I'm having difficulty figuring out the difference between (typically referred to as S.E):  A. The Standard Error  And  B. The Standard Deviation of the Sample Mean (typically referred to as s)  Are they the same thing?  OR   S.E is the standard deviation of the the mean of the sampling distribution similar formula as the standard deviation except you use n-1 instead of n in the denominator s is the standard deviation of a single random sample -- same formula as the standard deviation
I'm struggling to understand the difference between the standard error and the standard deviation. How are they different and why do you need to measure the standard error?
Another question I am struggling with:  Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined in a neighborhood of $a$ (except maybe $a$ itself) and suppose that, $\lim_{x \rightarrow a}\left(f(x)+\dfrac{1}{\left|f(x)\right|}\right)=0$. Find $\lim_{x \rightarrow a}f(x)$, and prove by definition, that this is indeed the limit.    Thank you all for your answers!!  But.. How can I prove by definition that this is indeed the limit? take $\epsilon&gt;0$, I have to show that there exists a $\delta&gt;0$ such that, $|x-a| \leq \delta$ implies $|f(x)+1| \leq \epsilon$. now, I do know that there exists an $\delta_1&gt;0$, such that $|f(x)+\frac{1}{|f(x)|}|&lt; \epsilon$.Or, $\frac{\epsilon}{k}$ for some constant $k$ But, how can I continue from there? Finding what's using all the space in *nix Which sequence of commands will tell me which files are the largest starting from a particular directory, including all sub directories? I want to know where all the space went.  Preferably just with regular unix'y commands.  If possible, try to keep it compatible with Sun OS 5.10 (perhaps in addition to regular Linux answers, not as a replacement, to keep it as general as possible.) Retrieving Current Location and Zip using Geolocation Button Nature of the quantum degeneracy pressure Install MS Truetype Fonts system-wide for all users I would like to install the MS Truetype Fonts that come through the package ttf-mscorefonts-installer, so I did:  sudo apt-get install ttf-mscorefonts-installer   Accepting the license, the process completes without any errors. The fonts end up in     /usr/share/fonts/truetype/msttcorefonts   as expected. Just to be sure, I also did a  sudo fc-cache -f -s -v   which among other things says in its output:     /usr/share/fonts/truetype/msttcorefonts: caching, new cache contents: 60 fonts, 0 dirs   So far, so good (and inline with question ) However, when I then open any font dialog (e.g. in OpenOffice or the font dialog of the terminal) I don't see all of the new fonts. For Arial, the standard form is missing, but bold and italic are there. Times New Roman is missing altogether.  I can use nautilus to browse to the font directory, double click the Arial.ttf file and see the font in all its beauty. I can even click the install button then and the font is copied over to my personal ./fonts directory and from there on visible to my apps. But it is not picked up from the system directory.  How do I get these fonts available system-wide for all users? How to locate a font in Ubuntu? Ubuntu desktop with 12.04 is about to crash.  Installed 13.04 on a new machine, want to transfer Book Antiqua font from old machine to the new one.  How do I find it on the old machine?  My /usr directory is empty. My fonts.conf file has these lines: &lt;!-- Font directory list --&gt; &lt;dir&gt;/usr/share/fonts&lt;/dir&gt; &lt;dir&gt;/usr/x11R6/lib/X11/fonts&lt;/dir&gt;  &lt;dir&gt;/usr/local/share/fonts&lt;/dir&gt; &lt;dir&gt;~/.fonts&lt;/dir&gt;  Don't know what this means, but my /usr directory has nothing in it. how can we show $\frac{\pi^2}{8} = 1 + \frac1{3^2} +\frac1{5^2} + \frac1{7^2} + …$? Let $f(x) = \frac4\pi \cdot (\sin x + \frac13 \sin (3x) + \frac15 \sin (5x) + \dots)$. If for $x=\frac\pi2$, we have $$f(x) = \frac{4}{\pi} ( 1 - \frac13 +\frac15 - \frac17 + \dots) = 1$$ then obviously :  $$ 1 - \frac13 +\frac15 - \frac17 + \dots=\frac{\pi}{4}$$ Now how can we prove that: $$\frac{\pi^2}{8} = 1 + \frac1{3^2} +\frac1{5^2} + \frac1{7^2} + \dots$$ Impact of hyperref, when varioref and cleveref are used In , someone suggested in a  that I should not cast the result of malloc. i.e., I should do this: int *sieve = malloc(sizeof(int) * length);  rather than: int *sieve = (int *) malloc(sizeof(int) * length);  Why would this be the case? I've just started to learn about the tensor product and I want to show: $$(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}.$$  Can you tell me if my proof is right:  $\mathbb{Z}/m\mathbb{Z}$ and $\mathbb{Z} / n \mathbb{Z}$ are both finite free $\mathbb{Z}$-modules with the basis consisting of one single element $\{ 1 \}$. So $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z})$ has the basis $\{ 1 \otimes 1 \}$.  Therefore, any element in $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z})$  is of the form $(ab) 1 \otimes 1$ and any element in $\mathbb{Z}/ \gcd(m,n)\mathbb{Z}$ is of the form $k 1 = k$ where $k \in \{ 0, \dots , \gcd(n,m) \}$.  I would like to construct an isomorphism that maps $ab$ to some $k$. Let this map be $ab (1 \otimes 1) \mapsto ab \bmod \gcd(n,m)$.  This is a homomorphism between modules: it maps $0$ to $0$ because it maps the empty sum to the empty sum. It also fulfills $f(a + b) = f(a) + f(b)$ because there is only one element, $a = 1$.   It is surjective. So all I need to show is that it is injective. But that is clear too because if $ab \equiv 0 \bmod \gcd(m,n)$  then both $a \equiv 0 \bmod n$ and $b \equiv 0 \bmod m$ so the kernel is trivial.  Many thanks for your help!! If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$. Was revising for a Linear Algebra when I came across this question.     "Given a Matrix $A$ over $\mathbb{R}^{n \times n}$, and $A^2 = 0$,   show that rank($A$) $\leq n/2$"   My attempt:  I was thinking of the theorem from Cayley-Hamilton but the matrix is a $n \times n$ matrix hence that's probably invalid.  Is there perhaps a significance of $A^2 = 0$? apart from a special case of the nilpotent matrix? Slow leak on front cruiser tire I ride my cruiser to the supermarket and back, using either a trailer or a basket over the front wheel for carrying groceries. The bike has developed a persistent slow leak on the front wheel, and I've replaced the tube without it helping. I can't see anything puncturing the tire, and the spokes aren't poking through the rim tape that I can see.   These are 26" wheels with low pressure (around 40 - 50 psi). What else can I look for that might be causing this? Is this correct? I thought it would be but when I entered it into wolfram alpha, I got a different answer.   $$\int (\cos^3x)(\sin^2x)dx  = \int(\cos x)(\cos^2x)(\sin^2x)dx  = \int (\cos x)(1-\sin^2x)(\sin^2x)dx.$$  let $u = \sin x$, $du = \cos xdx$  $$\int(1-u^2)u^2du = \int(u^2-u^4)du  = \frac{u^3}{3} - \frac{u^5}{5} +C$$  Plugging in back $u$, we get $\displaystyle\frac{\sin^3 x}{3} - \frac{\sin^5 x}{5}$ + C How can I create a coffee cream effect with Photoshop? Today I would like to create a coffee cup with the cream floral on it as a text by using Photoshop. It looks like you order a cup of hot cappuccino and the bartender will draw a flower with cream on the surface of coffee but now I want to make it like a text to present a tittle.     I just need to do the cream text effect, no need to draw the whole cup of coffee since i can't use some images. So anyone can tell me how to do this cream effect on Photoshop.  Thanks in advance for any feedback! How can I find the method that called the current method? When logging in C#, how can I learn the name of the method that called the current method? I know all about System.Reflection.MethodBase.GetCurrentMethod(), but I want to go one step beneath this in the stack trace. I've considered parsing the stack trace, but I am hoping to find a cleaner more explicit way, something like Assembly.GetCallingAssembly() but for methods. How to permanently turn off online results I don't like getting online results when searching in my dashboard, so I went to System Settings &gt; Security &amp; Privacy &gt; Search, and I turned the setting off:    But when I restart my machine, the setting is always reset to:    So how can I permanently disable it without it turning on again after a restart?  Information Update:  I have had a look at the post that this has been marked as a possible duplicate of, and I don't think that my question really is a duplicate of it, as the other question is asking about how to set it globally, and I am not, I am only wanting to set it for one user, so the fact that it is not global is not the problem, the problem is that it is not sticking for that one user, and just resetting to its default state after a restart. In fact, I don't want it to be set globally, and the answers from the other question only really explain how to do that.    OS Information:  Description:    Ubuntu 15.04 Release:    15.04   Package Information:  unity:   Installed: 7.3.2+15.04.20150420-0ubuntu1   Candidate: 7.3.2+15.04.20150420-0ubuntu1   Version table:  *** 7.3.2+15.04.20150420-0ubuntu1 0         500 http://gb.archive.ubuntu.com/ubuntu/ vivid/main amd64 Packages         100 /var/lib/dpkg/status I'm using an HP laptop running Windows 7 Home Edition.  The moment I reach the login screen the computer automatically types "ppppppppp" in the password place, luckily I have fingerprint login to counter that.  The moment I login, it keeps acting as if I'm typing 'p' repeatedly. I tried removing the button and cleaning under it but it was no use. I've tried restarting, uninstalling the keyboard driver, but it was no use.  I can't go to Control Panel because the moment I get there, it starts typing "pppppp" in the search field and I'm unable to find anything.  I'm typing this from my phone because going anywhere on my laptop seems impossible. Why couldn't Voldemort view the prophecy without Harry's help? Incorrect indentation for tag listing questions loaded via AJAX I am seeing a weird indentation for questions loaded via AJAX (i.e., by clicking the n question(s) with new activity link). The following screenshot was taken before I clicked the link:  And this is after the new question was loaded:  A browser refresh resolves the problem (as can be expected). It was tested with Chrome 88 and Firefox 85 on  (Catalina). How to prove or disprove following statement :     There are infinitely many primes of the form : $\lfloor \sqrt {3} \cdot n \rfloor $    Note: This is a problem I made myself.  There is a theorem that states :  $\lfloor nx \rfloor = \begin{cases}  n\lfloor x \rfloor, &amp; \text{if } 0 \leq \{x\} &lt; \frac{1}{n} \\  n\lfloor x \rfloor +1, &amp; \text{if } \frac{1}{n} \leq \{x\} &lt; \frac{2}{n} \\  n\lfloor x \rfloor +2, &amp; \text{if } \frac{2}{n} \leq \{x\} &lt; \frac{3}{n} \\  \vdots \\  n\lfloor x \rfloor +n-1, &amp; \text{if } \frac{n-1}{n} \leq \{x\} &lt; 1 \\ \end{cases}$  where $\{x\}$ is a non-integer part of $x$ .  Hence :  $\lfloor \sqrt{3}\cdot n \rfloor = \begin{cases}  n, &amp; \text{if } 0 \leq \sqrt{3}-1 &lt; \frac{1}{n} \\  n +1, &amp; \text{if } \frac{1}{n} \leq \sqrt{3}-1 &lt; \frac{2}{n} \\  n +2, &amp; \text{if } \frac{2}{n} \leq \sqrt{3}-1 &lt; \frac{3}{n} \\  \vdots \\  2n-1, &amp; \text{if } \frac{n-1}{n} \leq \sqrt{3}-1 &lt; 1 \\ \end{cases}$  How can I proceed from here ?