Show the expected number of packets to buy to collect a whole set of minifigures
calculating expected number of packets.
What's the current state of LaTeX3 (2020)? There are a few questions on this forum asking about why Latex3 is taking so long to come out. However, all of these are from the early 2010s. I'm just wondering what the current state of the project is and whether it's ready (or nearly ready) for prime time? Why do we say "It's time we ate" and not "It's time we eat"? Why do we use the simple past but not the present or future in the following expressions:  Don't you think it's time we went a little further Don't you think it's time we ate Don't you think it's time we went home Meaning of Conditional sentences with 'should' and 'happen to' How to make a file (e.g. a .sh script) executable, so it can be run from a terminal How can I install Whatsapp web app on Ubuntu? I found a similar question , but here I am asking for the WhatsApp desktop webapp so that I can directly access the application without opening a browser. I have a circuit board in production, primarily through-hole, to the tune of ~400 units per year. Each one is manually inspected and tested. The board has ten distinct voltage rails, four voltage feedback amps, three current feedback amps, a microprocessor with particular response characteristics, and a few other parts. That's a lot to check.  I'm considering building an automated test rig for all that. I think a PCB that feeds the appropriate input stimuli, observes the responses, and gives a go/no-go would save me a lot of time. I'm reasonably confident that this is done, commonly. However, the techniques used to do it are unknown to me.  I imagine a test rig with a circuit board of the same footprint as the unit under test (UUT). It would have a microprocessor, appropriate protection and scaling circuitry, and a large number of vertical pins sticking up from the board. Mount points would also stick up from the test rig for the UUT. When the UUT is mounted on top of the test rig, the bottom of the board makes contact with the vertical pins, allowing the test rig to feed stimuli to the UUT and observe responses.  This seems reasonable, and I have the idea I've seen something very much like this before. But the details elude me. What sorts of pins would be used on the bottom board? (Part numbers would be great!) Would I need to put specific kinds of test points on the UUT, or can I get away with just contacting the clipped pins protruding through the board? What other concerns might there be? How to define a command that takes more than 9 arguments I have a mathematical transformation that takes 16 parameters (grouped into 3+8+5) and would like to make a latex command for it, so that I can easily change the notation for it if the need arises.  As far as I know, both \def and \newcommand take a maximum of 9 arguments, is there any (recommended) way to extend this? When is it OK to shrink a Database? I know shrink is the devil: It reverses page order and is responsible for skin cancer, data fragmentation, and global warming.  The list goes on... That being said, say I have a 100 GB database and I delete 50 GB of data -- not on one table, but a general pruning of old data on a database wide level, covering 90% of the tables -- does this constitute an appropriate use case for shrinking the database?  If not, what are the appropriate steps to take to clean house after removing such a high percentage of data from a database?  I can think of two: Rebuild Indexes and Update Stats. What else? How to politely negotiate PhD funding offer? I have been fortunate enough to get accepted with 5 years guaranteed funding to my 2nd choice graduate school. While the guarantee is great, the funding is only for 9/months out of the year. Would it be rude to ask them to guarantee summer funding to me for the first year until I can write a grant to get on a RAship? I know I should be more appreciative, since not everyone gets funding, but just an increase of about ~3k a year would make living there so much better and I do not want to accept an offer at a lesser institution just because of money.  What would be the most polite way of doing this?  I want to make it clear that I am not trying to get more money for the sake of getting more money, moreover that I am just not comfortable with so little money and such a high living cost in that area.  Thank you for your time. Let $F,G,H$ be groups such that $F\trianglelefteq G \trianglelefteq H$.  I am asked whether we necessarily have $F\trianglelefteq H$. I think the answer is no but I cannot find any counterexample with usual groups. Is there a simple case where this property is not true? Calculate $$\int_C \frac{2xy^2dx-2yx^2dy}{x^2+y^2},$$ where $C$ is the ellipse $3x^2 +5y^2 = 1$ taken in the positive direction.   I tried to calculate the integral using green theorm.  now i need to build enclosier that doesn't enclose $(0,0)$ i am having hard time guessing what to build.  a circle and ellipse might be perfect but then the domain is not easy to write. can i have hint please ? Proving of Integral $\int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}dx = \ln\left(\frac{a}{b}\right)$ Prove that      $$ \int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}\,dx = \ln\left(\frac{a}{b}\right) $$   My Attempt:  Define the function $I(a,b)$ as  $$ I(a,b) = \int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}\,dx $$  Differentiate both side with respect to $a$ to get  $$ \begin{align} \frac{dI(a,b)}{da} &amp;= \int_{0}^{\infty}\frac{0-e^{-ax}(-x)}{x}\,dx\\ &amp;= \int_{0}^{\infty}e^{-ax}\,dx\\ &amp;= -\frac{1}{a}(0-1)\\ &amp;= \frac{1}{a} \end{align} $$  How can I complete the proof from here? Isomorphism between dual space and bilinear forms Studying from Roman's Advanced Linear Algebra, I want to prove that $$U^* \otimes V^* \cong (U \otimes V)^* \cong \hom(U,V;\mathbb{F})$$ The author proves that $U^* \otimes V^* \cong (U \otimes V)^*$ by showing that there is an unique linear transformation   $$\theta:U^* \otimes V^* \to (U \otimes V)^*$$  defined by $\theta(f \otimes g)=f \odot g$ where $(f \odot g)(u \otimes v)=f(u)g(v)$  I want to prove that $U^* \otimes V^* \cong \hom(U,V;\mathbb{F})$. I tried to use the universal property by fixing $f \in U^*, g\in V^*$ and defining $S: U^*\times V^* \to \hom(U,V;\mathbb{F})$ as $S(f,g)=F_{f,g}$, where $F_{f,g}(u,v)=f(u)g(v)$, but I got stuck. I have been given the task to rewrite a function as a sum of an even and an odd function. But when I went to analyze the parts of the original function I noticed that one part was neither even nor odd.   The original function was: $g(x) = (x + 1) / (x^2 - 3x + 4$)  After analyzing I realized that when using f(-x) to determine symmetry it produced a 'neither' result. How would I go about writing a function that is neither even nor odd as an even function. (It would have to be an even function as the (x+1) part of the original function is odd and since I need to write it as a sum of even and odd, the denominator would need to be even. Fréchet differentiability from Gâteaux differentiability Let $X$ be a Banach space and $\Omega \subset X$ be open.  The functional $f$ has a Gâteaux derivative $g \in X'$ at $u \in \Omega$ if, $\forall h\in X,$ $$\lim_{t \rightarrow 0}[f(u+th)-f(u)- \langle g,th \rangle]=0$$   How can I prove the following:  If $f$ has a continuous Gâteaux derivative on $\Omega$, then $f \in C^1(\Omega,\mathbb R)$. Let $x$ and $y$ be integers not congruent to $0$ modulo $p$ where $p$ is a prime. Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$.   I thought about proving this algebraically and by contradiction. So we say $x^2+y^2 = pz$ and have to show this is impossible. Environment variables in Mac OS X Update: The link below does not have a complete answer. Having to set the path or variable in two places (one for GUI and one for shell) is lame.  Not Duplicate of:     Coming from a Windows background where it's very easy to set and modify environment variables (just go to System Properties > Advanced > Environment Variables), it does not seem to be that straight forward on Mac OS 10.5. Most references say I should update /etc/profile or ~/.profile. Are those the equivalent of System Variables and User Variables? For example, where should I set my JAVA_HOME variable?    EDIT:  I want to be able to access the variable from the terminal as well as an app like Eclipse. Also, I hope I don't have to restart/logout to make this take effect. I added a user to a group, but group permissions on files still have no effect Cannot determine size of graphic I'm trying to include graphics in my Latex-file, which I compiled with latex+dvipdf on OS X. Latex however returns this error: "Cannot determine size of graphic"  My graphic is exported from PowerPoint, so I have tried both .pdf and .png. I get the same errors for both.  Sample code that doesn't work:  \begin{figure}[htb] \begin{center} \leavevmode \includegraphics[width=0.8\textwidth]{graph.png} \end{center} \end{figure}   If I have to manually set some measurements of the image, please tell me how I can find out those measurements. I need the image to take up about 80% of text width and be centered. Thanks. How to loop through a HashMap in JSP?