Are all uncountable infinities greater than all countable infinities? Are some uncountable infinities greater than other uncountable infinities? I recently finished a discrete mathematics class, and near the end of the semester, the prof (very superficially) touched on countable and uncountable infinities. His explanation of countable infinities began and ended with the example of the cardinal number set, while he explained that, for example, the set of all functions $f$ such that $f:\mathbb{R}\mapsto\mathbb{R}$ is, in fact, uncountable.  My question is threefold:   Does it make sense to say that any uncountable infinity is larger than any countable infinity? Does it make sense to say that certain countable infinities are larger than others? Finally, does it make sense to say that certain uncountable infinities are larger than others?   For example, is the set of all reals larger than the set of reals between any two integers? Can arithmetic-like rules be applied if this is true (addition of infinities makes them larger)?
What Does it Really Mean to Have Different Kinds of Infinities? Can someone explain to me how there can be different kinds of infinities?  I was reading "" by  and came across the concept of countable and uncountable infinities, but they're only words to me.  Any help would be appreciated.
After I read that companions don't drain the fusion core of a Power Armor when wearing it, I decided to try giving my armor to Nick so I'll always have it close by if I need it.  But I was wondering if there are any other drawbacks to that. For example, does the Armor get damaged or destroyed when the companion is under fire? Django templates offer the builtin tag cycle for alternating between several values at different points in a template (or for loop in a template) but this tag does not reset when it is accessed in a scope outside of the cycles definition.  I.e., if you have two or more lists in your template, the rows of all of which you'd like to use some css definitions odd and even, the first row of a list will pick up where the last left off, not with a fresh iteration from the choices (odd and even)  E.g., in the following code, if the first blog has an odd number of entries, then the first entry in a second blog will start as even, when I want it to start at odd.  {% for blog in blogs %}   {% for entry in blog.entries %}     &lt;div class="{% cycle 'odd' 'even' %}" id="{{entry.id}}"&gt;       {{entry.text}}     &lt;/div&gt;   {% endfor %} {% endfor %}   I've tried obviating this by patching with the resetcycle tag offered here:    to no avail.  (The code didn't work for me.)  I've also tried moving my inner loop into a custom tag, but this also did not work, perhaps because the compile/render cycle moves the loop back into the outer loop?  (Regardless of why, it didn't work for me.)  How can I accomplish this simple task!?  I'd prefer not to create a data structure in my view with this information pre-compiled; that seems unnecessary.  Thanks in advance. I submitted a manuscript last year to an IEEE journal, and the paper was accepted for publication. I recently submitted a new manuscript to the same journal, but, this time, the paper was rejected, and I was told that I should submit to another journal since my paper was not within the journal's scope. The two papers both target similar applications. Furthermore, the second paper uses some work in the first paper, making substantial improvements at solving a much more difficult problem. Based on the reviewer's comments (and my own personal assessment), I do not believe that the work presented in the second paper was "incremental" in nature. Far from it. Two of the three reviewers gave positive reviews. The third reviewer, however, mentioned the "outside of journal's scope" issue and the associate editor and editor-in-chief both sided with the 3rd reviewer, and the paper was stopped dead in its tracks.  I'm scratching my head trying to come to terms with how in the world my first paper could be accepted by this journal, but the second paper focused on the same application space and achieving substantial improvements on a much more difficult problem (backed up with measured results) could be rejected. Any ideas? Proof that there must exist infinitely many solutions to a homogeneous system of equations when more unknowns than equations i have a huuuuge problem.  Created a snapshot and image from my instance...instance has 1000GB of disk.  Now i need to reduce it but everytime i try to create an image or a snapshot from that, i get the same error "Disk need to be at least 1000GB".  There is a way to reduce disk size directly on instance? Some workaround? It has come to the point in my computing program now where I have 5 swinging pendulums that are all modified at once by slider values. These values are drawn onto the from and passed through the class pendulum. To change the values the update button is run, and their is a default option to use a setup that works. The values are also shown on screen through some simple labels. The zero button sets all values (except length) to zero.  The values for each pendulum drawn is:   String length angle Acceleration (angular) Angluar velocity Damping or friction Gravity   My calcuations for these values: (Bob = the weight at the end of the string, and there are X and Y Co-ordiantes for it.  bobX = originX + (Math.Sin(angle) * length);         bobY = originY +(Math.Cos(angle) * length);   Caclulate Acceleration:  Angular Acceleration = (-1 * gravity / length) * Math.Sin(angle);    Increment Velocity:          Angular Velocity += Angluar Acceleration;   Increment Angle          angle += AngularVelocity;   Friction action, pendulum eventually 0's          AngularVelocity*= damping;   Now my next task is to make the pendulums collide "realistically" and recreate a newtons cradle type effect. I've looked into elastic collision solutions on particle e.g. here is an excellent example:    I've also looked at the raw physics behind it  but my brain couldn't get to grips with the raw physics side of things.  So I'm wondering if anybody could help me figuring out the theory for the collisions.  My current idea is when the co-ordinates meet the velocity will simply change polarity e.g. +10 to -10 in the opposing pendulum. Am I correct here?  If you are into code I've posted the code on stack overflow How do I create a dynamic key to be added to a JavaScript object variable Prove using induction that from a set of $n+1$ numbers from $\{1,2,..2n\}$, at least one number will evenly divide another. Given a set of $n+1$ numbers out of the first $2n$ natural numbers, $\{1,2,\ldots,2n\}$, prove that there are two numbers in the set, one of which divides the other.  I can't tell if I'm reducing the problem in the right way.  Would it be something like:  Induction Hypothesis: A set of $n+1$ numbers from the first $2n$ natural numbers contains two numbers, one of which evenly divides the other.  Base Case: $n=1$, $\{1,2\}$, 2 is divisible by 1.  Consider a set of $n+2$ numbers, $A$, from the first $2n+2$ natural numbers. There are three cases:   Each number in $A$ is less than or equal to $2n$.  The proof follows from the induction hypothesis. One number in $A$ is greater than $2n$.  The remaining set of $n+1$ numbers must each be less than or equal to $2n$.  The proof follows from the induction hypothesis. $A$ contains both $2n+1$ and $2n+2$. I'm not sure how to solve this case.   I'm working through Udi Manber's Introduction To Algorithms: A Creative Approach for personal development, so in keeping with the spirit of the book, I'm trying to use induction.  Without induction, the question is identical to Event binding on dynamically created elements? Software to draw illustrative figures in papers I would like to have suggestions of good software for drawing illustrations in research papers. I already know about , but this works only on Linux and is at times, clunky when it comes to text. Moreover the resolution is not always perfect making it difficult to manoeuvre the objects. Besides it is tough to learn and master, with all its weird click procedures.   I would love to know about better alternatives. Not talking about graphs here, just block diagrams and explanatory illustrations. What is the difference between implication symbols: $\rightarrow$ and $\Rightarrow$? I do not understand the difference between $\rightarrow$ and $\Rightarrow$. Sometimes I see implication truth tables labeled with the former, sometimes with the latter.  Aren't they synonyms of logical implication or is there any difference? I'm just wondering how it is possible that IGN.com has this in its title tag:      Video Games, Cheats, Walkthroughs, Game Trailers, Reviews, News, Previews &amp; Videos at IGN   while on Google its title is simply "IGN".   How can one achieve such a thing? Does it use some special meta tag I can't see? I would like to know where gravity gets its energy to attract physical bodies?  I know that the law of conservation states that total energy of an isolated system cannot change. So gravity has to be getting its energy from somewhere, or else things like hydropower plants wouldn't be able to turn the power of the falling water into a spinning rotor.  Just to be clear, Lets create an example:  Lets say we have two objects with equal mass close to each other. So gravity does its job and it pulls each other closer, this gets turned into kinetic energy. This is where I'm lost. According to the law of conservation energy can't be created or destroyed and the kinetic energy comes from the gravitational pull so where does the gravitational pull gets its energy.  If that energy isn't being recycled from some where else then that means you have just created energy, therefore breaking the law of conservation. According to the official sources, is the radioactive green crystal that serves as Superman's Achilles' heel actually composed of the noble gas krypton, or is it supposed to be made out of some other substance? If so, what is that substance? HP PSC 1410 all scanner's images are black I have a HP PSC 1410 printer with scanner inside.  I've been scanning documents and images with it for the last couple of years.  Recently my scanner has stopped performing as excepted and i can't scan or copy images\documents.  My problem is that every images\document i'm trying to scan - the result from the printer is a black page.  I've been googling about this issue and couldn't find any helpful stuff beside replacing the whole printer - which is something i'm trying to avoid.  Any ideas ? I've read something about the shipping lock of the scanner but couldn't find any information about how to check it.. I am building a java server that needs to scale. One of the servlets will be serving images stored in Amazon S3.  Recently under load, I ran out of memory in my VM and it was after I added the code to serve the images so I'm pretty sure that streaming larger servlet responses is causing my troubles.  My question is : is there any best practice in how to code a java servlet to stream a large (>200k) response back to a browser when read from a database or other cloud storage?  I've considered writing the file to a local temp drive and then spawning another thread to handle the streaming so that the tomcat servlet thread can be re-used. This seems like it would be io heavy.  Any thoughts would be appreciated. Thanks. Allowing FTP access to a local user without allowing SSH/command line access Finding the radius of a third tangent circle Sorry if this is a foolish question, but I'm having difficulty understanding how to solve for $r_3$ in the following diagram...    According to WolframAlpha's page on tangent circles, the radius of $c_3$ can be calculated using the following formula $r_3=\frac{r_1 \times r_2}{(\sqrt{r_1}+\sqrt{r_2})^2}$, which can be simplified to $\frac{1}{\sqrt{r_3}}=\frac{1}{\sqrt{r_1}} + \frac{1}{\sqrt{r_2}}$.  To be completely honest, I'm having a hard time understanding understanding how this formula works (as I'm a very visual person). What exactly is happening here? help  displays information about builtin commands. What is the source of help? Does it maintain a database for builtin commands, or does it read some files of each builtin commands (similar to the manpage of each utilities)?  Sometimes I find its information seems to expand that by --help  $ cd --help bash: cd: --: invalid option cd: usage: cd [-L|[-P [-e]]] [dir]  $ help cd cd: cd [-L|[-P [-e]]] [dir]     Change the shell working directory.      Change the current directory to DIR.  The default DIR is the value of the     HOME shell variable.      The variable CDPATH defines the search path for the directory containing     DIR.  Alternative directory names in CDPATH are separated by a colon (:).     A null directory name is the same as the current directory.  If DIR begins     with a slash (/), then CDPATH is not used.      If the directory is not found, and the shell option `cdable_vars' is set,     the word is assumed to be  a variable name.  If that variable has a value,     its value is used for DIR.      Options:         -L  force symbolic links to be followed         -P  use the physical directory structure without following symbolic         links         -e  if the -P option is supplied, and the current working directory         cannot be determined successfully, exit with a non-zero status      The default is to follow symbolic links, as if `-L' were specified.      Exit Status:     Returns 0 if the directory is changed, and if $PWD is set successfully when     -P is used; non-zero otherwise.   I thought help extract the help part of the executable, but for a  python script , it isn't  $ help ./pdf-merge.py bash: help: no help topics match `./pdf-merge.py'.  Try `help help' or `man -k ./pdf-merge.py' or `info ./pdf-merge.py'.  $ ./pdf-merge.py --help usage: pdf-merge.py [-h] [-v] [--ask] [--output OUTPUT] [--title TITLE]                     [--author AUTHOR] [--keyword KEYWORD] [--pdftk PDFTK]                     [--gs GS] [--pdfmarks PDFMARKS] [--unicode]                     PDF [PDF ...]  Merge PDFs preserving bookmarks. Thanks to Larry Cai for suggesting that Unicode be supported and for discussion about the `--pdfmarks` option.  positional arguments:   PDF                  an input PDF to merge  optional arguments:   -h, --help           show this help message and exit   -v, --version        show program's version number and exit   --ask                pause for manual pdfmark tweaking   --output OUTPUT      name of the output PDF   --title TITLE        title of output PDF   --author AUTHOR      author of output PDF   --keyword KEYWORD    keywords for the output PDF   --pdftk PDFTK        path to the pdftk executable   --gs GS              path to the gs (Ghostscript) executable   --pdfmarks PDFMARKS  path to pdfmarks file. If not given, a temporary file                        is used. If given and the file is missing, execution                        will stop after the file is created (before the                        Ghostscript run). If given and the file exists, no                        attempt will be make to use pdftk to generate the mark                        file (I assume your input file is what you want).   --unicode            instead of merging PDFs, convert PDF-formatted unicode                        strings. For example `--unicode '&lt;FEFF03B103B203B3&gt;'                        \u03b1\u03b2\u03b3` Is it possible to get the progress of an XMLHttpRequest (bytes uploaded, bytes downloaded)?   This would be useful to show a progress bar when the user is uploading a large file. The standard API doesn't seem to support it, but maybe there's some non-standard extension in any of the browsers out there? It seems like a pretty obvious feature to have after all, since the client knows how many bytes were uploaded/downloaded.  note: I'm aware of the "poll the server for progress" alternative (it's what I'm doing right now). the main problem with this (other than the complicated server-side code) is that typically, while uploading a big file, the user's connection is completely hosed, because most ISPs offer poor upstream. So making extra requests is not as responsive as I'd hoped. I was hoping there'd be a way (maybe non-standard) to get this information, which the browser has at all times.