$\forall$:n∈${N}$ $\binom{2n}{n}$  $\ge  \frac{4^n}{2n+1}$ I tried with no any success...
How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}&lt;\binom{2n}{n}&lt;\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ &gt; 1 ?     Does anyone know any better elementary estimates?   Attempt.  We have $$\frac1{2^n}\binom{2n}{n}=\prod_{k=0}^{n-1}\frac{2n-k}{2(n-k)}=\prod_{k=0}^{n-1}\left(1+\frac{k}{2(n-k)}\right).$$ Then we have $$\left(1+\frac{k}{2(n-k)}\right)&gt;\sqrt{1+\frac{k}{n-k}}=\frac{\sqrt{n}}{\sqrt{n-k}}.$$ So maybe, for the lower bound, we have $$\frac{n^{\frac{n}{2}}}{\sqrt{n!}}=\prod_{k=0}^{n-1}\frac{\sqrt{n}}{\sqrt{n-k}}&gt;\frac{2^n}{\sqrt{4n}}.$$ By Stirling, $n!\approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$, so the lhs becomes $$\frac{e^{\frac{n}{2}}}{(2\pi n)^{\frac14}},$$ but this isn't $&gt;\frac{2^n}{\sqrt{4n}}$.
Do mankind and manmade activities/constructions have any effect on the rotation of the Earth? We walk or ride on our vehicles to our destinations daily. Does our movement have any effect on the rotation of the earth according to Newton's law? What will be the effect if we move all the peoples along with their vehicles at their maximum velocity in one line in one direction along the equator? How much effect will it make? Shape of confidence interval for predicted values in linear regression Integrate : $\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}dx$ $$\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}\ dx$$   My approach :   Dividing the denominator by $\cos^2x$ we get $\dfrac{x^2\sec^2x }{(x -\tan x)(x\tan x +1)}$ then  $$\int \frac{x^2\sec^2x}{x^2\tan x -x\tan^2x+x-\tan x}\ dx=\int \frac{x^2(1+\tan^2x)}{x^2\tan x -x\tan^2x+x-\tan x}dx$$  But I am not getting any relation between numerator and denominator so that I will take any substitution and solve further please suggest whether it is correct and how to proceed in this. Thanks. Stop setInterval call in JavaScript Prove that if $V$ and $W$ are three-dimensional subspaces of $\Bbb R^{5}$, then $V$ and $W$ must have a non-zero vector in common.  So far I got $V = \{v_{1}, v_{2}, v_{3}\}$ and $W = \{ w_{1}, w_{2}, w_{3}\}$ because they are 3-dimensional subspaces. Then $V \cup W = \{v_{1}, v_{2}, v_{3}, w_{1}, w_{2}, w_{3}\}$  My prof said that now I need to prove that they are linearly independent so this is a contradiction and the statement is wrong because we are in $\Bbb R^{5}$ and we have 6 elements on it.  I don't know how to express them as a linearly independent proof, so I hope anyone can give me a good explanation on it. Why do electrons move towards anode in the Photoelectric effect experiment? Consider a Photoelectric effect experiment apparatus as shown in Figure 11.1    The variation of photocurrent with the voltage applied across $\mathrm{A}$ and $\mathrm{C}$ is as shown in Figure 11.3.    My question –     Why is the current non-zero even if the voltage is zero?    Consider a situation in which the voltage across the plates is zero. Photons of sufficient frequency strike the outermost surface of the cathode, emitter plate $\mathrm{C}$. Electrons become free from the attraction of nucleus as their net energy becomes positive.  Why I think that there should be no net current?  As soon as photons strike the outermost surface of the cathode(assumed to be $1$ to $10$ atomic diameters thick), electrons become free. Now, as the electrons have kinetic energy, they'll come out of that outermost surface. Now, the electrons(with kinetic energy) should have an equal probability of going in every direction of the surface. So, around half of the electrons should go in the right direction towards the anode and strike it and half should go to the left, trying to penetrate the inner surfaces of the cathode plate.   Now, the electrons which will go to the right will face some field due to the space charge present in the tube but nevertheless, some electrons will definitely reach the anode plate. After they strike with the anode plate, they'll face some resistance but still, some electrons will reach the low resistance copper wires connected to the anode plate and milliammeter.  Now, consider the electrons which go towards left. They'll definitely face resistance from cathode plate but some will still reach the copper wire.  Now, as there is no potential difference applied between $\mathrm{A}$ and $\mathrm{C}$, there is no need of commutator and voltmeter. So, the ammeter is connected in series with the anode plates and cathode plates with copper wires.  Now, if there is steady state current in the circuit, electrons emitted to any one of the sides should dominate over the other. But the number of electrons emitted on both the sides is same, and all the electrons should face the same resistance as the current(electrons) should flow in a closed loop. But if the number and velocity of electrons coming from both the sides are same, there should be no net current. But there is. Why?  Edit –  From the answers of anna v and mmainville, it can be seen that the emission of electrons is dependent on the angle of emission of light.  But still, the question remains. If the electrons emitted towards anode have enough energy to complete the circuit and again reach the holes in cathode despite the resistance in between, the electrons emitted to the left should have enough energy to come to anode plate via copper wires and again get ejected through anode plate by the virtue of their kinetic energy and reach the cathode plate. In this case too, the current should be zero which is not true. Why? Characterize the groups $G$ for which the map $\iota: G \to G$, sending $x \mapsto x^{-1}$ for all $x \in G$, is an automorphism of $G$ I have trouble on this question: "Characterize the groups G for which the map $$\iota: G \to G$$ sending $$x \mapsto x^{-1}$$ for all $x \in G$, is an automorphism of $G$." From  post I have seen that $G$ must be abelian. Can anyone propose another characteristic of $G$ that I can derive from the given conditions besides abelian? Thank you so much!! Shell Script for logging into a ssh server I tried writing a shell script which can do automatic login into a ssh server using password which is mentioned in the script. I have written the following code:  set timeout 30 /usr/bin/ssh -p 8484 root@172.31.72.103 expect  {    "root@172.31.72.103's password"     {       send "password\r"     } }   This code is not running properly, still it is asking for the password. Can somebody please help me in solving this I just edited an answer on Unix&amp;Linux and when scrolling down, the semi-transparent right sidebar boxes "How to edit" and "How to format" kept their fixed positioning and overlapped with the page footer.     While posting this, I have seen several (not) duplicate questions about this issue, and although from my personal taste, I would prefer the boxes to stop scrolling before the footer is reached, I can accept the development teams decision to let it overlap.   However, on all those other posts, the boxes had an opaque background, so that the footer was hidden and the box content was readable. Here the boxes are transparent though, so the texts get mixed and are no longer well readable. "NG NS" on ticket after airport security I went through security at the Sacramento Airport with a friend. He got something written on his boarding pass "NG NS". I didn't have that. We both went through the same metal detector, neither beeped nor had random checks. Flying to Edmonton through Seattle connection. Early for flight.  I have tried searching for "what is tsa ng ns" on Google but nothing useful came up. Any ideas? If i try to update node revision like this:  $node = node_revision_load(59); $node-&gt;set('publish_on', null); $node-&gt;save();   or simple save:  $node = node_revision_load(59); $node-&gt;save();   I have a PHP memory error, (i have 1gb allocated) limsup of intersection of events as a subset of intersection of limsups I have an Asus UL30JT. It has Nvidia Optimus technology which I am not really interested in. I currently rarely use this laptop with it being connected to power. And I want to be able to play video at high res, play games and do some OpenCL coding.  In the BIOS I switched it to NVIDIA only (non windows 7). Previously, this worked just fine, using the nvidia driver.  After upgrading everything was broken. I have added nomodeset and blacklist.nouveau=1 as kernel options. The nvidia drivers install without error. However they are not found/used.  jockey-text -l kmod:nvidia_310_updates - nvidia_310_updates (Proprietary, Enabled, Not in use) kmod:nvidia_304_updates - NVIDIA binary Xorg driver, kernel module and VDPAU library (Proprietary, Disabled, Not in use) kmod:nvidia_313_updates - NVIDIA binary Xorg driver, kernel module and VDPAU library (Proprietary, Disabled, Not in use) kmod:nvidia_310 - NVIDIA binary Xorg driver, kernel module and VDPAU library (Proprietary, Disabled, Not in use) kmod:nvidia_304 - NVIDIA binary Xorg driver, kernel module and VDPAU library (Proprietary, Disabled, Not in use)   Trying to load the module manually fails as well.  sudo modprobe nvidia FATAL: Module nvidia not found.   Xorg.0.log show the following:  [    12.028] Loading extension GLX [    12.028] (II) LoadModule: "nvidia" [    12.037] (WW) Warning, couldn't open module nvidia [    12.037] (II) UnloadModule: "nvidia" [    12.037] (II) Unloading nvidia [    12.037] (EE) Failed to load module "nvidia" (module does not exist, 0) [    12.037] (==) Matched nvidia as autoconfigured driver 0 [    12.037] (==) Matched nouveau as autoconfigured driver 1 [    12.037] (==) Matched vesa as autoconfigured driver 2 [    12.037] (==) Matched modesetting as autoconfigured driver 3 [    12.038] (==) Matched fbdev as autoconfigured driver 4 [    12.038] (==) Assigned the driver to the xf86ConfigLayout [    12.038] (II) LoadModule: "nvidia" [    12.038] (WW) Warning, couldn't open module nvidia [    12.038] (II) UnloadModule: "nvidia" [    12.038] (II) Unloading nvidia [    12.038] (EE) Failed to load module "nvidia" (module does not exist, 0) [    12.038] (II) LoadModule: "nouveau" [    12.039] (WW) Warning, couldn't open module nouveau   I have installed the kernel headers, of the correct kernel. I verified that the kernel options are in the grub configuration. I have purged all nvidia packages and tried to reinstall (multiple times...)  Currently I am at a loss. I have checked the following question:  But nothing there worked for me. In the C and C++ programming languages, what is the difference between using angle brackets and using quotes in an include statement, as follows?   #include &lt;filename&gt;  #include "filename" How can I control HDD spin down time? I have 2 HDDs in my PC. Ubuntu is turning off the secondary HDD very quickly after about 15 minutes, which is short for me. I need to control this time. How can I do it?  I tried GNOME power management but did not find it useful. It is a common practice to apply PCA (principal component analysis) before a clustering algorithm (such as k-means). It is believed that it improves the clustering results in practice (noise reduction).   However I am interested in a comparative and in-depth study of the relationship between PCA and k-means. For example, Chris Ding and Xiaofeng He, 2004,  showed that "principal components are the continuous solutions to the discrete cluster membership indicators for K-means clustering". However, I have hard time understanding this paper, and Wikipedia actually .  Also, the results of the two methods are somewhat different in the sense that PCA helps to reduce the number of "features" while preserving the variance, whereas clustering reduces the number of "data-points" by summarizing several points by their expectations/means (in the case of k-means). So if the dataset consists in $N$ points with $T$ features each, PCA aims at compressing the $T$ features whereas clustering aims at compressing the $N$ data-points.  I am looking for a layman explanation of the relations between these two techniques + some more technical papers relating the two techniques. Spoilers from the Season 7 finale of Game of Thrones, "The Dragon and the Wolf".  In one of the final scenes, Jaime and Cersei are discussing Jaime's future with them, at which point       Cersei nods her head, signaling the kill order for 'The Mountain', yet 'The Mountain' pulled out his sword but why didn't he kill Jaime? How to calculate a heading on the earths surface? Given an initial position and a subsequent position, each given by latitude and longitude in the WGS-84 system.  How do you determine the heading in degrees clockwise from true north of movement? What is a suitable image resolution for canvas prints? I would like to create a 60 inch x 20 inch canvas print using an image with a resolution of 4759x1798 (240 pixels per inch) and I would like to know if the image quality will still be good at that size.  To generalise the question, what is the recommended maximum size of an enlargement at varying image resolutions?  e.g. 3 megapixel, 5 megapixel, 8 megapixel, etc Can I delete user directory after deleting user? The reason I ask is because I recently made the mistake of installing Oracle 12c on my sub-performance laptop. Noticing the excessive amount of resources this added up to on startup, I decided to uninstall it.  What I failed to see through, however, was the fact that it created a new Windows user account in order for the initial installation to work out. In my case, it was mandatory because I only had an administrator account and I didn't see the need to use the built-in account. So, after failing to successfully install 12c twice, I wound up creating not just one, but two Windows user accounts. This had to do with something in regards to how Oracle sets up the Oracle Home business and whatnot, but I have no idea what any of that detailed, so I just went ahead and did it.  After uninstalling what I could in regards to Oracle and the database, I decided the next thing to do would be to remove the created users.  Having Windows 10 Home installed, I couldn't use the local users &amp; groups view/tool to delete these from the GUI. SO, I did the next best thing and Googled a solution. In my case, I used the following command:  net user username /DELETE   When I executed the command, the console gave the response of having successfully deleted the account for both accounts specified.  To verify that this had worked, I went to the users directory located in %SYSTEMROOT%/Users/ and came to find that the user directories were still there.  What does this mean and if I should delete them, will anything bad happen?  Can I even delete these directories?  Regards,  --T.R.G