I know it has been asked before, but I'm still not sure if I understand the focal length issues of EF-lenses on APSC cameras in Canon system. I already have the EF-S 55-250 IS STM and I'd like to buy a longer telephoto - if I get the EF 70-200 will it give me effectively more range than EF-S 55-250 (200 x 1.6 = 320 mm)?
I have a Canon 7D, which has a crop factor of 1.6x.  When using a Canon EF-S lens I get the focal length stated on the lens, when using a Canon EF lens I get the focal length times 1.6 when used on the APS-C body. At least that's what I thought.  I just compared two lenses:   Canon EF-S 15-85mm f3.5-5.6 Canon EF 70-200mm f4   .  The top photo is shot with the 15-85, the bottom photo with the 70-200 - both lenses are set to 70mm. Now why do they look the same?  I thought I would get 112-320mm (70mm*1.6 crop factor). What did I misunderstand about the crop factor calculation? Lightroom also displays the photos as having "70mm" and not 112mm.
Simple series convergence/divergence: $\sum_{k=1}^{\infty}\frac{2^{k}k!}{k^{k}}$ Is the Hogwarts Express in the magical or non-magical realm? This question is related to  and explores an apparent inconsistency, given only the Harry Potter movies as source material.  The Hogwarts Express line seems to have an inconsistent existence between magical (invisible to ordinary humans aka muggles) and non-magical (muggle) realms; platform 9-3/4 is clearly hidden from muggles, but in Chamber of Secrets, when Harry and Ron take the car (in the muggle world) to catch up to the train, they make no explicit crossing into a magical realm, suggesting that the train is traveling the muggle world despite its departure from a magical platform.  In which realm (magical or muggle) does the Hogwarts Express exist? How is this apparent inconsistency resolved? What is the origin of "a cut above/below the rest"? Where did the phrases "a cut above the rest" and "a cut below the rest" come from? I'm supposing they came from the same source. I feel it something very obvious, but I don't know what. How to delete a row by reference in data.table? A random variable that induces a $\sigma$-algebra the same as the one in the sample space Independence of points from same underlying function I have a function $y=sin(x)$  I sample the points $ x_i=\{0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1\}$ and $$ y_i=sin(x_i) + \epsilon \  \ \epsilon \sim iid\ Normal$$  Now given only the set $\{x_i,y_i\}$ I want to uncover the underlying functional form $f(x)$ given as  $$y=f(x)+\epsilon \ \ \ \ \epsilon \sim iid\ Normal  $$  My question is : ARE $y_i$'s independent ?  My intuition is that $y_1$ and $y_2$ are not independent. This is because given a new ${x}$ say ${x_\star}$ we can use $y_i$'s to learn some information about ${y_\star}$.   Does the fact that the input points come from the same underlying function make them somehow correlated or posses commonalities ? and if not why do most methods (ex: Non-parametric splines, Gaussian process) use the $y_i$'s to get information and learn ${y_\star}$ ? Isn't the fact that if samples are independent then they can't learn new information from each other ? Are there critical hits for spells? Does the universe have a center? If the big bang was the birth of everything, and the big bang was an event in the sense that it had a location and a time (time 0), wouldn't that mean that our universe has a center?  Where was the big bang? What is there now? Are we moving away from the center still? Are these even valid questions? Let $g_n= {2^2}^n +1 $. Prove $g_0 · g_1 · · · g_{n−1} = g_{n} − 2$.    I'm not sure how to start this proof, if it should be done algebraically or if I should try to use a proof by induction. What is the right way to consistently mangle bash terminal? I'm on OSX 10.11.1 and occasionally my bash terminal gets mangled. It often happens when I accidentally cat a binary file. The result can be seen on the image below. The output becomes weird, and I can't type ascii characters anymore.    Even though this happens occasionally, I couldn't find a way to consistently reproduce the issue. Online search recommends doing cat /bin/*, but that works sporadically, only after a couple dozen tries.  I want to do this so I can find an easy solution how to handle this in tmux.  How do I consistently get bash to a "mangled" state? Is there maybe a magic unicode character that can do this? Problem requiring/including plugin files with plugin_dir_path( __FILE__ ) I am trying to write a WordPress plug-in. In my project I have three PHP files.   My plug-in file Widget for the plug-in Data installation   I need to install data table on plug-in activation. For that sake I wrote code to create table for my plug-in in data installation file. But I am not able to create data table using this file.  My code for include data installation file and widget file in my plug-in file is  define ( 'OTHER_FILES_PATH', plugin_dir_path( __FILE__ ) ); require_once( OTHER_FILES_PATH . '/data_installation.php' ); require_once( OTHER_FILES_PATH . '/plugin_widget.php' );   My problem is when I put my data installation code in my plug-in file it's working fine but when I put it in separate file (data installation) I am unable to create tables for my plug-in.  Please let me know if I am making any mistake. How to prove that exponential grows faster than polynomial? Contradiction between law of conservation of energy and law of conservation of momentum? Yesterday, while studying a simple question, a rather strange doubt hit my mind:     Consider a ball of mass $m$ moving with velocity $v$ that collides with a wall in a perfectly elastic collision and rebounds back with the same velocity. Taking the direction of initial motion of the ball as positive, the change in momentum of the ball is $-2mv$ and the wall is clearly at rest.   So, in order to obey the law of conservation of momentum, the wall must gain a momentum of $2mv$.  Now, this itself confused me a lot. I then applied the law of conservation of kinetic energy to this situation, and clearly the magnitudes of the initial and final velocities of the ball are equal, so kinetic energy of the ball is conserved and hence the final kinetic energy of the wall must be zero in order to conserve energy.  Now, this contradicts with the law of conservation of momentum, which says that the wall must gain momentum of $2mv$.  One more point: If we are talking about a wall then it is an important fact, which must be considered, that the wall is fixed on earth and so the mass of the wall is equivalent to that of earth, say $M$. If the law of conservation of momentum is applied now then we get velocity of the wall, say $V=2mv/M$ which is no doubt an extremely small value. But, the law of conservation of energy strictly says that the velocity of the wall must be exactly zero, not even the smallest value is allowed (theoretically speaking).   Please clarify this and correct me anywhere if I'm wrong. I have tried my best to make it clear what I want to ask :). When I have these facts,   French coins were not found in the ruin. German coins were not found in the ruin.   which of these sentences are grammatically correct?  1.     No French and German coins were found in the ruin.   2.     No French or German coins were found in the ruin.   3.     No french nor German coins were found in the ruin.   I think 1 is correct but I'm not sure.  By the way, I can also say     Neither German nor French coins were found in the ruin.   Also, when you have an additional fact   Italian coins were not found in the ruin   which of these statements are grammatically correct?  1.     No French, German and Italian coins were found in the ruin.   2.     No French, German or Italian coins were found in the ruin.   3.     None of French, German and Italian coins were found in the ruin.   4.     None of French, German or Italian coins were found in the ruin.   Thanks, After I deleted my Ubuntu partition, I rebooted my computer and I only get a prompt:   grub rescue &gt;   The command ls lists   (hd0) (hd0,msdos5) (hd0,msdos3) (hd0,msdos2) (hd0,msdos1) (hd1) (hd1,msdos1)   I check them one by one:  ls (hd0,msdos5)/ ls (hd0,msdos1)/   But I cannot find the right partition. Bounded sequence in a metric space How to make a different font and color of text in labels? What is the difference between would and will in the following examples?        How would you use this word in a sentence?     How will you use this word in a sentence? Reward for Passing Review tests? Just wanted to see what the community thought of adding a reward to passing the review tests? Maybe a 1 point reward?  I know seeing the tests in the review area has helped me be extra careful when reviewing, but I thought perhaps a little boost in incentive would be cool. I am wondering why matrix factorization techniques in the machine learning domain almost always expect the provided matrix to be non-negative. What is the advantage of this constraint?  Background: I want to use matrix factorization algorithms for a sparse user-item matrix containing positive and negative implicit feedback. Is there any another possibility to set interactions with negative indications apart from fields that denote that no interaction happened between the user and the item?