I am having trouble proving that the function $$f: N\times N \rightarrow N, \ \ f(i,j)=2^{i-1}(2j-1)$$ is surjective.   Work: I know that using the theorem in which $n$ is the product of prime numbers play an important role. For instance, $n=2^{i-1}p$, $p$ is the product of all prime factors of $n$. However, from here I don't know how to proceed. What makes it difficult for me is that how am I suppose to find an arbitrary $n\in N$ such that $f(i,j)=n$?
I am having trouble with the following problem:   $f: N\times N\rightarrow N$ and $f(i,j)=2^{i-1}(2j-1)$. Prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically equivalent.   Work: I am told that Euclid's Lemma will be useful to proving injection and also that for any positive integer $n$, $n$ is a product of prime numbers.   For injection, I set $f(a,b)=f(x,y)\rightarrow 2^{a-1}(2b-1)=2^{x-1}(2y-1)$.  Then I distributed across to get $2^ab-2^{a-1}=2^xy-2^{x-1} \rightarrow 2^ab-2^xy=2^{a-1}-2^{x-1} \rightarrow 2(2^{a-1}b-2^{x-1}y)=2^{a-1}-2^{x-1}$.   From here, I am unsure how to apply Euclid's Lemma to arrive at the injection result.  I am even more unsure how to prove surjection.
terminator: permission denied for sudo I'm trying to execute a string into my bash script. String contains variables (so I could substitute some CLI arguments depending on circumstances).  The problem is when I use variable containing --argument=value, everything breaks up, giving error:  Missing trailing-' in remote-shell command.   Example code:  #!/bin/bash  rsync_path="/usr/bin/rsync -azhrP" rsync_ssh_cmd="--rsh='ssh -o Compression=no'"  `$rsync_path /path/to/file $rsync_ssh_cmd username@localhost:/tmp`   If I just echo the last string from this code, it looks fine. But if i try to execute this string from script, it doesn't executes as expected.  I found a workaround - everything became OK if I try  eval "my-command-string"   instead of  `my-command-string`   or  (my-command-string)   Why does it so? How could I (should I really?) avoid to use eval?  I use bash 4.3.11(1)-release (x86_64-pc-linux-gnu) (Kubuntu 14.04.3). How to collapse unused parameters in moderncv? I am writing my CV using the moderncv package. I already have adjusted the CV to meet my needs. Therefore I amplified the maximum number of parameters possible in the \cventry command. However, now I have some ugly spacing in case I do NOT use all parameters (see second \cventry in the MWE for illustration):  Any suggestions to make the spacing collapse in case not all parameters are occupied? As you can see from the MWE I already have adjusted for \ifthenelse command. I use texlive 2014.  \documentclass[11pt,a4paper,sans]{moderncv} \moderncvstyle{banking} \moderncvcolor{blue} \usepackage[scale=0.7]{geometry} \name{John}{Doe} \title{Resume title} \address{street and number}{postcode city}{country} \phone[mobile]{+1~(234)~567~890}  \renewcommand*{\cventry}[9][.4em]{%  \begin{tabular*}{\textwidth}{l@{\extracolsep{\fill}}r}%  \ifthenelse{\equal{\bfseries #4}{}}{}{{\bfseries #4}} &amp; \ifthenelse{\equal{\bfseries #5}{}}{}{{\bfseries #5}} \\%   \ifthenelse{\equal{\itshape #3}{}}{}{{\itshape #3\ifthenelse{\equal{#6}{}}{}{, #6}}} &amp; \ifthenelse{\equal{\itshape #2}{}}{}{{\itshape #2}}\\%  \ifthenelse{\equal{#7}{}}{}{{#7}} &amp; \ifthenelse{\equal{#8}{}}{}{\\ {#8} &amp; \\}% \end{tabular*}%  }  \begin{document} \makecvtitle \section{Education} \cventry{Grade}{Degree}{Institution}{year--year}{M.Sc.}{Majors}{Majors}{} \cventry{}{Degree}{Institution}{City}{}{}{}{} \cventry{year--year}{Degree}{Institution}{City}{Grade}{Majors}{Majors}{} \end{document} I can't run Cisco packet tracer 7.1 on Ubuntu I have downloaded and extracted the packet tracer file, then I used this command as root  chmod +x install   then I ran (as root)  ./install    Then I used this command   ./set_ptenv.sh   but whenever I run this command   packettracer    it just doesn't start Least rank of a matrix Let $A$ be a $7\times 7$ matrix such that it has rank $3$ and $a$ be $7\times 1$ column vector.  Then least possible rank of $A+(a a^T)$ is? ($a^T$ is transpose of the vector)  Intuitively I think it's $2$ as $a a^T$ has rank $1$, and the most it could affect is one row of $A$. Solve $p_4(x) = x^4 −(2m + 4)x^2 + (m−2)^2 $such that $p_4$ is a product of two non-constant integer-coeficient polynomials I'm having trouble getting the starting idea for a problem I've been presented with:  I need to find values for m (integer) such that the following polynomial     $p_4(x) = x^4 −(2m + 4)x^2 + (m−2)^2$; $m$ is integer   can be described as the product of two non-constant integer-coeficient polynomials.  Any help/pointers/tips on how to go about this are greatly appreciated!  Edit: I meant polynomials with integer coeficients, but for some reason typed positive. "Happen to know" meaning "Do you happen to know is if she has a boyfriend?"   What does the above sentence means . I heard those line in the movie 'the social network' and I would like to know when do we use this phrase "happen to know" . Rendering fire animations with transparent background I'm creating a simple fireball sprite for my game and I want to render a fire animation.  I would like to get the upper images with transparent background, but when is set the transparency on under Render -> Film one of my frames will look like lower pictures.  You can see the difference of these two images. Run Ubuntu ISO live image from Windows BIOS - no USB or DVD Cannot authenticate without a password In Fallout 4 the Bobbleheads are back.  Once you pick them up they stay in your inventory with a weight of 0 and a value of 300. Will I need them later ? Do I lose the bonus if I sell them or drop them somewhere ? Same with the magazines.  Is there any disadvantage to selling them ? Why didn't James' love and sacrifice for Lily protect her? We know that Harry survived Voldemort's killing curse because Lily loved him so much that she sacrificed herself in an attempt to protect him. This resulted in a very powerful charm that pretty much rendered Harry immune to Voldemort (until Voldemort figured out the work-around). Then there's this passage describing that night:  [Voldemort] was over the threshold as James came sprinting down the hall. It was easy, too easy, he had not even picked up his wand ... &quot;Lily, take Harry and go! It's him! Go! Run! I'll hold him off -&quot; Hold him off, without a wand in his hand! ... He laughed before casting the curse ... &quot;Avada Kedavra!&quot; Harry Potter and the Deathly Hallows Chapter 17: Bathilda's Secret  So James Potter, fully aware that he was sacrificing himself (he said &quot;I'll hold him off,&quot; not &quot;I'll get rid of him&quot;) ran to face Voldemort because of his love for Lily and Harry. This seems to be very similar to how Lily tried to protect Harry from Voldemort even though she knew she would die. It seems to me that the same charm should have been cast upon Lily when James died, but just moments later Voldemort killed her too. Apparently James' sacrifice did not result in a charm that protected Lily, even though he did it from love for her and Harry. Why is this? Was something different in the circumstances? I keep running into a problem: How can I rotate an UV island by exactly 90 or 180 degrees? The same applies for scaling it to double in size, etc. Why does $(\frac{p-1}{2}!)^2 = (-1)^{\frac{p+1}{2}}$ mod $p$? Assume $p$ is prime and $p\ge 3$.  Through experimentation, I can see that it's probably true. Using Wilson's theorem and Fermat's little theorem, it's equivalent to saying $2^2 4^2 6^2 \cdots (p-1)^2 = (-1)^{\frac{p+1}{2}}$ mod $p$, but I can't figure out any more than that. VBA Hide Rows that contain N/A Take the following function:  DataTable go() {     return someTableAdapter.getSomeData(); }   When I set a breakpoint in this function, is there a possibility to inspect the returned value? go() is directly coupled to a datagrid in an .aspx page.  The only way to inspect the returned datatable is to use a temporary variable. However, that's a bit inconvenient. Isn't there another way? Why did the limit of power change from wizard/witch to wand? So in the first 5 or 6 books, it was very clear that the wand chooses the wizard and the power/level of the spells the wizard produces is based on the knowledge and skill of the wizard.  This is referenced by people constantly being telling Harry that he wasn't skilled enough to perform certain spells, or Bellatrix telling Harry that he needed to mean the Cruciatus curse for it to be effective.  However, in the 7th book, this entire idea is changed to the wand being the thing that's powerful and regardless of the wizard's/witch's skill or knowledge, he/she could still perform a very advanced spell if it's something the wand could handle.  Goyle accidentally creating a fire serpent is something that instantly comes to mind from the movies(I don't remember if that happened in the book).  So, my question is, what actually holds the power, the wizard/witch, or the wand? Magic leaves "smudges" and a sorceress becomes a book I could use some help identifying a book I read some time in the last year or so. The book goes between two different stories:  1) A student of magic finds a book in a cave that turns out to be a person transformed into a book. The book talks to the student and they share thoughts. At some point, the student runs away, chase ensues, and at the very end of the book, he manages to travel to another plane.  2) A girl in a religious society where magic is forbidden. She is the daughter of painters. She discovers she can see magic. Falls in love with some random painter, risks performing magic to not be pregnant, does it again so she can, and is arrested and sent to some prison thing.  The common element between the stories is that magic use leaves behind a smudge of some kind that can be visible.  Any assistance is appreciated. Is it possible to check where an alias was defined? How to make a declaration from a command? There are two forms of setting the type style: the declarative method e.g, \itshape and the command form, e.g, \textit. The command form takes an argument whereas the declarative form does not --- it just applies itself to whatever comes afterwards.  I would like to develop a declarative form, say \UppercaseIt, of the command form \MakeUppercase{&lt;text&gt;}. I have read, in source2e.pdf on CTAN the code for implementing \textit, \itshape and \MakeUppercase and can't claim that I understand any of them. I am after something like {\UppercaseIt some text} instead of \MakeUppercase{some text}.  I only have a code outline of what I'm after, since I don't know how to do it:  \documentclass{article}  \newcommand{\UppercaseIt}{%   %... magic }  \begin{document}  \MakeUppercase{some text}% Yields SOME TEXT  {\UppercaseIt some text}% Should yield SOME TEXT  \end{document}