why same regex expression gives two different results?
How do you use a variable in a regular expression?
How to find the determinant of this $3 \times 3$ Hankel matrix? Today, at my linear algebra exam, there was this question that I couldn't solve.   Prove that $$\det \begin{bmatrix}  n^{2} &amp; (n+1)^{2} &amp;(n+2)^{2} \\  (n+1)^{2} &amp;(n+2)^{2}  &amp; (n+3)^{2}\\  (n+2)^{2} &amp; (n+3)^{2} &amp; (n+4)^{2} \end{bmatrix} = -8$$   Clearly, calculating the determinant, with the matrix as it is, wasn't the right way. The calculations went on and on. But I couldn't think of any other way to solve it. Is there any way to simplify $A$, so as to calculate the determinant? How could I go about setting the 'above' and 'below' vspaces around the equation environment?  If the same could be done for the subequations or gather environments, this would be a bonus. Many spells have "At higher levels" that provides for more targets or more damage.  For example Magic Missile can be cast as a second level spell and shoot 4 missiles instead of it's 3 as a first level spell.  Can a spell without that verbiage, such as Firestorm be cast in higher level slots for the same effect as the lower level spell? I think every single one of us has experienced a dark dot in his/her sight when looking at a white surface such as a white ceiling, there's something in our eye that with a little focus we can see it, and when we move our eyes around, that thing floats around quickly in your sight. It's like a substance on eye's surface, I want to know what is it called. Hope I'm clear enough. I am a physics Master's student.  I have good grades, with CGPA above 9(/10)(However, all courses in my university are graded on a curve so the actual marks are significantly lower than 90%). This has been my marks throughout my undergraduate and graduate courses and I was happy because this was above average. Also, my summer projects in UG were usually computational and any difficulty I had in theoretical aspects I attributed it to my inexperience in advanced Physics.   However, recently I have started to work on pure theoretical project and while reading for the project I realized that I consistently miss some details from each topic. I understand the general idea, can re-derive most of the equations(with some effort) however when I discuss things with my professor or when I read further topics I realize that I have missed certain key-details. This is exactly similar to the case of me missing a few key questions in my exams and getting that 9 grade point instead of 10(or at-least I think so). I was not worried about this before but now I realize that there is some flaw in the way I study (but I do not know what).   Since I want to apply for Physics PhD I realized that I will need to have solid concepts and it would be best to correct my mistakes but I do not know how or even what could be the reason for my imperfect understanding.  Any direct suggestion would help or even certain general guidelines on what I can do.    Things I've tried recently -   1) I've started to written down almost everything I read. This is slightly laborious but is certainly helping. It also helps me to gloss over the things I read previously when I come back the next day.  2) I completely avoid smartphone when I'm library/study room so that I won't get distracted. Not sure if it helps me solve the above problem but certainly helps me study better in general.  3) I've tried to read the topics from several sources. The books I am reading is somewhat like a collection of research papers and hence does not have a exercises. So I though reading from various sources might help get better ideas but it's not really helping because most other sources are too high level as an introductory text. But this will probably help me in cases where it's practical to do this.  4) Somewhat similar to point (1)- I am trying reread, whenever I can, what I've done until that point. But not sure how much this will help me in filling the gaps. "May I have a glass of water" vs. "Can I have a glass of water": which is better, "may I" or "can I", when asking for something? What is the correct sentence when asking for something like while flying asking for a glass of water :     May I have a glass of water     or      Can I have a glass of water No such file or directory but I can see it! I'm trying to run a python script, on a headless Raspberry PI using winSCP and get the following error message:  Command '"./areadetect_movie_21.py"' failed with return code 127 and error message /usr/bin/env: python : No such file or directory.   When I try and run from terminal, I get:  : No such file or directory.   I try a similar python script, in the same directory, with the same python shebang, the same permissions and using the same user pi, and it works.  I also do a ls and I can see the file, so I don't know why it will not run. Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$. Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$. Is the converse true? I.e., if $g \circ f$ is invertible, does it follow that $f$ and $g$ are both invertible?  I know that if $f$ and $g$ are invertible, then they are both injective and surjective. Therefore, I assume from this that I have to show $g \circ f$ is injective and surjective as well, thus proving it is invertible. However, I don't know how to show $g \circ f$ is bijective based only on the information I have. I am given two charged particles of same charge at a distance of $r$. They initially apply force $F$.  Now an infinite dielectric (of dielectric constant $4$) of width $\frac{r}{2}$ is introduced between the particles. What will be the new force?   I find this problem confusing because I have only been told about forces when its either fully dielectric or fully vaccum given by Coulomb's law. How do we get forces when only partial space is dielectric? I'm trying to include a tabu inside a new definition of an environment but it fails if the postamble (enddef) of the new environment includes tabu's contents, so cells. With tabular or tabular* works. The error is:   ! Missing number, treated as zero. \tabu@0.H5   The MWE:  \documentclass{article} \usepackage{tabu} \newenvironment{testt} {\begin{tabu} to \textwidth{cccc}  11 &amp; 12 &amp; 13 &amp; 14 \\  21 &amp; 22 &amp; 23 &amp; 24 \\} {  41 &amp; 42 &amp; 43 &amp; 44 \end{tabu} }  \begin{document}  \begin{testt}  31 &amp; 32 &amp; 33 &amp; 34 \\ \end{testt} \end{document}   How to solve this? Why should the interface for a Java class be preferred? would report a violation for:  ArrayList&lt;Object&gt; list = new ArrayList&lt;Object&gt;();   The violation was "Avoid using implementation types like 'ArrayList'; use the interface instead".  The following line would correct the violation:  List&lt;Object&gt; list = new ArrayList&lt;Object&gt;();   Why should the latter with List be used instead of ArrayList? A net $\varphi : [0, \omega_1) \to M$ on a metric space $M$ converges $\iff \varphi$ is eventually constant I want to prove that      If $M$ is a metric space, then a net $\varphi : [0, \omega_1) \to M$ converges if and only if $\varphi$ is eventually constant. ($[0, \omega_1)$ is the set of ordinals less than $\omega_1$, where $\omega_1$ is the first uncountable ordinal, with the order topology).   $(\Leftarrow)$ is clear.  My trouble is with $(\Rightarrow)$. Suppose that $\varphi \rightarrow x$.  Let $U = \{ \alpha \in [0, \omega_1) \, \mid \varphi(\alpha) \neq x \}.$  If $U = \emptyset $ we are done, so assume $U \neq \emptyset$. Since $[0, \omega_1)$ is well ordered, there is a $\alpha_1 \in U$ such that $ \alpha_1 \leq \alpha \, \, \forall \alpha \in U.$  If $ U \setminus \alpha_1 = \emptyset$ we are done. If not, we repeat the procedure.  It is clear that if $card(U) = n$ for some $n \in \omega$ then we only have a finite set of ordinals $\{\alpha_1, \ldots, \alpha_n\}=U$ so $\varphi$ is constant for $\alpha &gt; \alpha_n.$  If $U$ is countably infinite, then $U = \{\alpha_n\}_{n \in \omega}$ and since $[0, \omega_1)$ is sequentially compact and the $\alpha_n$'s are countable ordinals  we obtain a convergent subsequence $\{\alpha_{n_k}\}$ of $U$ to some countable ordinal $\beta \in [0, \omega_1)$ $[\beta = \cup_{n_k \in \omega}\alpha_{n_k}???],$ hence $\varphi$ is constant for all $ \alpha &gt; \beta.$     But what if $U$ is uncountable?    Idea: If $U = [0, \omega_1),$ then since $\varphi \rightarrow x$, for every $n \in \omega$ there is $\alpha_n \in U$ such that $\alpha \geq \alpha_n \implies \varphi(\alpha) \in B_{\frac{1}{n}}(x).$ This is a countable sequence, so since $[0, \omega_1)$ is sequentially compact, there is a subsequence converging to a countable ordinal $\beta \in [0, \omega_1)$. Clearly, $\beta$ satisfies that $\phi(\beta) \in B_{\frac{1}{n}}$(x) for every $n \in \omega.$     We can conclude that $\varphi(\beta)= x$ right?   For if $\varphi(\beta) \neq x$, then taking $n$ such that $0 &lt; \frac{1}{n} &lt; d(x, \varphi(\beta)),$ we would have that $\varphi(\beta) \not \in B_{\frac{1}{n}}(x),$ contradicting the hypothesis.  I feel pretty confident about this, but If someone sees an error in the argument, please comment. Thanks. How do I get a slanted \sum symbol? I want to use a slanted \sum symbol denoting a different meaning from summation. How do I get a slanted \sum symbol? Using Illustrator CS6 to create shape out of multiple paths Recommendations for good Newtonian mechanics and kinematics books What are some good books for learning the concepts of Kinematics, Newton laws, 2D Motion of Object etc.? How to change from csh to bash as default shell I have csh as my default shell, as shown by echo $SHELL. I want to switch to bash as my default shell. I tried the following approaches to no avail:   With chsh I get:  chsh: can only change local entries; use ypchsh instead.  With ypchsh I get:  ypchsh: yppasswdd not running on NIS master host ("dcsun2").    I only have .chsrc in my home directory and I cannot find any .profile files in /etc. How can I change my default shell to bash? When I use ssh -X on my Mac (running OS X 10.6.7) to connect to my Ubuntu (11.04) box, I get the following warning:     Warning: untrusted X11 forwarding   setup failed: xauth key data not   generated Warning: No xauth data;   using fake authentication data for X11   forwarding.   Is there something I can do to make this warning go away?  If not, can I safely ignore it?  X11 forwarding seems to work fine, though I do see this message:     Xlib:  extension "RANDR" missing on   display "localhost:10.0".   Is that related to the warning?  (I'm guessing not.  If it's not, I'll file a new question about that.) How can I take browser screenshots at a higher resolution than my browser supports? Which one should should I use? For some reason I have always used "towards", but I see some people saying "toward", like :     A great deal of his work in economic   theory has been directed toward   strengthening the foundations of our   understanding of central banking and   social insurance--indeed, one of my   most advanced macroeconomic classes   was nothing but a set of changes rung   upon a particular model that Peter   Diamond had advanced at...   Is there a difference between the two? When should they be used? If $E$ has measure zero, then does $E^2$ have measure zero? Currently I'm just working through measure theory just to see if its something I would like to take.  Unfortunately I am stuck on this problem from Carothers.     If $m^*(E)=0$, then $m^*(E^2)=0$.   Where $m^*$ denotes outer measure and $$E^2=\{x^2:x\in E\}.$$  I toyed with the idea that $I_k &lt; 1 \Rightarrow I^2_k &lt; I_k$.  However I am at a loss as to how to set up a chain of inequalities (which is what I am assuming I need).