Calculate $$\lim_{n\to\infty}{\sqrt{5+\sqrt{5+...+\sqrt 5}}}$$   I have a similar example solved in my textbook so I tried to use the same technique though I'm not entirely sure what I'm doing.   $a_n:=\lim_{n\to\infty}{\sqrt{5+\sqrt{5+...+\sqrt 5}}}$  $a_1=\sqrt5$  $a_{n+1}=\sqrt{5+a_n}$  $L=\sqrt{5+L} \implies L=\frac{1+\sqrt{21}}{2}$  This is what I'm confused about, where did $L=\sqrt{5+L}$ come from? And after that I have to prove that $a_n$ is bounded above with $\frac{1+\sqrt{21}}{2}$, right?   So I use induction to prove that:   $a_1=\sqrt5\leq\frac{1+\sqrt{21}}{2}$  Let's assume that for some $n\in \Bbb N$ it's true that $a_n\leq\frac{1+\sqrt{21}}{2}$  Then $a_{n+1}=\sqrt{5+a_n}\leq\sqrt{5+\frac{1+\sqrt{21}}{2}}=\sqrt\frac{11+\sqrt{21}}{2}$   Well, this doesn't seem right. What did I prove here?   After that I proved that $a_n$ is strictly increasing, so I should now be able to find the limit only I don't know how to proceed.

Proposition: $$\sqrt{x + \sqrt{x + \sqrt{x + ...}}} = \frac{1 + \sqrt{1 + 4x}}{2}$$  I believe that this is true, and, using Desmos Graphing Calculator, it seems to be true.  I will add how I derived the formula in a moment, if you would like.  Working  I will be honest; all that I did was use the Desmos Graphing Calculator, and let $y = \sqrt{x + \sqrt{x + \sqrt{x + ...}}},$ let $x = 1, 2, 3, ...$, looked at the point at which the two graphs meet, and searched for the number in Google.  It turned up an interesting website, which you may access , which seemed to show a pattern.  I used this pattern to derive the formula that I stated earlier.


How can I remove duplicate items from a list if all I know is that the list elements can be ordered?  (I also don't care about the order of items in the list.)  Existing questions like  or  require the use of a set, i.e. require the items in the list to be hashable. In my case, hashability is not a guarantee.

Given a list of strings, I want to sort it alphabetically and remove duplicates. I know I can do this:  from sets import Set [...] myHash = Set(myList)   but I don't know how to retrieve the list members from the hash in alphabetical order.  I'm not married to the hash, so any way to accomplish this will work. Also, performance is not an issue, so I'd prefer a solution that is expressed in code clearly to a fast but more opaque one.


I looked in the Hegelian dictionary and got more confused.

I understand that Hegel intends to justify the history of mankind by stating that every town and society that has existed has been a test indirectly arranged by the 'world spirit' in order to eventually create a town in which people would live ethically, and that this process has been done in order for the 'world spirit' to become aware of its freedom. But what exactly is this 'world spirit'?   In the last quiz of a course I'm taking (I´ve taken this year a philosophy course at college), I wrote that the 'world spirit' was some kind of parasitic entity which inhabits every human being that has existed and is going to exist. Does this seem right?


$A$ and $B$ are two non-empty subsets of the real set. Given that  $$(x \in A \text{ and } y \in B) \implies x\leq y$$  Show that $\sup A$ and $\inf B$ exists and that $\sup A\leq\inf B$

If $A, B$ are two non empty sets of real numbers and for every $a$ from $A$ and $b$ from $B$     $a < b$,   how can I prove that      $\sup A \leq \inf B$?


Current State Deleting comments on a post, outside the flag queue, requires no prompt. Deleting comments on a flagged post (whether a comment or other flag) requires confirmation each time. This isn't a problem if someone flags all comments, but normally "too chatty" indicates other problem comments too. Issues with current state My workflow when moderating comment cleanup is to open the post in a new tab so I can more quickly delete and cleanup comments. This is annoyingly inefficient. Desired future state Remove confirmation in flag queue for deleting comments.

Ever since mods gained the ability to undelete comments, deleting them has become easier as the confirmation dialog was removed.  However, this is not the case in the mod queue — I still get a confirmation dialog for each comment I want to delete in the queue under a custom-flagged post (single-comment flags can be deleted just fine). And since this is the place from where I delete comments the most, it can be annoying.  In addition, the "purge comments" button, when invoked from the queue, refreshes it, so I have to scroll back down, remember which post I just purged, and mark the flag as helpful. Usually not a big issue but if there are multiple "obsolete comments" flags, then it can get confusing.  Could the workflow for comment thread moderation from within the queue be improved? Ideally, make comment deletion work like everywhere else on the site, and make the purge button AJAX-based like the comment deletes.


I was wondering how would I draw this graph using LaTeX and its tikz packages for my math class.

I need to make a cartesian plane for plotting the straight line joining the points (-3,1) and (6,-2). This is a very simple cartesian plane plotting. When I looked at the manual for tikz the graphics for cartesian planes are for more advance plotting, and what I need is very simple.  Something like this:


I have a checkboxlist from which I want to get check event.     the checkboxes are created dynamically.   I am not able to fire the event   what I tried so far is  $(".XAxisrowCheckbox").click(function () {              //Do stuff      });   I also tried with this       $("input.XAxisrowCheckbox").click(function () {              //Do stuff      });   but no luck for me. Tag goes like this:  <input type="checkbox" class="XAxisrowCheckbox">   Please suggest something.

I have a bit of code where I am looping through all the select boxes on a page and binding a .hover event to them to do a bit of twiddling with their width on mouse on/off.  This happens on page ready and works just fine.  The problem I have is that any select boxes I add via Ajax or DOM after the initial loop won't have the event bound.  I have found this plugin (), but before I add another 5k to my pages with a plugin, I want to see if anyone knows a way to do this, either with jQuery directly or by another option.


Prove that for $x>0$, $x + \frac1x \ge 2$ and equality holds if and only if $x=1$.   I have proven that $x+ \frac1x \ge 2$ by re-writing it as $x^2 -2x +1 \ge0$ and factoring to $(x-1)^2\ge0$ which is true because you cannot square something and it be negative.   Now I am stuck on the part where I have to prove equality to hold if and only if $x=1$. Any suggestions?

I was asked to prove that:   $$x + \frac{1}{x}\geqslant 2$$  for all values of $ x > 0 $  I tried substituting random numbers into $x$ and I did get the answer greater than $2$. But I have a feeling that this is an unprofessional way of proving this. So how do I prove this inequality?


For all uncountable set,  Is there an uncountable subset such that its complement is also uncountable?  How can I prove this?

Is it true that for any uncountable $S$, there exits two uncountable subsets $S_1,S_2 \subseteq S$ with $S_1 \cap S_2 = \emptyset$?   I can find no counter example, but no proof either.  I am aware of similar questions regarding the partition of uncountable subsets of the real line, and there are many examples of disjoints uncountable subsets of $\mathbb{R}$, say $R_1,R_2$.   It is not clear to me however whether this extends to any uncountable set. For uncountable sets $S$ with $S \succeq_{card} \mathbb{R}$, there exists an injection $f : \mathbb{R} \rightarrow S$ and choosing $S_1,S_2$ with $S_i = \{s\in S~|~ s = f(r_i) $ for some $ r_i \in R_i\}$, works.   But from what I understood of the continuum hypothesis, there might exist an uncountable sets $H$ with $\mathbb{R} \succeq_{card} H$. So using the above argument, one cannot infer that the property holds for every uncountable sets from observing that it holds for $\mathbb{R}$. Is this right? Are there other ways to prove or disprove the assertion?


I want to build a flow to trigger an email when a date in one of the fields is nearing. I have a field named Payment due date. is it possible to trigger an email 2 days prior to the Payment due date. So that I don't miss the payment due date and payment is not delayed. I tried using schedule in power automate. but what should I use while giving the path to the sharepoint list and selecting the sharepoint, I think if I use "when an item is created/modified" it will only trigger when an item is created/modified.   Thank you very much in advance!

I want to build a flow to trigger an email when a date in one of the fields is nearing. I have a field named Payment due date. is it possible to trigger an email 2 days prior to the Payment due date. So that I don't miss the payment due date and payment is not delayed. Any kind of assistance is appreciated. Thank you very much in advance!


In Avengers: Endgame, when Tony Stark steals the Infinity Stones from Thanos and inserts them into his own ad-hoc gauntlet, there isn't much of an energy outburst and Stark seems to be able to wield the gauntlet fairly easily:                                  By contrast, there is much more energy when Thanos gets all the stones:                                  The Hulk seems barely able to survive wielding them:                                  I understand that the actual snap killed Tony, but it seems he wielded the stones too easily. Is his nano-suit particularly good at buffering the power from the stones? Why didn't the gauntlet he designed previously for the Hulk do this?

In Avengers: Endgame, when Hulk initially uses the new Infinity Gauntlet, it takes a tremendous effort from him (in addition to almost burning his entire arm), in order to snap his fingers and restore the world as we knew it. Yet at the end of the movie, when Iron Man uses the Gauntlet, it seems very easy for him to snap his fingers.  How was Tony able to wield the stones with such ease when Hulk was clearly struggling with it?


I am currently running Windows 7. I'm not quite sure how to get this to boot into Ubuntu. I have the usb stick in drive, and I restart the computer, but nothing changes. I tried f12 before the windows startup prompt, still nothing. I downloaded 12.04 i386 using the steps in "Pen Drive USB Linux's installer" on this site. I can go into "devices" and browse files on the usb, but I can't get the option to "try Ubuntu". What am I missing?  Also, in the usb download, there was 12.04 i386 and 12.04 amd. What's the difference? What am I missing? Thanks.

I would like to see a full how-to guide on how to install Ubuntu.


List<Account> Acc// this list contains Account records. Acc.Sort() is not working  How to sort the records based on Created Date? Basically i need the recently record from the Acc list

I dont know how to sort a list of SObject Case.  I have a List<Case> newCases which i iterate over in for cycle, where i save Cases to my new List<Case> casesList.  for (Case newCase : newCases) {             if(newCase.Case_Content__c == '111' && newCase.Status == '24'){                 String cuid = newCase.CUID__c;                 cuidSet.add(cuid);                 casesList.add(newCase);                 casesList.sort(); //TODO sorting by CUID__c             }         }   What i need to do is, to sort this new list by field CUID__c. So the Cases with same CUID are together.  Thank you for any help :)


I am using Ubuntu 16.04 with an HiDPI monitor (3840x2160). Following an update earlier this week (which also broke the unity desktop) the display scaling now resets to 1 when resuming from sleep.  I was hoping that this would get corrected in the upgrade from 16.04.3 to 16.04.4 but that is not the case. When the unity desktop broke I ran the following commands which eventually corrected the issue of the missing launcher and top panel:  sudo apt clean sudo apt autoclean sudo apt install -f sudo apt install -f unity sudo apt install -f ubuntu-desktop sudo reboot   When I realized that display scaling was resetting to 1 after resuming from sleep, I tried the following commands:  cd .cache mv compizconfig-1 compizconfig-1_old reboot   and  sudo apt install --reinstall unity-settings-daemon reboot   However, neither of these approaches worked. Is this issue affecting anyone else?  Is there a way to prevent the display scaling from changing everytime the computer is inactive long enough to go to sleep?     Update: (7 March 2018)  I can't tell if this is truly a duplicate of the issue in Ubuntu 17.04 as indicated by david6. The problem with the scaling is the same, but only when using the AMD 17.50 drivers. When using the kernel's amdgpu driver the problem does not exist. Is this the same issue? Note using the latest AMD drivers released today (17.50-552542) does not fix the issue.      Update: (26 January 2019)  What an annoying problem.  Since the last update, subsequent upgrades resulted in the issue being present regardless of what driver I used.  After upgrading to 18.04 the issue was still there.  Presently my screen now goes blank but the computer does not inter a sleep state, judging by the whir of the fans, therefore there is no more scaling reset occurring.  This is fine for my desktop situation.

I've recently done a clean install of 17.04 on a machine with an Nvidia 750Ti card and a Dell 4K monitor. I'm using the Nvidia binary driver 381.09.  All this works great apart from the fact that the display scaling I set in the Ubuntu Screen Display settings always reverts back to 1 when the machine resumes after suspend. I need to set it at 1.5 otherwise everything goes tiny (BTW why doesn't Ubuntu scale to a sensible default with 4K monitors?)  Does anyone else get this? I can't find any mention of it elsewhere.  BTW I've tried the Nouveau driver and that doesn't wake up my screen at all after suspend. So that route's a non-starter.     Edit: Looks like it's been  Will monitor to see if anything happens.


I want all the benefits of dedicated schematic drawing software but I often don't like the way the diagrams actually look. I know it sounds trivial, but I find it easier to read nicely hand drawn diagrams than all-red wireframe type ones.  Is there a decent piece of software where for example I can customise the shape or colour of opamps/wires/transistors etc?

I was going to add a bit of information to my post on a previous day using schematics and some instructions. What programs are being employed for this purpose?  I mostly want to see what others are using and that I can easily use to give descriptive schematics.  In a perfect world, and I know this is a case of me wishing, it would be:   Free. Extremely easy to draw schematics in. Allows simple production of waveforms for the inputs/outputs.


I am triying to write my thesis in Latex. I want to create a title page. In my title page i have some space with the texts. I tried to give it with \vspace but it works between text not from top of the page. Can you suppose me a way to do this?

I'd like to, first thing before writing anything on a page, add some vertical space.  Minimal (non)working example:  \documentclass{article} \begin{document}  \vspace{250px} I want to be further down on the page :(  \vspace{250px} Haha! Sucker :D \end{document}     How can I get the first \vspace to work, just like the second one?


If $\Omega$ be a bounded open subset  of $C$, and $\phi :\Omega\to\Omega$ be a holomorphic function satisfied $f(z_0)=z_0,f'(z_0)=1$ then prove that $\phi$ is linear.   The hint told me to let $z_0=0$, and $\phi(z)=z+Cz^n+O(z^{n+1})$,  consider $\phi_k=\phi\circ\phi\circ...\circ\phi$ (k times), then prove $\phi_k(z)=z+kCz^n+O(z^{n+1})$, use Cauchy's inequality  and let $k\to\infty$ to get $C=0$.   I don't know how to prove  $\phi_k(z)=z+kCz^n+O(z^{n+1})$  and here is where i have trouble: assume that in $B(0,r_1)$ we have power series expansion of $\phi$: $\phi(z)=z+Cz^n+O(z^{n+1})$ to make sure we have composite $\phi(\phi)$, we need to restrict $z$ in $D=\{z\in B(0,r_1):\phi(z)\in B(0,r_1)\}$ and assume the maximal disc of $\phi(\phi)$ that has a power series expansion in $D$ is $B(0,r_2)$, then only in this disc we can have $\phi_2=z+2Cz^n+O(z^{n+1})$, next consider $\phi_3$ and so on.  Now we have a series of disc $B_k=B(0,r_k)$ that only in $B_k$ we have the power series expansion $\phi_k(z)=z+kCz^n+O(z^{n+1})$, so what if $r_n$ tend to zero?   We can't let $k\to \infty$.

This is an exercise in complex analysis:     Let $\Omega\subset{\Bbb C}$ be open and bounded, and $\varphi:\Omega\to\Omega$ a holomorphic function. Prove that if there exists a point $z_0\in\Omega$ such that   $$ \varphi(z_0)=z_0\qquad\text{and   }\qquad \varphi'(z_0)=1 $$   then $\varphi$ is linear.   I'm trying work out the case $z_0=0$ first, in which $$ \varphi(z)=z+\sum_{n=2}^{\infty}a_nz^2. $$ It suffices to show that $a_n=0$ for all $n\geq 2$. If let $$ \varphi(z)=z+a_2z^2+O(z^3) $$ then  $$ \varphi^k(0)=z+ka_2z^2+O(z^3), $$ and $$ \varphi^k(0)=0,\quad (\varphi^k)'(0)=1. $$ If one can show that $\{ka_2\}_{k=1}^{\infty}$ is uniformly bounded, then one at least has $a_2=0$. But I don't know how to go on. Any idea?


I've googled this but some of the questions seem a bit of of date. I'll ask a fresh one!  We would like to translate some of our webpages into Japanese. We want to have a MySQL table with 2 columns, English and Japanese.  I want to output the Japanese rows if a certain condition is met (such as a querystring value).  To test this quickly I have pasted some Japanese words into an HTML file and tried to preview in the browser but all I get is question marks.  町/市* - seems to be working on StackOverflow!  In my header is:  <meta charset="utf-8">   I was under the impression this was what I was meant to use for them to display correctly. Where am I going wrong?

I'm setting up a new server and want to support UTF-8 fully in my web application. I have tried this in the past on existing servers and always seem to end up having to fall back to ISO-8859-1.  Where exactly do I need to set the encoding/charsets? I'm aware that I need to configure Apache, MySQL, and PHP to do this — is there some standard checklist I can follow, or perhaps troubleshoot where the mismatches occur?  This is for a new Linux server, running MySQL 5, PHP, 5 and Apache 2.


I'm try to install ubuntu 14.04 Lts distribution in a machine Dell precision T1700 with Windows 8.1..The  LiveCD shows correctly the options of grub menu but, when I selected any option for installation, the error "ACPI PCC probe failed" appers and the screen keeps black finishing the process.  I already tried with other proposed solutions on tje dame error as ACPI disable motion in the grub menu and modify the graphics options multi-screen on the motherboard. The machine has all the draivers installed and also has a graphics card NVidia quadro K2200. Gread maid who can help me with this difficulty.

ACPI PCC Probe Failed is the error that pops up instead of the default GRUB loader after I updated my kernel to 3.19.  I searched around and couldn't find much information except that it is a known error and it is registered in kernel.org's launchpad . My question is, is anyone else facing it?, what are the impacts of this? And if only an update in the future can fix it, any suggestions or safety tips till then?


I couldn't find the notation for the external set of all positive unlimited real numbers.  It's like the symbol of infinity with a slash over it  I've tried \ninfty in fashion of \nequal  but it didn't work  you can find this notation in this book

My aim is to construct a relational symbol which is graphically equivalent the standard infinity sign (coded by\infty) crossed exactly in the middle of it. Yet I am not able to overcome something. To be more precise:    I would like to have slash sign moved exactly to the middle position of the infinity sign. The above was defined in the following way:  \documentclass{standalone} \begin{document} \newlength{\length} \settowidth{\length}{$\infty$} \newcommand\ninfty{\mathrel{\infty\hspace{-0.5\length/}}}   $\ninfty$ is not what I am aiming at.  \end{document}   So as you can see my idea was to define the length equal to the length of the infinity sign and then place slash sigh it halfway through it. As it can be seen above it was rather please quarter-way (backwards).  Could you please explain what I did wrong and how can I exactly determine the middle point (in terms of width) of a given mathematical glyph? Or a glyph in general?


I would appreciate your help in figuring out the correct phrasing / verb|adverb placement for the following phrase(s):   "Her talents will be best expressed..."   vs.   "Her talents will best be expressed..."  Are each of these phrases correct and comparable in meaning? Or is one more grammatically correct?  -Thanks!

In our application, when one deletes a room, a notification message pops up. It says "Your room has been deleted successfully". I have been taught that adverbs are often supposed to precede the verb, as in "Your room has been successfully deleted". However the original version with the adverb at the end does sound somehow appropriate to me, although I can't tell why.   Your room has been deleted successfully.    Your room has been successfully deleted.   Which version is preferable? Are they both grammatical?


Recently I saw a lot of marketing clips built like .  Mostly I'm talking about the words that jumps and become vertical/horizontal etc. (I'm not talking about the pictures, only the words)  I thought that it's probably complicated, but I saw it a lot so I thought perhaps there is some easy tool for that.  Am I right?  Thanks a lot!

I'm attempting to find motion graphics software to visually explain complex topics.  Here's an example of what I'm after:   I've looked at Houdini, After Effects, Boris, etc., but all of those products seem to be overkill for what I'm trying to do.   I'm just looking for a few standard shapes/images (and ability to import my own images) that I can move around and voice over.  (I'm using a PC)


I am sure there is something wrong i am doing but i just want someone to point it to me.  Why can't we say that $e^{\frac {\pi i}{3}} = \left(e^{\pi i}\right)^{\frac {1}{3}} = (-1)^{\frac {1}{3}} = -1$ Why do we have to calculate it by the formula as it is (which produces different value)  Another question is why can't we say that $ (-1)^{\frac {1}{2}} = \left(\left (-1\right)^{2}\right)^{\frac {1}{4}} = 1^{\frac {1}{4}} = 1$  Again i have to say that i know this is all wrong i just want to know why it is wrong


“I, Joe Smith, a Director of ABC Company, hereby swear and affirm that the original ‘Unanimous Consent of the Board of Directors By Way of Written Resolution of ABC Company, Passed on February 13, 2000 by circulation’ was presented to the below Notary Public for the State of Texas on ______________ , 20.  The copy attached is a true and correct copy of the original ‘Unanimous Consent of the Board of Directors By Way of Written Resolution of ABC Company, Passed on February 13, 2000 by circulation.’”

I have been/am being taught that end punctuation should always go inside quotes. For example, you are supposed to write:     Marvin thought it was "awful."   The problem is I do not see how does this make sense. Intuitively, I always wrote:     Marvin thought it was "awful".   as that makes more logical sense — you want a quote to be an exact replication of what somebody else said, so why should you add punctuation inside?  I always thought it made more sense to not touch the quote and add anything after or before if it must be added.  So, why should I put end punctuation inside quotes?


I was trying several method but none of them worked particularly. Also as this is a finite sum I thought of doing mathematical induction too. But it seems like there isn't a general term. I saw in a book that if you take the infinite series then there is the following relationship$$\sum\limits_{r=1}^{\infty}\frac{2r+3}{3^r(r+1)}=-3\left(-1+2\log\left(\frac{3}{2}\right)-\log\left(\frac{27}{8}\right)\right)$$ But I really cannot see how they obtained that either Appreciate your help

Find $$ \sum_{r=1}^n \frac{2r+3}{3^r (r+1)} $$   I don't understand how to split this fraction into partial fractions which I need to do to complete the rest of the question


Is it possible to see which posts I've downvoted?   I mean, I can see what is giving me reputation, and which questions I'm upvoting, but I couldn't find anywhere if we can see the downvotes.

The other day I was trying to find a question that I knew I had voted on. I remembered enough about it to search for it, and of course I could see once I got there that I had in fact voted on it, but I was wondering if there is any way to see all such questions. Like from my profile I can see what comments I've made, etc.


Evaluate the following indefinite integral. $$ \int {(\tan x)}^{\frac{1}{3}} dx $$ How could i do this integral ?  I tried multiplying the numerator by ${(\sec x)}^2$ and denominator by $1+{(\tan x)}^2$ but getting stuck after putting $\tan x=t$.

I know $\int \tan^{1/3}\theta  d \theta$ is a non integrable but wolfram alpha does it easily. And can anyone explain how a fuction is non integrable? My argument is that there has to be a function which represents the area under a graph. It's just that we do not know it. Please shed a bit light on it. Also, how can we solve $\int \tan^{1/3}\theta  d \theta$?


I am using an object's 3D position to apply a diffuse colour. I have a selection of objects that are different Z positions and hence have the colour ramp applied to them. As objects merge back into their final places I would like them to retain their colours.  The position of the plates when exploded will not be as they are now so I cannot just simply reduce the scale of the ramp when they contract back in. I need them to "remember" the value of their Z displacement for the colour ramp to use that data.  Basically I need a node (or some way to do it with what I have) that can be key framed to "freeze" it's output value.  H

I made a simple material that uses the 'object info' location to drive the color of the object. But when i animate my object now it, obviously, changes color while the objects moves through space.  How can i set the color to a fixed value using the INITIAL location of the Object info node?   Tnx RDG


I was bored today so i thought i would go back to learning to code in Java. I thought of making a easy math quiz program where you have two random numbers that you need to multiply with each other.  import java.io.BufferedReader; import java.io.IOException; import java.io.InputStreamReader; import java.util.Random;  public class Main {   public static void main(String args[]) throws IOException{     Random random = new Random();     int n1 = random.nextInt(30) + 5;     int n2 = random.nextInt(40) + 5;     int realanswer = n1 * n2;     String realanswer2 = Integer.toString(realanswer);     BufferedReader br = new BufferedReader(new InputStreamReader(System.in));     System.out.println("Answer these math questions:");     System.out.println("");     System.out.println(Integer.toString(n1) + " * " + Integer.toString(n2));     String answer = br.readLine();     if(answer == realanswer2){         System.out.println("CORRECT");     }else{         System.out.println("WRONG");         System.out.println("Answer is " + realanswer2);     }  }  }   My problem is that even if i write the right answer it says its wrong. I think the problem lies in the transition from int to String.  Any help appreciated.

I've been using the == operator in my program to compare all my strings so far. However, I ran into a bug, changed one of them into .equals() instead, and it fixed the bug.  Is == bad? When should it and should it not be used? What's the difference?


Suppose there exist two blocks A and B connected by a massless, inextensible cord. Both A and B have equal masses and are acted upon by equal forces in opposite directions. So, is the tension in the string the sum of the two forces, the resultant of the two forces or something else?        <-F--  A _________ CORD__________B   --F->

I'm going to provide my argument for why I think the tension in a rope should be twice the force exerted on either side of it.  First, let's consider a different example. Say, there is a person named A and a block in space. A pushes on the block with a force of 100 N. Then, the block will also push A with a force of 100 N by Newton's third law. Now, consider the case where instead of the block, there is a person B who is also pushing on A with a force of 100 N while A is pushing on him. A will experience a force of 100 N because he pushed on B, AND another 100 N because he is being pushed by B. Hence he will experience a force of 200 N. Similarly, B also experiences 200 N of force.  Now, back to the original problem. There are two people A and B in space with a taut rope (no tension currently) in between them. If only A is pulling and B is not, then I agree that the tension is equal to the force A exerts. This situation (in my opinion) becomes analogous to the above if B is also pulling. So, say both of them pull from either side with a force of 100 N. Then the rope at the end of B will pull B with a force of 100 N (this pull is caused by A). By Newton's third law, the rope will experience a pull of 100 N. But B is also pulling his end of the rope with 100 N. Therefore, the tension should be 200 N. Similarly, the end of the rope at A must pull A with 100 N of force (because B is pulling from the other side) and hence experience a force of 100 N itself by Newton's third law plus another 100 N because A is pulling on the rope.  Apparently, the answer is not this (according to my searching on the web). So, could anyone tell me why this reasoning is wrong? Thanks.  EDIT : So apparently people don't agree with my first example, leave alone the second. This is to the downvoters and the upvoters of the highest-rated answer: You all agree that if only A pushes B with a force of 100 N, then A and B both will get pushed by a force of 100 N in opposite directions, right? Then, in the case where B is also pushing with a force of 100 N, it doesn't make sense that the answer would be exactly the same. It doesn't seem right that no matter what B does, B and A will always experience the same force as they would have if B hadn't applied any force.  EDIT 2 : I'm going to provide here a link to a question that I posted:  According to the answer and the comments there, the reason as to why my first example is incorrect is different to the one provided here. So maybe you should all read the answer and the comments provided by the person and reconsider what you think.


As the title says, I want to change the "Show Applications" Grid Icon in ubuntu dock (17.10) with Ubuntu Icon like in Unity 8.I have found a similar question but the answer has no result. Thanks in advance.

I'm using Ubuntu with GNOME desktop.   I was wondering how to change the "Show Applications" button icon on the dash?


I've got absolutely no idea what this does and if I search something like "linux ./ command", Google just seems to ignore those two characters as they're puntuation. If someone can give me a simple answer, I'd be very grateful :)

Why do we use ./filename to execute a file in linux?  Why not just enter it like other commands gcc, ls etc...


I'm sure this must have been asked and answered many times, but I just can't seem to think of the right words for my search. I want to reference the following diagram, which is labeled '3.16.1', but I only get '3.16' in the reference. How can I control the number of 'levels'?

\documentclass{article} \usepackage{algorithm}  \begin{document} \setcounter{section}{54} \section{A section with no label} \begin{algorithm} \label{myalg} \caption{An algorithm with the label `myalg'.} \end{algorithm} Reference to myalg: \ref{myalg}. \end{document}   With this code, the reference givens 55 (the section number); if I move the label after the caption command, I get the correct reference (1).  Why should this be the case? I find it confusing: when thinking about an algorithm as an environment, you expect to be able to label it right after you began it. At the very least I would expect some kind of error or "??" to appear, rather than the previous environment or reference-able entity being used.


Is it true that when $n \geq 5$ there is a surjective homomorphism from the symmetric group $S_n$ to $S_{n-1}$? How come this is so? Does it have to deal with the subgroup $A_n$ being simple?

This has always been one of my most favorite exercises from Group Theory, and I was surprised to see that this hasn't been asked before. To repeat:     Characterize the natural numbers $n$ such that there is a surjective group homomorphism from $S_n$ to $S_{n-1}$.   I have a solution which I will post in a couple days (if someone doesn't recreate it), but I am more interested in seeing how other people would approach this problem. I am very interested in alternate proofs of this characterization.


The spells , , and  all allow you to make ranged attacks on subsequent rounds.  Wall of Light Until the spell ends, you can use an action to launch a beam of radiance from the wall at one creature you can see within 60 feet of it. Make a ranged spell attack. On a hit, the target takes 4d8 radiant damage.   Storm Sphere Until the spell ends, you can use a bonus action on each of your turns to cause a bolt of lightning to leap from the center of the sphere toward one creature you choose within 60 feet of the center. Make a ranged spell attack. You have advantage on the attack roll if the target is in the sphere. On a hit, the target takes 4d6 lightning damage.   Crown of Stars You can use a bonus action to send one of the motes streaking toward one creature or object within 120 feet of you. When you do so, make a ranged spell attack. On a hit, the target takes 4d12 radiant damage.  The spell sniper feat says:  When you cast a spell that requires you to make an attack roll, the spell’s range is doubled.  To use all of the spells effects, you are required to make an attack roll. That is, they have functions that don't work without a attack roll. Is that reading correct, and if so which of the ranges in each spell are effected?

Spell Sniper says:     When you cast a spell that requires you to make an attack roll, the    spell’s range is doubled.   Can I apply that to the part of Storm Sphere that says:     Until the spell ends, you can use a bonus action on each of your turns   to cause a bolt of lightning to leap from the center of the sphere   toward one creature you choose within 60 feet of the center. Make a   ranged spell attack. You have advantage on the attack roll if the   target is in the sphere. On a hit, the target takes 4d6 lightning   damage.   The spell itself does not require the caster to make an attack roll. But there is the option to make a spell attack. I'm not sure if the Spell Sniper feat could be applied here.  Would the lightning attacks of the Storm Sphere benefit from the Spell Sniper feat?


I have a table with 3 rows added by myself in html. i have a button "Add New Row" which appends a new row with "Edit" button inside last "td". now when i try to click on this newly generated "Edit" button, my script is not getting called. i don't understand what is wrong with my code snippet. I read about jquery "On" method it supposed to be able to generate an event but i don't know why it is not generating event in my code.    $(document).ready(function() {    $(".row").click(function() {       var markup = "
			<button id='myNewRow' class='btn btn-sm btn-mdb-color btn-rounded editbtn' >edit</button>
";       $("table").append(markup);    }); });  $(document).ready(function() {   $('.editbtn').on('click', function() {     console.log('sulabh')      var currentTD = $(this).parents('tr').find('td');     if ($(this).html() == 'Edit') {       $.each(currentTD, function() {         $(this).prop('contenteditable', true)       });     } else {       $.each(currentTD, function() {         $(this).prop('contenteditable', false)       });     }      $(this).html($(this).html() == 'Edit' ? 'Save' : 'Edit')    });  });

       Name       	       Position       	       Office

       Tiger Nixon	       System Architect	       Edinburgh	       <button id="edt" class="btn btn-sm btn-mdb-color btn-rounded editbtn">edit</button>

       Garrett Winters	       Accountant	       Tokyo	       <button id="edt" class="btn btn-sm btn-mdb-color btn-rounded editbtn">edit</button>

       Ashton Cox	       Junior Technical Author	       San Francisco	       <button id="edt" class="btn btn-sm btn-mdb-color btn-rounded editbtn">edit</button>

       Cedric Kelly	       Senior Javascript Developer	       Edinburgh	       <button id="edt" class="btn btn-sm btn-mdb-color btn-rounded editbtn">edit</button>

       Name       	       Position       	       Office

I have a bit of code where I am looping through all the select boxes on a page and binding a .hover event to them to do a bit of twiddling with their width on mouse on/off.  This happens on page ready and works just fine.  The problem I have is that any select boxes I add via Ajax or DOM after the initial loop won't have the event bound.  I have found this plugin (), but before I add another 5k to my pages with a plugin, I want to see if anyone knows a way to do this, either with jQuery directly or by another option.


I realize that this is a purely math question, which however has arisen in a physics computation. The reason to post it here is that I want a fast dirty answer. Not something cluttered with definitions. If the mods think that it isn't fit for the site feel free to remove it.  I am computing some derivatives of the matrix $U$ in chiral perturbation theory  $$U=e^{it^a\phi_a}$$  where $t^a$ are the Pauli matrices and $\phi_a$ are complex scalar fields. I want to take a derivative of this, for example  $$\partial_{\mu}U=\partial_{\mu}e^{it^a\phi_a}$$  since the exponent contains a matrix a am uncertain when applying the chain rule. Is the proper way to take the derivative  $$e^{it^a\phi_a}\partial_{\mu}(it^b\phi_b)$$  or  $$\partial_{\mu}(it^b\phi_b)e^{it^a\phi_a}$$

I'm asked to show that  $$\frac{d(\hat{A}\hat{B})}{d\lambda} ~=~ \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$$ With $\lambda$ a continuous parameter. Should I use the definition $$\frac{d\hat{A}}{d\lambda} ~=~ \lim_{\epsilon \to 0} \frac{\hat{A}(\lambda + \epsilon) - \hat{A}(\lambda)}{\epsilon}$$ applied to $\hat{A}\hat{B}$ like $$\frac{d(\hat{A}\hat{B})}{d\lambda} ~=~ \lim_{\epsilon \to 0} \frac{\hat{A}(\lambda + \epsilon)\hat{B}(\lambda + \epsilon) - \hat{A}(\lambda)\hat{B}(\lambda)}{\epsilon}$$ and do some algebra to get the RHS of the first equation, or I'm missing something? Another interesting derivative to pay attention to is: $$\frac{d}{d\lambda}\exp(\hat{A}(\lambda) )~?$$


I want to write down a sentence to express the fact that from the 1980's on computers got applied in many facets of our daily lives ... and I am wondering what sentence would be correct:  "As computing got more and more ubiquitous in people's daily lives, ..."  or  "As computing became more and more ubiquitous in people's daily lives, ..."  EDIT (CONCLUSION):  Thank you for all the comments! I have rebuilt the sentence as shown below:  "As computing started to become ubiquitous, affecting more and more aspects of people's daily lives, the demand for reliable programs that are proven to be correct increased"

Is the use of "get + adjective/participle" appropriate for formal writing (for example, scientific papers)?  I am thinking of usages analogous to    get fat  get inflated get sick   where the meaning is "become".  What about cases where the meaning is "make (sg)", such as    get (it) done get (her) dressed   Not being a native speaker, I cannot easily judge the level of formality of these constructs.


I'm doing a table using tabular and I am trying to write "ATP->AMP" but I can't quite figure out how to insert the arrow. I've looked and searched and I haven't found anything that can help.  Thanks in advance.

I know what my symbol or character looks like, but I don't know what the command is or which math alphabet it came from. How do I go about finding this out?


I've changed my password for SE and am trying to learn it. So I want to be logged out of SE when I close my browser or turn off the computer or similar, so that I learn the password by typing it lots. But when I log out and try to log back in, it logs me straight in without asking me for a password. Don't get me wrong, any other time that's a useful feature, but is there a way I can disable it, and a setting I can change so that I have to relog every time I leave my browser, or similar? I'm using Chrome, I cleared out the saved passwords and unticked "Offer to save your web passwords" in Chrome settings, but my password is still saved for SE.  I don't know enough about scripts to use a programmatic solution, so if there is no way I can do this through either site settings or Chrome settings, I'll grin and bear it.

The cookies for subdomain, 'openid.stackexchange.com' are not, as claimed, actually removed from the browser upon logging out of stackexchange.com, but they should be.  Applies to: * stackexchange.com, date: January 8, 2012  Desired result:   Upon logging out of Stack Exchange, Stack Exchange OpenID local credentials stored 'in your browser' (or stored anywhere on the user's computer) should actually be deleted, as promised. For example, Stack Exchange OpenID browser cookies actually should be deleted. Logging in again should require explicitly entering one's credentials again. This might be email address and password. It is misleading to claim that (all relevant) 'local credentials in your browser' are deleted, since anyone else sitting down at the same computer can merely click the login links and immediately be accepted as the same, previous user! (See procedure to duplicate, below.)   Procedure to duplicate the problem in FireFox:  Versions (Menu-Help-Troubleshooting Information-Application Basics-User Agent): * Mozilla/5.0 (Windows NT 5.1; rv:9.0.1) Gecko/20100101 Firefox/9.0.1   Start FireFox. In Firefox's cookies manager (Menu-Tools-Options-Privacy), click the button, 'Exceptions', enter stackexchange.com and stackoverflow.com, and each time, click the button, 'Allow for Session'. Click the button, 'Close'. Click the button, 'Show Cookies'. Type, 'Stack', select stackexchange.com and stackoverflow.com if they are there, and each time, click the button, 'Remove Cookies'. Close Firefox. Start Firefox. Close webmail notifier windows. Log out of Gmail. Wait ten seconds. Click the back button on the browser to make sure one is not still logged in to Gmail. Open a tab for,  . In , if you have it, allow the domains, 'stackexchange.com' and, 'sstatic.net' and refresh. Click the link at the top, 'login'. On the resulting page, click the box, 'Log in with Stack Exchange.' Enter one's Gmail email address and Stack Exchange password, which results in a successful Stack Exchange login. Click the link at the top, 'logout'. The resulting web page has an orange button labeled, 'logout', above which is presented the text, 'Clicking Log Out will clear all local credentials in your browser.' Click the orange button labeled, 'logout'.   One might think that one has completely logged out safely from Stack Exchange, with no possibility of simply clicking back in. But, not true!   Now the page has the link at the top, 'login'. However, the Firefox Cookie Manager still shows cookies under domains, 'stackexchange.com' and 'openid.stackexchange.com'. Click the link at the top, 'login'. On the resulting page, again click the box, 'Log in with Stack Exchange.' See the login unexpectedly succeed, without entering any email address or Stack Exchange password!


i am having lenovo g505s laptop with windows7(not preloaded),i want to install ubuntu alongside with windows 7, but it is not showing install alongside option, how do i get that? it is not detecting my os. i went from bios menu, theres no option like disable secure boot, boot mode :Legacy            UEFI  i have tried enabling both alternately still didnt worked . Need Help,  I Love Ubuntu ;

Whenever I startup from the LinuxLive USB, and try to install Ubuntu 12.04, it only gives me two options instead of the usual 3 or 4. My options are "Replace Windows 7 with Ubuntu" and "Something Else". Whenever I go into "Something else", it looks extremely confusing. How do I install it side-by-side while using the "Something else" section? These are my partitions.   /dev/sda /dev/sda1 ntfs (104Mb, System Required Partition) /dev/sda2 ntfs (1000097Mb, Windows 7 Partition)   Or should I try reinstalling my installer on a CD rather than USB and maybe then it will have the "Install alongside" function?


I'm tormented by this apparently simple question: If you toss a fair coin $7$ times in a row, what is the probability of getting an even number of heads? (please note: this is self-study and not a homework problem) I've seen a lot of questions here on "even number of heads" but they deal with biased coins, whereas I can't wrap my head around the seemingly simple problem stated above.  When I encountered this problem I thought like this: The possible scenarios where we can get an even number of heads are:   $2$ H, $5$ T $4$ H, $3$ T $6$ H, $1$ T   All we then need to do is add up the number of ways we can achieve these three outcomes, and divide by the total number of possibilities, which works out to be $\frac{\binom{7}{2} + \binom{7}{3} + \binom{7}{6}}{2^7} = \frac{63}{2^7} = 0.4921875$.  This is tantalizingly close to the answer given: $1/2$, but is not quite $1/2$.  I'd like two things here:   A simple explanation for why the final probability is exactly $1/2$ (I'm still in the early phases of basic probability, so wouldn't like an explanation that dives into length formulas or probability distributions). Why the reasoning I followed doesn't quite work.

A coin is tossed $n$ times. What is the probability of getting odd number of heads?   I started this chapter sometimes ago and faced in front of a tough problem. At first I started considering cases. Case-I : The probability of getting 1 head.Case-II : The probability of getting 3 head and so on. But there are many cases. So how can I solved this . Please help me. Thank you!


The humans became ringwraithes, because they were easily corruptible, I think.  The elves didn't because they made their own rings. Or at least I'm assuming that's the reason they didn't. As far as I know, Sauron actually never came into contact with the elve's rings so he could never corrupt them.  Why didn't the dwarves become ringwraithes?  Is it because dwarves are more resistant to magic then the other races? That's the case in the Warhammer universe, I'm not sure if it's the same in Middle Earth.

I was reading through some of the questions and answers here on the Rings of Power and I just wondered why the ring effects, affected different races in a slightly different way.  I know other questions ask and answer some of these things, but I just wondered why the Rings of Power caused greed in Dwarves for ever more amounts of treasure.  It's true that Dwarves loved mining and loved wealth and were built of sturdier stuff than men, but why did the Rings of Power not control Dwarves in the way the other Rings of Power controlled men and why did it produce greed and not invisibility?


In Skyrim on Xbox 360, I told my follower (a dark brotherhood initiate) to wait at the foot of the staircase in Folgunthur during the quest to get the fragment, and when I came back to get her, she was nowhere to be found. I've even waited for three days and haven't gotten a message that she got tired of waiting and returned home. Is there anything else I can do?

I had parked my follower at an inn with a bunch of valuable items that I didn't want to carry. I guess I have been out a bit too long as this message appears:      Your follower tires of waiting and leaves.    There have been other situations where my follower has mysteriously disappeared as well.  How can I find my follower when they get up and leave or disappear into thin air?


I was looking at the Iron Titan, and Tango said that all the machine has to be built at least 70 blocks from any other doors. I was wondering if it would be feasible to build it at say Y=120, and still be able to place doors on the ground.

In Minecraft, to avoid villagers getting confused with breeding mechanics, etc., any group of villagers (i.e. Iron Farm, Villager Spawner etc.) need to be placed >64 blocks away.   Does this apply vertically, or is it to do with Chunk mechanics?


I built , and I am having trouble with SEO for individual posts.  I implemented JSON-LD and have all necessary meta tags like (title, description, canonical url, og, twitter ...) for every page.  However, most posts are not showing up on google search.  For example,  does not show up on google search, even if I search with its h1 title "We now support multiple post drafts". See .    If you look at the search results, it shows a facebook post and pagination page on our site, but not the post itself.  What am I missing on my post page?  Should I have a sitemap for every post? If so, how do you keep track of sitemap if there are 10,000+ posts?  Sitemap:   robots.txt:

This is a general, community wiki  question and answer pair intended to address any questions   concerning the reasons a site or specific site contents do not appear   in search engine results.      If your question was closed as a duplicate of this question and you   feel that the information provided here does not provide a sufficient   answer, please open a discussion on .   My site (or specific pages on my site) is not appearing in search engine results.  Why isn't my content indexed and what can I do about it?


Find all functions $f:\Bbb R${0, 1} $\to \Bbb R$, such that $$f(x) + 2f(\frac{1}{x}) + 3f(\frac{x}{x-1}) = x$$ I first tried to substitute $x$ with $\frac{1}{x}$ which gave me, $$f(\frac{1}{x}) + 2f(x) + 3f(\frac{1}{1-x}) = \frac{1}{x}$$ Also by substituting $x$ with $\frac{x}{x-1}$ I got, $$f(\frac{x}{x-1})+ 2f(\frac{x-1}{x}) + 3f(x) = \frac{x}{x-1}$$ But this made things more complex and I could not reach my answer also I tried moving one of the terms to the right and then solving but this also led to nowhere. I am trying for a solution based on substitution. Can anyone show me the way?

Find all function $f:\mathbb{R}-\{0,1\}$ in $$f(x)+2f\left(\frac{1}{x}\right)+3f\left(\frac{x}{x-1}\right)=x$$   Attempt: put $\displaystyle x = \frac{1}{x}$, then $$f\left(\frac{1}{x}\right)+2f(x)+3f\left(\frac{1}{1-x}\right) = \frac{1}{x}$$  could some help me how to solve it , thanks


I know that dark matter rarely interacts with itself and with ordinary matter. but black holes produce a gravitational field strong enough to curve and drag space-time so shouldn't dark matter also get sucked in being a part of space-time.

I'm assuming that the answer to the question in the title is a resounding yes. Since Baryonic matter and dark matter interact via gravitational forces.  If this is the case how is information not lost if, as is presumed, dark matter does not interact via the electromagnetic force?


I'm attempting to import some data to my Google Sheet.  Using QUERY to reshape my data, an example code of what I am trying to import is as follows:  =query(importhtml("http://statfox.com/cbb/cbbteam.asp?teamid=DUKE&season=2019&log=1",  "table",8),"Select * where Col2 is not null order by Col1")   Unfortunately, in many cases where a team is an underdog and the line should be +#, the query is not returning the value.  If I remove the QUERY and use the following importhtml all values show up correctly:  =importhtml("http://statfox.com/cbb/cbbteam.asp?teamid=DUKE&season=2019&log=1","table",8)   Any insight into the matter or ideas on what might be causing this or solutions would be greatly appreciated.

I'm not sure what is happening in my Google Sheet query and could really use some help. I have  that demonstrates and explains the issue.   Basically, I'm using a query to pull data from a Google Spreadsheet that is populated by a Google Form. If the form submits only numbers (100), there is no problem. However, if someone submits a string ($100 or 100 dollars), the query will not display the string. It will only display form submissions that are 100% numbers.   Also, if the 1st submission is a string (not 100% numbers) the query will work.   Here is my query  =query(data!B2:CL,"select B, C, D, E, F, G, H, I, J, CI, BZ, CA, BA  where J = 'Industry Certification'",false)   Any ideas on why this is happening?  Also, I'm aware that I can limit the form to numbers only. I'm trying to avoid that and learn 'why' this is happening.


I'm trying to solve this integral by using complex analysis, but don't know clearly what to do next. Here is my approach:  $$\int_{0}^{2\pi}\frac{\cos(x)}{2+\cos(x)}dx = \int_{0}^{2\pi}\frac{\cos(z)}{2+\cos(z)}dz = \int_{0}^{2\pi}\frac{\cos(e^{ix})}{2+\cos(e^{ix})}dx$$

Calculate the integral for $\left|a\right|<1$  $$\int_{-\pi}^{\pi} \dfrac{\cos(n\theta)}{1-2a\cos(\theta)+a^2}d\theta$$  I'm supposed to evaluate this using method of residues, but the parameter a is confusing me. I'm not even sure what contour I should use. Any help would be very appreciated.


I would like to know how to remove particular key value pair in nested array object. But I need to get all the object. how to do in javascript.  In the below obj, remove the mon key value pair and fetch the obj in javascript    var result = getObj(obj, "mon"); getObj(arr, month){    return arr.filter(element=>         if (element != month){             return element         }     ); }    var obj =[   {id: 1, mon: "Dec", tot: 1000},   {id: 2, mon: "tues", tot: 2000} ]   Expected Output:  [   {id: 1, tot: 1000},   {id: 2, tot: 2000} ]

Say I create an object as follows: let myObject = {   "ircEvent": "PRIVMSG",   "method": "newURI",   "regex": "^http://.*" };  How should I remove the property regex to end up with new myObject as follows? let myObject = {   "ircEvent": "PRIVMSG",   "method": "newURI" };


I was watching a  on Banach-Tarski, which has a preamble section about Cantor's diagonalization argument and Hilbert's Hotel. My question is about this preamble material. At c. 04:30 ff., the author presents Cantor's argument as follows. Consider numbering off the natural numbers with real numbers in $\left(0,1\right)$, e.g. $$ \begin{array}{c|lcr} n \\ \hline 1 & 0.\color{red}50321642239817 \ldots \\ 2 & 0.0\color{red}7829136011205 \ldots \\ 3 & 0.31\color{red}11370055629 \ldots \\ 4 & 0.999\color{red}9261457682 \ldots \\ 5 & 0.0001\color{red}042507334 \ldots \\ \vdots & \vdots \end{array} $$ Then you could form a new real in $\left(0,1\right)$ not already in the list, e.g. $0.\color{red}{68281} \ldots$. Hence there are more reals than naturals. I have two questions about this:  Couldn't you run exactly the same argument (erroneously) for rational numbers in $\left(0,1\right)$? E.g. say I choose powers of $\frac{1}{2}$, giving:  $$ \begin{array}{c|lcr} n \\ \hline 1 & 0.\color{red}4999999999999 \ldots \\ 2 & 0.2\color{red}499999999999 \ldots \\ 3 & 0.12\color{red}49999999999 \ldots \\ 4 & 0.062\color{red}4999999999 \ldots \\ 5 & 0.0312\color{red}499999999 \ldots \\ \vdots & \vdots \end{array} $$ So $0.\color{red}{55555} \ldots$ is not in the list, suggesting that the cardinality of the rationals is greater than that of the naturals. But a different argument shows that their cardinalities are the same. So there seems to be something wrong with the diagonal argument itself?  As a separate objection, going back to the original example, couldn't the new, diagonalized entry, $0.68281 \ldots$, be treated as a new "guest" in Hilbert's Hotel, as the author later puts it (c. 06:50 ff.), and all entries in column 2 moved down one row, creating room?  Admittedly, you could diagonalize this expanded list again; but then you could also move the guests down again. So the argument does not seem to show that there's any fundamental problem, i.e. that you can't continue pairing off the reals with the naturals forever?

If we map every integer to a string that represents a rational number, and produce a number different from all the ones listed, we are essentially following Cantor's algorithm. But why does it not apply? Is it because we can't be certain that the number produced is a rational number?


TRUNCATE and DELETE commands does the same job, in both the cases data is manipulated, then why does the DELETE command comes under DML commands and TRUNCATE command comes under DDL commands?

What's the difference between TRUNCATE and DELETE in SQL?  If your answer is platform specific, please indicate that.


I have this really intelligent pregnant women stuck in the living quaters... I can't make her work  Anyone have any Solutions to this problem?

I am playing Fallout Shelter, have 3 kids in my shelter, all going well, thought I would make one more kid with big strength numbers and decent agility/perception (kids take on SPECIAL stats from their parents). All the women who are pregnant work in my shelter, apart from this one. She can not me put to work, can not be moved from the living shelter, can not be moved to a job via the clipboard short cut.  TL;DR One of the NPC females who is pregnant can not be put to work, can not be moved from the living shelter room, she just walks about as if she is carrying a mammoth inside her. I have NOT moved the shelter, have not let the dad die, have not done anything at all.


I need to create combobox with Field Set names. I know how to get fields for any Field Set, but how to get a list of all Field Set names. Map<String, Schema.FieldSet> canMap = Schema.SObjectType.Candidate__c.fieldSets.getMap(); Maybe there are methods like getValue or getNames, or something like that?

String fieldSetName = 'Account_FieldSet'; String ObjectName = 'Account';   I wanted get list of all fields from this fieldset.  I know if fieldset name is hardcoded in the apex then following syntax works -   SObjectType.Account.FieldSets.Account_FieldSet.getFields();   But here main issue is Object name and fieldset name will come at runtime.


I understand that the Announcer badge can be earned if the question is visited 25 times.  But is this only when I share a link to this question in one of my answers, or can I also share the link (with my user id somewhere to count the accesses) somewhere else, maybe Facebook?

Formerly List of all badges with full descriptions.  What are badge name's requirements? Why didn't I get badge name ? Which badges can I earn multiple times?    Jump to:                    Note: Some badges are awarded based on score. The term score means the total number of upvotes minus the total number of downvotes.  Any badge with a *** next to it is one of 20 badges that count towards the  displayed on a moderator candidate in an election. The candidate score is a total of 40 points: the first 20 are awarded based on the user's rep divided by 1000 and rounded down; the second 20 represents the total number of unique badges earned of the 20 that count. Note that if the same badge is earned multiple times, it will only count once towards the candidate score.  Any badge with a "(retired)" next to it is no longer awarded, but is retained by users who previously earned them. See  for more info on what it means for a badge to be retired.       Visit the  of the  on any site to see a complete list of badges that you can filter by earned, unearned, or type.


If I navigate to my site, it shows in the browser that my ssl key is correct for my site hosted at WP Engine. It is showing and validating my certificate. However, if I connect with openssl, it get's the underlying servers ssl key.  openssl s_client -connect mysite.com:443  How do I get the SSL command to resolve the ssl key using the domain name instead of the ip address only.  This isn't a WP Engine issue. I have reproduced it on Amazon servers as well.

When querying Sparkfun's CDN url using OpenSSL with the following command:  openssl s_client -showcerts -connect dlnmh9ip6v2uc.cloudfront.net:443  The common name returned in the certificate is *.sparkfun.com, which fails to verify, but if you load the host in Chrome, the common name shown is *.cloudfront.net  What is going on here?  This is causing a problem because the environment I am in proxies SSL via Squid SSL_Bump, which generates a certificate signed by my locally trusted CA for the domain. This works for all domains but the above, as the CN does not match as the new cert is generated using OpenSSL.  EDIT - I have verified the same occurs with OpenSSL on a server in a remote data centre that has a direct connection to the internet with no proxies or filtering involved.  EDIT - The issue is due to SNI, as accepted, but to fill out the information as to why it causes a problem with Squid and SSL_Bump:     This project will not support forwarding of SSL Server Name Indication   (SNI) information to the origin server and will make such support a   little more difficult. However, SNI forwarding has its own serious   challenges (beyond the scope of this document) that far outweigh the   added forwarding difficulties.   Taken from:


Iḿ really satisfied with ubuntu, but for job necessities I need Windows too.  The problem for me appears when I change from ubuntu to windows7, the time is incorrect and take a lot of time to adjust itself using internet services.    Related with the time: Iĺl try to explain myself:  I am using linux, but meanwhile I need to use windows and I restart the system to boot into windows. Thereś no problem with this, but the time is incorrect.  That is:  When I am using linux, the time is correct, but if I restart to use windows, windows time is incorrect

Dual boot system Windows XP Pro and Ubuntu 12.04.  I have the bios set for the correct time and Ubuntu set for US Eastern time.  Ubuntu will boot up and the time will be off by -4 hours.  If I correct the time in Ubuntu then when I boot up in Windows XP the time will be off by + 4 hours.  This is a fresh install of 12.04. I did not have this problem before reinstalling Ubuntu.


Let's say I have an alias on my system which I often use to tell me my system information, the alias is systeminfo, and it executes lsb_release -a which outputs:   No LSB modules are available. Distributor ID: Ubuntu Description:    Ubuntu 15.04 Release:    15.04 Codename:   vivid   Now because I use this very often, and I then copy and paste the information into my AskUbuntu questions, it would be very useful if upon execution of the command, for the output to not only be printed to the Terminal window, but also be copied to my system clipboard, so that I can save time, and not need to select the text and then copy and paste that here. So I was wondering how that could be achieved? How do I make it so that not only is it printed on screen, but also copied to the system clipboard? Preferably, I would like this question to be answered with a script which asks me if I want the information to be copied to my system clipboard in addition to being displayed on screen, but I can of course write up that script, so it would just be for convenience and to save time.  Information Update:  I have tried these, but they don't work for me:

For example,   Say I want to list the contents of a folder and directly paste them into a chat window for a friend to see.    I realize I could do ls > filename.txt to create a file (filename.txt) with those contents; I'd then have to open or print the file and manually select and copy the text block (which can be annoying/tedious.)  I clearly could also select and copy the output of ls directly from within the terminal window.  It would be much faster/easier to simply pipe standard output to the clipboard.  What terminal command allows me to do this?


I am trying to reproduce it by my own but I can not understand how to switch off vertical edges in render options inside of freestyle svg exporter? Any thoughts?  Many thanks in advance!

On Blender , a picture is shown with two Suzannes made of vertical/horizontal lines.    Click to enlarge  How to reproduce this effect with Blender Freestyle? Using modifiers? Custom shaders?


I know how to concact two strings, but this one will be a bit more complicated, i guess it's a duplicate but i could not find it after an hour of search so....  Suppose i have a table like this :  ExempleTable ____________________   ID   |  TEXT      | ___________________ |  1     |  'hello '  |  2     |  'how are '|  3     |  'you ?'   |   I need a select that would return a single line containing :  'hello how are you ?'   I thought something like this would exist :  CONCATTEXT( SELECT text FROM ExampleTable) as sentence   But no... maybe it's a GROUP BY matter ? I can't figure it out...

How do I get:  id       Name       Value 1          A          4 1          B          8 2          C          9   to  id          Column 1          A:4, B:8 2          C:9


If $y^y=x$, can y be expressed as a function of x?  Specifically, I am finding the solution to a PDE where the most general solution is $u=t^{-\frac{1}{2}} f(x,t)$ and  $$\LARGE f^f=Ce^{\frac{-x}{2\sqrt{t}}} $$ Any help will be appreciated!

Given that   $x^x = y$; and  given some value for $y$   is there a way to expressly solve that equation for $x$?


I need my enum to return a formatted string to the view, for example:   public enum status {     NotStarted,    InProgress, }    return: Not Started and In Progress. How I do it? Thanks (language C#)

My enum consists of the following values:  private enum PublishStatusses{     NotCompleted,     Completed,     Error };   I want to be able to output these values in a user friendly way though. I don't need to be able to go from string to value again.


My Colombia visa is on my previous passport. Since I have run out of pages, I have obtained a new passport. Can I still travel with both passports and enter the country?

If your visa is on an expired passport but still valid, can you still use it with your new passport, or does the old passport have to have a recent expiration date? If so, how do you transfer your visa to your new passport?


Problem Evaluate $$\lim\limits_{n \to \infty}\frac{1!+2!+\cdots+n!}{n!}.$$ Solution 1 Applying Stolz theorem, $$\lim_{n \to \infty}\frac{1!+2!+\cdots+n!}{n!}=\lim_{n \to \infty}\frac{(n+1)!}{(n+1)!-n!}=\lim_{n \to \infty}\frac{n!(n+1)}{n!\cdot n}=\lim_{n \to \infty}\frac{n+1}{n}=1.$$ Solution 2 \begin{align*} 1=\frac{n!}{n!}\leq \frac{1!+2!+\cdots+n!}{n!}&\leq \frac{(n-2)!(n-2)+(n-1)!+n!}{n!}\\&=\frac{n-2}{n(n-1)}+\frac{1}{n}+1\to 1(n \to \infty). \end{align*} Applying the squeeze theorem, $$\lim\limits_{n \to \infty}\frac{1!+2!+\cdots+n!}{n!}=1.$$ Any more solutions? Thanks.

How to evaluate the following limit? $$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}$$  For this problem I have two methods. But I'd like to know if there are better methods.  My solution 1:  Using Stolz-Cesaro Theorem, we have $$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}=\lim_{n\to\infty}\dfrac{n!}{n!-(n-1)!}=\lim_{n\to\infty}\dfrac{n}{n-1}=1$$  My solution 2:  $$1=\dfrac{n!}{n!}<\dfrac{1!+2!+\cdots+n!}{n!}<\dfrac{(n-2)(n-2)!+(n-1)!+n!}{n!}=\dfrac{n-2}{n(n-1)}+\dfrac{1}{n}+1$$


Lets say i have a string "53.430996", I want to convert it to a string and store all the values after the floating point I tried: #include <iostream> #include <bits/stdc++.h> #include <string> #include <stdlib.h> using namespace std; int main(){    char* convertme="53.430996";    double converted=atof(convertme);    cout << converted;    return 0; }  But the output is 53.431 but i need 53.430996 so I can later cast it to a long long with static cast < long long > (coord ∗ 100000)  So, I can get the value without the floating point  5343099

In my  I was printing a double using cout that got rounded when I wasn't expecting it.  How can I make cout print a double using full precision?


I would like to know how I can plot the graph of the following function in Latex or Tikz.  $f(t)=\frac{\int_{-\infty}^{t}\alpha(s)ds}{\int_{-2}^{-1}\alpha(s)ds}$,   where   $\alpha: \mathbb{R} \rightarrow \mathbb{R}$    is defined by,    e^{\frac{1}{(s+1)(s+2)}} in [-2,-1]    and zero outside of this interval.

Which is the best way to put function plots into a LaTeX document?


In Bernoulli principle we say that as velocity increases the pressure of fluid decreases but it contradicts kinetic theory of gas which says that as velocity increase-pressure increases. And this seems intuitive also. Please explain

I know the Bernoulli equation that states that the energy along a streamline is constant:  $$ \frac{v^{2}}{2}+gh+\frac{p}{\rho}=C $$  However I can't seem to figure out what exactly $p$ is in this equation. In every book I look and also on Wikipedia it only says "$p$ is the (static) pressure". I always considered $p$ to be the internal pressure of the liquid, i.e. the pressure exerted by the fluid on its surroundings (that includes covering fluid). But this raises a problem that I wasn't sure of how to solve.    Lets assume that the flow in the picture is an incompressible flow, like water. When the water flows from the wide to the narrow part of the tube, the speed $v$ increases, lowering the pressure $p$ (as $gh$ and $\rho$ remain constant). As this is an incompressible fluid, the volume cannot change. But if the pressure is lowered and the volume does not change, doesn't that mean that the temperature is lowered too? This just seems weird to me.  What goes wrong in this thought process?


I am writing a program that reads through a .CSV file that is located in a shared folder on a network. I am currently logged onto my computer with a user that can access this folder, so I can run the program without any issues. However, the program will be used by someone who doesn't have access to the folder, and so my colleague created a user with the credentials, for example, username:user1 and password:pass1 that the program should use to access the shared folder.  I would like to ask if someone knows how to implement this, that is, use the user my colleague created to access the shared folder, look for a specific file, and read through it line by line.  I emphasize that I only need to read the files, and not write in them.  Below is a part of the code that I use when reading from the file. Mind you that this is not all of the code; it only shows the parts where I am accessing the file or reading from it.  /* Directory location of the file */ var directory = new DirectoryInfo(@"\\Network\\Folder");  /* Get the latest available file */ file = (from f in directory.GetFiles()                  orderby f.LastWriteTime descending                  select f).First();    var reader = new StreamReader(File.OpenRead(file.FullName),                               Encoding.GetEncoding("iso-8859-1")); string line;  /* Read through it line by line */ while (!reader.EndOfStream) {     line = reader.ReadLine(); }

Is there a simple out of the box way to impersonate a user in .NET?  So far I've been using  for all my impersonation requirements.  Is there a better way to do it by using .NET Framework?  I have a user credential set, (username, password, domain name) which represents the identity I need to impersonate.


explain the difference between /bin and /usr/bin used in Linux?

I read this up on this website and it doesn't make sense.       When UNIX was first written, /bin and   /usr/bin physically resided on two   different disks: /bin being on a   smaller faster (more expensive) disk,   and /usr/bin on a bigger slower disk.   Now, /bin is a symbolic link to   /usr/bin: they are essentially the   same directory.   But when you ls the /bin folder, it has far less content than the /usr/bin folder (at least on my running system).  So can someone please explain the difference?


Let $R$ be a ring. An element $r \in R$ is called nilpotent if $r^n=0$ for some integer $n \ge 1$. Show that if $r$ is nilpotent in a ring with identity, then $1-r$ is a unit in $R$.   Proof. Recall that in any ring we have $(−a)(−b) = (ab)$ Thus in any ring with 1 (commutative or not) we have the following identities:  $1 −(−1)^na^n = (1 + a)(1 − a + a^2 − · · · + (−1)^{−1}a^{n−1})$  If $a^n = 0$ then we obtain an inverse $1 − a$  Now, I'm not so familiar with series and am not sure if this suffices. I also assume this can be shown with matrices, integers....as long as it is a ring

Let $R$ be a commutative ring with 1, we define  $$N(R):=\{ a\in R \mid  \exists k\in \mathbb{N}:a^k=0\}$$  and  $$U(R):=\{ a\in R \mid a\mbox{ is invertible} \}.$$  Could anyone help me prove that if $a\in N(R) \Rightarrow 1+a\in U(R)$?  I've been trying to construst a $b$ such that $ab=1$ rather than doing it by contradiction, as I don't see how you could go about doing that.


I got onto a horse in Skyrim (no DLC, Xbox 360). The horse was not stolen but I did not buy it. And whenever I dismount the horse it starts walking away. Why?   This is not a horse I have bought, I just found it in the wild, so this is not a duplicate of .

I've found a horse in Skryim and I ride it a bit to explore the wild but when I dismount, it wanders away and doesn't stay where I leave it. How do I stop it from doing this? Is there a special thing I forgot about horses? I didn't steal this horse - it was from an Imperial Fort that I've cleared.


Could you please tell me which one is correct?(and why?)  I sent an email to the company asking them for more information.  I sent an email to the company to ask them for more information.

In response to the long-term measures recommended by the School Board,   the then Principal initiated the Pledge Day on “Clean LA", to encourage   all schools to make the “Clean LA” commitment on that day.   I wonder if it sounds better to replace the to-infinitive clause with a participle clause:      In response to the long-term measures recommended by the School Board,   the then Principal initiated the Pledge Day on “Clean LA", encouraging   all schools to make the “Clean LA” commitment on that day.   because I think it is seldom to use to-infinitive clauses after a comma.


I'm getting issue to get my footer at bottom of page in all cases, or it's disappearing when scrolling up or down, or it's just showing at the top of the page... Tried to find out on here but impossible for me to get it.. I do apologize if this post is a duplicate from an existing one..   body {   display: flex;   flex-direction: column; }  footer {   background-color: rgba(255, 255, 255, 0.9);   height: 25px;   order: 4;   border-top: 1px black solid;   margin-top: auto;   text-align: center; }    Text of my footer     Not sure if Grid might help on such situation and make life easier? And thank you for any answer! :) Rgds, Erick

I have a simple 2-column layout with a footer that clears both the right and left div in my markup.  My problem is that I can't get the footer to stay at the bottom of the page in all browsers.  It works if the content pushes the footer down, but that's not always the case.


In …      I did not  type that. Where does it come from?

In     ... the number "5000" does not paste as such but instead as  5 0 U+200C U+200B 0 0   The HTML source looks like this:  perhaps the easiest would be \edef\tmp{\everypar{\widowpenalty\the\widowpenalty\relax}}\tmp\widowpenalty=-50&zwnj;&#8203;00 that may reset   Or (smaller snippet):  \widowpenalty=-50&zwnj;&#8203;00 that   So in the HTML, we have extraneous &zwnj; and &#8203; in between the two middle digits of 5000.  Upon pasting and trying out this code, this first led to a mysterious compiler error. The same issue happened within a (now deleted) comment thread below ; there, these or similar two characters were inserted after the letter "r" within the string \ref ("\ref").  Note that not all browsers wrap the same way. The above screenshot was for Firefox 19.0.2, in which it breaks between \relax}} and \tmp, with 5000 insidiously left intact on the second line but with the two special characters remaining in the middle. So that is why this took me completely by surprise. Other browsers might indeed only break on the invisible Unicode characters, such as Chrome 25:    Or on a smaller screen with the mobile site, multiple line breaks might show, possibly still leaving the Unicode characters unused:


So I've just started using Blender (2.74) and wanted to use a reference image.  So I made a reference image in Adobe Illustrator and saved it as JPG, then dragged into blender. It's not visible, even though I am in Orthographic Mode (Yes, I've tried googling it)  I have also gone to "properties" and enabled background image, and I have tried changing the axis of the image and then changed the camera to the axis. I could of course make a plane and place the image there, but it would be anoying every time I go into wireframe mode.  It's extremely frustrating! Can anyone help me out? Thanks for reading.  Screenshot:

How do I put a backdrop behind the model that I am creating so that I can compare the picture to the model while editing?


As you can see at the bottom of the anvil are a little bit more shape down there, I am sorry I forget what you call them, but if some one could help me and get rid of them, that would be nice.

I am currently trying to develop my modeling skills, and following a tutorial found on YouTube. The tutorial teaches how to create an anvil. At 19:10 or so into the video, a subdivision surface modifier is used to create a smooth, soft look.  However, when I do the same, mine looks very sharp and generally bad.  Any ideas why? I will include the blend file below, I appreciate your time and suggestions.    This is his when he subdivides, which is very different from mine when I do the same at the same settings. (I cannot post more pictures because I am new, but if you check the blend file I attached, you should be able to see the difference.   This is what it looks like when I subsurf without the base face.


I'm getting command not properly ended for the following script, can anyone tell me what I'm missing? Thanks  SELECT E_name, E_DOB FROM Empl ORDER BY E_DOB DESC LIMIT 3;

Is there a way to make an Oracle query behave like it contains a MySQL limit clause? In MySQL, I can do this: select *  from sometable order by name limit 20,10  to get the 21st to the 30th rows (skip the first 20, give the next 10). The rows are selected after the order by, so it really starts on the 20th name alphabetically. In Oracle, the only thing people mention is the rownum pseudo-column, but it is evaluated before order by, which means this: select *  from sometable where rownum <= 10 order by name  will return a random set of ten rows ordered by name, which is not usually what I want. It also doesn't allow for specifying an offset.


Windows 8  I really want to replace ubuntu with windows 8. I cant figure out how :O Can someone please help me?   And yes, i have bought windows 8. I have the ios file, but I also need help to create a bootable Usb stick.

I have absolutely no experience with Linux, and I desperately need to get my computer back up and running again with Windows.  How do I remove Ubuntu and reinstall Windows?    Editor's note: many of the answers are about removing Ubuntu from dual-boot but keeping Windows (which is a bit complicated), while other answers are about removing Ubuntu from single-boot (which is easy: basically just format the disk while installing Windows). The question as written is ambiguous between dual-boot or single-boot.


By considering z[sqrt[-2]] show that x^2+2=y^3 only has two integer solutions, (+/-5,3)  I can see that N(x+i Sqrt[2])=y^3, I think x+i Sqrt[2] is prime in z[i Sqrt[2]] so y^(3/2) must be x+i Sqrt[2] but I don't really see how any of this helps. I think I'm missing something.

It is known that the diophantine equation $y^{2}=x^{3}-2$ has only one positive integer solution $(x,y)=(3,5)$. The proof of it can be read from the book "About Indeterminate Equation" (in Chinese, by Ke Zhao and Sun Qi). But the method I have known of solving this problem is algebraic number theory.  My question:      Does there exists an elementary method for solving the diophantine equation $y^{2}=x^{3}-2$?   I have gotten an elementary method by the assumption that "$x$" is a prime. That is, we can get a conclusion from the equation that $3\mid x$. Since $x$ is a prime we get the only solution $(x,y)=(3,5)$. But if $x$ is not a prime, the answer seems to be difficult.  Thank you for your help.


The question is to evaluate  $$ \int_0^\infty \frac{dx}{(1+x^2)(1+x^c)} $$ for arbitrary $c\geq0$. Here are my attempts:  (The methods behave somewhat differently for $c=0$ but that case is trivial so I will assume $c>0$)  Integrate by parts: \begin{align*} \int_0^\infty \frac{dx}{(1+x^2)(1+x^c)}&=\arctan(x)\frac{1}{1+x^c}\Big|_0^\infty+\int_0^\infty\arctan(x)\frac{cx^{c-1}}{(1+x^c)^2}dx\\ &=0+\int_0^\infty\arctan(x)\frac{cx^{c-1}}{(1+x^c)^2}dx\\ &=\int_0^\infty\frac{\arctan(x^{1/c})}{(1+x)^2}dx. \end{align*} This looks just as bad, if not worse, as before...  Partial fractions: good for small values of $c$, but seems impossible as $c$ gets large.   Along the same line as partial fractions, but I tried the Ostragodski method, which only helped simplify things for $c=2$.   Try and show it doesn't depend on $c$:  For $c=1$, $c=2$, and $c=3$, the integral evaluates to $\pi/4$. My gut tells me this isn't a coincidence, so define  $$ f(c)=\int_0^\infty\frac{dx}{(1+x^2)(1+x^c)} $$ so that  $$ f'(c)=\int_0^\infty\frac{-x^c\log(x)}{(1+x^2)(1+x^c)^2}dx. $$ It would be great if this evaluated to $0$ for arbitrary $c$, but I don't see a nice way to show that it does.  Any hints would be greatly appreciated. Thanks in advance.

Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$  is independent on $a$?


I am really confused. I have two separate tickets, both with AirAsia.  First flight is Bombay to Kuala Lumpur, 2.5 hour layover, and connection to the second flight, Kuala Lumpur to Manila.   I believe that when I arrive in Kuala Lumpur, I have to go through immigration to collect my luggage, and go back to immigration to check in my next flight.  In that case, I will need a visa. When I go to  to apply for an e-Visa, it asks for accommodation proof, bank statement and Malaysia address, none of which I have.  I am Indian and having Indian Passport  Another search gave information regarding , but doesn’t say whether I get it at the airport or the local embassy.  Update: To avoid any issue while travelling, I obtained a Transit Visa from local Malaysian Embassy in Kuwait.

I am an Indian passport holder travelling from China to India via Kuala Lumpur, on Air Asia tickets bought separately. The layover is 4 hours and 30 minutes.  Do I need a transit visa?


I tried different methods of integration, but I can not solve this problem.  $$\int \sin^n (x) dx$$

I am working on trying to solve this problem:     Prove: $\int \sin^n{x} \ dx = -\frac{1}{n} \cos{x} \cdot \sin^{n - 1}{x} + \frac{n - 1}{n} \int \sin^{n - 2}{x} \ dx$   Here are the steps that I follow in the example that I am reading:     $u = \sin^{n - 1}{x}$      $du = (n - 1) \cdot \sin^{n - 2}{x} \cdot \cos{x} \ dx$      $v = -\cos{x}$      $dv = \sin{x} \ dx$            $\int \sin^n{x} \ dx =  \sin^{n - 1}{x} \cdot \sin{x} \ dx$      $\int \sin^n{x} \ dx = \underbrace{\sin^{n - 1}{x}}_{u} \cdot \underbrace{-\cos{x}}_{v} - \int \underbrace{-\cos{x}}_{v} \cdot \underbrace{(n - 1) \cdot \sin^{n - 2}{x} \cdot \cos{x} \ dx}_{du}$      $\int \sin^n{x} \ dx = -\cos{x} \cdot \sin^{n - 1}{x} + (n - 1)\int \sin^{n - 2}{x} \cdot \cos^{2}{x} \ dx$      $\int \sin^n{x} \ dx = -\cos{x} \cdot \sin^{n - 1}{x} + (n - 1)\int \sin^{n - 2}{x} \cdot \left(1 - \sin^{2}{x}\right) \ dx$   Here is where I get lost. How did we go from $\int \sin^{n - 2}{x} \cdot \left(1 - \sin^{2}{x}\right) \ dx$ to $\int \sin^{n - 2}{x} \ dx - (n - 1) \int \sin^{n}{x} \ dx$? Even more specifically, where did $\sin^{n}{x}$ come from?     $\int \sin^n{x} \ dx = -\cos{x} \cdot \sin^{n - 1}{x} + (n - 1)\int \sin^{n - 2}{x} \ dx - (n - 1) \int \sin^{n}{x} \ dx$   I get this part.     $n\int \sin^n{x} \ dx = -\cos{x} \cdot \sin^{n - 1}{x} + (n - 1)\int \sin^{n - 2}{x} \ dx$      $\int \sin^n{x} \ dx = -\frac{1}{n} \cos{x} \cdot x \ \sin^{n - 1}{x} + \frac{n - 1}{n} \int \sin^{n - 2}{x} \ dx$   Could someone please explain what I am missing?  Thank you for your time.


In the triangle $ABC\,$ we denote by $O$ and $I$ the circumcenter and incenter respectively. The perpendicular bisectors of the line segments $IA$, $IB$ and $IC$ pairwise intersect thus defining the triangle $A_1B_1C_1$ Prove that $\vec {OI}=\vec {OA_1}+\vec {OB_1}+\vec {OC_1}$      Hi there. Im a high schooler new to olympiad geometry. Ive been unable to solve this problem despite repeated angle chasing could someone give a solution?Note I dont know stuff like projective geometry and hard lemmas so a simple proof would be most appreciated.


The front room of our rented house has 8 down-lighter spotlights on one of those dimmers that you use a silver disc to turn on and off and supposedly dim. Beside the switch is another separate dimmer Switch, same type, that powers two dimmable other wall lights. Since being in, the lights seem to have their own mind, and on occasion, the main 8 will turn themselves fully on, then dim, then go off. We noticed that all the bulbs were 50w halogen. Even the TV remote with at times turn them all on. We changed all bulbs for 5w led bulbs. Now, we can turn them on, but turning off, a different bulb will stay on. Changes bulb each time we turn on or off. The bulbs are dimmable. Checked wiring and have attached how the two dimmers are wired. Any ideas please? House built in 80s. Oh, and the lights in question also have a remote (for varilight?) which seems to do bugger all.

I have two LED lamps in my corridor, and when I switch them off, they keep emitting light (not very strong, but it's annoying at night).  What can it be? How can I fix it?


I have two different sets of data (multiple shapefiles in each), when I import dataset 1 lets call it, into ArcMap it displays this error message:  I then use Define Projection tool on dataset 1 which adds its geographic as well as projected coordinate system which is exactly the same as dataset 2 yet they plot in different locations within ArcMap, but they should overlay with each other:   I have also tried to Define a coordinate system with the Define tool followed by the Project tool on dataset 1 but that doesn't work either. Both datasets should be plotting in the same location. Any solution to this?

I put 4 different layers into my ArcMap document, but only one of them will show up on the map! I should be seeing the other 3 layers as well. When I do a Zoom to layer, I can see one layer, but I cannot figure out how to make the other 3 show up at once on top of each other.  Before I opened ArcMap, I used ArcCatalog where I right-clicked 3/4 of the shapefiles and  clicked Properties → Coordinate system → Import and then clicked on my Trees layer, which has the coordinate system I want the other 3 to have. But when I opened ArcMap the 4 layers did not overlap, nor could I move them above or below each other.


So, I have to fill this black soul gem for Falion, so I killed a few Draugr and used Soul Trap on them within the 60 seconds but it didn't fill it?  I emptied out all of my other empty soul gems and still nothing happened? Does it have to be a human that I have to kill? I don't think I have long until I'm a 'Stage 4' Vampire?

Recently I enchanted a weapon with a fire damage enchantment. I found a better weapon and wanted to do the same thing with that weapon. However, this time, whenever I tried to select an enchantment, the game would say something like "charge not sufficient for this enchantment".  I figured that somehow the soul gems need to be charged with souls before I can  use them. However, since then, I feel like I've killed at least a couple of dozen enemies, yet the message stays the same.  I feel like I haven't understood something about the soul gems. How do they work?


this should be simple but i cant find the right anwer  I am new to git I added a bunch of new files (not staged) to my git tree i want to just delete them without also deleting the git ignore files in that directory any commands? the   git clean -nd   command tells me it will erase all my .git ignore files as well?

How do you delete untracked local files from your current working tree?


Not exactly a duplicate since this is answering a specific instance popular in social media.  You might have seen the viral posts about "save a penny a day for a year and make $667.95!" The mathematicians here already get the concept while some others may be going, "what"? Of course, what the challenge is referring to is adding a number of pennies to a jar for what day you're on. So:  Day 1 = + .01 Day 2 = + .02 Day 3 = + .03 Day 4 = + .04   So that in the end, you add it all up like so:  1 + 2 + 3 + 4 + 5 + 6 + ... = 66795   The real question is, what's a simple formula for getting a sum of consecutive integers, starting at whole number 1, without having to actually count it all out?!

Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$.  How? What's the proof? Or maybe it is self apparent just looking at the above?   PS: This problem is known as "The sum of the first $n$ positive integers".


This is my first post so apologies if there are any community rules that I overlooked. If I write something in displayed math within an enumerate environment this is not centered with respect to the enumerate environment but with respect to the whole text. The following image shows what I mean:  The code for reproducing the above is as follows. I use the geometry, enumitem and parskip packages but similar situation occurs without them as well. I include them in the code since I would like a solution compatible with those packages. \documentclass[a4paper,12pt]{amsart} \usepackage[margin=1in]{geometry}  \usepackage{enumitem} \usepackage{parskip}   \begin{document}  \begin{enumerate} \item This is the first item. I am just writing some nonsense to fill up the line. Now for a long displayed formula \[1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1\] \end{enumerate} This is some text following the enumerate environment to indicate that the formula is aligning with respect to this text rather than within the environment. \end{document}  I find the above rather ugly and I prefer instead the following image:  This was produced by adding \hspace{0.5\labelwidth} \hspace{0.5\labelsep} \hspace{0.5\leftmargin} inside the displayed math. My two questions are:  Am I wrong in preferring the second image to the first? What is the best way of obtaining the second image?

I have equations inside an environment which has a larger left-margin than the main text (analogous to an enumerate/itemize environment, implemented with the changepage package's adjustwidth environment). Just as equations in enumerate/itemize behave, the equations are centered with respect to the entire text, rather than just the width of the container environment. I want the latter behavior.  I'm trying to accomplish this by wrapping all display-math in minipages. Namely, I'm modifying the mathdisplay environment in amsmath.sty as follows:  \makeatletter \let\oldmd\mathdisplay \def\mathdisplay#1{%   \newline   \begin{minipage}{\textwidth-\@totalleftmargin}%     \oldmd{#1}% } \let\oldendmd\endmathdisplay \def\endmathdisplay#1{%   \oldendmd{#1}%   \end{minipage} } \makeatother   This is almost working, but there are some issues with spacing which I don't understand how to fix. Here's how the the normal \[ ... \] and numbered \begin{equation} ... \end{equation} look with the above modification:      My main confusion is why the numbered version is behaving so differently from the unnumbered. But all in all I want to fix this so that the spacing above and below is even as it usually is. Any help is appreciated, thanks.  Edit: Here's a compilable example:  \documentclass{amsart}  \usepackage{lipsum} \usepackage{calc} \usepackage{changepage}  \makeatletter \let\oldmd\mathdisplay \def\mathdisplay#1{%   \newline   \begin{minipage}{\textwidth-\@totalleftmargin}%     \oldmd{#1}% } \let\oldendmd\endmathdisplay \def\endmathdisplay#1{%   \oldendmd{#1}%   \end{minipage} } \makeatother  \setlength{\parindent}{0pt}  \begin{document}  \lipsum[1]  \begin{adjustwidth}{20pt}{} \lipsum[2] \[ x \mapsto \hom(-,x) \]  \lipsum[3] \begin{equation}   y \mapsto \hom(y,-) \end{equation} \lipsum[4] \end{adjustwidth}  \lipsum[5]  \end{document}


Let $X$ have the pdf $f_X (x;\theta)=1/(2\theta)$ for $-\theta<x<\theta$, zero elsewhere where $\theta>0$ Is the statistic $Y=|X|$ a sufficient statistic for $\theta$?    My book claims it is but I do not see it. A transformation argument shows that the distribution of $|X|$ is uniform with $f_X (x;\theta)=1/\theta$ for $0<x<\theta$. Thus the conditional probability  $$\frac{f(x_1;\theta)f(x_2;\theta)\ldots f(x_n;\theta)}{f_{|X|}(|X|;\theta)}$$  does depend on $\theta$ and therefore, $|X|$ cannot be a sufficient statistic for $\theta$.  Is there something that I am doing wrong here?

So, I know that $\max(-X_{(1)},X_{(n)})$ is a sufficient statistic for the parameter $\theta$. But can I also say that $(X_{(1)},X_{(n)})$ are jointly sufficient for the parameter $\theta$ ?  In other words, can a single parameter have jointly sufficient statistics?


I have a string:   'PROTEIN\t1\t0\ttr|Q6ZN40|Q6ZN40_HUMAN\t\t-1\t;;;;;'.    And I need to capture the text between two |. So it would be 'Q6ZN40'.   I tried this code for that:   re.match('|(.+?)|', string).group(0)   but it gives me an empty string: ''.   What is wrong with my code ? I presume that the regex is not correct but I don't understand why..

I am tired of always trying to guess, if I should escape special characters like '()[]{}|' etc. when using many implementations of regexps.  It is different with, for example, Python, sed, grep, awk, Perl, rename, Apache, find and so on. Is there any rule set which tells when I should, and when I should not, escape special characters? Does it depend on the regexp type, like PCRE, POSIX or extended regexps?


Let $T^2=S^1 \times S^1$ be a torus, where $S^1 \subset \mathbb{R}^2$ is the unit circle. Is $T^2$ connected? If so, how does one show this?

I'm trying to prove the next result:     Let $X$ be a connected covering space of the torus   $T:=\mathbb{R}^2/\mathbb{Z}^2$. If $X$ is compact then It is   homeomorphic to a product of two circles.   Probably the most direct way of proving this is to show that there are only $3$ possible connected covering spaces of $T$, to know: $\mathbb{R}\times\mathbb{R}$, $\mathbb{R}\times S^1$ and $S^1\times S^1$ (the last one is the only one which is compact, so the result follows). As far as I know this can be deduced from the classification theorem of covering spaces (see ), however I need to prove first:   $\mathbb{R}^2$ is the universal covering space of $T=\mathbb{R}^2/\mathbb{Z}^2$ The fibers of this covering by the subgroups of $\pi_1(T)$ are precisely those $3$ spaces (up to conjugacy).   I'd appreciate any hints to prove these two facts, or any ideas to prove the main result in a simpler way. Thanks in advance.


I want to know how the standard environment variables given by the linux OS like PATH, HOME are set automatically. In which file are these read from. There should be some file from which these variables are set when a particular user login.

I've read in a couple of places that the PATH is set in /etc/profile or the .profile file that's in the home dir.  Are these the only places that the path is set in? I want a better understanding of it.  In the /etc/profile file, as the following comment says "system-wide .profile file for the Bourne shell". Does that mean that profile files are the main configuration files for bash?  In that file I don't see the PATH var being set at all. In the .profile file in the home directory there's this line:  PATH="$HOME/bin:$PATH"   That's resetting PATH by the looks because it's concatenating the already set $PATH string with $HOME/bin: right? But if etc/profile and ~/.profile are the only files setting PATH where is $PATH coming from in that line of code if it's not defined in /etc/profile?  Can someone experienced please give a broad and detailed explanation of the PATH variable? Thanks!


Looking for a short story where two incorporeal beings meet, and one shows the other something they've been working on: the ability to manipulate matter. At the end of the story,   the character builds a replica of a human being, and it becomes clear the characters themselves were once human.  I thought this one was an Asimov, but my searching has proven fruitless so it's probably a completely different author.

I read in an old sci-fi anthology a very short story, about two sentient clouds/jets of energy having a conversation and one of them eventually remembering it used to be a human being. The story was about 2 pages long and from an anthology from around the 50's if I had to guess.  Any idea on the name of the story?


I have two Mac Book Pro's laptop with me.  In one of the laptop, I am getting the following popup prompting when I try to change the login options.  It is prompting me like this:    In my another Mac Book Pro, it's simply showing the prompt like this. So I guess, it should show the same prompt message in first Mac Book Pro too.    How to get administrator name to login into the first laptop?  I tried with possible usernames & password (I have never changed my password) that I am aware of but they are not working.     How to find/get the administrator name back so that I can log in with my password.

On my mac I have 2 accounts and the both are normal account. So in this moment I don't have a Administrator account, only one that was in the past a administrator, but now is not.  How I can have again a Administrator account?


I see my status below from     Now I can understand what others terms mean here like week rank, total reputation and week reputation. But what does change mean here?

What does the Change field represent on the  for SO? Refer to the highlighted areas in the image below:   Rank: the user's ranking for the selected time period (week, month, quarter, year) Change: +/- change representing... ??? Total Reputation: the user's total reputation [Time Period] Reputation: the user's reputation for the selected time period (week, month, quarter, year)  So what is the Change field measuring?