Prove that $[0,1] \approx  [0,a)$ where $a$ is a positive real number.  I know I need to fine a bijective function function between the two sets. The $a$ in the second set is throwing me off. Origninally I was thinking of doing $x=\frac{1}{a}$ that would get me $[0,x)$. But i am unsure if this is the correct route. Also how do I address the open interval on $a$ as oppose to a closed interval?
How to define a bijection between $(0,1)$ and $(0,1]$?   Or any other open and closed intervals?   If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that's how you're supposed to do it): I make a function $f : (-1, 2)\rightarrow (-5, 4)$ of the form $f(x)=mx+b$ by \begin{align*} -5 = f(-1) &amp;= m(-1)+b \\ 4 = f(2) &amp;= m(2) + b \end{align*} Solving for $m$ and $b$ I find $m=3\text{ and }b=-2$ so then $f(x)=3x-2.$  Then I show that $f$ is a bijection by showing that it is injective and surjective.
How can I execute `date` inside of a cron tab job? I want to create a log file for a cron script that has the current hour in the log file name. This is the command I tried to use:  0 * * * * echo hello &gt;&gt; ~/cron-logs/hourly/test`date "+%d"`.log   Unfortunately I get this message when that runs:  /bin/sh: -c: line 0: unexpected EOF while looking for matching ``' /bin/sh: -c: line 1: syntax error: unexpected end of file   I have tried escaping the date part in various ways, but without much luck. Is it possible to make this happen in-line in a crontab file or do I need to create a shell script to do this? "Pandora's Box" metaphors The majority of definitions give the same meaning - "Pandora's box" is a synonym for "a source of extensive but unforeseen troubles or problems."  Are there any other metaphors with the same meaning? Radius of Convergence of power series of Complex Analysis I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be greatly appreciated!  (i) $\displaystyle\sum_{n=0}^\infty \bigg(\dfrac{1}{2n}\bigg)z^{3n}$.  (ii) $\displaystyle\sum_{n=0}^\infty(2^n+i)(z-i)^n$.  (iii) Show that the radius of convergence of $\displaystyle\sum_{n=0}^\infty z^{n!}$ is $1$ and that there are infinitely many $z \in \mathbb{C}$ with $|z|=1$ for which the series diverges. App Store asks for old Apple ID when installing updates I got a MacBook that a colleague of mine had used previously. It runs OS X 10.10.2. I created a new administrator account for myself and turned off the "admin" flag on the old account.  I also entered my iCloud / Apple ID account, so that it syncs my calendars and contacts correctly, and even in the App Store, I am signed in with my correct account.  But when I try to install pending software updates (that is, OS X system updates!), the window that pops up has the the e-mail address of my colleague filled in and greyed out—and I can't change it.     Note that I'm signed in with a different (correct) account in the App Store. When I go to Featured and click Account on the right, it has my correct e-mail address filled in:    How do I make the App Store "forget" the old Apple ID?  I  where it just says "it's not possible", and that apps are glued to the original account the Mac was set up with, but that doesn't make a lot of sense, especially in a company where laptops may be handed over from one person to another. When I go to , my tablet is listed, but not my phone. I had signed up just as soon as ADM was available, and at that time both devices were listed. Now, for the last several days, I can only see my tablet. (For what it's worth, both devices appear in the "My Devices" section of the Google Play Store.)  I have done the following while troubleshooting:   Ensured that the Android Device Manager settings in Google Settings are both on. I have even toggled them off and back on again Ensured that Location Services are on Ensured that all sync services for my account are toggled on and work Have connected to the Play Store several different times Ensured Google Play Services is at the latest version Made sure I'm using the right account. (In fact, I removed the other secondary accounts from the phone) Power-cycled the phone several different times   I could understand if ADM couldn't find the phone due to it being off, out of range, poor signal, etc., but ADM would at least display the phone and tell me it couldn't find it.  So, any suggestions to get my phone to show up on Android Device Manager?  I would like this to be working before something happens to the phone. ( and all that.)  Although it shouldn't make any difference:   Tablet: Nexus 7 (2012) with Android 4.3 Phone: Verizon Galaxy Nexus with Android 4.2.2   Addendum:  There's apparently some (relatively new) advice on the :   Open  Google Settings from your device's apps menu. Touch Android Device Manager. Uncheck Allow remote factory reset. Go to your device's main Settings menu, then touch Apps > All > Google Play services. Touch Clear Data. Note that this action doesn't remove personal data. Go back to Google Settings and select Allow remote factory reset. Restart your device.   I have followed these steps, but it doesn't appear to have worked. How to determine the value of a domain name? I'd like to register a second-level domain with the .com extension and I noticed that it has already been registered and the owner put it for sale and let the possibility to make an offer in their site or by contacting them.  I saw that domains are generally sold for extremely different prices, so I'm confused.  Are there any general criteria to decide how much I should offer to purchase a domain? Checking for null before event dispatching... thread safe? Something that confuses me, but has never caused any problems... the recommended way to dispatch an event is as follows:  public event EventHandler SomeEvent; ... {     ....     if(SomeEvent!=null)SomeEvent(); }   In a multi-threaded environment, how does this code guarantee that another thread will not alter the invocation list of SomeEvent between the check for null and the invocation of the event? Qt signaling across threads, one is GUI thread? Looking for a classic sci-fi short story about a boy who discovers pods growing in a field For years, I've been looking for a short story about a boy who discovers alien pods in a field that are growing replacement people. If I remember right, he sees the pod creature eat his mother and then become her, and no one believes him. The story ends with him going in to see his dad, one of the few real people left until that point, and seeing something (an empty pod maybe) that lets him know his dad has been taken and he's next. I'm thinking that the title was something like Father Figure, but I may be remembering wrong.  The story was part of several collections of classic science fiction stories that we had on our bookcase when I was growing up, so I'm guessing that the story could be from the 70's or any time earlier. Question: Can anyone shed some light on this formula? I can't find any information on it.  It has three corollaries that I also need to understand: How can author deal with late reviews? How do you sort du -sh /dir/* by size? I read one site that said use | sort -n but that's obviously not right. Here's an example that is wrong.  [~]# du -sh /var/* | sort -n 0       /var/mail 1.2M    /var/www 1.8M    /var/tmp 1.9G    /var/named 2.9M    /var/run 4.1G    /var/log 8.0K    /var/account 8.0K    /var/crash 8.0K    /var/cvs 8.0K    /var/games 8.0K    /var/local 8.0K    /var/nis 8.0K    /var/opt 8.0K    /var/preserve 8.0K    /var/racoon 12K     /var/aquota.user 12K     /var/portsentry 16K     /var/ftp 16K     /var/quota.user 20K     /var/yp 24K     /var/db 28K     /var/empty 32K     /var/lock 84K     /var/profiles 224M    /var/netenberg 235M    /var/cpanel 245M    /var/cache 620M    /var/lib 748K    /var/spool "Your Sites" in Android left sidebar shown in unpredictable order How to perform the modular reduce of Rijndael's finite field I am trying to understand how to calculate the modular reduction of Rijndael's finite field.  The example on  page says that {53} • {CA} = {01}, because (11111101111110 mod 100011011) = {3F7E mod 11B} = {01} = 1.  I am able to get to the point where I have 3F7E mod 11B, but cant seem to figure out how the reduction works.        11111101111110 (mod) 100011011      ^100011011             1110000011110       ^100011011             110110101110        ^100011011             10101110110         ^100011011             0100011010            ^100011011 &lt;-- here, right shifted two places?             00000001   From this example, why does it right shift two places at one point?  I also found an example that 0x1b*0xaa equals 0x8c. Trying to solve this myself I get to the point: 0xE0E mod 0x11B   111000001110 ^100011011   11011010110  ^100011011    1010111010   ^100011011     010001100    ^100011011      10010111   This makes it 0x97 What do I need for extreme macro photography? Is $29$ the only prime of the form $p^p+2$? I washed and dried my passport by mistake, do I need a new one? Around two years ago I washed and dried my passport by mistake at the laundry room. My ID, my old US student visa and my stamps from european countries are still readable but the pages are not "flat" anymore and the front cover is kind of erasing (some parts of the Brazil symbol is gone).   In April I'll travel to US for a conference, I'll need to get a new US visa (B1/tourism visa) but I'm worried if my passport is not "good to go" anymore because it was "washed and dried". Apparently there is an electronic chip in it and I don't know if it is still working.  Should I get a new passport or can I save some money and time and use this one? It expires only in 2019. This might be subjective, I don't know. However, it's a serious problem for me. My electrical bill is outrageous. It's three times as large as my next-door neighbor, and four or five times the size of my neighbor's across the street. It's worse in the summer (I live in central TX), but I've had the air conditioner inspected, and while the house is right at the maximum capacity for our unit, the unit should still be able to take care of the load. That also doesn't explain why our winter bills are larger than the comparables.  What I would like to do is check the amount of current drawn by each running appliance and calculate where all my money is going. I just don't know how to go about doing this. Ever since upgrading to Windows 10 1903 the other day, my CPUs have all been pegged at 100%.  "Antimalware Service Executable" is using 60%, and "Microsoft OneDrive (32 bit)" is using 35% (with the other 5% being spread amongst everything else running).  After running like this for a day or so my computer case (Surface Tablet) is smoking hot.  How can I find out what's causing this problem?  Is there logging I can review for those services?  Known issues with 1903 that I just can't find on Google? Story about a deadly alien creature on another planet that has some chemical in its body that makes it faster I can not remember if it was a short story or a novel, but I am leaning towards it being a novel. There is a colony of humans and I can't remember if they crash-landed or not.   There is a creature that they come to discover after it has killed a few people. The creature has some kind of chemical in its body that is like a super-adrenaline. One part is I remember is one of these creatures getting very hot from using up its reserve of this chemical and it dived into a river or pond and it started to steam. Something like that.  It has to be at least 10 years since I read the story.