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I see many posts on both this forum and elsewhere the discuss the use of a wide angle on a crop sensor.  For example, they discuss how at 14mm lens is 14mm on a 35mm film or full frame, but with a 1.6x crop sensor it is an effective 28mm.  My question is, is a 14mm on a crop sensor the same image as a 28mm on a full frame.  Or does "effective" include some other conotation.  I read that a 14mm can create distortion on a full frame and some vignette, so people seem to say using it on a crop takes out some of those edge problems. My confusion is how the lens is actually taking in the extra image.  Reading kenrockwells blog I learned that wide angle keeps lines straight, while fish eye actually skews the lines.    So with this as my understanding, I am still a bit confused how a lens that is curved in order to get a greater angle of view, will be the same as a cropped version of a less curved lens.  Or does "effective" only refer to the objects to the sides but not to the distortion of the image.  Or, if it is different, is it so small that for photography it does not matter?  A technically description would be welcome as long as a basic understanding of the situation.  Thanks!  Edit: To be more clear after the comments,  I am specifically talking about the end resulting 2d projection from 3d space.  I have read both those previous answers.  the first one is closer to what I am talking about.  However, it is still confusing for me.  I have experience with 3d modeling and projection matricies, which might be confusing me with this.  But for example, i dont understand the images having a 50 mm lens being drawn Starting from the sensor, so it shows a different fov for crop and full frame . If the lens is the same, it takes the same amount of information from the world, which makes a ray from a certain degree off the last lens projected into a smaller space, there is no projection from the sensor, so drawing the line from the sensor doesnt make sense to me.  The distance of the sensor to the rear of the lens must have some impact on the light projected but is not represented in the images.  Further, again from that first answer it says cropping is the same as zoom, but from what i understand from how perspective works, it is different, since a wide angle will project lines differently than a small fov to the same size, so cropping the center of a wide angle and zooming into something is very different.  This is easily recreated in an opengl application with varying fov projection matrix, i imagine in painting class people learn this as well.  A fisheye would be a good example, since if cropping was the same as zooming, a fish eye lens would keep the center exactly like a normal lens and then a gradient weighted towards the outside would rapidly create a warped perspective, but from what I see, it is even.  To me those images just look like its comparing cropped with full frame in regards to orthogonal projections. Can polymorph and true polymorph change a PC's race and sex? prove that : $ i = \sqrt {-1}\ $ i have a pretty nasty question. i was glancing through a few olympiad papers and stumbled upon this question:  prove that  $ i = \sqrt {-1}\ $.  i tried the conventional methods namely euler's formula  but could not figure what to do next. how do you actually proceed further? i have checked a few books and a proof is not available.are there any proof available for this How many levels are there for Steam Trading Card badges? With the new Steam Trading Cards, I decided to craft the badge for CS:GO. I got this:    I then noticed that this is only a level 1 badge, and there was a level 2 badge to get. So I crafted it too:    Now it says there's a level 3 badge to get next. How many levels are there in total? I have a scenario where I have multiple threads adding to a queue and multiple threads reading from the same queue. If the queue reaches a specific size all threads that are filling the queue will be blocked on add until an item is removed from the queue.  The solution below is what I am using right now and my question is: How can this be improved?  Is there an object that already enables this behavior in the BCL that I should be using?  internal class BlockingCollection&lt;T&gt; : CollectionBase, IEnumerable {     //todo: might be worth changing this into a proper QUEUE      private AutoResetEvent _FullEvent = new AutoResetEvent(false);      internal T this[int i]     {         get { return (T) List[i]; }     }      private int _MaxSize;     internal int MaxSize     {         get { return _MaxSize; }         set         {             _MaxSize = value;             checkSize();         }     }      internal BlockingCollection(int maxSize)     {         MaxSize = maxSize;     }      internal void Add(T item)     {         Trace.WriteLine(string.Format("BlockingCollection add waiting: {0}", Thread.CurrentThread.ManagedThreadId));          _FullEvent.WaitOne();          List.Add(item);          Trace.WriteLine(string.Format("BlockingCollection item added: {0}", Thread.CurrentThread.ManagedThreadId));          checkSize();     }      internal void Remove(T item)     {         lock (List)         {             List.Remove(item);         }          Trace.WriteLine(string.Format("BlockingCollection item removed: {0}", Thread.CurrentThread.ManagedThreadId));     }      protected override void OnRemoveComplete(int index, object value)     {         checkSize();         base.OnRemoveComplete(index, value);     }      internal new IEnumerator GetEnumerator()     {         return List.GetEnumerator();     }      private void checkSize()     {         if (Count &lt; MaxSize)         {             Trace.WriteLine(string.Format("BlockingCollection FullEvent set: {0}", Thread.CurrentThread.ManagedThreadId));             _FullEvent.Set();         }         else         {             Trace.WriteLine(string.Format("BlockingCollection FullEvent reset: {0}", Thread.CurrentThread.ManagedThreadId));             _FullEvent.Reset();         }     } } The question is $$\text{If }\, p, \ q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad \text{Prove that } 1979\mid p.$$ So my solution went like this:  $$S=1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}\Rightarrow S=\sum_{k=1}^{1319}\frac{1}{k}-2\sum_{k=1}^{659}\frac{1}{2k}=\sum_{k=660}^{1319}\frac{1}{k}$$ Now this is equivalent to finding the sum of the roots of the equation $$\left(x-\frac{1}{660}\right)\left(x-\frac{1}{661}\right)\dots \left(x-\frac{1}{1319}\right)=0$$. Rewriting this equation, we get $(660x-1)(661x-1)\dotsc(1319x-1)=0$ (we can discard the $\displaystyle \frac{1319!}{659!}$ since $1979$ is a prime). Calculating the sum of the roots, we get the required value to be equal to $(1320+659)\times 330$ which is obviously divisible by $1979$ (again discarding the coeffecient of $x^{660}$).  Now my question is: Using the same method of converting the problem into finding the sum of roots of a polynomial, we get from the original equation something like this: $$(x-1)\left(x+\frac{1}{2}\right)\dots \left(x-\frac{1}{1319}\right)=0 \Rightarrow (x-1)(2x+1)\dotsc(1319x-1)=0$$ So, the sum of roots would be $-1+2-3+4-\dotsc+1318-1319=-660$ But this is not divisible by $1979$. Why does the former solution work while the latter doesn't? Is there a flaw in my reasoning? Note: I know that the question can be solved by some other methods. Please only explain my given doubts and don't instead give another solution  I realised my mistake after some hours of pondering over it. (In the 1st equation) To get the required sum I would need to take calculate $\displaystyle -\frac{\text{coeffecient of }x^{659}\,}{\text{coeffecient of }x^{660}}$. However what I've done is calculate this:  $\displaystyle -\frac{\text{ coeffecient of }x}{\text{coeffecient of }x^{660}}$ which is actually equal to $\text{sum of roots taken }659 \text{ at a time} $. It was just a coincidence that the value I calculated was divisible by $1979$. The correct way, indeed, would be through the link supplied Since reading the answers to my question about  I've been good about giving my meat time to defrost in the refrigerator. However, my plans for tonight have changed and I'd like to defrost some ground beef that is completely frozen for dinner this evening. What are my options to do so quickly and safely? Nested sequence of non-empty compact subsets - intersection differs from empty set I have the following proposition:  "If $(S_n)$ is a sequence of compact subsets of a metric space such that  $S_1 \supseteq S_2 \supseteq ... \supseteq S_n$  Then $\bigcap^\infty_{i=1} S_i$ $\neq$ $\emptyset$"  My lecture notes provide the following proof, whose last step I don't understand:    Proof: Assume by way of contradiction that $\bigcap^\infty_{i=1} S_i$ = $\emptyset$. Let $V_n := (S_n)^c$ for each $n$. The class $\mathbb{C} :=\{V_n: n \in \mathbb{N} \}$ is an open covering of $S_1$. Since $S_1$ is compact, $\exists I$ finite, $I \subset \mathbb{N}$, s.t $\{V_i: i \in I \}$ covers $S_1$. Let $n:=max(I)$. The class $\{V_1, ..., V_n \}$ also covers $S_1$ and we get the following contradiction   $\emptyset = \bigcap^n_{i=1} S_i = S_n$    So, I don't get the part from "Let $n:=max(I)$..." onwards, especially the contradiction. I have tried applying DeMorgan's Law and the only way the last equalities makes sense is if the union of $\{V_1, ..., V_n \}$ equals the whole metric space, a property I haven't been able to understand from the steps of the proof. This is a corollary to the finite intersection property, so it must have a connection but again I can't quite get it.  Thanks in advance I have seen many programs consisting of structures like the one below  typedef struct  {     int i;     char k; } elem;  elem user;   Why is it needed so often? Any specific reason or applicable area? Access user fields? $account = \Drupal::currentUser() loads the user object into $account, but how do I access custom fields I've created on the user object (via the admin UI)?  With dpm($account) I can't see any of my custom fields. Near the beginning of the definition of the amsmath environments aligned and gathered (and also mathtools' multlined) there's an explicit thin space \,. This often  has to be cancelled  with  \!, as has been observed  various  times:         I'd like to know why this explicit \, is there &ndash; what would go wrong without it?    From amsmath.dtx:  \newcommand{\start@aligned}[2]{%     \RIfM@\else         \nonmatherr@{\begin{\@currenvir}}%     \fi     \savecolumn@ % Assumption: called inside a group %    \end{macrocode} %    The \cs{null} here is to keep the \cs{,} glue from causing the %    invocation of the clause in \tex/'s built-in tag placement %    algorithm that can cause an equation to be shifted all the way over %    to the margin. %    \begin{macrocode}     \null\,% When a user is suspended, their account is reduced to 1 rep and all of his privileges are stripped. However, they can still join chat rooms.   This leads to various forms of possible abuse (all of which I've seen by suspended users):   Join/leave spams Expletives in profile messages Hold one sided conversations with users of the room via    Suspended users cannot be kicked, and even moderators can do very little (aside from completely removing the account, which is often overkill).  My proposal is to extend  for suspended users to one of two levels:   Prevent them from being shown in the user list at all. Strip their profile profile picture and description from the profile view visible from chat rooms (not from profile pages). I am writing a document using LaTeX and generating PDF.  Can anybody tell me how I can generate the PDF metadata, e.g. author or title, from LaTeX?  On Windows XP, I am using MikTex's texify to generate the PDF.  On linux, I am using pdflatex to generate the PDF.  Any answers for either platform would be much appreciated! How do I use large bitmaps in .NET? I'm trying to write a light-weight image viewing application. However, there are system memory limitations with .NET.  When trying to load large bitmaps (9000 x 9000 px or larger, 24-bit), I get a System.OutOfMemoryException. This is on a Windows 2000 PC with 2GB of RAM (of which 1.3GB is used up). It also takes a lot of time to attempt loading the files.  The following code generates this error:  Image image = new Bitmap(filename); using (Graphics gfx = this.CreateGraphics()) {     gfx.DrawImage(image, new Point(0, 0)); }   As does this code:  Stream stream = (Stream)File.OpenRead(filename); Image image = Image.FromStream(stream, false, false); using (Graphics gfx = this.CreateGraphics()) {     gfx.DrawImage(image, new Rectangle(0, 0, 100, 100), 4000, 4000, 100, 100, GraphicsUnit.Pixel); }   Also, it is enough to do just this:  Bitmap bitmap = new Bitmap(filename); IntPtr handle = bitmap.GetHbitmap();   The latter code was intended for use with GDI. While researching this, I found out that this is in fact a memory issue where .NET tries to allocate twice as much as is needed in a single contigous block of memory.    I know from other applications (Internet Explorer, MS Paint etc.) that it IS possible to open large images, and rather quickly. My question is, how do I use large bitmaps with .NET?  Is there anyway to stream them, or non-memory load them? Clearing NSUserDefaults I'm using +[NSUserDefaults standardUserDefaults] to store application settings. This consists of roughly a dozen string values. Is it possible to delete these values permanently instead of just setting them to a default value? Incompatible Software for next OSX? I understand the move past Snow Leopard meant losing Rosetta, and compatibility with any PPC apps.   Since there are some apps that can't be replaced (for example, old years of TurboTax that I might need to reference) I have a drive to boot Snow Leopard, and one for Yosemite. I was considering the jump to El Capitan, when I saw the question  which got me wondering the question -     Before installing a next generation OS, is there any way to find, in   advance, which apps I have that would be orphaned by the move?   Edit, the answer below from Jean Valjean was what I needed. One app concerning me appeared as follows:    It might be tedious to search many apps, but at least this gives me a chance to view the top ones that will cause me grief. Animating eye texture in a head mesh I'm trying to animate a mesh with different eye textures but it's not working.     This is a texture made by me to animate but I still couldn't figure out how to do it. I've seen tutorials on the internet but they don't work to me. Is the harmonic oscillator potential unique in having equally spaced discrete energy levels? An instructor has an exam of 5 questions and a question bank of 50 questions. Are there $50\choose 5$ or $(50)_5$ possible exams? I made a simple FM transmitter following  and it worked! Now I need some help to understand how.  My coil is made of 1 mm copper wire has 6 turns and 8 mm wire diameter. According to this online calculator:  this means that: L = 3.9e-7 H  I did not have any 5 pF trim capacitor so I used a 3 pF capacitor instead.  Am I correct in assuming that the frequency this will transmit on can be found by looking at the L1 C2 as a LC oscillator and using the frequency formulae:  fR = 1/(2*pi*sqrt(LC))  ?  If so I should get fR = 147 MHz. Which is strange since I received the signal on 91.5 MHz on my radio.  In a LC circuit I understand that the signal decays exponentially over time. That is hopefully not happening here. Why? Because of C4 maybe?  Also what is the purpose of C1?  If I want to increase the range of this FM transmitter somewhat can I do this by simply replacing R2 with a 470 Ohm resistor?  Are there free simulator programs that let me see the resulting spectrum at the antenna?