2) + \ldots + (1) = \frac{n(n-1)}{2}$$  Is this correct?  For (b). Since there's equality, it will be $$n + (n-1) + ... (1) = \frac{n(n+1)}{2}$$  Is my approach correct? Any help is appreciated. Thanks in advance. Literature: 'Why' at the beginning of sentences I’m currently reading George Martin’s A Song Of Ice And Fire novels in English. As a non-native speaker (I’m German), I stumbled upon some grammatical constructs that I’ve never seen before, one of them being the Why at beginning of sentences.  Examples:   Tyrion gave the rotting head a second look. Why, it almost looks as if those lips are smiling. Did he use that word? Why, the boy has a singer's soul . . . though if you believe that song, you may well be dimmer than the first Reek.   These kind of sentences are obviously different from sentences like Why is the sky blue?  I’d like to know if that construct is archaic, or could be used in everyday language, and how one could rephrase that kind of sentences. Can it be rephrased at all? To me it seems like the Why is just used to give the sentence a kind of “cynical” taste. Why my U.S. B1 Visa got refused under 214 (b)? I'm a US citizen and I've already gotten my Tier 4 visa to study in the UK in my passport, however, it looks like the start date on the visa and the letter to present to the Immigration Officer has allowed me entry an extra week on top of the usual month. My school starts on September 24th, but my visa states permission to enter on August 17th. I've read that you are only allowed a month before the beginning of school, but my visa is saying differently!  Should I assume its wrong and enter on the 24th or is it okay for me to enter on the 17th?   Thanks. Why use definition lists (DL,DD,DT) tags for HTML forms instead of tables? What is so special about speed of light in vacuum? Can matter really fall through an event horizon?