Find the equations of all tangent lines to the curve $x^2 + 4y^2 = 8$ that pass through the point (−4, 0).
Tangent lines to ellipse from external point
How are Star Wars blasters properly reloaded/charged/etc.? I've been looking into the inner workings of blaster weapons primarily using Wookieepedia. While most of it I understand, I still can't figure out how many rounds a gun would fire before needing the ammunition component replaced/reloaded. Is there any solid answer?  I'm someone who likes to design and draw his own weapons and such, but likes to keep to how the technology of an era is in mind. While Star Wars weapons don't really have magazines (and you don't really see them reloading during battle), other people's work still has magazines and treat ammunition consumption as though it's pretty much the same as our modern era.  I guess that I just find it hard to believe, with the technology of the Star Wars Universe, that characters would need to carry lots of ammunition (hence why clones/storm troopers/battle droids don't visually seem to carry very much).  Example: DC-15A     "The weapon's tibanna gas cartridge carried enough gas for up to 500 shots, depending on the power settings of the weapon, while the charge pack lasted about 50 shots"   Source:   I've read in several places that this can be seen in two different ways.  1. Which ever ran out first, had to be replaced before the weapon could fire again. 2. The weapon contained two different sources of ammunition (so I guess you could switch between the two) Prove that for every $a, b\in R\setminus\{0\}$ is correct this inequality:  $$\frac{|a+b|}{1+|a+b|}&lt;\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}.$$ Prove that $\int f\ d\lambda = \int_{a}^{b} f(x)\ dx,$ for any $f \in \mathcal R[a,b].$ Theorem $:$ Let $f : [a,b] \longrightarrow \Bbb R$ be a Riemann integrable function. Then $f \in L_1[a,b]$ and $$\int f\ d\lambda = \int_{a}^{b} f(x)\ dx.$$  The proof given in my book is as follows $:$     At the fag end of the proof $f$ has been proved to be measurable which uses the completeness of the measure space $\left ([a,b], \mathcal L \cap [a,b], \lambda |_{[a,b]} \right ).$ Then it has been stated that $f \leq \Phi_n,$ $\forall$ $n$ and hence $f \in L_1[a,b]$ since $\Phi_n \in L_1[a,b]$ by the proposition $5.4.3 \ $(i) which states the following $:$  Let $f \in \Bbb L$ and $g \in L_1[a,b].$ If $|f(x)| \leq g(x)$ for a.e. $x(\mu).$ Then $f \in L_1[a,b].$  I don't know how to use this proposition to claim that $f \in L_1[a,b].$ Also I don't understand the way dominated convergence theorem used here. What I know about dominated convergence theorem is as follows $:$  Let $(X, \mathcal S, \mu)$ be a complete measure space. Let $g \in L_1(X)$ and $\{f_n \}_{n \geq 1}$ be a sequence in $\Bbb L$ such that $|f_n(x)| \leq g(x)$ a.e. $x(\mu).$ Let $f_n(x) \to f(x)$ as $n \to \infty$ a.e. $x (\mu).$ Then $f \in L_1(X)$ and $$\int f\ d\mu = \lim\limits_{n \to \infty} \int f_n\ d\mu.$$  From here how to conclude that $$\Psi_n\ \bigg\uparrow\ f \implies \int f\ d\lambda = \lim\limits_{n \to \infty} \int \Psi_n\ d\lambda.$$ Please help me in this regard. Thanks in advance. How to control the coordinates in Tikz & Matrix? I use Tikz with Matrix because I need to draw arrows and write on and beside the matrix. However, I don´t control the coordinates well other than using trail and error. For instance in the matrix below I have put an X at (0,0) and it is not in the centre of the matrix. However, if the matrix is 4x4 than it is in the centre. Is there any convenient way to know the coordinates? \documentclass[border=1cm]{standalone}  \usepackage{tikz} \usetikzlibrary{matrix}   \begin{document}  \begin{tikzpicture} \matrix(m) [matrix of math nodes, nodes in empty cells, nodes={minimum size=1cm, outer sep=0pt, text height=1.5ex, text depth=.25ex}] {  36     &amp;   &amp; 32    &amp;   &amp; 33    &amp;   \\         &amp;   &amp;       &amp;   &amp;       &amp;   \\  10     &amp;   &amp; 8     &amp;   &amp; 7     &amp;   \\         &amp;   &amp;       &amp;   &amp;       &amp;   \\  12     &amp;   &amp; 17    &amp;   &amp; 16    &amp;   \\         &amp;   &amp;       &amp;   &amp;       &amp;   \\  23     &amp;   &amp; 15    &amp;   &amp; 16    &amp;   \\     &amp;   &amp;   &amp;       &amp;   &amp;       &amp;   \\ };  \draw (m-1-1.north west) rectangle (m-8-6.south east);    % Lineas horizontales \foreach \i in {2,4,6,8} {   \draw (m-\i-1.south west) -- (m-\i-6.south east); }  % Lineas verticales \foreach \j in {2,4,6} {   \draw (m-1-\j.north east) -- (m-8-\j.south east); }  % dual ui \node [font=\color{red}] at (3.5,3) {$\ge 10$}; \node [font=\color{red}] at (3.5,1) {$\ge 8$}; \node [font=\color{red}] at (3.5,-1) {$=4$}; \node [font=\color{red}] at (3.5,-3) {$=3$};  % dual vj \node [font=\color{red}] at (-2.5,-5) {$\le12$}; \node [font=\color{red}] at (-0.5,-5) {$=12$}; \node [font=\color{red}] at (1.5,-5) {$=6$};  \node [font=\color{red}] at (0,0) {X};  \end{tikzpicture}     \end{document} I'm having troubles with one of the problems in the book Introduction to Commutative Algebra by Atiyah and MacDonald. It's on page 11, and is the last part of the second question.  Given $R$ a commutative ring with unit. Let $R[x]$ be the ring of polynomials in an indeterminate $x$ with coefficients in $R$. We say that a polynomial $f = \sum\limits_{i=0}^n r_ix^i \in R[x]$ (with coefficients $r_0, r_1, \ldots, r_n$) is primitive if $\langle r_0,r_1,...,r_n\rangle = R$, i.e., the ideal generated by the coefficients of $f$ is $R$. Let $f$ and $g$ be two polynomials in $R[x]$. Prove that $fg$ is primitive iff $f$ and $g$ are both primitive.  The $\Rightarrow$ part is easy. Say, $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$; then $fg = \sum\limits_{i=0}^{m+n} c_ix^i$, where $c_k = \sum\limits_{i + j = k}r_is_j$. Since $fg$ is primitive, there exists a set of $\{\alpha_i\} \subset R$, such that $\sum\limits_{i=0}^{m+n} \alpha_ic_i = 1$, to prove $f$ is primitive, I just need to write all $c_i$'s in terms of $r_i$'s, and $s_j$'s, then group all $r_i$ accordingly, rearranging  it a little bit, and everything is perfectly done. And the proof of the primitivity of $g$ is basically the same. The $\Leftarrow$ part is just so difficult. Say $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$ are both primitive, then there exists $\{\alpha_i\}; \{\beta_i\} \subset R$, such that $\sum\limits_{i=0}^{n} \alpha_ir_i = 1$, and $\sum\limits_{i=0}^{m} \beta_is_i = 1$. At first, I thought of multiplying the two together, but it just didn't work. So, I'm stuck since I cannot see any way other than multiplying the two sums together. I hope you guys can give me a small push on this. Thanks very much in advance, And have a good day. "He added" vs. "Did he add" (inversion after negative adverb) If I'm standing at the equator, jump, and land 1 second later, the  Earth does NOT move 1000mph (or .28 miles per second) relative to me,  since my velocity while jumping is also 1000mph.   However, the Earth is moving in a circle (albeit a very large one),  while I, while jumping, am moving in a straight line.   How much do I move relative to my starting point because of this? I  realize it will be a miniscule amount, and not noticeable in practise,  but I'd be interested in the theoretical answer. If $A|B$ and $B|A$ then prove $A=\pm B$   So far I have $A|B \iff AX=B$ and $B|A \iff BY=A$ with $X,Y \in \mathbb{Z}$  Not sure how to finish, any help. "strlen(s1) - strlen(s2)" is never less than zero How to prove if $n$ is prime and $n | a^2$ then $n | a$? My professor assigned this for homework but I don't understand how to connect the dots. He suggested using the fact that $\gcd (x,y) \cdot \operatorname{lcm} (x,y) = xy$ but I'm not sure how that's relevant. I read a book a while ago in which a device is invented which opens up "tunnels" back through time. These tunnels cannot be traveled through, but can be used to view what happened in the past. I believe it starts out narrating the invention of the device, then skips ahead to after it is widely in use. There is another device created, a cloak of sorts,  which hides the user from the view of someone using the tunnel device. Does this ring a bell with anyone? Non numbered chapter* header issue From      Glueballs are predicted by quantum chromodynamics to be massive, notwithstanding the fact that gluons themselves have zero rest mass in the Standard Model. Glueballs with all four possible combinations of quantum numbers P (parity) and C (c-parity) for every possible total angular momentum have been considered, producing at least fifteen possible glueball states including excited glueball states that share the same quantum numbers but have differing masses with the lightest states having masses as low as 1.4 GeV/c2 (for a glueball with quantum numbers J=0, P=+, C=+), and the heaviest states having masses as great as almost 5 GeV/c2 (for a glueball with quantum numbers J=0, P=+, C=-).   Rather than going through a list of possible mechanisms that unfortunately I know next to nothing about, such as can the mass be attributed to virtual quarks, or binding energy between the gluons, I would rather leave the question as in the title to find out as much as I can.  Also, although the SM is firmly established, would the discovery of Glueballs buttress it further?  My apologies for not knowing more about the interior of hadron like particles or if the answer is readily available (or worse, blindingly obvious). How to compute the following integral?  $$\int\log(\sin x)\,dx$$    Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we find $F$ explicitly? Failing that, can we find the definite integral over one of intervals where $\log (\sin x)$ is defined? What Constitutes Spam? Over the course of the past several months, there have been one or more users who have been spamming about an XML parsing alternative named &quot;VTD-XML&quot;. I have occasionally assisted the moderators by pointing these answers out to them, and they have responded by deleting the answers, and the offending user (which hasn't stopped this person from immediately creating another user account and continuing). Now, there is a user , who claims to be the actual author of VTD-XML. In my mind, several of his answers, like his  to  looks a great deal like the original spam.  has asked me why I consider his answer to be spam. I decided to do so in public, in an attempt to get him to engage with this community, assuming he's not another sock puppet of the original spammer.  My reason is simple: he's been answering with one or two lines of text, followed by the exact same link to the exact same site that the spammer was using:  VTD-XML may well be worth a look as it is 1/3~1/5 the memory usage of DOM   Now, maybe I'm wrong. Maybe this does not constitute spam. Maybe Mr. Zhang can explain to us why we should not treat these posts as spam. Let's see.  Somehow, I missed the fact that in the same question, on July 31 2009:  how about vtd-xml  http://vtd-xml.sf.net   This one was from a user named &quot;Jimmy Zhang&quot;, which has since been deleted. Should I not be suspicious? Additionally, as pointed out by , he's answering a question about Delphi by posting a link to a site that doesn't even mention Delphi! Throughout the nonsense with the sock puppets, I kept asking for some real engagement from the sock puppet. It all just made VTD-XML look bad that instead of truly answering questions, we just kept getting incessant spam against old XML questions. Appeal Mr. Zhang, if you're not just desperately spamming StackOverflow because your product isn't selling, then please answer appropriate questions in a way that will actually help someone use your product to solve their problems. I think the following is true:  If $q$ is coprime to $a$ then $a\mid (nq-1)$ for some $n\in\mathbb Z$.  I have started to first reducing $n\in\mathbb Z$ to $n\in\{1,\ldots,a-1\}$ and then considering all the cases $q = ka + j$ for $j=1,2,3,\ldots,a-1$. I stopped at $j=3$ and found that this is kinda messy. Does anyone have an elegant proof for this (maybe using some results from number theory)?  [Note: question has been generalized from prime $q\nmid a$ to $\,q$ coprime to $a$] Where is the operator panel in Blender 2.8? The tutorial I read is based on 2.79. So I'm not sure if it's still the same name in 2.8. But I can't find anything providing similar functionalities. QGtkStyle was unable to detect the current GTK+ theme When I start any Qt app, I get this message in the terminal, and the Qt app doesn't use the same theme as my Gtk apps:  QGtkStyle was unable to detect the current GTK+ theme.   How do I make QGtkStyle detect my current GTK+ theme?  It seems that LXDE is using a settings daemon, and there is no gtkrc-2.0 file in my home folder. I need to set weights for some edges from a python script. I have tried looping through object.data.edges and setting bevel_weight values but I can't get it right. It only works if I have first edited some edge's bevel weight by hand from the 3d-view's transform window. But I would like to get it work from my script without any manual hacking.  Steps to reproduce:  Start Blender with default cube and switch to Scripting view.  Try to set weights for all edges:  for edge in C.object.data.edges:     edge.bevel_weight = 1.0   Verify that values changed:  for edge in C.object.data.edges:     edge.bevel_weight   Switch into edit mode in 3d-view and open up transform window by pressing n. Select edges to see that the change did not work (bevel weights are all zero). Switch back to object mode and read values again in scripting console to see that they were reset to zero:  for edge in C.object.data.edges:     edge.bevel_weight   Go back to edit mode and change some edge bevel weight straight from the transform window by hand. Go back to object mode and change values again from the scripting console:  for edge in C.object.data.edges:     edge.bevel_weight = 1.0   This time it works, bevel weights are 1.0 when you go back to edit mode and look from the transform window.   Any ideas? My final need is to apply a bevel modifier to only some set of edges and that's why I'm trying to mark them by using weights. Using vertex group for bevel modifier does not work in my situation, since I need to specify edges, not vertices, which to bevel.   I'm using Blender v2.71. apt install "*" method in Ubuntu 20.04 does not work