Is there a Latex symbol to represent the inverse angle of a complex number? Since, from what i see, this isn't very well known, I'll post a picture of my math book which uses it below (Engineering Mathematics - Stroud). Thanks.
I am trying to draw a partial box around one or more words within a paragraph--that is, a rectangle with one or more sides missing. I have read the documentation for the fancybox package and it doesn't seem to be able to do what I want. I was able to produce thus ugly hack:    That's just a vertical bar | followed by a word with \overline and \uline. The trick is to connect the vertical and horizontal lines so that you have a rectangle missing just one side.  Is there an easy, general way to do this, perhaps with Tikz? I don't know enough about that package to be able to do this myself, but I'm wondering if there might be other solutions too.
Capitalization of names that begin lowercased, at the beginning of a sentence Definition of injective function Best way to save a ordered List to the Database while keeping the ordering I was wondering if anyone has a good solution to a problem I've encountered numerous times during the last years.  I have a shopping cart and my customer explicitly requests that it's order is significant. So I need to persist the order to the DB.  The obvious way would be to simply insert some OrderField where I would assign the number 0 to N and sort it that way.  But doing so would make reordering harder and I somehow feel that this solution is kinda fragile and will come back at me some day.  (I use C# 3,5 with NHibernate and SQL Server 2005)  Thank you Group of order $8p$ is solvable, for any prime $p$ Consider the following question:     Show that a group $G$ of order $8p$ is solvable, for any prime $p$.   I am kind of stuck, but here are my first attempts:  I chose the series of subroups $G&gt;H_8&gt;H_4&gt;H_2&gt;H_1=\{e\}$ where $H_k$ has order $k$. All of these subgroups exist due to Sylow's Theorems. We have the quotients $H_8/H_4\cong H_4/H_2 \cong H_2/H_1 \cong \mathbb Z/2\mathbb Z$, so $G&gt;H_8\rhd H_4\rhd H_2\rhd H_1=\{e\}$. Only the factor $G/H_8 \cong \mathbb Z/p\mathbb Z$ is causing me a headache, because although its of prime order (and hence abelian) I don't know whether $H_8$ is normal in $G$ (unless $p=2$).  If $p\ne 2$, then the number $k$ of Sylow $2$-subgroups is $1$ mod $2$. Since $k$ divides $|G|$ we might have $k=1$ or $k=p$. If we had $k=1$ then $H_8$ would be the only Sylow $2$-subgroup (and hence normal in $G$). But how could we show this?  Or is there even an easier way to approach this problem?  $\ $  edit: I am going to try a different approach: The case $p=2$ is clear. When $p=3$ or $p=7$ the group has order 24 or 56 and since the smallest simple non-abelian group has order 60, the group must also be solvable in these exceptional cases (as Mariano pointed out).  In any other case we get a Sylow $p$-subgroup $H_p$ of order $p$, which is is normal in $G$. The quotient $G/H_p$ has order $8=2^3$; and prime power order implies solvable. So $G/H_p$ is solvable and $H_p$ is also solvable. And since the composition factors of $G$ are those of $H_p$ together with those of $G/H_p$, we conclude that $G$ is solvable iff $H_p$ and $G/H_p$ are solvable, which is the case. Hence $G$ is solvable if $|G|=8p$. Can I install pantheon terminal in Ubuntu 14.04 or newer? Is it possible to install Pantheon terminal from elementary OS on my Ubuntu 14.04 or newer LTS releases? If yes, how can I? Where did the Q come from, before the Q Continuum? In the Star Trek Universe where did the Q come from? I mean before they were Q in the Q Continuum? In Star Trek Voyager you learn that the race is somewhat intolerant of individualism (in the episode of the Q Civil War) but Q also suggest they have always been Q. Is that statement true, have the Q always been the Q? or were they a race that evolved into omniscience? Adjectives and nouns to describe an answer that doesn't answer the question? Define the content of a polynomial (over an arbitrary commutative ring $A$) to be the ideal generated by its coefficients, denoted $c(f)$. I want to show that   $$c(fg) = c(f)c(g).$$      (I was told this is true.)    What I was able to show was that $c(fg) \subseteq c(f)c(g)$ (this is obvious), and that their radicals are the same. My reasoning for the latter was as follows: let $f = a_0 + \dotsb + a_nx^n,$ $g = b_0 + \cdots + a_mx^m,$ and consider the matrix  $$ \begin{pmatrix}  a_0 b_0 &amp; \cdots &amp; a_0b_m\\ \vdots &amp; \ddots &amp; \vdots\\ a_nb_0 &amp; \cdots &amp; a_nb_m \end{pmatrix} .$$  Then $c(f)c(g)$ is generated by the entries of this matrix, while $c(fg)$ is generated by the sums along the diagonals. Let $\mathfrak p$ be a prime ideal of $A$ not containing $c(f)$ or $c(g)$, and let $i,\,j$ be minimal such that $a_i,\, b_j \notin \mathfrak p$.  Then all the terms in the generator of $c(fg)$ corresponding to the coefficient of $x^{i+j}$ are in $\mathfrak p$ except for $a_ib_j$, so that $c(fg) \not\subset\mathfrak p$. It follows that $c(fg) \subseteq \mathfrak p \Longleftrightarrow c(f)c(g) \subseteq\mathfrak p$, and hence $\sqrt{c(fg)} = \sqrt{c(f)c(g)}$.     Can I go all the way and show that in fact $c(fg) = c(f) c(g)$? Is it okay to have a multi-language deck? I want to buy a few booster pack from a local shop that only gets them in French. Is it generally acceptable to show up to a tournament with cards in different languages? In the final episode of season 6 we see that Lyanna Stark is whispering something to Ned. What is she saying? Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$ So I want to show that $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ and determine its Galois group.   My thoughts are as follows:  Define $\alpha := \sqrt{2+\sqrt{2}}$. Then it is easily shown that $\alpha$ satisfies $\alpha^4-4\alpha^2+2=0$.  Define $f(x) := x^4-4x^2+2$. Then $f$ is irreducible over $\mathbb{Q}$ by Eisenstein with $p=2$.  So we have that $f$ is the irreducible polynomial for $\alpha$ over $\mathbb{Q}$.   Further $|\mathbb{Q}(\alpha):\mathbb{Q}|=4$.  For $\mathbb{Q}({\alpha})$ to be Galois, it must contain all roots of $f$.   Define $K=\mathbb{Q}(\alpha)$ for convenience.  Define $\alpha := \alpha_1$.  Since $f$ has only even powers, we know that $-\alpha := \alpha_2$ is a root, and therefore contained in $K$ since $K$ is a field.  We note that the other two roots are $\alpha_3=\sqrt{2-\sqrt{2}}$ and $\alpha_4=-\sqrt{2-\sqrt{2}}$.  So in order to show $K$ is Galois, it must be shown that $\alpha_3$ and $\alpha_4$ lie in $K$.  Now $\alpha_1^2=2+\sqrt2$ and so $\sqrt2 \in K$. Thus $-\sqrt2 \in K$ since $K$ is a field.  Can somebody explain why $\alpha_3$ and $\alpha_4$ lie in $K$?  Next we are to determine the Galois group of $K$.  Assuming $K$ is Galois, since it has degree $4$ over $\mathbb{Q}$ (shown earlier), we know that its Galois group has size $4$. There are only two groups of size $4$, namely $V_4$ and $C_4$, the Klein four group and the cyclic group of order $4$.  How do we determine which of these choice is in fact the Galois Group? How should we deal with duplicate answers? Why do statisticians like "$n-1$" instead of "$n$"? If a creature with Multiattack is affected by Haste, can it make two Multiattacks? It's a fairly straightforward question.   A creature has a Multiattack option, and it also can cast the Haste spell. While under the effect of Haste, can it use the second Action to take another Multiattack Action?  My main concern is the clause of the Haste spell that limits the number of attacks made from taking the Attack Action. This is a different Action, though.   Then there is the concern of being able to use Multiattack at all, because Haste says what kinds of Actions you can take. How do I check the integrity of a storage medium (hard disk or flash drive)? I've run across (personally or by proxy) a bunch of storage media that exhibited physical problems during their life. Hard disks, solid state drives and other flash storage media all fail after some time. The situation is worse with USB flash drives and flash cards because of the large amount of suppliers with inferior flash cell and flash controller quality.  How can I reliably detect if a storage drive suffers from physical damage?  Note that this question isn't about file system integrity (which can be checked with ). How to get a list of appendices? How to get a list of appendices like the list of figures and list of tables? How to set it on a separate page, like the list of figures and list of tables? Distribution of the maximum of two correlated normal variables Say I have two standard normal random variables $X_1$ and $X_2$ that are jointly normal with correlation coefficient $r$.   What is the distribution function of $\max(X_1, X_2)$? How to align content of a div to the bottom Euler's equations of rigid body motion from least action principle I would like to derive Euler's equations of rigid body motion from least action principle.   Suppose we are in free space so we have no gravity so Lagrangian is equal to kinetic energy.   $$ L = T = \int_M \rho(x)(x\cdot \Omega_B)^2 d^3x $$  where $\rho$ is density, $\Omega_B$ is angular velocity bivector in reference body frame and $M$ is the rigid body.  The action principle says(if I haven't messed up something) that   $$ 0 = \frac{d}{ds}\bigg|_{s=0}\int_0^T \int_M \rho(x)[x\cdot (\Omega_B(t)+s \delta \Omega(t))]^2 d^3x $$  for every $\delta \Omega$ with zero ends i.e. $\delta \Omega(0)=\delta \Omega(T) = 0$. I should get these equations:  $$ I(\Omega_B) - \Omega \times I(\Omega_B)=0 $$  where  $$ I(\Omega) = \int_M \rho(x) (x\wedge (x\cdot \Omega)) d^3x. $$  But I can't see where I would get the time derivative of $\Omega_B$ from the action principle. Can someone tell me what I'm doing wrong? I guess there is some trouble with that $\delta \Omega$ can't be completely arbitrary. TTY doesn't wait for password I upgraded from 16.04 to 18.04. When I press Ctrl+Alt+F3 I get the terminal which prompts for the username. When I type my username and press Enter, it doesn't wait for my password (its as if I pressed Enter without entering anything). Even if I enter something, the text shown in plaintext and then it keeps on saying my password is wrong. After few times, the screen is cleared and it prompts for my username again and the same loop goes on.