What creates the physical difference in ageing when travelling close to the speed of light?
What is the proper way to explain the twin paradox?
Can not unlock my Android device as have no internet connection? Defining starred versions of commands (* macro) The phrase for when: working in a group to divide responsibility and reduce malicious intent \pagestyle{empty} doesn't seem to work for page with \maketitle I am taking an abstract algebra course and I am really interested about the topic.  So much so that I spend most of my time reading supplementary materials.  I consequently know a lot more theorems than the ones covered in class.    In a recent quiz, I used a theorem in one of my proofs that was not mentioned in the lecture since.  My professor gave me partial marks for my answer for the reason that we didn't cover this theorem in class even though the proof was completely valid!  I can't really see where he is coming from.  How would someone be able to use a theorem correctly if he doesn't know its proof?  I was quite baffled by his comment.   Do you agree with this?  EDIT: The professor just Emailed me and said that after thinking for a while, he decided to award me the full mark for the question. He also mentioned that he didn't want discourage me from studying the subject (since he saw most of my previous quiz grades were full marks) and I was passionate about it, but kindly requested that I include any proof for a theorem that I cite from now on in his exams. He also sent a broadcast message to the whole class indicating this. Create Trigger that updates the same table before (or after ??) the Insert I have just started using triggers and one thing that I would need to do is to create a trigger that updates the same table that's inserting the row.  TABLE:  CREATE TABLE `city` (   `city_id` smallint(5) unsigned NOT NULL AUTO_INCREMENT,   `city` varchar(50) NOT NULL,   `country_id` smallint(5) unsigned DEFAULT '0',   `country` varchar(150) DEFAULT NULL,   `last_update` timestamp NOT NULL DEFAULT CURRENT_TIMESTAMP ON UPDATE CURRENT_TIMESTAMP,   PRIMARY KEY (`city_id`),   KEY `idx_fk_country_id` (`country_id`),   CONSTRAINT `city_ibfk_1` FOREIGN KEY (`country_id`) REFERENCES `country` (`country_id`) ON UPDATE CASCADE ) ENGINE=InnoDB AUTO_INCREMENT=6009 DEFAULT CHARSET=utf8;   Trigger:  DELIMITER $$ CREATE TRIGGER `populate_country_id` BEFORE INSERT ON `city` FOR EACH ROW BEGIN  IF NEW.country_id = 0  THEN     UPDATE city     INNER JOIN country B ON city.country = B.country     SET NEW.country_id = B.country_id;  END IF;  END $$ DELIMITER ;   MySQL is just returning: Can't update table 'city' in stored function/trigger because it is already used by statement which invoked this stored function/trigger.  Is there any way I can make this work?  Thanks everyone! Solving the Kolmogorov forward equation for transition probabilities Let $\lambda \mu &gt; 0$ and let $X$ be a Markov chain on $\{1,2\}$ with generators  $$ Q = \begin{pmatrix} -\mu &amp; \mu \\ \lambda &amp; -\lambda \end{pmatrix}$$  Write down the forward equations and solve them fro the transition probabilities, $p_{i,j}(t), i,j = 1, 2$.  The answers say:  "We have that   $$p_{11}' = -\mu p_{11} + \lambda p_{12}, \, \, \, p_{22}' = -\lambda p_{22} + \mu p_{21}$$  where $p_{12} = 1 - p_{11}$ and $p_{21} = 1 - p_{22}$. Solve these subject to $p_{ij}(t) = \delta_{ij}$, the Kronecker delta, to obtain that the matrix $P_t = \{p_{ij}(t)\}$."  The bits I don't understand are these  1) How do they get those equations in the first place? For example, why have they added $\lambda p_{12}$ when the coefficient for $p_{12} = \mu$?  2) How do you solve these equations? What do you do with that Kronecker delta to solve them? Hint on how to prove the identity $\sum_{0 \leq k \leq n } \binom{k}{m} = \binom{n + 1}{m + 1}$ Positioning of decorations on parallel lines Is the Schrödinger equation derived or postulated? I'm an undergraduate mathematics student trying to understand some quantum mechanics, but I'm having a hard time understanding what is the status of the Schrödinger equation.  In some places I've read that it's just a postulate. At least, that's how I interpret e.g. the following quote:     Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger.    -- Richard Feynman   (from the Wikipedia entry on the Schrödinger equation)  However, some places seem to derive the Schrödinger equation: just search for "derivation of Schrödinger equation" in google.  This motivates the question in the title: Is the Schrödinger equation derived or postulated? If it is derived, then just how is it derived, and from what principles? If it is postulated, then it surely came out of somewhere. Something like "in these special cases it can be derived, and then we postulate it works in general". Or maybe not?  Thanks in advance, and please bear with my physical ignorance. Say I'm flying from Sydney, to Los Angeles (S2LA), back to Sydney (LA2S).  During S2LA, travelling with the rotation of the earth, would the flight time be longer than LA2S on account of Los Angeles turning/moving away from our position?  Or, in the opposite direction, would the flight to Sydney be faster since the Earth turns underneath us and moves Sydney closer?  ===   Please ignore jet stream effects and all other variables; this is a control case in an ideal environment. By "dramatically" I suppose I mean a delay of 1 hour or more. Any way to list similar commands? When you write the command name wrong, bash often does this:  septi@norbert:~$ good No command 'good' found, did you mean:  Command 'gold' from package 'binutils' (main)  Command 'gmod' from package 'gmod' (universe)  Command 'goo' from package 'goo' (universe)  Command 'god' from package 'god' (universe)  Command 'geod' from package 'proj-bin' (universe)  Command 'gord' from package 'scotch' (universe) good: command not found   Or sometimes it does this:  septi@norbert:~$ nftp No command 'nftp' found, but there are 23 similar ones nftp: command not found   Is there any way to ask bash to show these 23 similar commands for me? And, is there a way to show similar commands, including those that aren't yet installed, instead of running the application, ftp for example? "Interested with the course" or "interested in the course"? Interested in/with the course   Which preposition is correct? in or with, or both. Is RSA key size the size of private key exponent? I have implemented a key pair generation scheme for RSA algorithm. I have taken the length of private key exponent as RSA key size, but then I've got to know that RSA key size is the size of the modulus $n$. Is it wrong to take RSA key size as the size of the private key exponent?  To generate a private key of 1024 bits, I increase the size of private key exponent regardless the modulus $n$. Do opposite categories always exist? Let $G$ be a locally compact Hausdorff Abelian topological group.  Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure.  Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$.  Define the function from $G$ to $\mathcal{L}^1(G)$ that assigns to each $y$ in $G$ the function that takes $x$ in $G$ to $f(xy^{-1})$.  I.e. it assigns the $y$-translate of $f$.  Is this a continuous function of $y$ into $\mathcal{L}^1$?      On p.94 3.10 of  it is asserted that it is.  The usual proof of this sort of thing in cases where $G=\mathbb{R}$, say, is you use density of compactly supported continuous functions, and then dominated convergence.  The former survives, but dominated convergence does not work on nets.  In fact, a limit of a net of measurable functions need not be measurable. I have opened Disks program today and noticed that there is FAILING in the Assessment column for the SMART status.     What does this error mean? Is it serious problem?  I have Lenovo G50-45 laptop, disk is SEAGATE ST1000LM014-SSHD-8GB(LVD3) 1TB SSHD with 8GB NAND Flash (original disk in this latop).  Update 2017-09-24:  The disk has died yesterday. 1 year and 3 months have passed since 1st hint of problem. About 4 months ago I had few issues with it not booting, but I was able to make another clean install. Since yesterday the disk is non-responsive. As this was long expected I had backup of all important data.(!) The disk was about 2 years old, so that was quite premature death. Can I use lens brand X on interchangeable lens camera brand Y? What factors influence the spring constant? For certain springs we have been taught that Hookes law, $F=-kx$ works. For such springs, what factors determine k. For instance, k is proportional to length. Can k be expressed in terms of dimensions, mass and other fundemental quantities, or is k a fundemental property of the spring itself? let q be an odd prime and p a prime dividing $1+n+\dots+n^{q-1}$. Prove that $p = 1 \mod q$ or $p = q$.  I don't actually know how to start, i thought of using fermat's little theorem or an application of eulers formula, but it didn't really give me any insights.   Any hints?