Give a bijection between two sets
How to define a bijection between $(0,1)$ and $(0,1]$?
Shannon confusion and diffusion concept I read the document(not the whole document) from Shannon where he speaks about the concepts of confusion and diffusion. I read in many places(not in the document but around the internet) that confusion is enforced using substitution. Diffusion is enforced using permutation/transposition. Ciphers must use both of them because either confusion or diffusion alone are not enough. I read that a substitution cipher can apply by itself confusion(only). Permutation/Transposition applies by itself diffusion(only). It's precisely the last case that bothers me: Can a permutation/transposition cipher by itself apply diffusion?  Shannon explains diffusion as a property that spreads statistic properties of text all over the text preventing statistic analysis. It's frequently translated to: an alteration to a plaintext symbol affects many cipher text symbols. Assuming permutation of bits or characters, how can diffusion be achieved by simple permutation? I mean, if you permute bits, there will be no other bits affected. But if you consider a symbol a character and you permute bits, a change in a character will affect many other characters. It's a question of what is a symbol.  I also read versions of this diffusion concept where the point was just changing bits order just to avoid pattern analysis of the text. But, where is the avalanche effect in simple permutation?  So, what is the correct definition and the implications of diffusion?  I hope you understand what is troubling me.  Thank you once more How do I force TeX to completely use the 1st column? Can any linear transformation be represented by a matrix? Use $\cal L$ to denote a linear transformation on some vector space. We know any matrix $\bf{A}$ can be viewed as a linear transformation by defining $\cal L:= \cal L(\bf{v})= Av$ where $\bf{v}$ is a vector. I am curious is any linear transformation can be represented by a matrix? If so, why? If not, can anyone help provide a counterexample?  In particular, projection of vectors onto a vector $\bf{b}$ is a linear transformation defined by ${\cal L(\bf{v})}=({\mathbf{v}} \cdot \frac{{\mathbf{b}}}{{|{\mathbf{b}}|}})\frac{{\mathbf{b}}}{{|{\mathbf{b}}|}}$. Can this linear transformation be represented by a matrix?  Thank you! My network includes machines running Linux and others running Windows. And my machine is running Linux. Film where aliens looking like humans are waiting in an airport closed due to snowstorm I saw the movie 6 - 7 years back on cable tv, from the looks of it it seemed to be made in the late 1990's or early 2000's. Unfortunately I could not see the complete movie. The story goes like--   Some people are waiting at a small airport because snowstorm activity has cancelled all flights in and out. There are two children in the group as well (one boy and a girl). Somehow they discover there are aliens looking like humans among them. They decide to check passengers through baggage scanners (checking the skeletal structure via x-ray). They find and kill 2 aliens.   and the big twist:      5. They then realize the two children are also aliens.   any ideas what the name of the movie is ??? 'innerText' works in IE, but not in Firefox When an account is deleted, don't auto delete its negatively scored posts on the meta site Meta is different. Negatively scored questions are not always bad questions, and sometimes they have merit and draw good answers, especially discussions.  When an account (in the main site) is deleted, either by self deletion or by a moderator, all questions with score of -1 or less are being deleted regardless of their answers, both in the main and meta site.  Can you please change this behavior so that the questions on the per-site meta will survive and won't be auto-deleted? \pdfendlink ended up in different nesting level than \pdfstartlink Is a transit visa required to pass through Bangkok Airport? I know to secure safety it is now not allowed anymore to open apps that were not downloaded from the mac app store.  But surely there is a workaround for this to open an app that you know is 'good' but just isn't in the app store. Preferably allowing only that app and not allowing all downloaded apps to be opened.    How do I open an app that is not downloaded from the app store in mountain lion? Number theory counting sums of squares modulus This is an extra question from an old examination paper:     VI. Let $n&gt;1$ in $\mathbf{Z}$ and let $r(n) = \#\{(a,b)\in \mathbf{Z}^{2};n=a^{2}+b^{2}\}$ Let also $n=n&#39;n&#39;&#39;n&#39;&#39;&#39;$ where $n&#39;,n&#39;&#39;,n&#39;&#39;&#39; \in \mathbf{N}$ and   $p|n&#39; \Longrightarrow p\equiv 1\mod 4 ; p|n&#39;&#39; \Longrightarrow p=2 ; p|n&#39;&#39;&#39; \Longrightarrow p\equiv 3 \mod 4$   for $p\in \mathbf{N}$ prime.   i) Show that if $n&#39;&#39;&#39;$ is not a square, then $r(n)=0$.  ii) Show that if $n&#39;&#39;&#39;$ is a square and $n&#39; = 1$, then $r(n)=4$  iii) Show that if $n&#39;&#39;&#39;$ is a square and $1&lt;n&#39;=p_{1}^{e_{1}}\ddots p_{k}^{e_{k}}$ where $p_{1},...,p_{k}$ are different primes in $\mathbf{N}$ and $e_{1},...,e_{k} \in \mathbf{N}$, then it holds that $r(n)=r(n&#39;)=4(e_{1}+1)\ddots (e_{k}+1)$    I am completely dumbstruck and can't see how to begin (and neither did an older student who took this exam and whom I asked for hints). Help is greatly appreciated. "He was neither seer nor prophet" How would you explain the absence of an article? Here's a fragment from Jack London's Star Rover:     Wordsworth knew. He was neither seer nor prophet, but just ordinary man like you or any man. What he knew you know, any man knows. But he most aptly stated it in his passage that begins "Not in utter nakedness, not in entire forgetfulness.   To the best of my — admittedly sketchy — knowledge of grammar, the second sentence should read neither a seer nor (a) prophet, but just an ordinary man. Even if we accept the seer and the prophet as "poetic entities" ("father and son," "robber and robbed," "man to man," or whatever), logic would still dictate that ordinary man should be preceded by an an.  It is a well-known fact that Jack London's usage wasn't always up to par; his grammar, on the other hand, was always top-notch.  Logically, the sentence is grammatically incorrect (correct me if I'm wrong: no pun intended).  Illogically, it reads smoothly: intuitively, a native speaker understands that there's nothing wrong here.  However, those to whom English is not the first language cannot be expected to be intuitive in this case and would want an explanation. Any suggestions? Cyclic Group Generators of Order $n$ I'm making a mini game (bomb squad) and the armor needs to be enchanted. I know that armor can be enchanted through command blocks and that is what I intend to do. I am wondering if it is possible to enchant armor twice in one command block or should I need multiple command blocks as I have done in the past? How does Cypher exit/enter the Matrix without an operator? Efficient list intersection algorithm Given two lists (not necessarily sorted), what is the most efficient non-recursive algorithm to find the intersection of those lists? There is a bar of chocolate of size $N$ x $M$. Little square $(x, y)$ is poisoned. There are 2 players taking turns. In each move they can cut the bar along any of the horizontal or vertical lines, then the part which does not contain $(x, y)$ is being eaten. The player who is left only with square $(x, y)$ loses (beacause it is poisoned). Which player has the winning strategy? I guess that this game is equivalent to game of Nim due to Sprauge-Grundy theorem, but I don't know how. Number of normals from a point to an ellipse From a typical point $P$ inside an ellipse, how many points $Q_i$ on the ellipse have $PQ_i$ normal to the ellipse?  Someone asked me at school many years ago but I don't think I worked it out. I want to set a from address in a salesforce SingleEmailMessage, but can't see a way to do this and there is no standard method described in the documentation.   I want the email to be sent from doNotReply@blahblah.com, however, the email address of the user editing the record (which triggers the email send) is used as from address.  I'm using the following code:  mail.setReplyTo(fromaddress); mail.setSenderDisplayName(fromaddress);   But my code only changes the display name, not the email address.  Full code:   public static void sendSingleMail(id objId, ID templateId, string fromaddress) {      Messaging.SingleEmailMessage mail = new Messaging.SingleEmailMessage();     mail.setReplyTo(fromaddress);     mail.setTemplateId(templateId);     mail.setTargetObjectId(objId);     mail.saveAsActivity = false;     mail.setReplyTo(fromaddress);     mail.setSenderDisplayName(fromaddress);      ErrLogger.logger('Email being sent to :');     ErrLogger.logger('objId found:  ' + objId);     ErrLogger.logger('templateId found:  ' + templateId);     ErrLogger.logger('fromaddress found:  ' + fromaddress);      Messaging.sendEmail(new Messaging.SingleEmailmessage[] {mail}); } Problems with \printbibliography I am writing my thesis and need to do this with apa citation style.  I used the following code in my main document:  \usepackage[backend=biber,style=apa,sorting=none,natbib=true]{biblatex} \addbibresource{library.bib}  \usepackage[autostyle=true]{csquotes}    and in the end of my main document..  \printbibliography[heading=bibintoc,[title={References}]   However I always get this error:     "Undefined control sequence.   \printbibliography[heading=bibintoc,[title={References}]"   Furthermore in my references table, I get (yearmonthday) instead of (2015)e.g. and the references are not alphabetically sorted.