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What Is Energy? Where did it come from? Some Git commands take commit ranges and one valid syntax is to separate two commit names with two dots .., and another syntax uses three dots ....  What are the differences between the two? If the matrix $A^3 = 0$, is $A$ singular? What's the best way to version CSS and JS URLs? What are the chances of getting $K$ identical outcomes in a row in a series of $Y$ independent events? Each event has two possible outcomes: a and b.  Each event is independent.  Outcome a has $X\%$ chance of happening.  If we run $Y$ number of such events, what are the chances of getting $K$ outcomes a in a row, at least once?  To put it in practical terms imagine a series of coin tosses. Each coin toss has two possible outcomes: a = heads and b = tails. Outcome a = heads has a chance of $50\%$. If we run $Y = 100$ number of coin tosses, what are the chances of getting $K = 10$ outcomes a=heads in a row, at least once?   I'm using the practical example only so I can express the idea more clearly. I'm interested in finding out the general formulas and computations so that I can run the calculations for any $X$, $Y$, or $K$.  I would also be very interested in knowing how the probabilities would change if the events would not be independent. What if the more outcomes a we have in a row the bigger the chance of getting an outcome a in the next event? What if it's the other way around, and the more outcomes a in a row we have the lower the chance of getting another a? Was V'ger responsible for the creation of the Borg? I remember reading somewhere a theory that  (the being that the Voyager probe had become in Star Trek: The Motion Picture) was somehow responsible for the creation of the Borg.  Is this canonical?  Where does the theory come from?  And how exactly did it go down? How to install Android Studio on Ubuntu? Sending file of custom python script with all parameters for popup interface already defined? I have a python script in ArcGIS 10 that when clicked, has a popup for a user to choose parameters. I would like to send this file to someone else. While I could send them the python code and have them setup the parameters - this method repeats work I have already done.   I would like to send someone one file that contains all of this information so they can just recieve my file, add it to a toolbox, and then have it all work. I am wondering where the file resides that has both the python code and parameters / what is the export process/ file type I should be looking at? (All the searches I do for "export script" seem to refer to exporting a model to python code...) Proof Verification : Prove -(-a)=a using only ordered field axioms I need to prove for all real numbers $a$, $-(-a) = a$ using only the following axioms:     Thanks to many members of the Mathematics Stackexchange Community, I have the following proof worked out:  Theorem: The Additive Inverse Identity is Unique  $ ( \forall a,b,c \in \mathbb{R} )(a+b=0) \land (a+c=0) \\ ( \forall a,b,c \in \mathbb{R} )(a+b=a+c) \\ ( \forall a,b,c \in \mathbb{R} )(b=c) \\ \text{QED} \\ $  Theorem $\textbf{a} \cdot \textbf{0 = 0}$  $ \begin{align} a \cdot 0 &amp;= a \cdot 0 \\ a \cdot (0 + 0) &amp;= a \cdot 0 \\ a \cdot 0 + a \cdot 0 &amp;= a \cdot 0 \\ a \cdot 0 + a \cdot 0 + (-a) \cdot 0 &amp;= a \cdot 0 + (-a) \cdot 0 \\ a \cdot 0 &amp;= 0 \\ &amp;\text{QED} \\ \end{align} $  Theorem: $-\textbf{1} \cdot \textbf{(a) = (}-\textbf{a)}$  $ \begin{align} a \cdot 0 &amp;= 0 \\ a \cdot \left[ 1 + (-1) \right] &amp;= a + (-a) \\ 1 \cdot a + (-1) \cdot a &amp;= a + (-a) \\ -1 \cdot (a) &amp;= (-a) \\ &amp;\text{QED} \\ \end{align} $  Theorem: $-\textbf{(}-\textbf{a) = a}$  $ \begin{align} 0 &amp;= 0 \\ -a \cdot 0 &amp;= 0 \\ -a \cdot \left[ 1 + (-1) \right] &amp;= 0 \\ -a \cdot 1 + -a \cdot (-1) &amp;= 0 \\ -a + \left[ -(-a) \right] &amp;= 0 \\ -(-a) &amp;= a \\ \text{QED} \\ \end{align} $  Does everything look alright? Have I missed anything?  Also--why is it necessary to show that the additive inverse is unique?  Many thanks in advance. Why do I get "Exception; must be caught or declared to be thrown" when I try to compile my Java code? I am having problems with the su command. I know my password and I am typing it correctly but su indicates authentication failure.  So I checked on the internet and then went into recovery mode and changed my username's password to what I was entering before.  Even now entering the same password on su gives me authentication failure.  Please suggest what am I doing wrong?? Is the use of "one of the" correct in the following context? Let $f$ be a holomorphic function in $H={\{z\in\mathbb{C}}:\Re(z)&gt;0\}$ such that it is continuous in $cl(H)$, $|f(z)|\leq1$ $\forall z\in \delta H$ and $\lim_{|z|\to\infty}f(z)=0 $. Prove that $|f(z)|\leq1$ $\forall z\in H$. I thought about taking the bounded set $H\cap D(0,R), R&gt;0$. This way we could use the maximum modulus principle to prove it but I do not really know how to relate it with the limit since I would need $|f(z)|\leq1$ $\forall z\in H\cap D(0,R)$, and I do not have it on $D(0,R)$. Hello everyone interested. Here's a seemingly simple problem.  Suppose $G$ is a complete, simple graph with vertices $\{1,2,..,n\}$, $n\geq k\geq 3$.  How many cycles of length $k$ can we have on $G$?  The statement seems simple but I have trouble putting everything together.  So I say fix a vertex and start forming cycles with length k. But I get nowhere... What is the proper approach to this problem since counting is not easy? Is there a relevant theorem that I am unaware of?  Thanks for any help... Reason for strange "canyons" of dust on the cooling platform I took apart my cooling platform Something like this :  And I found something like this inside :       I wonder why around the hollows are the "canyon like" structures.  PS: I also drop this platform on the floor, maybe this could be a reason ?   EDIT :  Everybody seems (except Brandon Enright answer) to ignore the word "dust" in this question. Those structures are not in the plastic itself. It is dust ON the plastic.  Proof :     Completely wiped out with my finger : What should be done with questions that have been resolved as a comment instead of an answer? I have had several business loans and understand that this is not by any means typical.  I noticed that for a year my payments were not decreasing the principal on the loan. When I asked the bank why, I found out that at some point my payments were getting applied to principal only. Now I am paying interest off that was not paid for 6 months or so.  My banker states that he can change all the interest payments to principal payments. I am confused to why this would be. If I make all my principal payments first then would the bank lose a substantial amount of interest?  This has happened on 2 of the 3 loans I have looked at so far.  If they are willing to change all the payments to principal only, would this be beneficial to paying less interest ultimately? I assume there would be an interest account accruing somewhere that would need to be paid off at the end of the loan.  I understand that typically you have a loan balance that would include interest and principal. My balance owed did not change when I continued to make same payment? It gets applied to interest. The balance on the statement must not be a true balance. There must be interest owed above the balance showed but why would they do this to start with? They applied full payments to principle only first in order for interest to get to be more than several payments Please add TeX rendering on the Android app MathJax isn't currently working on the Android app, which makes it totally useless for the 'technical' sites like  or .  For example, in the Android app:    Same post in a (mobile) browser:     Please add this functionality! How to get Microsoft Search to index common file types like .pdf and .pub I'm trying to get Windows Search (4.0) to index PDFs and .pub, .doc, etc. files.  The files are on an XP Pro PC acting as a file server. But I don't have MS Office, etc. installed (no need, trying to keep it "slim and trim") but WS 4.0 is doing the indexing of those files and not indexing .pdf and .pub.  I've found a few  (from Foxit PDF reader) rated 3/5 and only support WS 2.0 or 3.0.  Looking for other solutions and also a way to index .pub files. How to disable copy in shared folder I am trying to make folder that everyone can view and can't copy,modify.    For all answers.I thinking about specially ms office files. I tried following methods  Software  - copy protect    - anti copy    This method can't protect save as and upload  MS office macro    Users can enable,disable macros.  MS office restriction    This is best choice but can't protect from file uploading.   Almost I found the solution. Way is combination of MS office restriction and restrict access from browsers using firewall or antivirus. But I don't exact how to do this.