Simplify an expression with complex numbers
Complex number operations: telling Mathematica variables are real
Pre & post increment operator behavior in C, C++, Java, & C# org.apache.tomcat.util.bcel.classfile.ClassFormatException: Invalid byte tag in constant pool: 15 How does the bounty system work? Cardinality of a totally ordered union Interview: How to ensure that a thread runs after another? Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$. In Burton's Elementary Number Theory, when introducing absolute pseudoprimes (also called Carmichael numbers), the author states that they are composite numbers $n$ such that $a^{n-1} \equiv 1 \pmod n$ for all integers $a$ with $\gcd(a,n) = 1$.   When showing that 561 is an absolute pseudoprime, he then concludes that $a^{561} \equiv a \pmod{561}$ for all $a$.   I searched a bit on the internet, and it seems that these two are actually equivalent definitions of a Carmichael number. However I have not been able to find a proof; the fact that the implication in the title of the post is stated so plainly in my textbook makes me think it is quite simple to see, but I can't seem to understand the argument. I am trying to import an Excel table with around 10,000 XY points into ArcMap 10.2. I am using State Plane NAD 1983 Alabama West (US Feet) as the Coordinate System in my map data frame. I then add the excel table to my Table of Contents in the map then select Display XY Data. Next, I choose the same coordinate system as my data frame then select OK. Once it creates the Events layer for my points they are not in the correct location.   Should I switch everything in my map to WGS84 or is there a solution for my points to be added onto my current map with the state plane coordinate system? How to execute multiple statements in a MATLAB anonymous function? I'd like to do something like this:  &gt;&gt; foo = @() functionCall1() functionCall2()   So that when I said:  &gt;&gt; foo()   It would execute functionCall1() and then execute functionCall2().  (I feel that I need something like the )  EDIT:  functionCall1 and functionCall2 are not necessarily functions that return values. Did Spike get his reflection back with his soul? How did Voldemort get his body back? In the first book it seems that Voldemort has been reduced to nothing but a ghostly wisp. But IIRC by the fourth book he somehow has a body that is carried around by Wormtail. In the first book he needs someone to inhabit, by the fourth, he can't really move around, but he does have a body back which can hold a wand.  How did this transition happen? Is $\frac{x^2+x}{x+1}$ a polynomial?  Fist question can be: on which field/ring or etc? In basic, let's take over $\mathbb R$.  Actually, it is $x$ if $x \ne -1$.  Can we say again it is a polynomial, or a near-polynomial or what?   Can we say it is a polynomial over $\mathbb R - \{-1\}$? But there is structure here for $\mathbb R-\{-1\}$.  This question can be academically, not sure, but I want to ask it in the level of primary school, then academically. Prove or disprove that $f(x) = \sin x/x$ is uniformly continuous over the interval $(0,1)$? I am thinking of $\lim_{x \to 0} \ {f(x)} = 1$ but I am confused as $x=0$ is not in the domain and also I want to write an $\epsilon$-$\delta$ proof. So any help is much appreciated!  Thanks! How to make putty to not break my session after some time? color glitches and weird texture in render When I look in my preview render everything looks fine (for not having finished lighting) but when i actually render it the color of my doughnut icing glitches and the texture of the table doesn't appear. Sitemaps - one per subdomain or one for the base domain? I have more than 5 subdomains on a project, say developer, www, staff and some more, what is the best practice in creating sitemaps?   All subdomains that share a same sitemap at base domain (example.com/sitemap.xml) ? or different sitemaps for different subdomains?(developer.example.com/sitemap.xml and www.example.com/sitemap.xml)   Since the subdomains are several parts of the main application, what are the pros and cons of each, and what should I consider while choosing the one over the other? (one sitemap per subdomain or one for the whole site) Black artifacts when looking through many transparent objects? I have the following text:    A   Hello   world   B   Hello   world   C   Hello   world   I know that I can replace Hello by Hi using sed:  sed 's/Hello/Hi/g' -i test   but this replace each Hello with Hi:    A   Hi   world   B   Hi   world   C   Hi   world   what I really want is to replace only the Hello after B:    A   Hello   world   B   Hi   world   C   Hello   world   so I have tried this:  sed 's/"B\nHello"/"B\nHi"/g' -i test   but nothing happened, How can I do this?  Note: There are some white-spaces on the beginning of each line of the file. I need an approach to allow delegated admins (i.e., non-Administators) to add members to Public Groups and Collaboration Groups (the GroupMember and CollaborationGroupMember objects).  I added a custom object (UserGroupRequest__c) to capture which Users should be added/removed to which Groups.    I've also built a batchable class to process the requests.  However, GroupMember is a Setup object, and CollaborationGroupMember is not.  So this doesn't work if I combine the operations in a single batchable class.  Should I instead be using 2 batch classes, calling one from the finish method of the other?    Any recommendations on how best to proceed?  Am I way off base?    Would a trigger on the Request object perhaps be better? I'm guessing that there may be a permission problem if this is attempted via a trigger as the delegated admins won't be able to add users to public Groups.  Also, I'm guessing the trigger approach would run into the same mixed DML issue - on Insert of the Custom object, I would need to Insert/Delete from GroupMember and from CollaborationGroupMember, and then perform a 2nd update on the Custom object to indicate that processing was completed... Column count doesn't match value count at row 1 Can I get SSD rw performance while keeing data security if I combine SSD & HDD in btrfs RAID1 I have 2 storage devices; classical slow HDD (750GB, /dev/sda) and faster SSD (128GB, /dev/sdb). Currently I have installed Ubuntu &amp; Mint on same btrfs partition on SSD (/dev/sdb5). My btrfs pool consists of /dev/sdb5.  What I want to achieve is data replication on HDD while keeping SSD performance: Something like this:  For read case it would be ideal for procedure to go like this:   read from SSD check checksum if it verifies.. awesome, give me the data if not, fetch data from HDD and perform error correction through duplicate data   I'd manually run (some tool I'm not sure which one... btrfs check --repair perhaps?) every month or so to check HDD checksums and correct them via SSD data against bitrot.  For write case:   write to SSD sync to HDD when write requirements are low, thus not slowing down the system.   Is this possible and how would I do it?  Bottom line is, I like btrfs and would like it to duplicate data across HDD &amp; SSD drive automatically while keeping SSD excellent performance.   Using SSD as write cache is not an option since it's missing data duplication and security.