I have this problem that I have been stuck on for a while. I know that I can write the elements of $D_4$ as products of the identity, rotation by $90$ degrees and a vertical reflection and I have already written the $8$ elements of $D_4$ as their cycle notation is $S_4$ but have no idea what to do from there and would really appreciate the help!
Is $D_n$, the dihedral group of order $2n$, isomorphic to a subgroup of $S_n$ ( symmetric group of $n$ letters) for all $n&gt;2$?
My web application has a login page that submits authentication credentials via an AJAX call. If the user enters the correct username and password, everything is fine, but if not, the following happens:   The web server determines that although the request included a well-formed Authorization header, the credentials in the header do not successfully authenticate. The web server returns a 401 status code and includes one or more WWW-Authenticate headers listing the supported authentication types. The browser detects that the response to my call on the XMLHttpRequest object is a 401 and the response includes WWW-Authenticate headers. It then pops up an authentication dialog asking, again, for the username and password.   This is all fine up until step 3. I don't want the dialog to pop up, I want want to handle the 401 response in my AJAX callback function. (For example, by displaying an error message on the login page.) I want the user to re-enter their username and password, of course, but I want them to see my friendly, reassuring login form, not the browser's ugly, default authentication dialog.  Incidentally, I have no control over the server, so having it return a custom status code (i.e., something other than a 401) is not an option.  Is there any way I can suppress the authentication dialog? In particular, can I suppress the Authentication Required dialog in Firefox 2 or later? Is there any way to suppress the Connect to [host] dialog in IE 6 and later?    Edit Additional information from the author (Sept. 18): I should add that the real problem with the browser's authentication dialog popping up is that it give insufficient information to the user.  The user has just entered a username and password via the form on the login page, he believes he has typed them both correctly, and he has clicked the submit button or hit enter. His expectation is that he will be taken to the next page or perhaps told that he has entered his information incorrectly and should try again. However, he is instead presented with an unexpected dialog box.  The dialog makes no acknowledgment of the fact he just did enter a username and password. It does not clearly state that there was a problem and that he should try again. Instead, the dialog box presents the user with cryptic information like "The site says: '[realm]'." Where [realm] is a short realm name that only a programmer could love.  Web broswer designers take note: no one would ask how to suppress the authentication dialog if the dialog itself were simply more user-friendly. The entire reason that I am doing a login form is that our product management team rightly considers the browsers' authentication dialogs to be awful. How do I suppress the "New release '12.10' available" message? I was editing a curve for hair, and pressed W to bring the specials menu and Subdivide it, but instead everything but the curve I had active dissapeared, and my cursor turned into a circle like the one from the circle select mode.  I'm guessing they changed the specials menu and I'm inside some unknown to me function, but I'm not sure how to turn everything back to normal. What is *.pkinfo file? What is a *.pkinfo file?   Google Search results talk about Package Info files and how they contain map documents and layers, so I don't see how this is different from a .mpk file.    I also have not found anything about how to create a *.pkinfo file. Transcendental Extensions. $F(\alpha)$ isomorphic to $F(x)$ Let $E$ be an extension field of $F$ and $\alpha \in E$. Then $\alpha$ is transcendental over $F$ if and only if $F(\alpha)$ is isomorphic to $F(x)$, the field of fractions of $F[x]$.   This was a theorem in an abstract algebra textbook with a very brief proof. Can someone please explain why this theorem holds? I'm having difficulty grasping the concepts at hand.  Thanks! Consider the subset $B=\{(x,y):x^2+y^2\leq 1\}\ \subseteq \mathbb{R}^2$. Show that $\vert B \vert = \vert \mathbb{R}^2 \vert$.  It is easy to see tht $f:B\rightarrow\mathbb{R}^2$ defined as $f(x,y)=(x,y)$ is injective.   However, in the book I'm working out of (Book of Proof by Richard Hammack), the other injection noted is  $g:\mathbb{R}^2\rightarrow B$ defined as  $$g(x,y)=\left(\frac{x^2+y^2}{x^2+y^2+1}x,\frac{x^2+y^2}{x^2+y^2+1}y\right).$$  In the book I'm asked to verify this is indeed an injective function.  My approach was to assume $g(x,y)=g(m,n)$ and through basic algebra I was able to deduce  $x(m^2+n^2+1)(x^2+y^2)=m(x^2+y^2+1)(m^2+n^2)$  and  $y(m^2+n^2+1)(x^2+y^2)=n(x^2+y^2+1)(m^2+n^2)$  but how does this show $(x,y)=(m,n)$? Or is this the wrong approach? I've been looking for the name of a sci-fi romance book about space travel but haven't been successful so far. I came across it a few years ago, read the plot on its back cover but didn't read the book. It is about two spacecraft pilots, a man and a woman, who travelled at light speed (or maybe FTL, I can't remember) in separate ships. Every time they return to Earth, centuries have passed on Earth because of time dilation. They meet each other for a few times on Earth in different eras, but don't know when, if ever, they will meet again. I don't know how it ends.   If anyone knows which book I'm talking about, could you please tell me the name? Thanks a lot! We know that continuous version of $\sum$ is $\int$, but, can there be a continuous version of $\Pi$? For equal cost, why would you select the MacBook Air over the MacBook Pro? I can see that the 13" version of MacBook Pro and MacBook Air is same $1199, but the Pro is more feature-loaded than Macbook Air. Why would anyone buy MacBook Air? What's the catch in this?    I was thinking of buying a MacBook for development purposes, and worried if there's any problem in the Pro version, because it comes with more features (like more storage).  Edit:    Thanks for the answer guys, I bought a MacBook Pro a week ago and already loving it. Why does friction play the role of centripetal force during the turning of a car? What makes frictional force the centripetal force of a car turning along a curve? As friction is the opposing force and acts anti-parallel so there is no component of frictional force towards the center,right? Then how can frictional force be centripetal force? How to Uninstall Blender Software I installed Blender for a project working on and it's way more than I need. I'd like to uninstall it, but can't figure out how to do so. It's not showing up in Programs (using Windows 9) and I can't find instructions on how to remove it in the Blender documentation. Any assistance would be greatly appreciated. Thanks. Easy way to determine the virtualization technology of a Linux machine? C# vertical label in a Windows Forms How to create a fancy table of contents for Beamer How can I create a fancy-looking table of contents for a Beamer presentation?    \documentclass{beamer} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc}  \begin{document}  \frame{ \tableofcontents }  \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{}  \end{document} GeoReference tiff image into the Google map? I want to Geo-reference a tiff image contain a Map for a specific region, I need to add this region(image) and Geo-reference it on the Google map, SO I can see on the map for the region I select the image which I have inserted.  the image also need information such as the Latitude and Longitude of corners and size, and the pixel per meter measurement.  How to do the correct Geo-reference for that ? Suppose $m,n&gt;0$, $U$ an open subset of $\mathbb{R}^{m+n}$ and let $f: U \to \mathbb{R}^n$ be continuously differentiable.  Is it possible for $f$ to be injective?    My thinking is that continuous differentiability+injectivity suggests that this is sort of getting at a converse to the Inverse Function Theorem.  We know the Jacobian is not invertible at any point in $U$, and the question is asking if this implies that $f$ cannot be invertible in any neighborhood contained in $U$. network connection over thunderbolt bridge between linux and mac I know that if we solve the Maxwell equation, we will end up with the phase velocity of light being related to the permeability and the permittivity of the material. But this is not what I'm interested in - I want to go deeper than that. We know that the real speed of light is actually not changing, the decrease in speed is just apparent. Material is mostly empty, the light will still travel with $c$ in the spacing. The rare atoms will disturb the light in some way. So I am interested in how the atoms affect the light. Photon absorption-emission theory Some textbooks that I read explain it in a way kind of like this: In a material the photons are absorbed by atom and then re-emitted a short time later, then they travel a short distance to the next atom and get absorbed&amp;emitted again and so on. How quickly the atoms in a material can absorb and re-emit the photon and how dense the atoms decides the apparent speed of light in that material. So the light appears slower because it has a smaller “drift speed”. Interference theory But recently I realize an alternative explanation: Atoms respond to the light by radiating electromagnetic wave. This “new light” interferes with the “old light” in some way that results in delayed light (advanced in phase), this can easily be shown by using simple phasor diagram. Consequently effectively the light covers a smaller phase each second, which gives the impression of a lower phase velocity. However the group velocity is changing in a complicated way. I think that the first explanation does not explain the change in phase velocity of light. if we consider light travelling into a slab of negative refractive index non-dispersive material, let’s say the light is directed perpendicular to the slab. The phase velocity’s direction will be flipped, but group velocity’s direction in the material will not change. Only the second explanation can explain the flipped phase velocity direction. I guess that the velocity that we get in the first explanation is actually belongs to the group velocity. It makes sense to me that the front most of the photon stream determines the first information that the light delivers. So the question is What really cause the phase velocity of light to be decreased?  &quot;drift velocity&quot; of photons (they aren't the same photons, they are re-emitted all the time) phase difference between absorbed and emitted light something else  And also, I still don't really understand detailed explanation of the absorption-emission process for small light's wavelength (for large lambda compare to the atoms spacing, the photons will be absorbed by the phonons). The dispersion relation that we know is continuous and also some material is non-dispersive, therefore the absorption process must occur in all frequency for a certain range. So definitely it doesn't involve the atomic transition, otherwise it will be quantized. My guess is that the relevant absorption process gets smooth out by the dipole moment. What makes the spectrum continuous? EDIT: link for dispersion relation: The mysql extension is deprecated and will be removed in the future: use mysqli or PDO instead Presently, how much information can a quantum computer store, in how many qubits? What restrictions are there and how does it vary across realizations (efficiency of data storage, ease of reading and writing, etc)?