I have the following sum $\sum\limits_{i=1}^{\lceil \sqrt n \rceil}\sqrt i$.  This sum is $\leq n$, but what is a good bound and what is the method to compute this type of sums?
By looking at an integral and bounding the error?
"Not possible" and "Impossible" When we say,      It is not fair.   or     It is unfair.   I'm not sure enough to say whether both of the sentences have the same meaning or not though superficially, there is no difference between them but if we say,     Something is not possible.   or     Something is impossible.   then there is really an observable difference in my first language. Hence, the question - is there a difference between the preceding two sentences in English? Change the passphrase of an ecryptfs directory Let $(X, \mathcal{T})$ be a metrizable space. Prove that $X$ Lindelöf   implies that $X$ is separable.   My attempt:  Let $d$ be the metric that induces the topology $\mathcal{T}$.   For every $n \geq 1$, consider the open cover: $$\{B_X(x,1/n)\mid x \in X\}$$  Because the space is Lindelöf, this implies that for every $n \geq 1$, there is a countable (or finite) subcover given by $$\mathcal{B}_n:= \{B_X(x,1/n) \mid x \in I_n, |I_n| \leq |\mathbb{N}|\}$$  Now, define $D:= \bigcup_{n\geq 1} I_n$. Let $y \in X$. Let $\epsilon &gt; 0$. Choose $m$ a positive integer so large that $1/m &lt; \epsilon$.  Because $y\in X =\bigcup\mathcal{B}_m$, it follows that there exists $x \in D$ such that $y \in B_X(x,1/m)$, meaning that $d(x,y) &lt; 1/m &lt; \epsilon$, such that $x \in B_X(y, \epsilon) \cap D$. Hence, $y \in \overline{D}$ and it follows that $D$ is a dense countable set in $X$, such that $X$ is separable.  Is this correct? Let a,b,c,d be real or complex numbers with absolute value less than $1$. Is the following true:  $$|ab-cd| \leq |a-c| +|b-d|?$$ Convert integer to hexadecimal and back again How do I print all lines of a file with duplicate values in a certain column How to use GROUP BY to concatenate strings in SQL Server? What is the range of the validity of Coulomb's law? What is the smallest and biggest distance in which  is valid?  Please provide a reference to a scientific journal or book. Just saying that this law is valid from this range to that range is not enough. Which experiments did result to these ranges? How can I make the map::find operation case insensitive? Since my previous questions were not answered I will try to ask again more precisely.  I previously had the Nvidia driver installed and working correctly.  Then the ubuntu software update did something which broke it.  In fact, it completely de-configured it so that the standard display driver is used when the computer reboots.  I would like to re-enable the existing Nvidia  driver without re-installing the Nvidia driver.  How do I do that?  This machine is no longer connected to the internet.  NOTE: I am not asking how to disable automatic updates in this question.  I asked that elsewhere (and it wasn't answered).  NOTE: I am not asking how to stop the Ubuntu software updater from breaking the Nvidia driver the next time it updates the system.  That would be nice to know, but I asked that already and did not receive the correct answer either.  NOTE: Please don't make this as duplicate simply because someone else asked a question about Ubuntu and the software updater.  Try to read the actual question and see how it's different and try to realize how annoying it is when users have to spend hours of their own time to correct the problems that other people causing by incorrectly designing software updaters. PDF video Embedding Problem I have embed an mp4 video in the PDF file as given below:  \documentclass{beamer} \usetheme{Singapore} \usepackage{media9} \usepackage{hyperref} \begin{document}  \begin{frame}\frametitle{Title}  \includemedia[ width=\paperwidth,height=0.7\linewidth, activate=pageopen, flashvars={aaa.mp4 &amp;autoPlay=true} ]{}{C:/Users/name/Desktop/aaa.mp4}  \end{frame}  \end{document}    There is no compile time error in the code but after running it this massage comes along a  blank pd f file....  "To view the Flash @R Technology content in this pd f file, please install the required version of Flash Player"  I would like to add here that I already have installed the latest version of Flash Player(Adobe Flash Player 12 ActiveX). So,Can anyone tell what version is required to be installed to obtain Video in pd f file. Is it possible to find a smooth function $p$ such that $$p(x)=0 \text{ if } x\le -1$$ $$p(x)=0 \text{ if } x&gt; 0$$ $$0\le p(x) \le 1 \text{ if } -1&lt;x&lt;1$$ $$p(x) = 1 \text{ if } -\frac{1}{2}\le x \le \frac{1}{2}$$ Hint: The following function has continuous derivatives of all orders $$f(x) = 0 \text{ if } x\le 0$$ $$f(x) = e^{-\frac{1}{x^2}} \text{ if } x&gt;0$$  My thought: Considering the left-half of the function $p$, I tried something like $e^{-\frac{1}{\tan^2(x\pi)}}$. The shape looks good but I am not sure about the smoothness at $-1$ and $-\frac{1}{2}$. For $x=-1$, since $\tan$ is infinitely differentiable so we have smoothness there. But for $x=-\frac{1}{2}$, I am not sure about the smoothness after &quot;gluing&quot; with $y=1$. Sync +- 70 000 records from CSV into drupal I have a CSV file with approx. 70.000 records that I want to import into Drupal 7. Once this is done the client will provide a new CSV file on a daily base that has to sync the updated rows to Drupal.   I wrote a import with  and the  module, but in both cases it takes like forever to import the 70.000 rows. Is there any other solution to import the data faster in Drupal 7? $r$ primitive root of prime $p$, where $p \equiv 1 \mod 4$: prove $-r$ is also a primitive root question is as follows:  Let $p$ be a prime with $p \equiv  1 \mod 4$, and $r$ be a primitive root of $p$.  Prove that $-r$ is also a primitive root of $p$.  I have shown that $-r^{\phi(p)} \equiv 1 \mod p$.  What I am having trouble showing, however, is that the order of $-r$ modulo $p$ is not some number (dividing $\phi(p)$ that is LESS than ($\phi(p)$).  By way of contradiction, I've shown that the order of $-r$ cannot be an EVEN number less than $\phi(p)$.  But my methodology does not work for the hypothetical possibility of an ODD order that is less than $\phi(p)$.  Any and all help appreciated.  Happy to show methodology for any of the parts I have managed to do, if requested. What are the alternatives for storing client side encrypted data? I want to store client side encrypted data in Drupal 7, so that it is encrypted on the client end, and stored and transmitted in encrypted form. The password is never transmitted or stored on the server. The only modules I have found which do this are  and  but they are for Drupal 6. I need something that is user specific, so that different users can use different keys, rather than a single key or a set of keys used on the server end. Are there any similar modules for Drupal 7? Where can I find medicine? I know it is possible to be stung by bees in Animal Crossing: New Leaf.   I learned that the hard way when I was trying to find my furniture tree. Now, I have an eye swollen shut and all of my towns people are making fun of me. They're all being kind enough to inform me that I need Medicine, but they seem to be enjoying themselves at my expense too much to tell me where to actually find the Medicine.  Where can i get medicine? I'd really like it if my townspeople stopped making fun of me. How to determine if a type implements a specific generic interface type How many ounces is a British "tin" of tomatoes? If I have a British recipe that lists "1 tin chopped plum tomatoes," how many ounces is that? In British English, does "tin" usually refer to a specific size? In the US, cans generally come in 14 and 28oz sizes - which one is likely closest to what the recipe means? When does decision tree perform better than the neural network? Take the following script:  #!/bin/sh sed 's/(127\.0\.1\.1)\s/\1/' [some file]   If I try to run this in sh (dash here), it'll fail because of the parentheses, which need to be escaped. But I don't need to escape the backslashes themselves (between the octets, or in the \s or \1). What's the rule here? What about when I need to use {...} or [...]? Is there a list of what I do and don't need to escape?