Story that includes a description: "Two concentric circles, intersecting at three points"
Children's book with "square triangles" in parallel universe?
I was just thinking about this recently, and I thought of a possible bijection between the natural numbers and the real numbers.  First, take the numbers between zero and one, exclusive.  The following sequence of real numbers is suggested so that we have bijection.    0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.01, 0.02, ... , 0.09, 0.10, 0.11, ... , 0.99, 0.001, 0.002, ... , 0.999, 0.0001, etc.  Obviously, this includes repeats, but this set is countable.  Therefore, the set of all numbers between zero and one is a subset of the above countable set, and is thus countable.  Then we simply extend this to all real numbers and all the whole numbers themselves, and since the real numbers, as demonstrated above, between any two whole numbers is countable, the real numbers are the union of countably many countable sets, and thus the real numbers are countable.    Please help me with this.  I understand the diagonalization argument by Cantor, but I am curious specifically about this proof which I thought of and its strengths and flaws.  Thanks. Countries with no visa requirements on its visitors An elementary proof of $\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \frac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2$ Question on groups of order $pq$ Let $G$ be a group of order $pq$, $p&gt;q$  and $p$, $q$ are primes. Then prove that  If $q\mid p-1$ then there exists a non abelian group of order $pq$. Any two non-abelian groups of order $pq$ are isomorphic.   I have proved that if $q\not\mid p-1$ then $G$ is cylic . But how to prove this one I have no idea. Any kind of hint is very much welcome. This problem is in Herstein book, page 75. Recommendations for good Newtonian mechanics and kinematics books What are some good books for learning the concepts of Kinematics, Newton laws, 2D Motion of Object etc.? How to find the joint PDF? Ley X,Y two random variables with density $f(x,y)=8xy \  \text{if} \   0&lt;x&lt;y&lt;1$  Find the joint distribution F(x,y).  I find that $F(x,y)=2x^2y^2-x^4\ \text{  if}  \ 0&lt;x&lt;y&lt;1$ but I dont know how to find the didtribution in other points. Pages documents created on Yosemite cannot be opened by Pages on Yosemite after latest update (5.5.2) Seemingly every single document that I created in the previous version of Pages (I update regularly whenever I see that a new update is available) on this same MacBook Pro with Yosemite will now not open with Pages after the latest update (5.5.2). I am still on Yosemite and nothing has changed aside from updating Pages, Numbers &amp; Keynote.  All of my documents are stored on my internal hard drive. When I attempt to open them I get a message stating the following:     This document can't be opened because it's too old. To open it, save it with Pages '09 first.   As I mentioned above, however, these documents were created in whichever version was right before the current version on the same computer with the same operating system (Yosemite).  Can anybody help? These are time sensitive documents. For now, I may try opening them with my iPad, but wow, if I knew this was going to happen I certainly would not have updated. .NET Attributes: Why does GetCustomAttributes() make a new attribute instance every time? How do I do a SELECT * INTO [temp table] FROM [stored procedure]? Not FROM [Table] and without defining [temp table]?  Select all data from BusinessLine into tmpBusLine works fine.  select * into tmpBusLine from BusinessLine   I am trying the same, but using a stored procedure that returns data, is not quite the same.  select * into tmpBusLine from exec getBusinessLineHistory '16 Mar 2009'   Output message:     Msg 156, Level 15, State 1, Line 2   Incorrect syntax near the keyword   'exec'.   I have read several examples of creating a temporary table with the same structure as the output stored procedure, which works fine, but it would be nice to not supply any columns. Making an MFC application international I've got several large MFC applications here, and converting them into any other format is out of the question.  We're expanding into other markets, and would like to make the apps work in other languages and cultures.  So far, I've found occasional references as to what to do with Visual C++ version 6, with one mention that later versions of MFC have additional relevant features.  Searching MSDN gives me instructions on how to convert the apps to Unicode, which we already did.  I found nothing on MSDN on how to make languages multilingual once they're in Unicode, only a few older things using Google, and one book about internationalization using VC++6.  (We're using Visual Studio 2008 now, on XP and Vista.)    I make no claims for the strength of my Google-fu, and would be happy to be directed to things I've missed.  Is it reasonable to use the methods of VC++6, or should I use later features?  If I should use features later than that, where can I find some reference to them?  Is there something I should worry about other than setting the locale, converting all strings to resources, and duplicating resources in different languages?  I can find more things about .NET internationalization, but I'm inexperienced in .NET.  What will I find there that's applicable to VC++ and MFC?  Edit:  I just ran into difficulties trying to put Unicode strings into the String Table resource.  (As far as I can tell, Unicode strings need to be entered into the .rc file with a text editor, in L"0x0034" form.  I haven't found a resource editor way to do that.)  Any tips on that?  Any other resource I can use in VS2008 using VC++ and MFC?  Edit:  Somebody on a Microsoft forum suggested that I open the offending .rc file in Notepad, and save it in Unicode.  Unfortunately, this doesn't seem to fix the dialogs, although the stringtable seems to be working in Japanese. Where to declare environment variables? What are correct places for:   Global environment variables meant to affect all users? User-specific environment variables? A No-Nonsense Introduction to Quantum Field Theory I read this quite a while back, part of my parents library (they're in their 50s now) I think the book was probably published in the 70s or 80s, my copy was paperback. I don't recall the cover art, but I have vague recollections of a satellite on the cover and maybe blue with grid-lines below it?  I've made concerted efforts to find it again but to no avail. My parents don't remember it either, but mum's an avowed book-aholic and buys a lot of books, reads them once and forgets about them, so that's not so surprising.  The key parts I remember:   Both the Soviets and Americans were working on psychic techniques and technologies, with real and tangible results including Remote Viewing. MK-ULTRA was explicitly namedropped in relation to the American psychic program. The Soviets and Americans both had orbiting superweapons, One was a microwave laser (or something very similar) and the other used slivers of the brains of psychics to blow people's brains up in a wide area. A super-psychic (male) took offence to the state of the cold-war and took control over the orbital weapons by manipulating the onboard computers psychically. There's a gruesome description of a psychic weapon being tested on monkeys in a lab, the POV character throws up afterwards in horror. I think there might have been more than one POV character. There's an extended description of the psychic orbital weapon as experienced by people in its area of effect. Similar to the monkeys, it builds up, starting with alzheimer-esque hand-shaking and building up to seizures and screaming before their brain liquifies fully and either comes out the nose and ears or outright blows the skull open. (this is one of the more vivid recollections I have of the book) I remember that the super-psychic was working wholly alone and did most of his psychic stuff from motels halfway around the world from events. I believe the super-psychic also killed other psychics explicitly. The weapon satellites had code-names Both weapons were used on cities in the course of the story I vaguely recall an office building getting targeted specifically at one point. Does JPA support mapping to sql views? I'm currently using Eclipselink, but I know now days most JPA implementations have been pretty standardized. Is there a native way to map a JPA entity to a view? I am not looking to insert/update, but the question is really how to handle the @Id annotation. Every entity in the JPA world must have an ID field, but many of the views I have created do not conform to this. Is there native support for this in the JPA or do I need to use hacks to get it to work? I've searched a lot and found very little information about doing this. A particle experiences an acceleration described by  $$ a=kx^{-2} $$ where x is the displacement from the origin and k is an arbitrary constant.  To what value does the velocity v of the particle converge to as x approaches infinity if the particle starts at some point x0?  If I approach this problem with energy, then $$ W = \int F \mathrm{d}x $$ $$  = \int_{x_0}^\infty mkx^{-2} \mathrm{d}x $$ $$  = mk(-\infty^{-1}+{x_0}^{-1}) $$ $$ W = K = mk{x_0}^{-1} $$ $$ \frac{1}{2}mv^2 = mk{x_0}^{-1} $$ $$ v = \sqrt{2k{x_0}^{-1}} $$  How would I solve this problem with pure kinematics? (there appears to be some sort of cyclical dependency where acceleration affects velocity, velocity affects displacement, and displacement affects acceleration)  Likewise, two particles experience accelerations described by $$ a_1=k_1x^{-2} $$ $$and $$ $$ a_2=-k_2x^{-2} $$ where x is the distance between the two particles  What two velocities do the particles reach as x approaches infinity if the two particles are initially separated by some x0? I need to know the ip address of another machine in a lan network.  What is the Windows command for getting ip address from machine name and vice versa? How can I get the page title? Show that the countable product of metric spaces is metrizable Running programs in the background from terminal How do I run a program in the background of a shell, with the ability to close the shell while leaving the program running? Lets say my UI is having problems or for some reason, I need to boot up a program from the terminal window, say, nm-applet:  nm-applet   When it's started, it occupies the foreground of the terminal window.  Is there any simple way to run the program in the background without needing to leave the terminal open or have it occupy the whole terminal?  On that note, I did find a way to run programs from the terminal and have it allow for other inputs, by appending an ampersand (&amp;) to the command as such:  nm-applet &amp;   But this isn't much use as any processes started in the terminal are killed once the terminal is closed. Out of Bag Error makes CV unnecessary in Random Forests?