I'm interested in showing the inequality $$ \cos\left(\frac{1}{n}\right) &gt; 1-\frac{1}{2n^2}$$ For all $n$ positive integer. I've tried to proceed by induction and it's easily seen (assuming properties of trigonometric functions) that it holds for $n=1$ but I get stuck on showing it for $n+1$. Once $\cos(x)$ is decreasing on $[0,1]$ we have $$\cos\left(\frac{1}{n+1}\right) &gt; \cos\left(\frac{1}{n}\right) &gt; 1- \frac{1}{2 \, n^2}.$$ But I don't know how to get it from here.
Show that   $$\displaystyle1 - \frac{x^2}2\leq\cos x\quad\forall x \in \mathbb{R}$$   Let $f(x) = \cos x - 1 + \frac{x^2}2$; then we need to show that $f(x) \geq 0\quad\forall x \in \mathbb{R}$.  Since $$\lim_{x \to \pm\infty} f(x) = +\infty$$ and $$\operatorname{im}f'' = [0, 2]$$ (as $f''(x) = 1 -\cos x$), it remains to show that $x = 0$ is a minimum for $f(x)$, but I cannot do that because I would need to solve $f'(x) = x -\sin x = 0$, which I cannot. I think it would suffice to show that plugging in $x = 0$ we have that $f'(x) = 0$, and combining it with the fact that the function $f$ is convex, we would be done.    Is it enough? What's a better proof?  EDIT: It just occurred to me that there is a way to solve $f'(x) = 0$. Just observe that $f'(x)$ is an odd function and that $f''(x) \geq 0$ implies that $f'(x)$ is monotonically increasing. Since an odd function, if defined at $x = 0$, must be $0$ there, $x = 0$ is a solution to the equation and it's the only one.
Margin top in inline element I was just watching a mega factory video and wondered why they use an AC motor which requires a power inverter instead of DC which may be powered directly from their DC battery? Introducing an inverter means more cost (weight, controller, etc).  Are there any reasons for that? What are the differences between an AC and DC motor that may have lead to this decision? Also does anyone know what kind of motor is used in other electric cars? I haven't seen the question stated precisely in these terms, and this is why I make a new question.  What I am interested in knowing is not the definition of a neural network, but understanding the actual difference with a deep neural network.  For more context: I know what a neural network is and how backpropagation works. I know that a DNN must have multiple hidden layers. However, 10 years ago in class I learned that having several layers or one layer (not counting the input and output layers) was equivalent in terms of the functions a neural network is able to represent (see Cybenko's ), and that having more layers made it more complex to analyse without gain in performance. Obviously, that is not the case anymore.  I suppose, maybe wrongly, that the differences are in terms of training algorithm and properties rather than structure, and therefore I would really appreciate if the answer could underline the reasons that made the move to DNN possible (e.g. mathematical proof or randomly playing with networks?) and desirable (e.g. speed of convergence?) Fibonacci addition law $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$ How to make the taskbar blink my application like Messenger does when a new message arrive? I was up late working with my red LED headlamp on and when I was looking at the black part of my LCD (Apple Retina Macbook Pro) screen I noticed this interesting diffraction pattern. I'm confused because all I can think of that would cause this are Fourier and Fraunhofer patterns which I thought required more advanced optics the light to be collimated. One friend suggested it might be the polarizer. Any thoughts?  Unfortunately, the picture doesn't show this, but along the x and y axes there were discrete points not a continuous streak. Sorry for the horrible photo quality. I was wondering about the reason of the choose of the X as symbol used to represents the "Close application" operation. The probability of two dice summing to k is simple enough, make a diagram of the throws,  $$ \begin{array}{c}  &amp;|&amp;1&amp;2&amp;3&amp;4&amp;5&amp;6\\\hline 1&amp;|&amp;2&amp;3&amp;4&amp;5&amp;6&amp;7\\ 2&amp;|&amp;3&amp;4&amp;5&amp;6&amp;7&amp;8\\ 3&amp;|&amp;4&amp;5&amp;6&amp;7&amp;8&amp;9\\ 4&amp;|&amp;5&amp;6&amp;7&amp;8&amp;9&amp;\color{green}{10}\\ 5&amp;|&amp;6&amp;7&amp;8&amp;9&amp;\color{green}{10}&amp;11\\ 6&amp;|&amp;7&amp;8&amp;9&amp;\color{green}{10}&amp;11&amp;12\\ \end{array} $$  and the probability of a sum is just the length of the corresponding diagonal divided by $36$. For instance, $\mathrm{P}(\mathrm{sum} = 10) = \frac{3}{36} = \frac{1}{12}$. However, it's not as easy to draw a diagram for three dice or more, so how would a general formula look? Dual-boot boot menu does not show up after installing Ubuntu 15.10 alongside Windows 10 I have installed Ubuntu 15.10 alongside Windows 10 with UEFI. To install Ubuntu, I chose the option install alongside Windows 10 or something similar to this. Then I created a new partition for Ubuntu and installed it. After installation, the boot menu did not show up. Initially I thought Ubuntu has not been installed, but when I plugged in the USB drive and wanted to install Ubuntu I saw an option of reinstalling Ubuntu on my machine. So, I found out that Ubuntu is installed. Pressing F8 and F12 also does not help. Can anyone help me bring up the GRUB boot menu? In Windows, I also entered the command bcdedit /set {bootmgr} path \EFI\ubuntu\grubx64.efi in cmd, but still the boot menu does not show up.  Reply (Himanshu) : Just boot in bios and, if you can, add boot option with name say ubuntu and path EFI/ubuntu/shimx64.efi. No need for live PCB or anything. Move the boot option to top. This is for dell GUI bios but I assume it works for all. Or you may want to see  or , where you can enable Windows Boot Loader for one boot or forever as you wish, and boot Ubuntu from it. If you want now, you can use the installed Ubuntu terminal to use the commands update-grub to use GRUB instead. (Not enough reputation to answer properly btw.) Conditional \endinput I want to prevent processing rest part of a package in case if it is loaded not from lualatex. For some reason I cannot load ifluatex package, so I borrowed definition of \ifluatex from that package and invented the following chunk of code:  \expandafter\ifx\csname luatexversion\endcsname\relax % = not \inluatex   %\typeout{^^J********************** not LuaTeX^^J}   \PackageError{babel}{     ^^J\space\space * Option russianu is intended for LuaTeX only.              ^^J\space\space * Option russian  is used instead.                     }   \relax   \input{russianb.ldf}     \let\@next@action\endinput \else   %\typein{^^J********************** LuaTeX^^J}   \let\@next@action\relax \fi \@next@action   It seemed to work but Is there a better solution. In particular, it is desirable to avoid introduction of \@next@action because it is used the only time.  Just to explain: my goal is to hack babel with option russian to make it working painlessly with LuaLaTeX without any additional hacks (as described, eg, in  and ) Allowing all/different file type uploads Testing random variate generation algorithms Which methods are used for testing random variate generation algorithms? Writing test code for wrapper class I created a controller which looks like this  public with sharing class EventRegistrationController {      // A CONTROLLER FUNCTION     // FUNCTION RETURNS A LIST OF cSession      public List&lt;cSession&gt; getSessions() {            //DO STUFF       }     //A PUBLIC CLASS WITHIN CONTROLLER    @TestVisible public class cSession      {         public Session__c con {get; set;}         public Boolean selected {get; set;}          //This is the contructor method.          public cSession(Session__c c) {              con = c;             selected = false;         }     }  }   I need to write a test code for the controller, how can I create a list of cSession and call getSession function? Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$? Given three matrices $A$, $B$ and $C$ such that $ABA^T C$ is a square matrix, the derivative of the trace with respect to $A$ is:  $$ \nabla_A \operatorname{trace}( ABA^{T}C )  = CAB + C^T AB^T $$  There is a proof . However, I was not able to understand the second line of the proof. it says something like this:  $$ \nabla_A \operatorname{trace}( ABA^T C )= \nabla_\bullet \operatorname{trace}(f(\bullet)A^{T}C) + \nabla_\bullet \operatorname{trace}(f(A) \bullet^T C) $$  Where  $f(A) = AB$  I tried searching the proof or any hint using google, but I couldn't fine anything. Could anyone please help me with this second line. I cannot sleep. I am obsessed with this proof. First of all, I know that light does indeed travel slower in a medium like air or water, but that's because the photons are bouncing off of the medium's particles and in different directions so the light beam as a whole is slowed down (or at least that's the classical explanation I've read somewhere).  But my question is about light traveling freely in the vacuum. What if an astronaut during a spacewalk switches on a flashlight towards the opposite direction he is moving. Will that light travel slower than normal?  If not, is there another way to make light move slower? For example, can you make a beam of light move so slow that you can visually perceive it being affected by gravity? Convert continuous numeric values to discrete categories defined by intervals I'm looking for a time and memory efficient solution to calculate a  in C. I need to avoid dividing because I'm on a PIC 16 which has no dedicated division unit.  At the moment, I just store all values in a  and simply store and update the sum each time a new value arrives. This is really efficient, but unfortunately uses most of my available memory... Does opacity:0 have exactly the same effect as visibility:hidden If so, does it effectively deprecate the visibility property?  (I realize that Internet Explorer does not yet support this CSS2 property.) What EXACTLY cause Voldemort's soul piece to split off when he killed Lily and tried to kill Harry? Combinatorial proofs: having a difficult time understanding how to write them out Can someone explain how combinatorial proofs work?  I've included an example questions that's been giving me a hard time.  Any insight on the topic would be great.  $$\sum_{k=1}^{n}k{n \choose k} = n2^{n-1}$$  Another one:  $${2n \choose 2} = 2{n \choose 2} + n^2$$  The solution says:  LHS: Pick two people from a pool of n men and n women. RHS: Pick two women, pick two men, and pick one of each.  But I don't see how this applies.  How does $n^2$ show that we're picking "one of each?"  Thanks for the help :)