Suppose that $v$ and $w$ are unit vectors. If the angle between $v$ and $\hat{i}$ (the unit vector in the positive $x$ direction) is $\alpha$ and that between $w$ and $\hat{i}$ is $\beta$, prove that   $$\cos(\alpha − \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$$
If two vectors $a$ and $b$ make angle $\alpha$ and $\beta$ with the $x$-axis, prove, using vectors, that:   $$\cos(\beta - \alpha) = \cos \alpha \cos\beta + \sin\alpha \sin\beta$$   I tried taking components of each vector along the plane, but that didn't get me anywhere. I tried taking component of vector $b$ along vector $a$, but I also got stuck there. Can anyone point me in the right direction?
I am currently working my way through "A Concise Introduction to Pure Mathematics" by Martin Liebeck, and have been stuck on an excersise for a couple of days now. The question reads:  "Show that for an integer n $\geq2$, the period of the decimal expression for the rational number $\frac{1}{n}$ is at most $n-1$."  I've been trying to solve this without making any substantial progress. And yes, I have seen explanations that when performing a long division the decimal expression either terminates to $0$ or eventually repeats. But I haven't seen any proof of this that I've been satisfied with, even less do I know how to formalise this type of proof. I've tried to write out long division with arbitrary digits, but that obviously turns out to be too much of a hassle and basically impossible to draw any conclusions from. Why are neural networks becoming deeper, but not wider? In recent years, convolutional neural networks (or perhaps deep neural networks in general) have become deeper and deeper, with state-of-the-art networks going from 7 layers () to 1000 layers ( in the space of 4 years. The reason behind the boost in performance from a deeper network, is that a more complex, non-linear function can be learned. Given sufficient training data, this enables the networks to more easily discriminate between different classes.  However, the trend seems to not have followed with the number of parameters in each layer. For example, the number of feature maps in the convolutional layers, or the number of nodes in the fully-connected layers, has remained roughly the same and is still relatively small in magnitude, despite the large increase in the number of layers. From my intuition though, it would seem that increasing the number of parameters per layer would give each layer a richer source of data from which to learn its non-linear function; but this idea seems to have been overlooked in favour of simply adding more layers, each with a small number of parameters.  So whilst networks have become "deeper", they have not become "wider". Why is this? Loading external GeoJSON file into Leaflet map? I would like to load a geoJSON (polygon) file into my leaflet map. I have seen examples where geoJSON is embedded into the javascript code but I can't find any examples showing how it is done with an external file.   &lt;head&gt;     &lt;meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no"&gt;     &lt;link rel="stylesheet" href="http://cdn.leafletjs.com/leaflet-0.5/leaflet.css" /&gt;     &lt;script src="usStates.geojson" type="text/javascript"&gt;&lt;/script&gt;     &lt;style&gt;         html, body, #map {             height: 100%;         }         body {             padding: 0;             margin: 0;         }     &lt;/style&gt; &lt;/head&gt;  &lt;body&gt;     &lt;div id="map" style="height: 100%"&lt;/div&gt;     &lt;script src="http://d3js.org/d3.v2.min.js?2.9.3"&gt;&lt;/script&gt;     &lt;script src="http://cdn.leafletjs.com/leaflet-0.5/leaflet.js"&gt;&lt;/script&gt;      &lt;script type="text/javascript"&gt;       var map = L.map('map').setView([38.57, -94.71], 4);      L.tileLayer('http://{s}.tile.cloudmade.com/9067860284bc491e92d2342cc51d47d9/998/256/{z}/{x}/{y}.png', {attribution: 'Map data &amp;copy; &lt;a href="http://openstreetmap.org"&gt;OpenStreetMap&lt;/a&gt; Imagery © &lt;a href="http://cloudmade.com"&gt;CloudMade&lt;/a&gt;'}).addTo(map);      var featureStyle = {         "color": "#ff7800",         "weight": 5,          "opacity": 0.2     };      L.geoJson(usStates).addTo(map);      &lt;/script&gt;  &lt;/body&gt; I saw this keyword for the first time and I was wondering if someone could explain to me what it does.   What is the continue keyword? How does it work? When is it used? Further anonymize deleted accounts by changing the username in close/delete notices When an account is deleted, the username in the user's posts and comments is changed to a generic user&lt;userID&gt;. But the username stays the same on close and delete notifications on posts the user had voted to close / delete1. I think it shouldn't, especially for users who used their real name as their username, if they chose to distance themselves from the site and remove their accounts, it makes sense that they wouldn't really want their name plastered on close and delete notifications.   A perhaps easy way to avoid the actual username remaining visible would be to change it to the generic user&lt;userID&gt; (or anything generic enough, like "anonymous") prior to deletion, and wait a bit for the generic username to propagate all over the site. If I'm not horribly misreading , this was part of the process in the old days.   This could be automated or added as a suggestion to the .  Glaring example can be seen , where the deleted user also posted a comment where their name is properly anonymized:        1 I've seen it on close notifications and self deleted answers. I'm not sure it happens on deleted questions, or deleted answers that aren't self deletions, and I don't have a way to verify it - other than stumbling on them accidentally Is $f(x,y)=(xy)^{2/3}$ differentiable at $(0,0)$? I looked at the graph of $f(x,y)=(xy)^{2/3}$ at the origin, and no matter how far I zoom in it has four corners along $x=\pm y$, so it doesn't look like it is differentiable as $z=0$ doesn't look like a tangent plane.  I also found that its partial derivatives are not continuous at $(0,0)$, e.g.  $$ \lim_{(ky^2,y) \to (0,0)} \frac{\partial f}{\partial x} = \lim_{(ky^2,y) \to (0,0)} \frac{2}{3}x^{-\frac{1}{3}}y^{\frac{2}{3}} = \frac{2}{3}k^{-\frac{1}{3}} $$ (although $\frac{\partial f}{\partial x}(0,0) = \frac{\partial f}{\partial y}(0,0) = 0$)  I also tried to prove/disprove its differentiability by definition, and for it to be differentiable I need $$ \lim_{(x,y) \to (0,0)} \frac{f(x,y)-0}{\sqrt{x^2+y^2}} =0 $$ but then I got stuck. Am I even on the right track?  [UPDATE] I need to do the same thing for $f(x,y)=(xy)^{\frac{1}{3}}$... I think the answer is that $(xy)^{2/3}$ is differentiable but $(xy)^{1/3}$ is not, according to my professor but I'm not sure Big-O proof showing that t(n) is O(1) Let t(n) = $\sqrt{n^2+100n} - n$  Show that $t(n)$ is $O(1).$  I tried solving this using the formal definition of O( ) but am not getting anywhere. Quite confused on how to tackle this question. How to hide a folder in sdcard in android Why did JJ Abrams recreate the Star Trek Timeline? In the 2009 version of Star Trek, Nero and the Romulans come back through time via Blackhole and kill Kirk's dad and the original Star Trek timeline is altered  creating a whole new reality.  Why would Abrams not just stick to the timeline given by Roddenberry instead of altering it? When I add LIMIT 1 to a MySQL query, does it stop the search after it finds 1 result (thus making it faster) or does it still fetch all of the results and truncate at the end? How can I change the registered name in Windows 7 to my own? how do I change the registered name in Windows 7 to my own name? How do I activate Dauntless Cathar's ability? has the following ability:     {1}{W}, Exile Dauntless Cathar from your graveyard: Put a 1/1 white Spirit creature token with flying onto the battlefield. Activate this ability only any time you could cast a sorcery.   Please clarify. If the card is IN my hand, how can I exile if FROM the graveyard if it is not there? Did the writers mean SEND it to graveyard, or actually exile it? Confused, the writing does not make sense. Conformally Map Region between tangent circles to Disk What is true for a ring with exactly two right ideals Bibliography with Metadata: I am constructing a bibliography of as much of the prior research in a field I am working in. I would like to use BibTeX to print the citation in the pdf, and then I would like to add the details of the paper after the citation. Only then would I like to print the next citation. For example:  [1] Jahan M. Dawlaty and Yunquing Chen. Measurement of the optical absorption of graphene in the terahertz. Applied Physics Letters, 2008. Research Group: Cornell Pump Wavelength: 500 nm. Probe Wavelenth: 7 microns. Relaxation Time: 50 fs. [2] Paul George and Dmitry Vekslar, Measurement of Ultrafast Carrier Dynamics in Graphene. Applied Physics Letters, 2009.  . . . and so on. Does anyone know how this could be done? What is the difference between private and protected members of C++ classes? What is the difference between private and protected members in C++ classes? I understand from best practice conventions that variables and functions which are not called outside the class should be made private—but looking at my MFC project, MFC seems to favor protected. What's the difference and which should I use? Thus is a picture of both of my hard-drives:    My C Drive and my D drive. I want to put Ubuntu onto my D Drive. Now my question here is if i do install onto my D Drive then how should I do it in order to retain use of the drive. I intend to partition about 60 gigs from it. Now after doing that I should still have about 830 gigs left. I want to still be able to use that in Windows. So how and will I be able to retain use of the drive after Installing? I have installed once before and I believe i did it wrong because not only could Windows not recognize my D drive but I also lost all the data on it.   My computer is a Lenovo Y700 It is a UEFI (I believe i got that right). The habitual past "would" versus "simple past" tense One can use the habitual past 'would' to describe past events. For example one could say,      Jim would spend weekends at the farm. He would cut weed and water the plants.   One can also use the simple past.       Jim spent weekends at the farm. He cut weed and watered the plants.   Would the use of habitual past be considered "weak" writing or prose, as opposed to the simple past?  How does the simple past in the following sentence "He would cut weed and water the plants." without any preceding time expression e.g. "spend weekends"  make a distinction between a repeated action/habit and a one-time action of cutting weeds and watering plants? Finding matrix $B$ is not zero matrix where $AB= 0$ The square roots of different primes are linearly independent over the field of rationals