Continuous mapping problem
A continuous mapping is determined by its values on a dense set
Links in [link text][number] format don't work properly if they follow a tag using the new [tag:tagName] format. Here's the result of [tag:python] [gold badge page][1]:      [gold badge page]   Putting the tag inside the link is worse. Here's the result of [[meta-tag:bug] some text][2], where [2] is google.com:        The part of the link outside the tag just disappears silently.  EDIT: To be more precise, the entire link to [2] disappears. The markup that is produced contains (invalid) nested anchor tags, as shown here from Firefox's "View Source":  &lt;a href="http://google.com" rel="nofollow"&gt;&lt;a href="/questions/tagged/bug" class="post-tag required-tag" title="show questions tagged 'bug'" rel="tag"&gt;bug&lt;/a&gt; some text&lt;/a&gt;   So the tag is still a link, but it is a link to the tag page, which is the behavior it would have if the link to [2] didn't exist at all. The user's desired behavior in this case is... ambiguous, to be charitable.  EDIT 2: I briefly included markup from Firebug here, but it turned out to be irrelevant. It has now been removed to improve readability. See  if you're interested. Why didn't Sauron find Bilbo when he put on the ring like Frodo? In Westeros there are three coins:   Copper Stars (The Seven-Pointed Star is the symbol of the Faith of the Seven.) Silver Stags (The Stag is the sigil of house Baratheon.) Gold Dragons (The Dragon is the sigil of house Targaryen.)   Was the Stag already on the silver coin before Robert became king?  If so, why? Was house Baratheon that important?  If not, what was on the silver coin during the reign of the Targaryen kings?  Why is the sigil of house Targaryen still on the gold coin after Robert took the crown?  Robert hates the Targaryens so much, even murdering innocent children is OK for him.   Keeping this in mind I find it especially strange, that he doesn't change the sigil on most valuable coin. Android/Java - Date Difference in days In 1D why are vectors, such as, e.g., velocity $v$ and acceleration $a$, not written with a vector arrow? In my textbook instantaneous velocity and acceleration are wrote only as $v$ and $a$ in the chapter of One dimensional motion. Instead, in two dimensional motion chapter they are written with the "arrow" symbol upon them, $\vec{v}$ and $\vec{a}$.  Why if they are both vectors they don't have an arrow in that particular chapter? I just can't get it. What Hogwarts classes are taught to two houses at once? How to stop people from using my domain to send spam? I receive Mailer Daemon messages saying certain emails fail. My domain is itaccess.org which is administered by Google apps. Is there any way I can identify who is sending emails from my domain, and how they are doing it without me creating an account for them?  Delivered-To: 7e949ba@itaccess.org Received: by 10.142.152.34 with SMTP id z34csp12042wfd;         Wed, 8 Aug 2012 07:12:46 -0700 (PDT) Received: by 10.152.112.34 with SMTP id in2mr18229790lab.6.1344435165782;         Wed, 08 Aug 2012 07:12:45 -0700 (PDT) Return-Path: &lt;whao@www20.aname.net&gt; Received: from smtp-gw.fsdata.se (smtp-gw.fsdata.se. [195.35.82.145])         by mx.google.com with ESMTP id b9si24888989lbg.77.2012.08.08.07.12.44;         Wed, 08 Aug 2012 07:12:45 -0700 (PDT) Received-SPF: neutral (google.com: 195.35.82.145 is neither permitted nor denied by best guess record for domain of whao@www20.aname.net) client-ip=195.35.82.145; Authentication-Results: mx.google.com; spf=neutral (google.com: 195.35.82.145 is neither permitted nor denied by best guess record for domain of whao@www20.aname.net) smtp.mail=whao@www20.aname.net Received: from www20.aname.net (www20.aname.net [89.221.250.20])     by smtp-gw.fsdata.se (8.14.3/8.13.8) with ESMTP id q78EChia020085     for &lt;7E949BA@itaccess.org&gt;; Wed, 8 Aug 2012 16:12:43 +0200 Received: from www20.aname.net (localhost [127.0.0.1])     by www20.aname.net (8.14.3/8.14.3) with ESMTP id q78ECgQ1013882     for &lt;7E949BA@itaccess.org&gt;; Wed, 8 Aug 2012 16:12:42 +0200 Received: (from whao@localhost)     by www20.aname.net (8.14.3/8.12.0/Submit) id q78ECgKn013879;     Wed, 8 Aug 2012 16:12:42 +0200 Date: Wed, 8 Aug 2012 16:12:42 +0200 Message-Id: &lt;201208081412.q78ECgKn013879@www20.aname.net&gt; To: 7E949BA@itaccess.org References: &lt;20120808171231.CAC5128A79D815BC08430@USER-PC&gt; In-Reply-To: &lt;20120808171231.CAC5128A79D815BC08430@USER-PC&gt; X-Loop: whao@whao.se From: MAILER-DAEMON@whao.se Subject: whao.se:  kontot avstängt - account closed X-FS-SpamAssassinScore: 1.8 X-FS-SpamAssassinRules: ALL_TRUSTED,DCC_CHECK,FRT_CONTACT,SUBJECT_NEEDS_ENCODING      Detta är ett automatiskt svar från F S Data - http://www.fsdata.se      Kontot för domänen whao.se är tillsvidare avstängt.     För mer information, kontakta info@fsdata.se      Mvh,     /F S Data      -----    This is an automatic reply from F S Data - http://www.fsdata.se    The domain account "whao.se" is closed.   For further information, please contact info@fsdata.se    Best regards,   /F S Data Grub rescue prompt, repair grub Why do prisms work (why is refraction frequency dependent)? It is well known that a  can "split light" by separating different frequencies of light:    Many sources state that the reason this happens is that the  is different for different frequencies.  This is known as .  My question is about why dispersion exists.  Is frequency dependence for refraction a property fundamental to all waves?  Is the effect the result of some sort of non-linearity in response by the refracting material to electromagnetic fields?  Are there (theoretically) any materials that have an essentially constant, non-unity index of refraction (at least for the visible spectrum)? It is quite easy to show that (by using residue theorem) $$\int_0^{\infty} \frac{1}{1+x^n}dx = \frac{2 \pi i^{1+2/n}}{n(e^{2 \pi i / n} - 1)} $$ for $$n \ge 2$$  Is there any possibility to simplify $$\frac{2 \pi i^{1+2/n}}{n(e^{2 \pi i / n} - 1)}$$ or it is best result?  Thanks in advance! How was this answer posted after this question was closed? Question:   Answer:   The question was closed at 11:57:57 (according to the hover-over-"x-mins-ago" bar) and the answer was posted at 12:16:35. It's a fairly lengthy answer, so it's possible that it was started to be written before it was closed but I swear I've started to answer a question then I got a banner along the top saying "this question has been closed" and the answer was no longer post-able. Why does "touch -a" also set the ctime? Is it touch(1) from coreutils-8.4-37.el6.x86_64 or my brain which is broken?  $ touch abc $ LANG=C stat abc   File: `abc'   Size: 0               Blocks: 0          IO Block: 4096   regular empty file Device: fd04h/64772d    Inode: 10485773    Links: 1 Access: (0664/-rw-rw-r--)  Uid: (3060399/ nodakai)   Gid: (  418/   quant) Access: 2016-10-14 18:42:06.189751847 +0800 Modify: 2016-10-14 18:42:06.189751847 +0800 Change: 2016-10-14 18:42:06.189751847 +0800 $ touch -a abc $ LANG=C stat abc   File: `abc'   Size: 0               Blocks: 0          IO Block: 4096   regular empty file Device: fd04h/64772d    Inode: 10485773    Links: 1 Access: (0664/-rw-rw-r--)  Uid: (3060399/ nodakai)   Gid: (  418/   quant) Access: 2016-10-14 18:42:17.374235446 +0800 Modify: 2016-10-14 18:42:06.189751847 +0800 Change: 2016-10-14 18:42:17.374235446 +0800 $ touch --help | grep 'access time'   -a                     change only the access time   As you see, it was not only atime but also ctime which was updated by touch -a !?!?  The filesystem is ext4 over LVM if that makes any differences. I was watching , and it got me thinking about when optional parameters are available in C# what is going to be the recommended way to declare methods that do not need all parameters specified?  For example something like the FileStream class has about fifteen different constructors which can be divided into logical 'families' e.g. the ones below from a string, the ones from an IntPtr and the ones from a SafeFileHandle.  FileStream(string,FileMode); FileStream(string,FileMode,FileAccess); FileStream(string,FileMode,FileAccess,FileShare); FileStream(string,FileMode,FileAccess,FileShare,int); FileStream(string,FileMode,FileAccess,FileShare,int,bool);   It seems to me that this type of pattern could be simplified by having three constructors instead, and using optional parameters for the ones that can be defaulted, which would make the different families of constructors more distinct [note: I know this change will not be made in the BCL, I'm talking hypothetically for this type of situation].  What do you think? From C# 4.0 will it make more sense to make closely related groups of constructors and methods a single method with optional parameters, or is there a good reason to stick with the traditional many-overload mechanism? What did Auggie's father give to his wife in the movie Wonder? There is a scene roughly in the last third of Wonder where Auggie's parents Nate and Isabel (Owen Wilson and Julia Roberts) spend an evening together while the kids are out. Nate has a surprise gift for his wife. She opens the box just enough to see it, laughs, and closes it again. So I asked myself the question:  What's in the box? I'm developing a multilanguage software. As far as the application code goes, localizability is not an issue. We can use language specific resources and have all kinds of tools that work well with them.  But what is the best approach in defining a multilanguage database schema? Let's say we have a lot of tables (100 or more), and each table can have multiple columns that can be localized (most of nvarchar columns should be localizable). For instance one of the tables might hold product information:  CREATE TABLE T_PRODUCT (   NAME        NVARCHAR(50),   DESCRIPTION NTEXT,   PRICE       NUMBER(18, 2) )   I can think of three approaches to support multilingual text in NAME and DESCRIPTION columns:   Separate column for each language  When we add a new language to the system, we must create additional columns to store the translated text, like this:  CREATE TABLE T_PRODUCT (   NAME_EN        NVARCHAR(50),   NAME_DE        NVARCHAR(50),   NAME_SP        NVARCHAR(50),   DESCRIPTION_EN NTEXT,   DESCRIPTION_DE NTEXT,   DESCRIPTION_SP NTEXT,   PRICE          NUMBER(18,2) )  Translation table with columns for each language  Instead of storing translated text, only a foreign key to the translations table is stored. The translations table contains a column for each language.  CREATE TABLE T_PRODUCT (   NAME_FK        int,   DESCRIPTION_FK int,   PRICE          NUMBER(18, 2) )  CREATE TABLE T_TRANSLATION (   TRANSLATION_ID,   TEXT_EN NTEXT,   TEXT_DE NTEXT,   TEXT_SP NTEXT )  Translation tables with rows for each language  Instead of storing translated text, only a foreign key to the translations table is stored. The translations table contains only a key, and a separate table contains a row for each translation to a language.  CREATE TABLE T_PRODUCT (   NAME_FK        int,   DESCRIPTION_FK int,   PRICE          NUMBER(18, 2) )  CREATE TABLE T_TRANSLATION (   TRANSLATION_ID )  CREATE TABLE T_TRANSLATION_ENTRY (   TRANSLATION_FK,   LANGUAGE_FK,   TRANSLATED_TEXT NTEXT )  CREATE TABLE T_TRANSLATION_LANGUAGE (   LANGUAGE_ID,   LANGUAGE_CODE CHAR(2) )    There are pros and cons to each solution, and I would like to know what are your experiences with these approaches, what do you recommend and how would you go about designing a multilanguage database schema. I have a question about inheritance in Java.  I have two classes A and B , and class B, inherits from A:  public class A {      public A() {          System.out.println("Hi!");      } }   public class B extends A {      public B() {          System.out.println("Bye!");      }       public static void main(String[] args) {          B b = new B();      } }   When I run program B, the output is:  Hi! Bye!      Question : why the constructor of class A is invoked, when I create and object of class B ?    I know that B inherits everything from A - all instance or class variables, and all methods, and in this sense an object of B has all characteristics of A plus some other characteristics defined in B. However, I didn't know and didn't imagine that when I create an object of type B, the constructor of A is also invoked. So, writing this:  B b = new B();   creates Two objects - one of type B, and one of type A.  This is getting interesting,   can somebody explain why exactly this happens? Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups? I think the answer is yes.   Sketch of the proof Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Let $\{e_\lambda:\lambda\in\Lambda\}\subset\mathbb{R}$ be its Hamel basis. Then $\{(e_{\lambda_1},e_{\lambda_2}):\lambda_1,\lambda_2\in\Lambda\}\subset\mathbb{R}^2$ is a Hamel basis of $\mathbb{R}^2$. Consider set theoretic bijection $i:\Lambda\to\Lambda\times\Lambda$ define map $\varphi:\mathbb{R}\to\mathbb{R}^2$ by its action on element of Hamel basis of $\mathbb{R}$ by equality $\varphi(e_\lambda)=(e_{i(\lambda)_1},e_{i(\lambda)_2})$. This is an isomorphism between $\mathbb{Q}$-vector spaces $\mathbb{R}$ and $\mathbb{R}^2$.  Question Could you tell me is this proof correct, and if it is not, where is the mistake?  EDIT: Thanks to GAJO we found that this proof is wrong. The correct one basis is  $$ \{(e_\lambda,0):\lambda\in \Lambda\}\cup\{(0,e_\lambda):\lambda\in\Lambda\} $$ So we had to look at bijections of the form $i:\Lambda\to\Lambda\coprod\Lambda$ and the desired linear operator is defined by equalities $\varphi(e_\lambda)=(e_{i(\lambda)},0)$ if $i(\lambda)$ lies in the first copy $\Lambda$ and $\varphi(e_\lambda)=(0,e_{i(\lambda)})$ otherwise. First, this question is motivated by the imprecise question: Is there a sensible notion of parity (evenness and oddness) for real numbers?   Here are some properties a notion of parity for $\mathbb{R}$ should have:    It should be an equivalence relation on $\mathbb{R}$ with exactly two equivalence classes.  Each equivalence class should be dense in $\mathbb{R}$. There should be some kind of symmetry between the two equivalence classes. (This is intentionally imprecise.)   If we divide $\mathbb{R}$ into rationals and irrationals, this lacks symmetry, since the rationals have measure zero and are countable, while the rationals have infinite measure and are uncountable.   So we could require that each equivalence class be uncountable and/or have positive (or infinite) measure. Then take one equivalence class to be the rationals, union with countably many fat cantor sets with positive measure, and the other class its complement. Then both classes are now dense, uncountable, and have infinite measure. But again, symmetry is lacking. One of these classes is "uncountable everywhere" and has "positive measure everywhere" while the other does not.   I propose some definitions: A subset $A \subset \mathbb{R}$ is uncountable everywhere if for any open interval $(a,b)$, the intersection $A \cap (a,b)$ is uncountable. A subset $A \subset \mathbb{R}$ has positive measure everywhere if for every open interval $(a,b)$, the intersection $A \cap (a,b)$ has positive measure. We can see immediately that having positive measure everywhere implies being uncountable everywhere, since a positive measure set must be uncountable.   Finally, my questions:    Is there a partition of $\mathbb{R}$ into two sets that are both uncountable everywhere?  Is there a partition of $\mathbb{R}$ into two sets that both have positive measure everywhere?  Is there a partition of $\mathbb{R}$ into two sets that split each interval $(a,b)$ into two parts of equal measure?   Potential generalizations for extra credit: What about partitions of $\mathbb{R}$ with $n$ equivalence classes, where $n \in \mathbb{N}$, or even with countably many, or even uncountably many equivalence classes?  Note: If you've seen these definitions of uncountable everywhere or positive measure everywhere somewhere under a different name, please let me know. I've never found anything where other people were thinking about these notions.  EDIT: By "measure," I mean Lebesgue outer measure, so we don't have to worry about anything being measurable. Geometry with Trigonometry Integrating the inverse of a function. $f(x)$ is a strictly increasing, and continuous, function on $[0,+\infty)$. Consider the integral,  $$\int_0^a f^{-1}(x) dx.$$  Reasoning  The interval $[0,a]$ is "reflected" across the line $y=x$ onto the interval $0 \leq y \leq a.$  $f(x)$ and $f^{-1}(x)$ are symmetric across the line $y=x$. So, the integral is equivalent to the area bounded by $y = 0$, $y=a$, and $f(x)$.  So, "integrating $f$ along an interval on the $y$-axis" is the same as integrating $f^{-1}$ along the same interval on the $x$-axis?  Question  Is my reasoning correct?