There is a long line of people waiting outside a theatre to buy tickets. The theatre owner comes out and announces that the first person to have a birthday same as someone standing anywhere before him in the line gets a free ticket. Where will you stand to maximize your chance?
At a movie theater, the whimsical manager announces that a free ticket will be given to the first person in line whose birthday is the same as someone in line who has already bought a ticket.  You have the option of choosing any position in the line. Assuming that you don't know anyone else's birthday, and that birthdays are uniformly distributed throughout the year (365 days year), what position in line gives you the best chance of getting free ticket?
How to split 7.1 soundcard outputs to 4 stereo outs with ALSA Does tourist visa (90 days) apply after a long-term visa ends in Schengen countries? Simulation software to solve high school level mechanics problems I'm a physics teacher and I would like to find a simulation software that could be used to give exact numerical answers to high school level problems in Newtonian mechanics.  I want the students to first calcultate a numerical answer to a physics problem and then use the simulator to visualize the problem in order to understand it better and verify that the simlator gives the same answer that they got when doing the calculation themselves.  Here are some examples of what kind of problems I want the students to be able to solve with the simulator:   A slope with a specific angle of inclination and a ball starts rolling at a certain point. Determine the speed, angular momentum and total mechanical energy of the ball after a specific time t. A ladder of known mass and length is leaning against the wall at a specific angle. The contact between the ladder and the wall is frictionless. Calcultate the x and y components of the force between the floor and the ladder.   I would like the software to have the following features:  - Simulation in 2D. No 3D simulations are necessary and the software should at least have the option of doing simulations only in 2D.  - Position all objects exactly by writing numbers (just positioning the objects using the mouse is not precise enough).  - Show animations of the time evolution of the system.  - Show forces, velocities and accelerations as arrows.  - Give exact numerical results so that the students can use the software to verify their calculation.  Ideally the software would have a graphical user interface, but if no suitable software with graphical interface exixts, I could also consider a software with text-based inputs. I would consider both free and commercial software.  My question is: Is there any software that would meet these requirements and if there is, where could I get it? Covariance of Euler-Lagrange equations under change of generalized coordinates Positive semidefinite cone is generated by all rank-$1$ matrices. Why is $A^TA$ invertible if $A$ has independent columns? How can I understand that $A^TA$ is invertible if $A$ has independent columns? I found a similar , phrased the other way around, so I tried to use the theorem  $$ rank(A^TA) \le min(rank(A^T),rank(A)) $$  Given $rank(A) = rank(A^T) = n$ and $A^TA$ produces an $n\times n$ matrix, I can't seem to prove that $rank(A^TA)$ is actually $n$.  I also tried to look at the question another way with the matrices  $$ A^TA = \begin{bmatrix}a_1^T \\ a_2^T \\ \ldots \\ a_n^T \end{bmatrix}   \begin{bmatrix}a_1 a_2 \ldots a_n \end{bmatrix} = \begin{bmatrix}A^Ta_1 A^Ta^2 \ldots A^Ta_n\end{bmatrix} $$  But I still can't seem to show that $A^TA$ is invertible. So, how should I get a better understanding of why $A^TA$ is invertible if $A$ has independent columns? I'm trying to continue the numbering of examples in enumerate environments in beamer. For example, I'd like one slide to feature   Foo Bar   and the next slide   Zip Yadda   My strategy so far has been to define a counter that saves the enumerate counter at the end of the first environment and can then be queried to set the enumerate counter at the start of the second enumerate environment. (This is essentially the proposal also found at )  That works wonderfully so long as I don't try to uncover elements of the slide sequentially. The problem, of course, is that as each of the pdf-pages that make up the slide is created, the counter is advanced. Here's a MWE.  \documentclass{beamer}  \setbeamercovered{highly dynamic}  \newcounter{saveenumi} \newcommand{\seti}{\setcounter{saveenumi}{\value{enumi}}} \newcommand{\conti}{\setcounter{enumi}{\value{saveenumi}}}   \begin{document}  \begin{frame}[&lt;+-&gt;]   \begin{enumerate}   \item foo   \item bar%     \seti   \end{enumerate} \end{frame}  \begin{frame}[&lt;+-&gt;]   \begin{enumerate}     \conti   \item zip   \item yadda%     \seti   \end{enumerate} \end{frame}  \end{document}   As the second slide is progressively uncovered (pages 3 and 4 of the resulting PDF), the counter at the beginning of the enumerate environment is advanced each time. So page 3 of the PDF has the right numbering, (3) and (4) respectively, while the fully revealed page has the wrong numbering. Pictures follow:   on one of the slides    on the following one Convergence of $\int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx$ I need to show that $$ \int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx $$ converges if and only if $\alpha + 1 &lt; \beta$. I used substitution and integration to get $$ \lim\limits_{t\to\infty}\int\limits_1^tx^\alpha\sin(x^\beta)dx \overset{u = x^\beta}{\underset{du = \beta x^{\beta-1}}{=} }\lim\limits_{t\to\infty}\left(\frac{1}{\beta}\int\limits_1^{t^\beta}u^{\frac{\alpha + 1 - \beta}{\beta}}\sin{u}du\right) $$ If $\alpha + 1 -\beta &lt; 0$, then we have a product if a function with bounded anti-derivative and a differentiable decreasing function which tend to 0, and from Dirichlet's criterion, the integral converges.   For the converse, I used integration by parts $$  = \frac{1}{\beta}\lim\limits_{t\to\infty}\left(\left.-u^{\frac{\alpha + 1-\beta}{\beta}}\cos{u}\right|_1^{t^\beta} + \frac{\alpha+1-\beta}{\beta}\int\limits_1^{t^\beta}u^{\frac{\alpha + 1-2\beta}{\beta}}\cos{u}du\right) $$ For $\alpha + 1 -\beta\geq 0$, I get that the first term indeed diverges, but this is not enough to show total divergence. How can one show it? This is something I don't particularly understand. I get that for Windows, activation exists to prevent software piracy, but you don't have to pay for iOS. Perhaps activation exists to prevent Android devices from using iOS or to prevent jailbreaks and downgrades? Anyone know the real reason why? 3.8 GPA, but 3 Fs and 1 D on transcript Do I need a Japanese transit visa when arriving at Narita and departing from Haneda? I'm arriving from New Delhi, India (Indian passport) with JAL at Narita on 11th July at 07:30 and departing for San Francisco with JAL at 00:05 on 12th July from Haneda. Do I need a transit visa as I'll be switching airports and spending a few hours at JAL City Haneda Airport Hotel?  Previously I had connecting flights from Narita to San Francisco and obtained shore passes to relax at Narita Nikko Hotel. Now I'm travelling with my elderly Mom and do not want complications at Narita immigration or later in the day while going from Narita Airport to Haneda airport. Please advise. TikZ \foreach loop with macro-defined list Count if two criteria match while ignoring duplicates I have a simple Sheet with 2 cells that I need to countif 2 conditions match and treat  duplicates as 1.  Column A    Column B Male        closed Female      invalid # Male        exception Male        closed Male        closed Female      new   I would like a formula that counts Male closed and return 1 result (ignoring duplicates) Some people (me) would prefer the straight GNOME experience as opposed to the Unity style desktop offered in Ubuntu 17.10.  How can I get the standard GNOME desktop and interface? Which is correct?        She was told that it was against the law to steal.      Or        She was told that it is against the law to steal.        Background information: The original context is that I was relating an incident via email, where my colleague told a customer that some of our actions were part of the organisation's policy. The policy is still valid, but the statement I used to relate the incident in my email went like this:    "Mr Jay was informed that it was our policy to do this."   And I received feedback that it should go like this instead:    "Mr Jay was informed that it is our policy to do this." I've just installed QGIS 3.2.2 and after installation, tried to use the Plug in Manager to install the OpenLayers plugin.  But it's not in the list to install.    Any ideas? I'm trying to fill out the skills section on my first character, but when I look at the character sheets in the Starter Set as examples of how to make a character, the skills that are circled have higher modifiers than their Dexterity or Strength or whatever should make them.  I've filled out all skills but those. What do I add to the ability modifier to get those numbers?   My abilities are Strength 13, Dexterity 14, Constitution 11, Intelligence 15, Wisdom 8, and Charisma 8. Proficiencies are Arcana, Athletics, History, Investigation and Stealth. My saving throws are Strength and Dexterity.   I just started D&amp;D and have the Player's Handbook, DM's Guide, Monster Manual, and the Starter Set. When working with an image in Photoshop, I'd like to have a simple keyboard shortcut, like in , that automatically does:   Export > Export As or Export > Save For Web  Resize to biggest dimension = 600px (either width or height) Add suffix _600.jpg to filename   How to do that?  I'm not sure how to edit a recorded macro to do that (add the suffix and choose 600px for the biggest dimension).  Example: you open myphoto.png, you press F6 and then you automatically get myphoto_600.jpg. Fix terminal after displaying a binary file How can I display "recent folders" in the sidebar of "Save as" dialogue? I was using 14.04 and upgraded to 16.04 yesterday.   In 14.04 I had recent folders on top of the sidebar of the Save as dialogue when I saved something from the net (Firefox if that helps). But that feature disappeared in 16.04 and I really want to have it back.   Here is a picture of what I'm trying to explain:  (the save as dialogue in 16.04 without the recent folders on top)