Asymptotic expansion for $u_{n+1}=u_n+\ln(u_n)$
What is a good asymptotic for $f_n = f_{n-1}+\ln(f_{n-1})$?
Well, this is an exercise problem from Herstein which sounds difficult:   How does one prove that if $|G|&gt;2$, then $G$ has non-trivial automorphism?   The only thing I know which connects a group with its automorphism is the theorem, $$G/Z(G) \cong \mathcal{I}(G)$$ where $\mathcal{I}(G)$ denotes the Inner- Automorphism group of $G$. So for a group with $Z(G)=(e)$, we can conclude that it has a non-trivial automorphism, but what about groups with center? Macros for code annotations Google isn't surfacing Events rich snippets in SERP when JSON-LD Structured Data is valid Google confirms our site has thousands of valid MusicEvent structured data objects presented using JSON-LD, as indicated in the Structured Data section of Webmaster Tools. However, rich snippets for the events do not show up in any search result pages. The Rich Cards tab in Webmaster Tools says:     We did not find any structured data for rich cards on your website.   The structured data is valid with no errors in the Structured Data Testing Tool. We do generate some warnings occasionally for missing endDate and price fields. Comparing our code to a survey of other sites whose rich snippets do appear, we are not doing anything unique in terms of our markup.  The data has been there for months, in fact over the months we've tried also using Microdata format. The Structured Data section indicates that many pages were crawled this week.  Besides SDTT validation, what else can I check to troubleshoot this issue? A story about a guy who helped humanity by being held in a prison I remember a story I have read buy could not remember the author the name or anything apart from the basic plot. The story is about some future where humanity has to fight with some aliens (not 100% sure). The story tells about a guy who was some kind of activist and who was in a prison two or more times and somehow helped humanity by being held in a prison (can't remember details). I think the story starts by saying that there is a statue in his honor that is not very well known but he did more to help humanity than famous politicians.   The story is most likely written by some famous English-speaking author, though I read it in a translation. How do I create files using the lines in a .txt file? For example I have a file called fileNames.txt.  cat &gt;fileNames returns  abet  able  abacus   How do I use fileNames to create abet, able, and abacus as .txt files?  I tried using pipes with cat and touch. If $\tan(\frac{\pi}{4}+\frac{y}{2})=\tan^3(\frac{\pi}{4}+\frac{x}{2})$,then prove that $\sin y=\frac{\sin x(3+\sin^2 x)}{(1+3\sin^2 x)}$ If $\tan(\frac{\pi}{4}+\frac{y}{2})=\tan^3(\frac{\pi}{4}+\frac{x}{2})$,then prove that $\sin y=\frac{\sin x(3+\sin^2 x)}{(1+3\sin^2 x)}$    $\tan(\frac{\pi}{4}+\frac{y}{2})=\tan^3(\frac{\pi}{4}+\frac{x}{2})$ $\frac{1+\tan\frac{y}{2}}{1-\tan\frac{y}{2}}=(\frac{1+\tan\frac{x}{2}}{1-\tan\frac{x}{2}})^3$ Applying componendo and dividendo rule on both sides, $\tan\frac{y}{2}=\frac{\tan\frac{x}{2}(3+\tan^2\frac{x}{2})}{(1+3\tan^2\frac{x}{2})}$ I am stuck here. Tracking email bounces, opens, clicks When should I use rich black? How does the size of insert affects the rate of Homologous Recombination in yeast? When performing genetic knockouts in yeast using homologous recombination to replace a target gene sequence via a vector DNA, does the region between the flanking regions in the vector have to be the same length as the site of recombination in the host cell's gene? How to make a multiboot USB key? I want to cut my 8 GB USB key into several partitions to use  for a Windows XP (maybe d before).   I also want to put the  remix pack into it such that it has the second bootable OS. I want to tweak and modify it a little because if I can I want to switch Ubuntu 9.04 to the  version.  How to do it easily?  IMP: in short I want to make a dual-boot USB key with Windows XP SP3 and Ubuntu 8.04.3 LTS. SVN repository backup strategies I'm new to SVN and I'd like to know what methods are available for backing up repositories in a Windows environment? Editing/Creating files in C:\Program Files\WindowsApps The spells that we know about in the Harry Potter world all seem to be either in Latin, derived from Latin, or resembling Latin.  This makes sense for the European World, but there were likely witches and wizards in Africa, Asia, and the Americas.  These cultures would have developed independently, and the magical people in them would likely have still found ways to cast their spells.  Do cultures with languages that did not derive from Latin use the same words and phrases that are related to Latin for their spells?  Or did they have other phrases based on the languages in their cultures? Influence of charged particle's own electric field on itself I read this in my textbook: A charged particle or object is not affected by its own electric field.  Since I find this completely unintuitive and my mind is yelling "wrong! wrong! how could a particle even distinguish between its own field and the external fields?" I would really like to hear an explanation about why this is true, if it is.  (A corollary question that comes to my mind after thinking about this is... What about gravity and mass considering the analogous situation? And the other two forces?) Where did Harry's parents get all of their money from? Is it a good practice to ask before exiting an application (with nothing to save)? Every now and then, I come across applications that, when clicking on the main window's "X", ask me whether I'm sure that I want to quit the application.    I'm not talking about asking me to save unsaved documents, but about applications that aren't document related at all.  This always bothers me, since I feel totally interrupted in my workflow. I was already thinking two steps ahead of what I want to do next, and this message box abruptly pulls 100% of my attention.  My question:  Are there any valid use cases where a non-document-centric application should ask me whether I'm sure that I want to exit the application? Entering the Schengen area 12 days before my visa expires I have a Vietnamese passport. I'm planning to visit Vietnam on July 7th and be back in the Czech Republic (in Schengen area) on August 29th. My visa is a multiple-entry, long-term, D type visa (for study purposes) and the expiration date is September 10th. I am afraid that the boarding officer will not allow me to enter EU since the arrival is so close to the expiration date.   Is it OK for me to enter Schengen on August 29th when I have only 12 days left until visa expiration? In my book, the following are described to be the steps that one must follow if he wishes to find the direction of static friction force acting on an object:  &quot;(1) Draw the F.B.D(free body diagram) of the object in question with respect to the other object on which it is kept. (2) Include the pseudo force also, if the contact surface is accelerating. (3) Decide the direction of the resultant force and resolve this force into two components; one along the surface of contact and the other along the normal to this surface. (4) The direction of static friction is opposite to the component of the result force along the contact surface.&quot;  Firstly, are these steps correct? Will I always obtain the correct answer by following these steps? Secondly, consider the following problem:  In questions like these, when I try to find the resultant force and then assign the direction to static friction, I find that my answer is incorrect! Are there some other steps to follow when I know that the object is definitely accelerating in one direction or another? Edit 1: I approached the above question by doing exactly what my book suggests; first, I identified all the forces acting on the smaller block, being the normal force applied by the block A and its weight mg downward. Adding these two forces, I get another force that actually acts on the body like so:  I wonder if I'm not supposed to take either of the contact forces into consideration when drawing the F.B.D to figure out the direction of static friction acting on the body. Also, are there other steps for finding the direction of kinetic friction? Edit 2: From Judge's answer, I have been able to figure out my mistake, it being that I completely missed step (3) and didn't resolve the resultant force into components! Sheesh. Sorry! Much thanks in advance! :) Regards. I'm trying to install , an open-source Microblogging application on my Windows development server using XAMPP as per the . The website cannot find PEAR, and throws the below errors:  Warning: require_once(PEAR.php) [function.require-once]: failed to open stream: No such file or directory in C:\xampplite\htdocs\laconica\lib\common.php on line 31 Fatal error: require_once() [function.require]: Failed opening required 'PEAR.php' (include_path='.;\xampplite\php\pear\PEAR') in C:\xampplite\htdocs\laconica\lib\common.php on line 31   PEAR is located in C:\xampplite\php\pear phpinfo() shows me that the include path is .;\xampplite\php\pear  What am I doing wrong? Why isn't the PEAR folder being included? Do local flows of left-invariant vector fields satisfy $\Phi_X^t\circ L_x=L_x\circ \Phi_X^t$? Given $G$ a lie group and $X$ a left-invariant vector field. Let $\Phi_X^t$ be the local flow of $X$. Why can we conclude that $\Phi_X^t \circ L_x=L_x \circ \Phi_X^t$? Thanks!