curved arrows with tikz package
How to draw arrows between circled coefficients of polynomials using tikz
Lets say I have a function defined as $f(x) = \sqrt {x^2}$. Common knowledge of square roots tells you to simplify to $f(x) = x$ (we'll call that $g(x)$) which may be the same problem, but it isn't the same equation. For example, say I put $-1$ into them:  $\begin{align} f(x) &amp;= \sqrt {x^2} \\ f(-1) &amp;= \sqrt {(-1)^2} \\ f(-1) &amp;= \sqrt {1} \\ f(-1) &amp;= 1 \end{align}$  $\begin{align} g(x) &amp;= x \\ g(-1) &amp;= -1 \end{align}$  thereby, we conclude that $f(x)$ and $g(x)$ do not produce the same results even though they are mathematically the same. This is also shown when we try to graph the functions:  :    :    :    From this, we can see that given $f(x) = \sqrt {x^2}$, when simplified is not the same as $f(x) = x$. So, is there any way to prove that $y = \sqrt {x^2}$ is not the same as $y = x$ for negative values, but is infact the same as $y = \mid x \mid$? How to ignore the first line of data when processing CSV data? add transistor model to LTspice I'm trying to add the mmbt3904lt1g model to LTspice. - I've copied its data from the MMBT3904LT1.LIB.txt in the standard.bjt library. - I've added an NPN to my schematic then I've changed its value to mmbt3904lt1g  - through the Edit->spice directive I've pasted the path to this library I thought it was all to use this device but no simulation is achieved.. I get only: Can't find definition of model "MMBT3904LT1g"  where I'm doing wrong ? What is the difference between "information on", "information of", "information about" "Information on something", "information of something", "information about something" — these three usages all sound having the same meaning for me. Are there any differences? When are attributive nouns plural? Sorry for the title, it is not very evident and intuitive but I really do not how to tell it better...  Well, you know, several times, or better, many times, we use this form:  If I want to say:      "development of special weapons was   the first point in Hitler's   program..."   I will say this (a better form):      "special weapon development was..."   OK...  is it      "special weapon development"   or      "special weapons development" (note   the plural...)   what's the grammar rule in order to understand how to use this very used form? How can I evaluate $\int_{-\infty}^{\infty} e^{-x^2} dx$ without using polar coordinates Can Gmail automatically base "send from" address on the "to" address? This is an attempt to gather together the various questions about time that have been asked on this site and provide a single set of hopefully authoritative answers. Specifically we attempt to address issues such as:   What do physicists mean by time? How does time flow? Why is there an arrow of time? Hi I have to proof that for $\Omega$ a nested family of odd subsets of $V$ with $V$ finite, we have that $|\Omega|\leq\frac{3}{2}|V|$. I can prove that for certain constructions of $\Omega$, it is satisfies $|\Omega|\leq \frac{3}{2} |V|$, the problem I encounter is that I have to proof that my construction of $\Omega$ is optimal. (Since I have to prove that this holds for every family of odd subsets) Can anyone give me a hint on how to proof that every $\Omega$ satisfies this equation.    $\Omega$ is nested if for all $A,B\in\Omega$ either $A\subseteq B$, $B\subseteq A$, or $A\cap B=\varnothing$; a set is odd if its cardinality is odd. I checked this  but didn't fully understand it.   I know that the volume of a right circular cone is $V = \frac{1}{3}\pi x^2h$  I know that I must take the first derivative and set it equal to zero, which will find the maximum.   My problem is how to deal with the variable $h$, the height?   How can I rewrite that in terms of $x$ or $r$.      I have tried to implicitly differentiate this equation, but that wasn't helpful for me. One of my columns is called from. I can't change the name because I didn't make it. Am I allowed to do something like SELECT from FROM TableName or is there a special syntax to avoid the SQL Server being confused? How to prove Fibonacci sequence with matrices? Is it normal for BibTeX to replace similar author names with "------"? In my .bib file, I have two entries with exactly the same author names:  @article{Seshadrinathan2010A-Subjective-St,     Author = {K. Seshadrinathan and R. Soundararajan and A. C. Bovik and L. K. Cormack}, ....  @article{Seshadrinathan2009Study-of-Subjec,     Author = {K. Seshadrinathan and R. Soundararajan and A. C. Bovik and L. K. Cormack}, ....   When cited after each other, and typeset with the IEEEtran bibliography style, the .bbl file reads:  \bibitem{Seshadrinathan2010A-Subjective-St} K.~Seshadrinathan, R.~Soundararajan, A.~C. Bovik, and L.~K. Cormack, ``A   subjective study to evaluate video quality assessment algorithms,''   \emph{SPIE Proceedings Human Vision and Electronic Imaging}, 2010.  \bibitem{Seshadrinathan2009Study-of-Subjec} ------, ``Study of subjective and objective quality assessment of video,''   \emph{IEEE Transactions on Image Processing}, 2009.   As you can see, the author names have been replaced with ------. It looks like this in the final PDF:    Is this normal behavior? apt-get cannot connect to Ubuntu archives Once a post has been edited, you can go into the revision list, and click "View Source".  If a post has not been edited (and I don't have the edit privilege) is there any way to see the raw Markdown?  The reason I ask I because of  - it's impossible to see the source markdown, and how it's different from the rendered Markdown.  Not exactly a huge issue, just curious! Experimental observation of matter/antimatter in the universe Ordinary matter and antimatter have the same physical properties when it comes to, for example, spectroscopy. Hydrogen and antihydrogen atoms produce the same spectroscopy when excited, and adsorb the same frequencies. The charge does not make a difference in the potential (regardless if it's generated by a proton or an antiproton) nor in how the positron behaves in this potential (being its mass equal to the mass of an electron)  How can astronomy evaluate if a far galaxy is made of matter or antimatter, given that from the spectroscopy point of view, they behave in the same way? In other words, how do we know that an asymmetry exists between matter and antimatter in the universe? Rewiring a switch controlled receptacle to a ceiling light When editing really long code blocks (which should definitely be refactored anyway, but that's beyond the scope of this question), I often long for the ability to collapse statement blocks like one can collapse function blocks.  That is to say, it would be great if the minus icon appeared on the code outline for everything enclosed in braces.  It seems to appear for functions, classes, regions, namespaces, usings, but not for conditional or iterative blocks.  It would be fantastic if I could collapse things like ifs, switches, foreaches, that kind of thing!  Googling into that a bit, I discovered that apparently C++ outlining in VS allows this but C# outlining in VS does not.  I don't really get why.  Even notepad++ will so these collapses if I select the C# formatting, so I don't get why Visual Studio doesn't.  Does anyone know of a VS2008 add-in that will enable this behavior?  Or some sort of hidden setting for it?  Edited to add: inserting regions is of course an option and it did already occur to me, but quite frankly, I shouldn't have to wrap things in a region that are already wrapped in braces... if I was going to edit the existing code, I would just refactor it to have better separation of concern anyway. ("wrapping" with new methods instead of regions ;) Do we need a global test before post hoc tests? I often hear that post hoc tests after an ANOVA can only be used if the ANOVA itself was significant.    However, post hoc tests adjust $p$-values to keep the global type I error rate at 5%, don't they?  So why do we need the global test first?  If we don't need a global test is the terminology "post hoc" correct? Or are there multiple kinds of post hoc tests, some assuming a significant global test result and others without that assumption? Why xargs can't work with tail -f? I want tail -f /var/log/syslog | grep it with pattern "arpwatch" and send every line to myself via jabber: xmpp username@jabber.server using xargs  tail -f /var/log/syslog | grep arpwatch | xargs sendxmpp username@jabber.server   no working.  but tail  /var/log/syslog | grep arpwatch | sendxmpp username@jabber.server   works well.  I think it's some fundamental thing about xargs and tail -f that I don't understand.