k of n aides to n ministers leak information; ministers only know whether other ministers' aides leaked (not their own) I know the relationships between the ministers in the below problem can be represented by a $K_n$ graph since each one of the n ministers has connections to the n-1 other ministers/ministries they represent. However, I can't seem to figure out where to go next. Here is the problem:  In a certain government there are n cabinet ministers. Each of these has a senior aide. Exactly k of these n aides leak important information to the press. None of the ministers know whether their own aide leaks information or not. However,because of interdepartmental rivalries, every minister has many sources in the other ministries, and thus is aware of all the leaks outside of her ministry. There is an unspoken law in this government that as soon as any minister has irrefutable proof that their aide leaked information, they must re the aide at their first press conference. Every ministry has one press conference per day at exactly 8:00 A.M. For a long time no one is fired, because no minister is aware of the leaks, if any, made by their own aide. One afternoon a newspaper runs a story, and announces that news are leaked by some of the top aides. Exactly k days later, all the guilty aides are fired. What happened?
Spivak Chapter 2, problems 27 (and 28) To be honest, I have no idea how to even start this problem. I'm sorry I don't have any work to show, but I'm just at a blank. Help?  Chapter 2: Problem 27:  University B, once boasted $17$ tenured professors of mathematics. Tradition prescribed that at their weekly luncheon meeting, faithfully attended by all $17$, any members who had discovered an error in their published work should make an announcement of this fact, and promptly resign. Such an announcement had never actually been made, because no professor was aware of any errors in her or his work. This is not to say that no errors existed, however. In fact, over the years, in the work of every member of the department at least one error had been found, by some other member of the department. This error had been mentioned to all other members of the department, but the actual author of the error had been kept ignorant of the fact, to forestall any resignations. One fateful year, the department was augmented by a visitor from another university, one Prof. X, who had come with hopes of being offered a permanent position at the end of the academic year. Naturally, he was apprised, by various members of the department, of the published errors which had been discovered. When the hoped-for appointment failed to materialize, Prof. X obtained his revenge at the last luncheon of the year. &quot;I have enjoyed my visit here very much&quot;, he said, &quot;but I feel that there is one thing that I have to tell you. At least one of you has published an incorrect result, which has been discovered by others in the department.&quot; What happened the next year?&quot;   Chapter 2: Problem 28:  After figuring out, or looking up, the answer to Problem 27, consider the following: Each member of the department already knew that Prof.X asserted, so how could his saying it change anything?
Ever since adding an iTunes gift card to my account, I'm getting verification requests like crazy. I can't even download free apps. It takes me to my payment info and when I try to select none, I continually get a message under the payment info that says "you must verify your payment info to continue"  I don't want a credit card on file and would really like the none option to work. My ID shows I still have iTunes credit. Let $p(t) =  a_0+\dots +a_nt^n, a_n\neq 0,$ a polynomial with degree $n&gt;0$. Prove that $p:\mathbb{R}\rightarrow \mathbb{R}$ is not bounded, but for any bounded subset $X\subset \mathbb{R}$ the restriction $p\vert X$ is bounded.   I want to prove this in very elementary terms. I mean, without using the notion of limit or even continuity. I can only show this with "strong" results. Is it possible with elementary terms? Solving trigonometric equations of the form $a\sin x + b\cos x = c$ Suppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 &lt; x &lt; 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos x = 2$ where $0&lt;x&lt;2\pi$.  How do you solve this equation without using the method that moves $b\cos x$ to the right side and squaring left and right sides of the equation?  And how does solving $\sqrt{3}\sin x + \cos x = 2$ equal to solving $\sin (x+ \frac{\pi}{6}) = 1$ When using zero as a quantifier, is it correct to use the singular form on the object of the quantifier, or the plural form?  It sounds confusing when I put it that way, but what I mean is: Which is correct?   Your password expires in 0 days. Your password expires in 0 day.   Essentially I suppose I'm asking, does "singular" mean "one" or "the opposite of more than one", as zero is not "plural" in the traditional "more than one" sense?  I'm pretty sure "days" sounds correct, but I can't be sure. Position an exampleblock in front of content I need to position an example block:  \begin{exampleblock}{}    Proposed and implemented \end{exampleblock}   Without disturbing other frame content. Something like:  -------------------- ------------- -------------------- -------------------------- -----------    --/============\--------- -----|            |-- -----| Proposed   |-------- -----| and        |-------- -----| implemented|------ -----\============/ -------------------- --------------------------    -----------------   Let it overlay the text behind it. And not to be taken into account during layouting.  Update The Claudio Fiandrino's way is not apropriate to me, because, i need an Itemize environment working inside, and it would be great, if block style is the same as theme-style. How to compress / reduce the size of JPEG photos for archiving? I have 50,000 high resolution JPEG photos, where a couple of them might occasionally be needed, about once a year.  I wanted to zip them to save disk space, except that zipping gives no space benefit - so trying to reduce the images disk usage using winzip, winrar or 7zip was not successful.  Is there any software or algorithm similar to zip to compress image size on hard disk for storage without loosing any image information? I am hearing my screen What is $\mathbb{E}[W(s)\mathrm{e}^{W(S)}]$ where W(S) is a standard Brownian Motion? I know that since it is a Brownian motion that $W(s)\sim{N(0,s)}$. One approach I tried, was to see that $W(s)-W(0)\sim N(0,1)$ and to let $X = W(s)$ and then solve $\mathbb{E}[X\mathrm{e}^X]$ but I end up with some awful integral. I was wondering whether there was an easier way? Or perhaps a simple way to solve this integral: $\int_{-\infty}^{\infty} \frac{1}{\sqrt{2{\pi}s}}x\mathbb{e}^{x}\mathbb{e}^{-\frac{x^2}{2s}}$ Putting char into a java string for each N characters Longtable: page breaks within row possible? I am editing a longtable. Some of my cells contain a lot of text. Page breaks seem to happen only where table rows end/begin. I'd like, however, for the rows to span pages; it's fine with me if, for example, the first few lines of a row are at the end of one page and the rest of the row's lines are on the next page. I moved postgresql's data directory by following the following steps:   Stop postgres cp -a source_data_directory destination_data_directory export PGDATA=destination_data_directory Changing data directory to destination_data_directory within postgresql.conf pg_ctl start   But when I run: "pg_ctl start" I get the following error. Can someone please help me figure out the error.   server starting postgres@Server:/usr/lib/postgresql/9.1/bin$ postgres cannot access the server configuration file "/home/destination_data_directory/postgresql.conf": No such file or directory My 8 year old son won't fall asleep We put him to bed at 9pm and he stays up til 1am or later. When I go to check on him he is always playing with his hands and sometimes talking. I ask him why he is awake and not trying to sleep and he says it's because he just wants to play. He never comes to tell us that he is having a hard time sleeping or that there is anything wrong. The weird thing is that he used to be the first one to fall asleep out of 3 boys and the staying up started a few months ago. He does this every night and on top of keeping us awake, he is now waking up his brothers.   About a month or so ago, we decided to punish him by grounding him from video games. He doesn't want to be grounded, but he still keeps doing it. Nothing we do seems to be helping and it seems to be getting worse. I have talked to him about why it is so important to get enough sleep and why it's even more important for kids. He also gets plenty of exercise and time to play during the day.   I'm really starting to wonder if he is doing it just to be defiant. If anyone has any suggestions I would greatly appreciate it.  The problem is that he is being disruptive to everyone in the house and it has been EVERY night for the last 3 months.   He is tired and emotional during the day, and it is causing his brothers to be tired during the day while they are at school.   Because he is tired during the day he doesn't really want to be as active as he used to be. However, I do still encourage him to be active. We have always had a bedtime routine. We have asked him many different times if he is just having trouble sleeping, if he is uncomfortable, and if there is something wrong. He always tells us that he just wants to play.   At the beginning of all this he told me that he wants to be the one to decide when he goes to sleep and when he wakes up. Which is why I would even ask if it was out of defiance.   We really didn't want to get to the point of punishing him, but nothing else seemed to help and it was only after he started keeping his brothers awake.   It's not like we haven't asked him what he thinks would help or what would help him sleep better. He ALWAYS says that he just WANTS to stay up and play because it is fun. We know it is important for him to get enough sleep, and I'm worried because I don't think he is (and he's preventing the rest of us from getting enough, too). Do I need Intel Smart Response when installing Ubuntu? Execute shell commands in Python How to use find command to search for multiple extensions I can get all jpg images by using:    find . -name "*.jpg"     But how can I add png files to the results as well? How to format numbers as currency strings I would like to format a price in JavaScript. I'd like a function which takes a float as an argument and returns a string formatted like this: &quot;$ 2,500.00&quot;  What's the best way to do this? Is there a way to replicate Nikon D3100 live view on my laptop screen? I own a Nikon D3100 and I need a way to see the live view of my camera on my laptop screen.  There must definitely be a way (I am not just aware of it). I have searched all around the web sites... I could not get a solution.  The Nikon Camera Control Pro 2 does not support D3100, and "Camera Control 5.1" connects and shoots remotely from a laptop but does not show live-view nor shoot in RAW.  There is a way of connecting my camera to a TV or monitor directly via an HDMI cable and getting the live view, but I want it displayed to my laptop screen. What is the command to truncate a SQL Server log file? I need to empty an LDF file before sending to a colleague.  How do I force SQL Server to truncate the log? SQLite - UPSERT *not* INSERT or REPLACE Is there some clever way to do this in SQLite that I have not thought of? Basically I want to update three out of four columns if the record exists, If it does not exists I want to INSERT the record with the default (NUL) value for the fourth column. The ID is a primary key so there will only ever be one record to UPSERT. (I am trying to avoid the overhead of SELECT in order to determine if I need to UPDATE or INSERT obviously) Suggestions?  I cannot confirm that Syntax on the SQLite site for TABLE CREATE. I have not built a demo to test it, but it doesn't seem to be supported. If it was, I have three columns so it would actually look like: CREATE TABLE table1(      id INTEGER PRIMARY KEY ON CONFLICT REPLACE,      Blob1 BLOB ON CONFLICT REPLACE,      Blob2 BLOB ON CONFLICT REPLACE,      Blob3 BLOB  );  but the first two blobs will not cause a conflict, only the ID would So I assume Blob1 and Blob2 would not be replaced (as desired)  UPDATEs in SQLite when binding data are a complete transaction, meaning Each sent row to be updated requires:  Prepare/Bind/Step/Finalize statements unlike the INSERT which allows the use of the reset function The life of a statement object goes something like this:  Create the object using sqlite3_prepare_v2() Bind values to host parameters using sqlite3_bind_ interfaces. Run the SQL by calling sqlite3_step() Reset the statement using sqlite3_reset() then go back to step 2 and repeat. Destroy the statement object using sqlite3_finalize().  UPDATE I am guessing is slow compared to INSERT, but how does it compare to SELECT using the Primary key? Perhaps I should use the select to read the 4th column (Blob3) and then use REPLACE to write a new record blending the original 4th Column with the new data for the first 3 columns? A question about special relativity theory We have two sets: set No.1 and set No.2 as in this picture:   The observer is fixed to set No.1 . He sees set No.1 motionless and observes set No. 2 approaching with velocity 100,000 m/s.   Each set has one lamp and two, so called, touchers. Each set is designed so that if both touchers are touched simultaneously the lamp is turned on otherwise it remains off.  Set No. 2 is approaching set No. 1 so that each toucher in each set will be touched twice by the touchers of the other set.  The observer on set No. 1 observes the distance between touchers in each set 10 meters. He thinks the lamps on set No. 1 will be turned on because touchers in it will be touched simultaneously.  My question: Will the lamp on set No. 2 be turned on? How if an electrical current flowing from one toucher in a set to the other proves simultaneous touching?