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Let $(X,d)$ be a metric space and for $x,y \in X$ define $$d_b(x,y) = \dfrac{d(x,y)}{1 + d(x,y)}$$ a) show that $d_b$ is a metric on $X$  Hint: consider the derivative of $f(t)$ = $\dfrac{t}{1+t}$  b) show that $ d$ and $ d_b $ are equivalent metrics. c) let $(X,d) $ be $(\mathbb{R}, |\cdot|)$ Show that there exists no $ M&gt;0$ such that $ |x-y| $ $\leq$ $Md_b$ $(x,y)$ for all $ x,y$ $\in$ $\mathbb{R}$  I have calculated $f(t)$ and $f'(t)$ and from this I know $f(t)$ is an increasing function as $f'(t)$ is strictly positive. But I don't know where to go from here or how to do parts b) and c) Is it possible to identify the font used in a specific document/picture?  Answers to this question should identify:   Possible methods to do this (perhaps one answer per method) and adequately describe how to use it (as opposed to merely stating it); Ways of finding the identified fonts, if possible (free or not); and Any prerequisites associated with the method used, if required (for example, "In order to use method X, your document has to be in format Y").   This question is meant as an FAQ, based on an . Its aim is to facilitate the community with the general procedures involved in font identification. Similar cases are solved on a per-usage basis on 's  tag. In System Info in 11.10, the graphics driver wasn't detected. I use ACER Aspire 4750 with Intel Core i3-2340M and Intel HD Graphics 3000 graphic card  How to install the driver for my Ubuntu? How can tangential acceleration from a radial force be explained? Openshift application url: example.rhcloud.com  Custom domain: example.com  I came across this  and added the redirection rule. However, I can not access example.rhcloud.com now. My custom domain is www.example.com.  To avoid duplicate content problems, I tried this redirection.   Here is my .htaccess file:  RewriteEngine on   # Uncomment the following lines to force HTTPS #RewriteCond %{HTTP:X-Forwarded-Proto} !https  #RewriteRule .* https://%{HTTP_HOST}%{REQUEST_URI} [R,L]   # WordPress Defaults RewriteBase / RewriteRule ^index\.php$ - [L] RewriteCond %{REQUEST_FILENAME} !-f RewriteCond %{REQUEST_FILENAME} !-d RewriteRule . /index.php [L]  # Redirect Https traffic to Http RewriteEngine on  RewriteCond %{SERVER_PORT} ^443$ [OR] RewriteCond %{HTTPS} on RewriteRule  ^(.*)$ http://%{HTTP_HOST}%{REQUEST_URI} [R=301,L]   # Redirect non-www to www RewriteEngine on  RewriteCond %{HTTP_HOST} !^www\. [NC] RewriteRule ^(.*)$ http://www.%{HTTP_HOST}/$1 [R=301,L]    # Redirect openshift rhcloud domain to custom domain RewriteEngine on  RewriteCond %{HTTP_HOST} ^example.rhcloud.com$ RewriteRule (.*)$ http://www.example.com/$1 [R=301,L]   Why is example.com not redirecting to ww.example.com? How do I fix this?  Also, I see that since I have https to http redirection and non www to www redirection, my blog has been loading slow. How can I reduce it to lesser number of rules?  Edit:  In chrome:   In Firefox:     The page isn't redirecting properly, Firefox has detected that the server is redirecting the request for this address in a way that will never complete.     This problem can sometimes be caused by disabling or refusing to accept cookies. Choosing equipment options at character creation So if I'm correct, when you chose your starter gear and wealth you can either choose to roll some dice depending on your class and buy everything from that or you can choose the starter gear from your class and add the background items (including some gold). Here are my 2 questions:   If I choose the gear from my class, do I have to take all the equipment from only one of the (a) or (b) choices, or can I freely mix-and-match choices from the two sets? Can you spend the extra gold you get from your background on extra equipment (e.g. some soap and some fishing equipment) or do you have to start with that gold in its pouch?   The reason I ask is that I have recently created a level 1 wood elf druid and I mixed-and-matched the gear from the class, so that I chose a shortbow as a "(b) simple weapon" for the first point and a "(a) Scimitar" at the second point. My DM wasn't so happy about this, as he can't understand why I'm mixing (a) and (b).  He also wasn't so happy with me spending my 10gp from my outlander background. I decided that my druid wouldn't lay so much worth in 10gp compared to a bunch of stuff I got for that (e.g. some soap and some fishing equipment and some other useful things) and he seems to think that I'm supposed to just start with the gold instead. Continuous and bounded imply uniform continuity? I have some thoughts on this. First, I want to say I am no expert on cryptography, I just know some stuff, and I took a cryptography class in University. I am very interested in this topic.  I understand that many security experts out there might get angry just by the discussion of these things, but I hope you can spare me :)  My Questions:  Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing prime keys up to 2^2048 ?  I understand this would take an immense amount of time to do, but it would only be done once.  Given the public key, you would be able to look up what the two primes are, and hence retrieve the private key instantly.  When I consider who might break such a code, I am primarily thinking about NSA and other Governments that might have spent years in research and have the resources and interest in doing so.  Breaking an 256 AES would then be easier, as AES keys are often generated using the RSA private key.  Would it be possible to build such an index ? How long would that take in permutations, and perhaps with the fastest computer in the world ?  I hope we can have a nice chat about this. We are just discussion theory here. I hope you can offer me some insight on this topic.  Thanks! doGet and doPost in Servlets It is said in many textbooks that alpha decay involves emitting alpha particles, which are very stable. Indeed, the binding energy () is higher than for $Z$-neighboring stable isotopes. But the binding energy is lower than, for example, ${}^9\mathrm{Be}$ (). My question is why aren't other nuclear compounds ejected from heavy nuclei, e.g. ${}^9\mathrm{Be}$?  The   $$e^{-\frac{4\pi}{\hbar}\frac{Ze^2}{4\pi\epsilon_0}\frac{1}{v_\alpha}}$$ decreases exponentially in $Z$, so it explains intuitively why lower-$Z$ particles would tunnel more often. Specifically, it would explain why we would see ${}^9\mathrm{Be}$ emission ${e^{-2}}\simeq 0.14$ times as often compared to ${}^4\mathrm{He}$ emission. Also, the particles need to form in the nucleus prior emissions; but with a similar binding energy per nucleon (~7.08 MeV for ${}^4\mathrm{He}$ vs. 6.47 MeV for ${}^9\mathrm{Be}$) and higher total binding energy for the ${}^9\mathrm{Be}$ nucleus, I would expect that its formation in the same order of prevalence as the alpha particle (according to Ohanian, between 0.1 and 1 alpha particles are in existence at any moment in time).  Can anyone explain this? A reference to an article/textbook would be preferable.  EDIT Same goes for ${}^{16}\mathrm{O}$ which is also a double magic isotope, as 'anna v' pointed out. For it, the Gamow factor is smaller by $e^{-4}\simeq 0.02$, and emission should still be viable. VMware Linux Server -- how can you tell if you are a vm or real hardware? An interesting question.  I have logged into a Linux (most likely SuSE) host.  Is there some way that I can tell programmatically that I am a VM host or not?    Also assume that the vmtools are not installed. Rules for refreezing food How do you return from 'gf' in Vim I am using Vim for windows installed in Unix mode. Thanks to this site I now use the gf command to go to a file under the cursor.  I'm looking for a command to either:   return to the previous file (similar to Ctrl+T for ctags), or   remap gf to automatically launch the new file in a new window. Can I permanently block an update (non-driver) that will not install on my machine on Win10 pro? I have an update to Visual Studio 2010 that will NOT update and has failed hundreds of times. Manually installing it also fails, and no source has been found for manually getting this very old patch to install on Windows 10. I believe the problem is that Microsoft simply hasn't re-tested KB2890573 at all and it doesn't install on Win10 ver 1511 at all, anymore.  I do not want to use the MS utility that will postpone this update for a while. I want to permanently block it so it does not appear in the windows update list ever again. Migrate site from dev environment to production server - 404 errors We completed development on our dev environment and I copied the files over from the dev server to the production server through FTP.   Then through back up and migrade module, I made a back up of the SQL database. I created a new database through Plesk with the same name, username and password. I then imported the SQL database into the new server through phpmyadmin. I go to the URL and get the Site Maintenance message (which is correct, I placed the site in maintenance mode before copying the files over) but cannot navigate to any sub pages. Generally I can add /user or /admin to the end of the url and get the login screen. However all I get are 404 errors. Is there an unsubdivide function in Blender? After updating to 2.81, my previous works (done in 2.80) doesn't show colors This image is what's supposed to look like in Edit Mode/Object Mode(although I still can make these screenshots even after the update by pressing F12 which is rendering) but the following is what my donut looks like now in Blender.  It shows no shadows and colors. The only thing I did is updating blender to 2.81 from 2.80. And I don't know what but something seems off with blender. Sometimes I can't click any objects so I just pressed tab and then now it's all clickable. Seems like I was in edit mode, but to me, it looked exactly like object mode. I don't know what is wrong with it. How do I make my colors show up again? How does the NSAutoreleasePool autorelease pool work? As I understand it, anything created with an alloc, new, or copy needs to be manually released. For example:  int main(void) {     NSString *string;     string = [[NSString alloc] init];     /* use the string */     [string release]; }   My question, though, is wouldn't this be just as valid?:  int main(void) {     NSAutoreleasePool *pool;     pool = [[NSAutoreleasePool alloc] init];     NSString *string;     string = [[[NSString alloc] init] autorelease];     /* use the string */     [pool drain]; } Printing a .NET DataGridView I am fairly new to .NET and C#, but I have a DataGridView that I would like to print.  What would be the best way to go about doing so? Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer Let $a,b$ be positive integers.  When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square.    Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second root is determined in two different ways from the expansion $(x-r_1)(x-r_2) = x^2 - (r_1 + r_2)x + r_1 r_2$.)  Now suppose we have $a,b$ such that $k$ is an integer but not a square, by the investigation about roots we have that the second root is a nonzero integer since $k,b,a$ are integers and $k \not = b^2$, futhermore it is positive which is easily seen from its defining equation.  WLOG assume $a \ge b$ so that the second root is strictly smaller than $a$. This leads to a decent, replacing $a$ with the second root.    Proof 2 (Don Zagier): Apply reduction theory (specifically, Sätze 1 and 2 of Section 13 of my book on quadratic fields) to the quadratic form $x^2 + kxy + y^2$, which is the unique reduced quadratic form in its equivalence class.    Note that Proof 2 is pretty much the same as Proof 1 when written out in explicit detail, but I could not read Zagier's book because I cannot read German.  I would like to know more approaches to this and other alternative proofs of this result if possible! Thanks in advance.  I would also be interested in related problems (especially easier ones of a similar nature) and texts which cover the reduction theory in English.