Now I have a binary tree which is  How would I go about proving binary tree with $n$ leaves has exactly $2 n - 1$ nodes ?
How would I prove that any binary tree that has n leaves has precisely $2n-1$ nodes ?  Given that a binary tree is either a single node "o" or a node with the left and right subtrees contains a binary tree        o  /       \ binary  binary tree    tree   Any help would be appreciated !
Is there a definition of  and proper explanation of it?  It is said "Electric charge is the physical property of matter that causes it to experience a force when close to other electrically charged matter."  How is it though that matter can get charged?  Defining charge as the property of feeling a force with other charged matter seems circular.  What is charge?  Is there a non-circular definition / explanation? How can I backup my RAW photos while travelling without Internet access? This question seems similar to , but my needs are different :   I shoot in RAW exclusively I'm looking for a 100% offline solution: while traveling, Internet won't be accessible the "laptop" solution is too heavy for me: I'm looking for a lightweight solution the simplest, the better: one device to rule them all   EDIT:  I'm using good old Compact Flash cards (yes, the bigger ones), so I need a device that can read it. If it can read from other cards, this question might be interesting for more people than myself  - that's the whole purpose of the website.    Also asked by :  Portable Storage Device while traveling  I need storage device which is able to read CF cards and make a backup of my photos while traveling. I do not want to carry laptop (it's bulky) and tablet (too little storage) but looking into some other solutions.   So far I found:   Epson PXXXX - price is blocker for me Jobo Giga Vu - same as above Nexto DI - bit better, but still a lot Hyperdrive Colorspace UDMA - this looks the best for price/options Wolwerine PicPac - cheap, bad opinions DigitalFoci PhotoSafe - same as above Ex-Pro® Picture 2 Drive - didn't find much about it   So ladys &amp; gents what do you use? I was trying to do some of these questions to check my understanding about the topic, but I'm not sure if they're correct. Here are my answers.   1) Suppose $f$ is continuous on $[0,1]$. Must there be a nondegenerate closed subinterval $[a,b]$ of $[0,1]$ for which the restriction of $f$ to $[a,b]$ is of bounded variation?   No, because $f(x)$ might be a constant function. In this case, $f(x)$ will not be the difference of two increasing functions...and so will not be of bounded variation.  2) Let $f$ be the Dirichlet function, the characteristic function of the rationals in $[0,1]$. Is $f$ of bounded variation on $[0,1]$?   No, because we cannot express it as the difference of two increasing functions.  3) Let $f$ be a step function on $[a,b]$. Find a formula for its total variation.  If we partition $[a,b]$ in such a way that for each $[x_{i-1}, x_i]$ (which is part of the partition), $x_{i-1}$ and $x_i$ each correspond to the midpoint of a step. I know this is not a formal answer, but I'm just trying to check my understanding. Making a div vertically scrollable using CSS Can we use media9 to show a video on a PDF but the video is not embedded to the PDF? I have video tutorials that I want to show in my presentation with beamer class. I don't want the video files get embedded to the PDF because they will make the PDF size very very huge.  If you know how the videos can be played on the HTML with a reference rather than embedding the video file to the HTML file, I want something like this to be applied to my PDF presentation. I just want to reference the video URL from my PDF presentation.   Shortly speaking, can we use media9 to show a video on a PDF but the video is not embedded to the PDF?  Uploading those videos to a web server might be difficult for me because internet connection is very slow and sometimes not available in the presentation room. :-) $\operatorname{Ext}$ and injectives, respectively projectives What is the difference between Shrink Database and File? DBCC ShrinkDatabase() DBCC ShrinkFile()    Do I need to run both DBCC commands in order to shrink the database? What is the difference between these two above? Lost passport with Schengen visa My passport was stolen in Prague on my way to Vienna. I filed a police report and our embassy swiftly acted by giving me a temporary passport for my onward journey to Munich and Rome. Upon exit from Rome, the immigration ask for the Schengen visa, when I told him that I lost my passport, he did not ask for any document to support my claim, he just stamped my new passport with exit. I am not sure if on record, I am officially out of the Schengen area or not. I am scared that without the lost visa, it appears that I am still in the area and might reach the 90 day limit and will be denied when I re-apply as it would appear I broke the rule. What to do? Markdown change: Intra-word emphasis now works There are tons of questions I could maybe post this as an answer to, but before choosing the best one, I'll just post this as an announcement question that can be linked elsewhere. The short story From now on, intra-word emphasis is possible in our Markdown version. That means you can now write  The &quot;V&quot; is for Model**View**Controller. That's un*fricking*believable.  and it renders like this:  The &quot;V&quot; is for ModelViewController. That's unfrickingbelievable.  This only works for asterisks, not for underscores. It works in posts and comments. The reasons There are two big reasons for this. For one thing, this is more or less the behavior of , which we eventually . &quot;More or less&quot; because CommonMark's emphasis behavior is still in flux, and supports a few things that we don't support yet, but at least with this change we're closer. The other reason is languages that don't even have the concept of &quot;words separated by spaces&quot;. See  and  for two examples. This change allow posts in those languages to emphasize words without having to resort to HTML tags. To use an example from the first linked post,  **北京**和**上海**是直辖市。   now renders like this:  北京和上海是直辖市。  Possible issues In the 99% cases, this should only make things better; I've only found very few posts where this change causes unintended bolding/italicizing. And even in those cases, note that the rendered versions of questions and answers are not re-rendered until a post is edited, so this will not affect anything in posts retroactively. Unlike posts, comments are rendered , so this immediately applies to all comments. In most cases this doesn't change anything or actually fixes things. Should you discover a rare case where this breaks an important comment badly, we can always edit it to fix. There's one side effect that I'd like to mention: If you find a great quicksort implementation on GitHub that has a bug that causes a crash when sorting Windows versions, and the author doesn't want to fix it, you may be inclined to say  f**k this s**t   (i.e., &quot;fork this sort&quot;). With intraword emphasis, this would however look like this:  fk this st  So you would have to escape the asterisks (one of them would actually be enough). I'm not worried about this case too much, since if you must put censor-worthy words into a post, you might as well have to jump through a little hoop. Some details This being Meta, there are probably lots of people interested in a more detailed description of the new behavior (don't feel bad if you're not one of them; just skip this section). Here are the major rules regarding intraword-emphasis:  As mentioned, it only applies to asterisks, not underscores. Underscores  this restriction was in place at all; asterisks are much less of a problem.  If a single or double asterisk is supposed to be interpreted as intra-word emphasis, it cannot have punctuation on one side; it actually must be inside a word. An example of an edge case this solves is  It's a web*-based solution. *see Wikipedia for a definition of &quot;web&quot;  where the asterisks after &quot;web&quot; could cause unintended italicizing if there was a second asterisk in the same paragraph somewhere.  For italics, the &quot;inside a word&quot; part of rule 2 is even a bit stricter. For bolding, any word or digit character is allowed; for italics, digits don't work either. That's because the asterisk is often used as a multiplication symbol, so  The game worked in 320200 and 640480 resolution.  wouldn't look right without this restriction.  A minor addition that I made that's only tangentially related to intra-word emphasis is that a sequence of more than three underscores or more than three asterisks will no longer take part in any emphasis. This shouldn't really cause any problems, but fixes some (e.g. the underline in the second blockquote in , while not really visible, is a weird combination of bold and italic underscore characters; after this change, it would just be (more) underscore characters). This behavior also matches CommonMark. How to get past a forgotten iPhone passcode? Are virtual particles only a fictive tool in equations? There is no "action at a distance" in nature. Attraction of a piece of iron by a magnet, attraction between distant electric charges of opposite sign, have to be mediated by something. The  are proposed as an explanation.  If they have an observable effect, it seems like they must exist. If so, why do we call them virtual? It can't be that they have short lifetimes, as there are short-lived particles (resonances) that we don't call "virtual."  On the other hand, some people say that the so-called virtual particles only exist on paper, or that they are just a technique to help us handle our equations.  So what is the meaning of the word "virtual" here? Do these particles really exist, or not? How to pronounce decimals? Something like:  0.1 0.23 0.999   How to pronounce them correctly in English? $\nabla U=0 \implies U=\mathrm{constant}$ only if $U$ is defined on a connected set? Rings' second isomorphism theorem I am thinking about the proof of the second isomorphism theorem, and something isn't very clear to me.   Let $R$ be a ring, $S\subset R$ a subring and $I\subset R$ an ideal. We have the natural homomorphism $f:R\rightarrow R/I$. The theorem states that $\mathrm{Im}(S)=(S+I)/I$. My question is why not simply $\mathrm{Im}(S)=S/I$?  I understand that it is not true (for starters, $S/I$ need not be a subring), but I cannot explain that to my self in a convincing way. By 18 years of age, I want a brand new car that's $43,668 I'm from Arizona and I will be 18 in 4 months. I make about $1,200 a month from my job. I have $600 in my bank account and I don't plan to  spend any of it until I'm  18.     I'm looking to get a brand new Jeep Wrangler Rubicon that costs about  $43,000. The down payment is 10% (about $5,000) with an APR of  4.28%. The duration is 60 months with a monthly payment of  $800.  I don't have any credit. My friend suggested that I ask my parents to put my name on one of their credit cards so that I can build some credit. I'm not sure how that works. Can someone suggest a good way of going about this so I can get this car on or around my birthday?  My mom does finance work  for a big company and she told me that I should get a cheaper car. I really want this one and I'm willing to work for it. I'm getting a second job to get more money rolling in. My mom said that the only way that this would work is if I have a down payment of $15-20k. She also said that not everyone qualifies  for a car like this one. What can get me to qualify ASAP when I turn 18? this is my first question on PhysicsSE (I'm already an user of MathSE).   I'm a mathematics students trying to understand Faraday's law, that is   $$\varepsilon= -\frac{d \Phi_B}{dt}$$  where $\varepsilon$ means electromotive force and   $$\Phi_B=\iint \mathbf{B}\cdot d\mathbf{S}$$   means flux of magnetic field. As my textbook points out, there is an interpretation problem here: if the change in magnetic flux is due to movement of the conductor, then free charges in it are subject to Lorentz force, which then causes a current. On the contrary, if the conductor holds steady in a changing magnetic field, the induced current must be explained in terms of an electric field $\mathbf{E}$, described by the equations   $$\begin{cases} \nabla \cdot \mathbf{E}=\frac{\rho}{\varepsilon_0} \\ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}\end{cases}.$$  Question Do those equations hold if we have a moving conductor in a stationary magnetic field? I guess not: this would mean $\mathbf{E}=\mathbf{0}$. How to solve this? I would like to protect myself from unreasonable demands during my work day.  One of those is telling people that I will not respond to any emails before 3pm.  I'd like to set up a rule in outlook that replies to all emails before 3pm with a specific message. I'd like this to be done every day, no matter the date.  How do I set this up?  I've scoured Google best I can so far, and I think, based on a Exchange forum, that a script might be needed for this.  But what I found so far was only server side scripts. I need something on my own machine.  references: datetime conversion to CDT Can anyone please help me how to convert the given time into CST timezone.  Input timezone will change dynamically, so i want to convert the given timzone into CST and then need to get the hour from that CST timezone in apex. An init-script does not get called on shutdown I have set up an init-script to control the state of a VirtualBox VM:  #!/bin/sh #chkconfig: 35 99 5 #description: vTiger virtual machine  ### BEGIN INIT INFO # Provides: vtigervm # Required-Start: $local_fs # Requider-Stop: $stop_fs # Default-Start: 3 5 # Default-Stop: 0 1 2 6 # Short-Description: Manage vTiger virtual machine # Description: Utility to start and stop vTiger virtual machine on VirtualBox ### END INIT INFO  start() {     echo -n "Starting vTiger"     echo     su myuser -c '/usr/bin/VBoxManage startvm "vTiger" --type headless'     echo "Started virtual machine" &gt;&gt; /var/log/messages }  stop() {     echo -n "Shutting vTiger down..."     echo     su myuser-c '/usr/bin/VBoxManage controlvm "vTiger" acpipowerbutton'     while [ ! -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; do         echo -n "."         sleep 1     done     echo "Done."     echo "Stopped virtual machine" &gt;&gt; /var/log/messages }  status() {     echo -n "Running VMs: "     su myuser -c '/usr/bin/VBoxManage list runningvms'     echo     if [ -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; then         RETVAL=3         else         RETVAL=0     fi     echo "Queried virtual machine status" &gt;&gt; /var/log/messages }  echo "Called virtual machine management script with: $1" &gt;&gt; /var/log/messages case "$1" in   start)         start    ;;    stop)         stop    ;;    restart|try-restart|condrestart|reload)         stop         start    ;;    status)         status    ;;    *)         echo $"Usage: $0 {start|stop|restart|status}"         exit 1    ;; esac  exit $RETVAL   The system is CentOS 6.5. If I manually run service [start|stop|status] vtigervm, it works as I expect it to. If the VM is running, $? after the service vtigervm status returns 0, and 3 if it is stopped. I installed it with chkconfig and it created among a few others rc5.d/S99vtigervm and rc0.d/K05vtigervm.  The problem  When I start the system it starts the "service", but on shutdown it does not even run the scripts.  grep "virtual machine" /var/log/messages shows:  *[machine starting]* Called virtual machine management script with: start Started virtual machine *[shutdown -h now]* *[machine stopped]*   What I expect:  *[machine starting]* Called virtual machine management script with: start Started virtual machine  *[shutdown -h now]* Called virtual machine management script with: status Queried virtual machine status Called virtual machine management script with: stop Stopped virtual machine *[machine stopped]*   File permissions &amp; etc:  # ls -lah /etc/rc0.d/ lrwxrwxrwx. 1 root root 13 7.2. 17:49 /etc/rc0.d/K05atd -&gt; ../initd.d/atd lrwxrwxrwx. 1 root root 18 9.2. 00:06 /etc/rc0.d/K05vtigervm -&gt; ../initd.d/vtigervm Where, how and when I can acquire HM's?