Is there a measurable subset E ⊆ R such that whenever a &lt; b are real numbers we have both $m(E ∩ (a, b)) &gt; 0$ and $m((a, b) -E) &gt; 0$ ?  This is an extra question on my real analysis class, but it is not graded, just for fun.   I was thinking about the "fat Cantor set"(), and play with it a little bit, but somehow I can still find "gaps" between real numbers. Can anyone give me some hint so I can carry on ?  Thanks !!!
I was wondering how one can construct a Borel set that doesn't have full measure on any interval of the real line but does have positive measure everywhere.   To be precise, if $\mu$ denotes Lebesgue measure, how would one construct a Borel set $A \subset \mathbb{R}$ such that $$0 &lt; \mu(A \cap I) &lt; \mu(I)$$ for every interval $I$ in $\mathbb{R}$?   Moreover, would such a set necessarily have to contain infinite measure?
Do Irrationals form a group? It's true that the irrationals do not form a group under addition or multiplication, but I want to find a binary operation "∗" such that ($\mathbb R \setminus \mathbb Q,∗$) is a group. It is possible by transpose of operation from $\mathbb R$, but I want a explicit example. Evaluate trigonometric function Can a homomorphic encryption scheme be made CCA2 Secure? Is it possible to modify a homomorphic encryption scheme so that it can be CCA2 secure?  From the definition of a homomorphic scheme, it seems that it is malleable, which would result in lack of CCA2 security. So, if we use the regular method to convert a CPA secure scheme to CCA2 secure scheme (using a hash of the message and a random value), will it still preserve its homomorphic property? Please correct me if my understanding is wrong. Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist I've been trying to solve the following problem:  Suppose that $f$ and $f&#39;$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\infty}f&#39;(x)$ exist. Show that $\displaystyle\lim_{x\to\infty}f&#39;(x) = 0$.  I'm not entirely sure what to do. Since there's not a lot of information given, I guess there isn't very much one can do. I tried using the definition of the derivative and showing that it went to $0$ as $x$ went to $\infty$ but that didn't really work out. Now I'm thinking I should assume $\displaystyle\lim_{x\to\infty}f&#39;(x) = L \neq 0$ and try to get a contradiction, but I'm not sure where the contradiction would come from.   Could somebody point me in the right direction (e.g. a certain theorem or property I have to use?) Thanks Why must an English learners should know Grammar first? Why must everyone understand Grammar first before learning to speak? How should one maintain and update a MiKTeX installation? Batch Rename Files in Folder Divisibility is transitive: $\ a\mid b\mid c\,\Rightarrow\ a\mid c$ As the title says, if a number is divisible by a number, is it always divisible by that number's factors?  An example being that $100$ is divisible by $20$, it is also divisible by $10, 5, 4, 2$ as well?  Does this always apply? (This is on NT based Windows like 2000, XP, Vista, 7)  On the command-line, the percentage characters are ignored:  wget "http://www.justitie.nl/images/Handleiding%20voor%20verwerkers%20persoonsgegevens_tcm34-3940.pdf"   So it correctly downloads this file (each %20 becomes a space):  "http://www.justitie.nl/images/Handleiding voor verwerkers persoonsgegevens_tcm34-3940.pdf"   But inside a batchfile, all the %20 are being interpreted as expanding parameter 2 (which is empty) resulting in this file to get downloaded (each %20 becomes 0):  "http://www.justitie.nl/images/Handleiding0voor0verwerkers0persoonsgegevens_tcm34-3940.pdf"   Is there a way to circumvent the percentage parameter expansion?  --jeroen How does the Green-Flame Blade cantrip interact with the tiefling feat Flames of Phlegethos? I work on different types of monitors (CRT, LCD) and all PCs have the same screen refresh rate as 60 Hertz. Recently I've read an article which claims that low refresh rates cause headaches. The writer states that 60 Hertz is a low refresh rate and should be increased to the maximum value that the monitor supports. If I increase this value to the maximum as the monitor can display (for example 75 Hertz), does it damage the monitor in the long term? Answers about the pros and cons of low &amp; high refresh rates would be highly appreciated. Error: Declaration of MyClass::start_lvl() should be compatible with that of Walker_Nav_Menu::start_lvl() I'm currently working on a WordPress site and whenever in the admin I go the Appearance > Menus page I get the following error:     ErrorException: Runtime Notice: Declaration of Walker_Nav_Menu_Edit::start_lvl() should be compatible with that of Walker_Nav_Menu::start_lvl() in wp-admin/includes/nav-menu.php line 203   It seems like some sort of PHP compatibility issue. I'm running PHP 5.3.10 on my local dev machine.  What would be the best way to remedy this problem? Stack Exchange iOS app crashes on start Upon clicking on SE icon, blue (starting) screen appears and then I am back to home screen, SE app simply won't start. This kind of behaviour never happened before (on any other app). I tried to remove it from &quot;task manager&quot;, reinstalling and logging back in; neither did work. In Settings -&gt; Stack Exchange -&gt; Notices I found a log message:  FILE DOESN'T EXIST, WILL CREATE: users/dev/buildagent/work/{id}/pods/target support files/pods/pods-ack...plist   App Version: 1.4.0 Device: iPhone7,2 OS Version: Version 9.1 (Build 13B143)   TLDR: app starts when cellphone has no internet access. I just noticed when I am on 3G (cellular internet) it crashes ASAP, but when I am on wifi that has no internet (for LAN etc.) app starts normally but no new content, when I turn off WiFi and iPhone switches to cellular; app crashes. Note: I turn off WiFi using control center - so I never leave the app. Another thing, when I force iPhone to use EDGE (slowest possible internet) app starts, and crashes after 5 seconds or so. I assume when app downloads some data from server.  Known workaround (works for me too)  I found a workaround: turn on Airplane Mode, launch the app, then turn Airplane Mode off. – NobodyNada MyClass[] array; List&lt;MyClass&gt; list;   What are the scenarios when one is preferable over the other? And why? I just upgraded from Ubuntu 10.04 to Mint 16. The installer did not detect my existing RAID 1 array, and I can't access it anymore.  My drives are:  SDA - First RAID Disk SDB - Second RAID Disk SDC - Non-RAID where the OS is installed.   The disks are full of stuff I want to keep, so I don't want to create a new RAID.   On the old machine, the two raid drives became MD0, which was a physical disk for Logical Volume Management.  How do I put the two halves together and get it to mount at boot? (I know how to mount single drives and how to add fstab entries for single drives, but the part I don't know is how to mount the two drives as one disk as a RAID. Proving that $G$ is a group if $a*x=b$ and $y*a=b$ have solutions. (Reference : Fraleigh, A first course in abstract algebra)     Prove that a nonempty set $G$, together with an associative binary operation * on $G$ such that $a*x=b$ and $y*a=b$ have solutions in G, $\forall a,b\in G$, is a group.   Because * is an associatie operation, we have to prove that: there exists $e\in G$ such that $e*a=a=a*e$ for any $a\in G$, and that for any $a\in G$ there exists $\alpha \in G$ such that $a*\alpha=e=\alpha *a$.  This is very confusing, first I tried to prove the identity exists. If we consider a=b, then $a*x=a=y*a$, and because of the hypothesis $x,y\in G$ always exists, now we have to show that $x=y$, but I just can't do this with out inverses, so I tried to prove that the inverses exists, however I have no idea how the identity looks like, I'm stuck.  And I tried to get unstuck, so I thought particular cases, first if $G$ had only one element, then what would happend if it had just two elements... but I remembered that we don't know if G is finite or not, and I think this makes things a little bit more complicated.  What can I do? This question is from [Number Theory George E. Andrews 1-1 #3].     Prove that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$$   This problem is driving me crazy. $$x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+\dots +xy^{n-2}+y^{n-1)}$$  $(x^n-y^n)/(x-y) =$ the sum for the first $n$ numbers and then I added $(xy^{(n+1)-2}+y^{(n+1)-1})$ which should equal $(x^{n+1}-y^{n+1})/(x-y)$ but I can't figure it out    This is a similar problem in the book and I tried this method but it wasn't working out      $\quad$Thereom $\bf1$-$\bf2$: $\,\,\,\,$ If $\,x$ is any real number other than $1$, then $$\sum_{j=0}^{n-1}x^j=1+x+x^2+\ldots+x^{n-1}=\dfrac{x^n-1}{x-1}.$$    $\quad$Remark: $\displaystyle\sum_{j=0}^{n-1}A_j$ is shorthand for $A_0+A_1+A_2+\ldots+A_{n-1}.$     $\quad$Proof: Again we proceed by mathematical induction. If $n=1$ then $\displaystyle\sum_{j=0}^{1-1}x^j=x^0=1$ and $(x-1)/(x-1)=1$. Thus the theorem is true for $n=1$.  $\quad$ Assuming that $\displaystyle\sum_{j=0}^{k-1}x^j=(x^k-1)/(x-1)$, we find that $$ \eqalign{ \sum^{(k+1)-1}_ {j=0}x^j &amp; = \sum^{k-1}_ {j=0}x^j+x^k=\dfrac{x^k-1}{x-1}+x^k \\ &amp;= \dfrac{x^k-1+x^{k+1}-x^k}{x-1}\\ &amp;= \dfrac{x^{k+1}-1}{x-1}. }$$   Hence condition $(\rm ii)$ is fulfilled, and we have established the theorem.    $\quad$Corollary $\bf1$-$\bf1$: $\,\,$ If $\,m$ and $n$ are positive integers and if $m&gt;1$, then $n&lt;m^n.$ Let $f$ and $g$ be entire functions and $g(z)\neq 0$ for all $z\in \mathbb{C}$. If $|f(z)|\le |g(z)|$, can we say $f$ is constant?  Liouville theorem says that an entire bounded function is constant, but g is not given bounded here. So I think it should be false. What else can we say about $f$? Is the intersection of a closed set and a compact set always compact? I am going through Rudin's Principles of Mathematical Analysis in preparation for the masters exam, and I am seeking clarification on a corollary.  Theorem 2.34 states that compact sets in metric spaces are closed. Theorem 2.35 states that closed subsets of compact spaces are compact. As a corollary, Rudin then states that if $L$ is closed and $K$ is compact, then their intersection $L \cap K$ is compact, citing 2.34 and 2.24(b) (intersections of closed sets are closed) to argue that $L \cap K$ is closed, and then using 2.35 to show that $L \cap K$ is compact as a closed subset of a compact set.  Am I correct in believing that this corollary holds for metric spaces, and not in general topological spaces? Helpful flags not getting updated after getting 'close question' privilege