Let $p$ and $q$ be distinct primes. Then find the number of positive integer solutions of the equation $$\frac{1}{x}+\frac{1}{y}=\frac{1}{pq}$$   We get $pq=\frac{xy}{x+y}$  Now $x+y$ must divide $xy$ as L.H.S. is a positive integer with two prime factors but how do we make sure the same on R.H.S. ?  Given options are $3$ or $4$ or $8$ or $9$.
For $x,y\in\mathbb{N}$ how many ordered pairs $\left(x,y\right)$ satisfy   $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{pq}$ where $p,q$ are distinct primes?
Evaluating the limit of a sequence given by recurrence relation $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Is my solution correct? Problem  The sequence $(a_n)_{n=1}^\infty$ is given by recurrence relation:   $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$.   Evaluate the limit $\lim_{n\to\infty} a_n$.  Solution   Show that the sequence $(a_n)_{n=1}^\infty$ is monotonic. The statement $$V(n): a_n &lt; a_{n+1}$$ holds for $n = 1$, that is $\sqrt2 &lt; \sqrt{2+\sqrt2}$. Let us assume the statement holds for $n$ and show that $V(n) \implies V(n+1)$. We have that $$a_n &lt; a_{n+1}.$$ Adding 2 to both sides and taking square roots, we have that $$\sqrt{2+a_n} &lt; \sqrt{2+a_{n+1}},$$ that is $a_{n+1} &lt; a_{n+2}$ by definition. Find bounds for $a_n$. The statement $$W(n): 0 &lt; a_n &lt; 2$$ holds for $n=1$, that is $0 &lt; \sqrt2 &lt; 2$. Let us assume the statement holds for $n$ and show that $W(n) \implies W(n+1)$. We have that $$0 &lt; a_n &lt; 2.$$ Adding two and taking square roots, we have that $$0 &lt; \sqrt2 &lt; \sqrt{2+a_n} &lt; \sqrt4 = 2.$$ The limit $\lim_{n\to\infty} a_n$ exists, because $(a_n)_{n=1}^\infty$ is a bounded monotonic sequence. Let $A = \lim_{n\to\infty} a_n$. Therefore the limit $\lim_{n \to\infty} a_{n+1}$ exists as well and $\lim_{n \to\infty} a_{n+1} = A$. (For $(n_k)_{k=1}^\infty = (2,3,4, \dots)$, we have that $(a_{n_k})_{k=1}^\infty$ is a subsequence of $(a_n)_{n=1}^\infty$, from which the statement follows.) We have that $a_{n+1} = f(a_n)$. That means that $A = \lim_{n\to\infty} a_n = \lim_{n \to\infty} {f(a_n)} = f(\lim_{n \to\infty} a_n) = f(A) = \sqrt{2 + A}$. Solving the equation $A = \sqrt{2 + A}$, we get $A = -1 \lor A = 2$. Putting it all together, we get that $A = 2$, because the terms of the sequence are increasing and $a_1 &gt; 0$.   Is my solution correct? We know that in high temperature, resistivity in metals goes linearly with temperature. As temperature is lowered, resistivity goes first as $T^5$ due to "electron-phonon" interaction, and then goes as $T^2$ as temperature is further lowered due to "electron-electron" interactions.   My question is, is there an intuitive way of understanding the above power-law dependence at each temperature range physically? How to wrap lines correctly inside algorithmic I am having problems displaying algorithms. The wrapping is really ugly and I can't, for the life of me, find a fix that doesn't involve manually tweaking the hell out of everything, to the point that I might as well scrap the crappy algorithm packages and write everything out in TeX.  Is there any way to properly indent the following  \documentclass{article} \usepackage{algorithm} \usepackage{algpseudocode} \begin{document} \begin{algorithm}    \caption{My pseudo code.} \begin{algorithmic}[1]    \While{this line is sooooooooooo long and boring and too much for algorithmic to handle}       \State look at this state, this state is just too long for algorithmic to display it properly        \While{again this line is sooooooooooo long and boring and too much for algorithmic too handle}          \State look at this state again, this state is just too long for algorithmic to handle, I'm just going to switch to Word       \EndWhile    \EndWhile \end{algorithmic} \end{algorithm} \end{document}​     so that the output is similar to If we have a complex scalar $\phi$ we know that the gauge-invariant interaction with $A$ is given by $A^\mu J_\mu$, where $J$ is the Noether current of the $U(1)$ symmetry of the Lagrangian $$ J_\mu\sim \phi^\dagger\partial_\mu\phi-\phi\partial_\mu \phi^\dagger \tag{1} $$  If we have a real scalar instead, $J=0$ and the field doesn't couple to $A$: real particles aren't charged.  My question is: what if we took $$ J_\mu\sim \phi\partial_\mu\phi \tag{2} $$ to couple $\phi$ to $A$? The interaction is Lorentz invariant and renormalisable, but it is not gauge-invariant, which is probably a bad thing. At what point would this theory break down?  This is probably very naïve, but I think that the Feynman rules for this theory are straightforward. So I guess the theory makes sense at least perturbatively. The theory is probably flawed at a more fundamental level, but I cant't seem to find where (some kind of gauge anomaly perhaps?) Visual Studio: How to break on handled exceptions? I would like Visual Studio to break when a handled exception happens (i.e. I don't just want to see a "First chance" message, I want to debug the actual exception).  e.g. I want the debugger to break at the exception:  try {    System.IO.File.Delete(someFilename); } catch (Exception) {    //we really don't care at runtime if the file couldn't be deleted }   I came across  notes for Visual Studio.NET:     1) In VS.NET go to the Debug Menu >>   "Exceptions..." >> "Common Language   Runtime Exceptions" >> "System" and   select "System.NullReferenceException"      2) In the bottom of that dialog there   is a "When the exception is thrown:"   group box, select "Break into the   debugger"       3) Run your scenario. When the   exception is thrown, the debugger will   stop and notify you with a dialog that   says something like:              "An exception of type "System.NullReferenceException" has   been thrown.              [Break] [Continue]"       Hit [Break]. This will put you on the   line of code that's causing the   problem.   But they do not apply to Visual Studio 2005 (there is no Exceptions option on the Debug menu).  Does anyone know where the find this options dialog in Visual Studio that the "When the exception is thrown" group box, with the option to "Break into the debugger"?  Update: The problem was that my Debug menu didn't have an Exceptions item. I customized the menu to manually add it. What are the proper uses of:   static_cast dynamic_cast const_cast reinterpret_cast C-style cast (type)value Function-style cast type(value)   How does one decide which to use in which specific cases? Generating DLL assembly dynamically at run time Currently I have some code that is being generated dynamically.  In other words, a C# .cs file is created dynamically by the program, and the intention is to include this C# file in another project.  The challenge is that I would like to generate a .DLL file instead of generating a C# .cs file so that it could be referenced by any kind of .NET application (not only C#) and therefore be more useful. init.m in Autoload directory is not read by parallel kernels in M11.2 How many stickers do I need to complete my FIFA Panini album? I'm playing the , which is an Internet adaption of the classic Panini albums that are usually published for the soccer world cup, European championship, and possibly other tournaments.  The album has placeholders for 424 different stickers. The purpose of the game is to collect all 424. The stickers come in packs of 5, which can be obtained through codes found online (or, in case of the classic printed album, bought at your local newsstand).  I make the following assumptions:   All stickers are published in the same quantity.  One pack of stickers does not contain duplicates.     How can I find out how many packs of stickers I need to acquire in order to be reasonably sure (let's say 90%) that I have all 424 unique stickers? Illustrator: Is there an efficient and accurate way to join two circles by their tangents (and keep it editable)? It's easy to join two circles if they are the same size and horizontal/vertical, into a pill shape:    If the circles are at an add angle then it's still possible but you can no longer rely on smart guides.     You can add lines roughly where the tangents should be and then use the scissors tool on the circles, cut out the inner area and then join the two crescents into one shape.    Is there a better way to do this? Is there a way to do this that is still editable afterwards too?  There are some other questions that cover the same area as this but don't cover editability or joining circles that are not in line with each other (where smart guides can't help). What are the differences between a multidimensional array and an array of arrays in C#? What are the differences between multidimensional arrays double[,] and array-of-arrays double[][] in C#?  If there is a difference, what is the best use for each one? How add 14 days on end date I have End Date column and I want to add 14 Days on it auto when created, can I do it by workflow? Can I do it via jQuery. If yes how? if no how I can do it? iPhone's Bluetooth - What? Why? How? I googled a lot and read many documents. But I can't understand the concept behind iPhone's Bluetooth technology.  In my knowledge, Bluetooth was used for:   Transferring files between devices Wireless communication for things like video gaming and earphones   Recently I tried to copy some of the photos from my old Android mobile to my iPhone 4S via Bluetooth. It's quite simple in my older phones to connect with other devices by Bluetooth for file sharing. But I can't pair other devices (even another iPhone :( ) with my iPhone 4S.   Copying songs/videos may affect their iTunes as well as their business. But what about images? Why is the usage of Bluetooth restricted and what's the use of it being able to transfer images? Is there any other way to use iPhone's Bluetooth to transfer files? Maybe using an app? But I don't want to jailbreak my device.   I'm confused about the restrictions and Apple's thinking. How does quantum trapping with diamagnets work? I just saw  demonstration by someone from a Tel Aviv University lab.  What they achieved there is mind blowing. I myself own a levitron that uses the Hall effect to levitate a magnet, the problem with that is the magnet must always be flat facing the Hall effect base, any unbalance will wreck the levitation. But in the case of the video you can see that he isn't constrained by such things, the disk can levitate and stay still at whatever angle and orientation. Instead of being levitated, the disk is simply being held still.   How does this happen? He referred to this as quantum trapping, but I've little idea about what that is....  Also, what material is the base in these demos? Is the circular track in the demonstration magnetic? I assume this is so. Is there any conceivable way for this technology to make objects levitate on any surface? How can I override the OnBeforeUnload dialog and replace it with my own? I need to warn users about unsaved changes before they leave a page (a pretty common problem). window.onbeforeunload = handler  This works but it raises a default dialog with an irritating standard message that wraps my own text. I need to either completely replace the standard message, so my text is clear, or (even better) replace the entire dialog with a modal dialog using jQuery. So far I have failed and I haven't found anyone else who seems to have an answer. Is it even possible? Javascript in my page: &lt;script type=&quot;text/javascript&quot;&gt;        window.onbeforeunload = closeIt; &lt;/script&gt;  The closeIt() function: function closeIt() {   if (changes == &quot;true&quot; || files == &quot;true&quot;)   {       return &quot;Here you can append a custom message to the default dialog.&quot;;   } }  Using jQuery and jqModal I have tried this kind of thing (using a custom confirm dialog): $(window).beforeunload(function () {     confirm('new message: ' + this.href + ' !', this.href);     return false; });  which also doesn't work - I cannot seem to bind to the beforeunload event. Fastest way to move Collection from Sharded MongoDB Cluster to a another Sharded MongoDB Cluster I have to move a MongoDB Collection (Size: 1TB+) from Sharded MongoDB Cluster to another[5 shards on both clusters]. During this operation, it is assumed that the accompanying program will be offline, so no need to worry about inconsistent data, but I need to minimize the downtime.  I've tried testing it with mongodump and mongorestore, but looks like that is taking ages to finish.  Please share your experiences for such kind of scenarios.  Note: Cluster has only one collection, so I am open at cluster level sync as well. Let f be an r-cycle in $S_n$. Given any $h \in S_n$, show that $hfh^{-1} = (h(x_1), h(x_2), \dots, h(x_r))$ Filter to change the content of 404 page I have created custom rewrite rules. These rules now go to 404 page as expected. I can use action hook template_redirect to change the template matching query_var.  But I don't want to create separate template files as the page structure depend on the current theme in use. Rather I just want to change the content and title. I have tried change the content using the_content filter, but it seems the filter doesn't even get called for 404 page.  I also tried loading index.php template and then using the_content filter, but it also doesn't work.  Is there any way to achieve this? Showing a function on $[0,1]$ is the zero function Why is a Hyperplane called a "Hyper"plane? I just had this curious question. In other fields, the word "hyper" is actually used to refer to something which is "over; beyond; above" as defined by Google. An example of such terms would be Hyperglycaemia, defined by Wikipedia as:     a condition in which an excessive amount of glucose circulates in the blood plasma.   Or Hypersonic, defined by Google as:     relating to speeds of more than five times the speed of sound.   So, my question is, what would be the motive to give a Hyperplane such a name when it is defined as :     a subspace of one dimension less than its ambient space