Undefined index
mysql_fetch_array()/mysql_fetch_assoc()/mysql_fetch_row()/mysql_num_rows etc... expects parameter 1 to be resource
Let $G$ be a finite abelian group s.t. it contains a subgroup $H_{0} \neq (e)$ which lies in every subgroup $H \neq (e) $. Prove that $G$ must be cyclic.   Also what can be said about $o(G)$ ?    I'm clueless about this problem from Herstein. I tried to think $G$ might be a solvable group and its subgroups are all normal (as $G$ is abelian) . But I'm unable to get further. Is there any way to recover unsaved notepad++ docs? I am trying to integrate one Salesforce org to other org by using Oauth setup, I wanted to use  use consumer key and consumer secret key in my REST class to connect with the org, after connection part completion, I want to access Get(), Put(), Post(), etc. Including the ability to set Oauth and Authorization headers needed for Salesforce Rest API calls.  Actually I'm new in REST implementation so can you please explain me by using code samples. Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$ For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$.   My attempt: Let  $$\begin{align*} f_n(x) &amp;= \frac{\ln\left(1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2\right)-\ln\left(1-2a\cos x+a^2\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(\displaystyle\frac{1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2}{1-2a\cos x+a^2}\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(1+\dfrac{1}{n}\left(\displaystyle\frac{2a-2\cos x+\frac{1}{n}}{1-2a\cos x+a^2}\right)\right)}{\frac{1}{n}}. \end{align*}$$ Now it is easy to see that $f_n(x) \to \frac{2a-2\cos x}{1-2a\cos x+a^2}$ as $n \to \infty$. $|f_n(x)|\le \frac{2a+2}{(1-a)^2}$ RHS is integrable so $\lim_{n\to\infty}\int_0^\pi f_n(x)dx = \int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2} dx=I'(a)$. But  $$\int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2}=\int_0^\pi\left(1-\frac{(1-a)^2}{1-2a\cos x+a^2}\right)dx.$$ Consider $$\int_0^\pi\frac{dx}{1-2a\cos x+a^2}=\int_0^\infty\frac{\frac{dy}{1+t^2}}{1-2a\frac{1-t^2}{1+t^2}+a^2}=\int_0^\infty\frac{dt}{1+t^2-2a(1-t^2)+a^2(1+t^2)}=\int_0^\infty\frac{dt}{(1-a)^2+\left((1+a)t\right)^2}\stackrel{(*)}{=}\frac{1}{(1-a)^2}\int_0^\infty\frac{dt}{1+\left(\frac{1+a}{1-a}t\right)^2}=\frac{1}{(1-a)(1+a)}\int_0^\infty\frac{du}{1+u^2}=\frac{1}{(1-a)(1+a)}\frac{\pi}{2}.$$ So $$I'(a)=\frac{\pi}{2}\left(2-\frac{1-a}{1+a}\right)\Rightarrow I(a)=\frac{\pi}{2}\left(3a-2\ln\left(a+1\right)\right).$$  It looks too easy, is there any crucial lack?  $(*)$ — we have to check $a=1$ here by hand and actually consider $[0,1), (1,\infty)$ but result on these two intervals may differ only by constant - it may be important but in my opinion not crucial for this proof. Why do we say "last night" and not "yesterday night"? As from object, is there a rational reason for saying "last night" rather than  "yesterday night", though you would say "yesterday morning" and "yesterday afternoon"? He is loved.   This is something that I've always kind of wondered.  In a sentence like this, is loved a verb or an adjective? Can it be considered either? EDIT: As of Java 8, static methods are now allowed in interfaces.  Here's the example:  public interface IXMLizable&lt;T&gt; {   static T newInstanceFromXML(Element e);   Element toXMLElement(); }   Of course this won't work.  But why not?  One of the possible issues would be, what happens when you call:  IXMLizable.newInstanceFromXML(e);   In this case, I think it should just call an empty method (i.e. {}).  All subclasses would be forced to implement the static method, so they'd all be fine when calling the static method.  So why isn't this possible?  EDIT: I guess I'm looking for answer that's deeper than "because that's the way Java is".   Is there a particular technological reason why static methods can't be overwritten? That is, why did the designers of Java decide to make instance methods overrideable but not static methods?  EDIT: The problem with my design is I'm trying to use interfaces to enforce a coding convention.  That is, the goal of the interface is twofold:   I want the IXMLizable interface to allow me to convert classes that implement it to XML elements (using polymorphism, works fine). If someone wants to make a new instance of a class that implements the IXMLizable interface, they will always know that there will be a newInstanceFromXML(Element e) static constructor.   Is there any other way to ensure this, other than just putting a comment in the interface? It is well know that $SO(n)$ is connected and $O(n)$ has two connected components: $O^+(n)=\{A\in O(n):\det A=+1\}$ and $O^-(n)=\{A\in O(n):\det A=-1\}$.   In what book can I find this property? I have multiple cameras in a scene. What I want to do is to change which one is active. This way I can view the scene from different angles. How would I do this? Name of a special matrix I have a matrix which is kind of symmetrical with the other diagonal, i.e., something like  $$A = \left[ \begin{array}{c c c c}    a &amp; b &amp; c &amp; d \\    e &amp; f &amp; g &amp; c \\    h &amp; i &amp; f &amp; b \\    j &amp; h &amp; e &amp; a  \end{array} \right]$$  Does this matrix have a special name in literature? What are it's properties?  And a matrix that is symmetrical by both diagonals   $$A = \left[ \begin{array}{c c c c}    a &amp; b &amp; c &amp; d \\    b &amp; e &amp; f &amp; c \\    c &amp; f &amp; e &amp; b \\    d &amp; c &amp; b &amp; a  \end{array} \right]$$  What's the name of it? Any interesting properties? Non-https links to other StackExchange sites in User profile when using https Formula across objects Regular expression to match a line that doesn't contain a word I know it's possible to match a word and then reverse the matches using other tools (e.g. grep -v). However, is it possible to match lines that do not contain a specific word, e.g. hede, using a regular expression?   Input:  hoho hihi haha hede   Code:  grep "&lt;Regex for 'doesn't contain hede'&gt;" input   Desired output:  hoho hihi haha GUI/Unity crashing in 16.04 LTS after updates 2018-01-04, compiz segfaults The latest updates (4/1/2018) make my laptop constantly crash the GUI/Unity:  (Intel Pentium su4100 processor)  syslog:  Jan  5 11:38:53 1810 kernel: [ 1850.327738] compiz[11911]: segfault at 0 ip 00007f6c7baf0c16 sp 00007ffd5ec71c60 error 4 in i965_dri.so[7f6c7b513000+82d000]   After login I end up with an empty desktop, just a wallpaper, GUI/Unity are in a crash loop:   No panel.  No dash.  Can't start a program. Launcher disappears. Sidebar disappears. System crash. Windows key not working. The key combination Alt+Tab not working.   Simply, the computer is not usable. Same for guest login. The only thing working: open up a terminal CTRL+ALT+T to examine log files.  I somehow forced  to let me log in and file the bug, but still lots of problems:   Dash not working Alt+Tab not working Sometimes windows all disappear for a few seconds     Any ideas for a solution or workaround?   UPDATE     If you have the same/ very similar problems, follow the link to my    report and mark there that you're affected too. Or file your own. Please leave a comment   if you wish, or upvote - this is a nasty problem - but an answer should   offer a fix or a workaround so users can use their computer again.    Without , Ubuntu can't fix issues. Find the maximum value of the expression ${\frac {x}{1+x^2}} + {\frac {y}{1+y^2}}+{\frac {z}{1+z^2}}$. Find the maximum value of the expression :   $${\frac {x}{1+x^2}} + {\frac {y}{1+y^2}}+{\frac {z}{1+z^2}}$$   where $x,y,z$ are real numbers satisfying the condition that $x+y+z=1$.   Taking $x=y=z=\frac {1}{3}$, I get the expression as $\frac {3x}{1+x^2}$, which is equal to $\frac {1}{1+{\frac{1}{9}}}$ or $\frac {9}{10}$.  How can I actually solve the problem without making unnecessary assumptions ? How to install gail and atk-bridge? Improving Review Queues - Design overview II: Changing review bans and other new features Differences between /bin, /sbin, /usr/bin, /usr/sbin, /usr/local/bin, /usr/local/sbin proof of functional completeness of logical operators If I know that the set of operators {∨, &amp; , ¬} is functionally complete, how do I go about proving/disproving the functional completeness of the following set of operators?  a) $\{\vee,\neg\}$  b) $\{\to,\neg\}$  c) $\{\to\}$  I have looked at the answer here for (b) : How do you spell the following date:  December 6 (AmE)  6 December (BrE)  Do you spell this date as 'December six' or 'December the sixth' in American English?  And in British English, do you spell it as 'Six December' or 'The sixth of December'?