What is the meaning of "formal" in math-speak? I have been encountering the word "formal" in several math books, and I don't understand what it means in context.  For example, Hatcher describes n-chains as "finite formal sums" on p. 105 and p. 108. What is a "formal sum?"  Are there "informal sums?"  Similarly, Steve Awodey on p.15 of "Category Theory" says opposite (or dual) categories are derived from other categories "with all the arrows formally turned around."  What does "formally" mean here?  This may be more of a language question than a math question, but I'd appreciate any clarification.
What does "formal" mean? I know the definition of formal power series, power series and polynomials. But what does the adjective "formal" mean? In , does it mean "9. Of or relating to linguistic or logical form as opposed to function or meaning" or maybe another one in the link?   Or does "formal" have some mathematical meaning which is other than usual dictionary meaning?
What makes some laser beams visible and other laser beams invisible? Let's say I have code like this:  $dbh = new PDO("blahblah");  $stmt = $dbh-&gt;prepare('SELECT * FROM users where username = :username'); $stmt-&gt;execute( array(':username' =&gt; $_REQUEST['username']) );   The PDO documentation says:     The parameters to prepared statements don't need to be quoted; the driver handles it for you.   Is that truly all I need to do to avoid SQL injections?  Is it really that easy?  You can assume MySQL if it makes a difference.  Also, I'm really only curious about the use of prepared statements against SQL injection.  In this context, I don't care about XSS or other possible vulnerabilities. I have file1.txt    this is the original text   line2   line3   line4   happy hacking !     and file2.txt    this is the original text   line2   line4   happy hacking !   GNU is not UNIX     if I do: diff file1.txt file2.txt I get:    3d2   &lt; line3   5a5   &gt; GNU is not UNIX     How is the output generally interpreted? I think that &lt; means removed but what do 3d2 or 5a5 mean?  If I do:    $ diff -u file1.txt file2.txt   --- file1.txt        2013-07-06 17:44:59.180000000 +0200   +++ file2.txt        2013-07-06 17:39:53.433000000 +0200   @@ -1,5 +1,5 @@    this is the original text    line2   -line3    line4    happy hacking !   +GNU is not UNIX     The results are clearer but what does @@ -1,5 +1,5 @@ mean? need to make salesforce app available in multiple language I want to make my appexchange app available in multiple language like Spanish, dutch etc. so that depend on the language of the user they can access the app.  Is it possible in any way..  any help will be appreciated.. Thanks Left and right subscript / superscript I am trying to put two subscripts at the left and right of a character. For example, something like:  _{t} p_{x}  where p is in the middle. How do you do this? 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 $\sin (x+ \frac{\pi}{6}) = 1$ I am using the Service Console in Lightning Experience, and using the Chatter Feed component on the Case layout to display updates to the Case.  If I use the "Log a Call" Quick Action from the Chatter publisher, these calls are displayed in the Chatter feed.  This is the desired behaviour.  However, if I insert a Log a Call record related to the Case any other way - e.g. using the Developer Console, or my CTI system's custom components - the call does not display in the feed.  If I query the record, it looks exactly the same as the record that is created with the Quick Action.  It just seems that the Chatter post is never made in this situation.  I have checked my Feed Tracking settings:   Task has Enable Feed Tracking and Subject checked Case has All Related Objects checked   How can I enable calls created with a custom component (part of a CTI system) to display in the case feed? Label rows of a matrix by characters Let $x$ be transcendental over $F$. Let $y=f(x)/g(x)$ be a rational function. Prove $[F(x):F(y)]=\max(\deg f,\deg g)$ Let $F$ be a field and $x$ transcendental over $F$. Let $y=f(x)/g(x)$ be a nonconstant rational function with relatively prime polynomials $f,g\in F[x]$. Prove $[F(x):F(y)]=\max(\deg f,\deg g)$.  My attempt: By replacing $y$ with $\frac{1}{y}$ if necessary we assume $\deg g\geq\deg f$. By the Euclid algorithm we assume $\deg g&gt;\deg f$. Then $\deg g=\max(\deg f,\deg g)$, let $n=\deg g$. Let $R(t)$ be this polynomial $$R(t)=yg(t)-f(t)=\frac{f(x)}{g(x)}g(t)-f(t)$$ Then $\deg R=\deg g=n$ because $n=\deg g&gt;\deg f$. The coefficients of $R(t)$ lie in $F(y)$ and hence $R(t)\in F(y)[t]$. Clearly $R(x)=0$.  Now I only have to prove $R(t)$ is actually the minimal polynomial of $x$ over $F(y)$. I believe the next step is to obtain a contradiction that $R(t)$ being reducible can lead to $f,g$ being not coprime, but I can't figure out how to do that.  P.S. A hint is preferred over a full answer. Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$ Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$.  Work so far:  $$x^8 - x = x(x^7 - 1) = x(x - 1)(x^6+x^5+x^4+x^3+x^2+x+1)$$  From here, I think the Zeros of an Irreducible over a splitting field Theorem would be the method. I am unsure how to proceed Looking at the properties for a Windows file I get two attributes, “Size” and “Size on disk,”  and "Size on disk" is always larger.  What do these two metrics mean? How to make the taskbar blink my application like Messenger does when a new message arrive? On my homepage, I'm using this method to hide my email from spam bots:  &lt;a href="admin [at] example.com"    rel="nofollow"    onclick="this.href='mailto:' + 'admin' + '@' + 'example.com'"&gt;Contact me&lt;/a&gt;   What do you think about it? Is it effective? What other methods do you know or use? If a function maps every connected set onto a connected set, is it necessarily continuous? I know the converse is true. How to find all the tables in MySQL with specific column names in them? Minecraft Test If A Chest Contains An Item I've seen a lot of questions for finding specific items in chests using commands using testforblock but how would you test if any block/item has been placed in a chest?  Thanks Adapt V-brakes bicycle to disk brakes I've been thinking in changing my v-brakes to disc brakes. Now that my rear v-brake can't brake anymore I think that it is time change it. I've already seen . My frame is prepared for disc and, as I don't want to change my brake lever, I'll go with mechanical brakes (does  look ok?);  The only problem is the rear wheel hub that has not the six holes for the disk. Is it ok if I just change the rear wheel hub and keep the front one? Word with the same meaning as "good moments" or "pleasant memories"? Is there a single word with the same meaning as "good moments" or "pleasant memories"? I'm looking for a word which describes good thing(s) that happened to someone. The only word I can think of is "joy".  Some examples might be:   Wayne has many pleasant memories of the days spent with the family. She could remember many good moments from the last weekend. Most commented post should be the first post in the blog WordPress default setting is, most recent post is the first post of the blog.  But, I need to "Most commented post should be the first post in the blog". ( Categories page as well as home page) I understand that the mechanics and design of a lightsaber aren't that complicated.  Do others besides the Jedi and Sith (or their minions) make or use lightsabers?  And if they don't, why not?