Are there parameters where a biased estimator is considered "better" than the unbiased estimator?
When is a biased estimator preferable to unbiased one?
What is the difference between the bind and live methods in jQuery? If $q$ is coprime to $a$ then $a\mid nq-1,\,$ so $q$ is invertible mod $a$ I think the following is true:  If $q$ is coprime to $a$ then $a\mid (nq-1)$ for some $n\in\mathbb Z$.  I have started to first reducing $n\in\mathbb Z$ to $n\in\{1,\ldots,a-1\}$ and then considering all the cases $q = ka + j$ for $j=1,2,3,\ldots,a-1$. I stopped at $j=3$ and found that this is kinda messy. Does anyone have an elegant proof for this (maybe using some results from number theory)?  [Note: question has been generalized from prime $q\nmid a$ to $\,q$ coprime to $a$] I edit a lot of posts every day. I often run across posts with 'Hi' and 'Thanks' on the top and the bottom of the post respectively. I also run across things like:     --User   Should these items be removed during the editing of the post by an editor? CSS two divs next to each other Is my MAC address public when browsing the Internet? SF Story about human adapting to alien environment I just installed a fresh install of 32-bit 12.04 LTS Pangolin today. When I tried to install a few applications, it was really slow. But when I download a .deb or .tar.gz file from on my web browser, it is a lot faster.  Computer specs:  HP Pavilion Slimline s7700n PC  [Memory]         873.7 MiB  [Processor]      AMD Athlon(tm) 64 X2 Dual Core Provessor 3800+ x 2  [Graphics]       GeForce 6105 LE/integrated/SSE2/3DNOW!  [Disk Size] 638.5 GB  I normally get 200-400 kbps.  This is the comparison. I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$.  Okay, fine.  You write down the action functional, require that it be a minimum (or maximum), and arrive at the Euler-Lagrange equations.  Great.  But now you want these Euler-Lagrange equations to not just be derivable from the Principle of Least Action, but you want it to be equivalent to the Principle of Least Action.  After thinking about it for awhile, you realize that this implies that the Principle of Least Action isn't really the Principle of Least Action at all:  it's the "Principle of Stationary Action".  Maybe this is just me, but as generous as I may be, I will not grant you that it is "natural" to assume that nature tends to choose the path that is stationary point of the action functional.  Not to mention, it isn't even obvious that there is such a path, or if there is one, that it is unique.  But the problems don't stop there.  Even if you grant the "Principle of Stationary Action" as fundamentally and universally true, you realize that not all the equations of motions that you would like to have are derivable from this if you restrict yourself to a Lagrangian of the form $T-V$.  As far as I can tell, from here it's a matter of playing around until you get a Lagrangian that produces the equations of motion you want.  From my (perhaps naive point of view), there is nothing at all particularly natural (although I will admit, it is quite useful) about the formulation of classical mechanics this way.  Of course, this wouldn't be such a big deal if these classical ideas stayed with the classical physics, but these ideas are absolutely fundamental to how we think about things as modern as quantum field theory.  Could someone please convince me that there is something natural about the choice of the Lagrangian formulation of classical mechanics (I don't mean in comparison with the Hamiltonian formulation; I mean period), and in fact, that it is so natural that we would not even dare abandon these ideas? I am trying to really understand why the gradient of a function gives the direction of steepest ascent intuitively.  Assuming that the function is differentiable at the point in question, a) I had a look at a few resources online and also looked at this   , a popular question on this stackexchange site. The accepted answer basically says that we multiply the gradient with an arbitrary vector and then say that the product is maximum when the vector points in the same direction as the gradient? This to me really does not answer the question, but it has 31 upvotes so can someone please point out what I am obviously missing?  b) Does the gradient of a function tell us a way to reach the maxima or minima? if yes, then how and which one - maxima or minima or both? Edit: I read the gradient descent algorithm and that answers this part of my question.  c) Since gradient is a feature of the function at some particular point - am I right in assuming that it can only point to the local maxima or minima? What's the best way of handling permissions for Apache 2's user www-data in /var/www? Has anyone got a nice solution for handling files in /var/www? We're running Name Based Virtual Hosts and the Apache 2 user is www-data.  We've got two regular users &amp; root. So when messing with files in /var/www, rather than having to...  chown -R www-data:www-data   ...all the time, what's a good way of handling this?  Supplementary question: How hardcore do you then go on permissions?  This one has always been a problem in collaborative development environments. How to fill down in Numbers without dragging? Here's a scenario:  Someone asked about "How to build a SQL query to get count and blah blah blah..."  His only tag is . I added the  and  tags to it.  Someone else answered with a nice SQL statement that has a "group by" statement which seems "logically inevitable".  I proceed to add a tag . I did that on the basis that someone who searches for  will be able to find this post useful, just as I did. My concern is, it wasn't described at all in the question, nor was it implied, unlike  and .  What is the community opinion?  The scenario above is a real one . And the one who answered it nicely is Jon Skeet. Is there any tool available for extracting the X, Y and Z coordinates from a .las file and exporting them to a .csv or similar file? Suspending with root on USB I have a laptop running Ubuntu 14.04 from a root file system on USB storage. This is not working well, because after wake up from suspend, ext4 will frequently try to write to the root file system before USB is ready.  Here is what I see in the kernel log when this happens, notice how I get a bunch of I/O errors on sda1, and then one second later the USB storage drive is finally detected by the kernel.  [ 2826.517419] wlan0: associated [ 2826.517452] IPv6: ADDRCONF(NETDEV_CHANGE): wlan0: link becomes ready [ 2827.575371] EXT4-fs warning (device sda1): ext4_end_bio:317: I/O error -5 writing to inode 1733735 (offset 0 size 0 starting block 12629950) [ 2827.575380] Buffer I/O error on device sda1, logical block 12629694 [ 2827.575400] EXT4-fs warning (device sda1): ext4_end_bio:317: I/O error -5 writing to inode 3148603 (offset 0 size 8192 starting block 12844470) [ 2827.575404] Buffer I/O error on device sda1, logical block 12844212 [ 2827.575411] Buffer I/O error on device sda1, logical block 12844213 [ 2827.575448] EXT4-fs warning (device sda1): ext4_end_bio:317: I/O error -5 writing to inode 3015015 (offset 0 size 90112 starting block 6588832) [ 2827.575453] Buffer I/O error on device sda1, logical block 6588576 [ 2827.575461] Buffer I/O error on device sda1, logical block 6588577 [ 2827.575465] Buffer I/O error on device sda1, logical block 6588578 [ 2827.575469] Buffer I/O error on device sda1, logical block 6588579 [ 2827.575473] Buffer I/O error on device sda1, logical block 6588580 [ 2827.575477] Buffer I/O error on device sda1, logical block 6588581 [ 2827.575481] Buffer I/O error on device sda1, logical block 6588582 [ 2828.857284] sd 0:0:0:0: [sda] No Caching mode page found [ 2828.857293] sd 0:0:0:0: [sda] Assuming drive cache: write through   At first there is no visible indication outside of the kernel log, that the problem has triggered, but if I let Ubuntu stay running beyond this point, then the file system will get errors and eventually switch to read only mode. At that point I have to reboot into recovery mode and run fsck.ext4 manually from a root shell in order to repair the file system.  Is there some setting I can change such that access to the root device after waking from suspend can be delayed until the USB drive is ready? What is the difference between a field and a property? Proving $\lim_{x\to 1} x^3=1$ with $\epsilon$-$\delta$ definition Problem: I need to formally prove that $$\lim_{x\to 1} x^3 = 1.$$ My work: This is what I have so far and I'm generally a bit stuck with these proofs from here onwards. Because$$-\epsilon &lt; x^3-1 &lt; \epsilon  =  | x^3-1 | &lt; \epsilon,$$ then $$ -\epsilon &lt; x^3-1 &lt; \epsilon$$ $$-\epsilon+1 &lt; x^3 &lt; \epsilon +1$$ $$ \sqrt[3]{-\epsilon+1}&lt; x &lt; \sqrt[3]{\epsilon+1}.$$ Hoping that what I have done so far is correct. Am I right in thinking that $$ \sqrt[3]{-\epsilon+1}&lt; x &lt; \sqrt[3]{\epsilon+1}$$ is giving me an interval where $x$ is going to give me a $f(x)$ value that falls within the distance $\epsilon$ from the limit on the $y$-axis ? Or is this interval smaller than the $\epsilon$-distance on the $y$-axis? Can you explain briefly the main concepts and command line tools used to manage file permissions? What was the point of testing carbonite-freezing technology on Han Solo? Darth Vader tested carbonite-freezing technology on Han Solo because he didn't want to kill Luke who was precious to the emperor. But, why was it necessary? Wasn't it already a tested and proven technology? Even Darth Vader (when he was Anakin) himself used it along with others decades ago (Star Wars: The Clone Wars S03E18). We know that in high temperature, resistivity in metals goes linearly with temperature. As temperature is lowered, resistivity goes first as $T^5$ due to "electron-phonon" interaction, and then goes as $T^2$ as temperature is further lowered due to "electron-electron" interactions.   My question is, is there an intuitive way of understanding the above power-law dependence at each temperature range physically? I came to know about , it's a website that allow search for flights by country. I mean you just enter from like "Germany" and to like "Belgium" and you get best offers from all airports between the two countries. That is great. And I was able to arrange a good-price trip to Brussels.   Now I want to arrange another trip to Turkey, but Whichbudget is not helping much this time. And I can't find any other websites that allow "search by country".   Update: Not duplicate to  as his question is broad to the extent of a state. But my question is broad to the extent of a whole country + less price + "Any Time". If anyone checks , he'll understand what I'm saying. Especially that whichbudget allows "Any Time" option, not just +/-3 days.  Does anyone know more similar websites?