How can I link those two separate objects by joining the mesh points? I tried F and Merge at Center but didn't work.
I'm trying to fill empty space in my mesh, and I looked at the knife tool page on the Blender Wiki, and I'm not sure if there's a way to make it so that a new edge is made using the knife tool. I took a screenshot, and hopefully it'll explain better than I can of what I'm trying to do. Oh and I'm using Blender version 2.71, by the way.
Rubik's Cube Combination Could anyone explain why the number of legal or reachable combinations of a $3\times 3\times 3$ Rubik's Cube is $1/12\mbox{th}$ of the total. I understood the logic behind the total number of combinations: $8! \cdot 2^{12} + 3^{12} \cdot 12!$. What I am not able to understand is why do we divide this quantity by $2 \cdot 3\cdot 2$. I don't have any idea about parity or group/set theory. Is there another possible explanation for it? I didn't know how to solve this problem.  How to prove $$\sum_{n=0}^{\infty}\sin(e\pi n!)$$ converges? I recently made my pen-drive bootable with Ubuntu 14.04. But I need to unboot my pen-drive now. I formatted my pen-drive once,But the size of the pen-drive still shows 2.25 MB. Whereas,this is an 8GB pen-drive. How can I get rid of this problem. I'm still using Windows 7. I've been using the beta iCloud Photo Library for a week and a bunch of photos got duplicated and I'd like to wipe all the photos out and start over.  I have backups of all the photos and have turned off iPhoto Cloud Library on all iOS devices. When I log into the web app - it still shows dozens of albums and I can delete all the photos, then go to deleted items and then purge them, but the ghost albums are still there messing things up.  I've turned off the library entirely and you can see there is a 30 day waiting period.     If I click on albums, there are pages of them and no delete button anywhere. If I turn on the library on an iOS device, data starts flooding back as if I wanted to recover the photos.  I don't want to recover anything, I want a clean wipe and clean slate.  Currently, there are 7807 photos in the deleted album and no way to mass delete them. I have 31 albums that show up in iCloud and no way to delete them.  Is there some way to accelerate this or must people wait a month for a corrupt library to eventually be purged to start over?  (I have tickets open with AppleCare after  and it's escalated to engineering, but I was informed that it might be a week or two before it gets seen :-( Perhaps someone has a better idea to clear things out or noticed something I've missed.) I've installed VM VirtualBox v. 4.3.8 on my Ubuntu 12.04 LTS (Lenovo G500S with Intel 1005M processor) and I can't configure any 64-bit guest (nor Windows, neither other OS). Inslalation of Win 8.1 64-bit under 32-bit profile obviously fails... My processor supports VT-x technology, but in my BIOS there is no enable/disable option for VT-x. Is there any way to force VirtualBox to support 64-bit guest ? How can I eliminate condensation on my toilet tank? My toilet is getting a great deal of condensation on the tank and dripping onto the floor. What can be done to address condensation on the exterior of a toilet tank? I used source code to build one package such as below:  ./configure --prefix=/usr --sysconfdir=/etc --localstatedir=/var --libexecdir=/usr/lib --with-package-name="Myplugin" --with-package-origin="http://www.ubuntu.org/" --enable-gtk-doc --disable-static make make install   But unfortunately, i discovered that its the latest version, and has lot of bugs, so i need to remove it/uninstall it. But how can i do so? I tried make clean; make uninstall but still i see it exist:  # pkg-config --list-all | grep Myplugin myplugin-.... $ ls /usr/lib/myplugin/libXYZ.so exist....   How do you remove this now? Significance of assumption in competition inequality questions Refer to the problem below (IMO 2009 Shortlist)     Let $a, b, c$ be positive real numbers such that $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = a + b+ c$$   Prove that    $$\frac{1}{(2a+b+c)^2} + \frac{1}{(2a+b+c)^2} + \frac{1}{(2a+b+c)^2} \leqslant \frac{3}{16} $$    In the solution given by the  (Pg 16), it states that      Without loss of generality, we choose $$a +b+c = 1$$.    Thus, the problem becomes    $$ \frac{1}{(1 + a)^2} + \frac{1}{(1 + b)^2} +\frac{1}{(1 + b)^2} \leqslant \frac{3}{16}(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})$$    Applying Jensen’s inequality to the function $$f(x) = \frac{x}{ (1+ x)^2}$$   ,which is concave for $0 ≤ x ≤ 2$   and increasing for $0 ≤ x ≤ 1$, we obtain   $$α \frac{a}{(1 + a)^2} + β \frac{b}{(1 + b)^2} + γ \frac{c}{(1 + c)^2} \leqslant (α + β + γ) \frac{A}{(1 + A)^2}$$   , where $A =\frac{αa + βb + γc}{α + β + γ}.$   Choosing $α = \frac{1}{a} , β =\frac{1}{b},$ and  $γ = \frac{1}{c}$   , we can apply the harmonic-arithmetic-mean inequality   $$A =\frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} ≤ \frac{a + b + c }{3} = \frac{1}{3} &lt; 1$$   Finally we prove:   $$ \frac{1}{(1 + a)^2} + \frac{1}{(1 + b)^2} +\frac{1}{(1 + b)^2} \leqslant ( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \frac{A}{(1 + A)^2} \leqslant ( \frac{1}{a} + \frac{1}{b} + \frac{1}{c})\frac{A}{(1 + \frac{1}{3})^2} = \frac{3}{16}(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) $$   However,  (Pg 4) differs by having $a +b+c=3$ and setting $f(x) = \frac{1}{16x} - \frac{1}{(x+3)^2}$ and allowing Jensen to resolve the rest.   The questions are as follows;   How do you choose before hand which $a + b+ c =$ to choose?  Why particularly does the assumption differ between the 2 solutions? How does one know before hand the choosing of $α = \frac{1}{a} , β =\frac{1}{b},$ and  $γ = \frac{1}{c}$?  Where and how did $A =\frac{αa + βb + γc}{α + β + γ}$ come from?   Any help would be much appreciated. Possible Duplicate:          In  the following claim was mentioned:  $R/I \otimes_R M \cong M / IM$  So I tried to prove it. Can you help me finish my proof? Thanks!    We recall that $M \otimes_R -$ is a covariant right-exact functor that it is exact if $M$ is flat and we observe that the following is an exact sequence: $$ 0 \to IM \xrightarrow{i} M \xrightarrow{\pi} M / IM \to 0$$  $R/I$ is an $R$ module since $R/I$ is a subring of $R$ closed with respect to multiplication from $R$ but it is not necessarily flat hence we only get exactness on one side:  $$  (R/I) \otimes_R IM \xrightarrow{id \otimes i} (R/I) \otimes_R M \xrightarrow{id \otimes \pi} (R/I) \otimes_R M / IM \to 0$$  Then $$ \mathrm{Im(id \otimes \pi)} = (R/I) \otimes_R M / IM  \cong (R/I) \otimes_R M / \mathrm{Ker} (id \otimes i) $$  so we want to show two things:   (i) that $\mathrm{Ker} (id \otimes i) = \{0\}$. To this end let $r + I \otimes im \in (R/I) \otimes_R  IM $ and assume $ id \otimes i(r + I \otimes im ) = r + I \otimes im = 0 + I \otimes 0$. I'm not sure how to proceed from here.   (ii) and that $M /IM \cong (R/I) \otimes_R M / IM $  Can you help me? Thanks. How to prove that the number $1!+2!+3!+...+n!$ is never square? How to increase battery life on Ubuntu 20.04 and what power saving software should I install, if any? I recently installed Ubuntu 20.04 LTS on my Lenovo Ideapad Flex 5 and realized that battery life in Ubuntu is not as good as Windows. The battery drains faster in Ubuntu. I have seen the questions in Ask Ubuntu about this, but I haven't found anything that regards Ubuntu 20.04 LTS. I've done some research and found out that the battery drain is because most vendors optimize their drivers for Windows and with Linux, you have to hope that the kernel drivers do the job (which they usually do). After that, I researched how to increase my battery life and almost all sources I found recommended installing tlp or powertop or something similar to these. But I have a few questions.  Does tlp, powertop, and similar software actually affect battery life?  If the answer to 1 is yes (in a positive way), what would you recommend I install (such as whether I should  install tlp or powertop by itself or in a combination)?  Are there any other ways to increase battery life in Ubuntu 20.04?   Note: I am a beginner to Linux and Linux distros and do not want to mess with anything advanced that may negatively affect the system. I may not be able to immediately know how to tweak any of the aforementioned software, so please suggest any tweaks if they are needed to get results. And if it helps to know, here are my system specs: OS: Ubuntu 20.04.1 LTS x86_64 Host: 81X2 IdeaPad Flex 5 14ARE05 Kernel: 5.7.17-050717-generic CPU: AMD Ryzen 7 4700U with Radeon Graphics (8) @ 2.000GHz GPU: AMD ATI 04:00.0 Renoir (These were given by neofetch) P.S: If you need anything else, just ask. Thanks in advance! I still do not understand what a  precisely is. Is it exactly the same as a particle? Or only an excited state?  And why does it make a peak in some diagrams? And which diagrams? How to link data to a polygon ArcMap 10.3 I have a map with different polygons (118). Each of this polygons represent diferent vegetation types.  For each of those polygons I have data of which species there is and how abundant they are (the abundance is measured with a scale from 1 to 3).  I would like to "link" (I am sorry if its not the correct way of name it) the data of which species and abundance there is in each of those polygons.   I would like afterwards to be able of selecting the different polygons containing the same abundance and species.  In the polygons I created there is no atributes associated other than (FID, shape and Id. the 3 of them created alone by the program when I drew them) and the tables I have in excel with the data I want to "link" to the polygons contain a column called species and another abundance.  As you can see, I'm pretty new using the program...    As I have tried to do what you all suggest in the comments down here, and I couldn't achieve what I was wanting to do (provably because of my bad explanation before...)  I leave a picture:  As you can see in the attribute table, that is how every attribute table of each polygon (which represents one area) looks like. On the right there is how I have my data in excel. Could you please suggest something I could do in order to add to the polygons attribute table the data belonging of species and its abundances (polygon number 1 has to have all the species (Acer mons, Acer pseu, Acer repens….) and its abundance (which are the numbers of the row Area1). How can I test the quality of a RNG? On electronics.stackexchange we had  about constructing a True Random Number Generator. Since the method relies on noise, which isn't deterministic, the only way to test the quality of the RNG seems to be empirical. Not being a statistician I suggested to test a long bit sequence for normality, but I have no idea what outcome is acceptable and what isn't. For instance counting single bits I guess we have a thumbs-up for a 499500/500500 distribution, but when is the ratio too much skewed to be acceptable? Same for 2-bit and longer sequences. Of course, if testing for normality is a Bad Idea™, I'd like to hear it, including better alternatives.    edit Diehard was mentioned a few times, but I'm not sure this answers my question. The normality test should give a normal distribution, with better bell curve approximations for longer sequences. Diehard seems also to have tests which either should result in normal or exponential distributions. But my question remains: how do I judge the results? Just by looking at the curve and discern a bell curve in it? To get back at my first example, a 499500/500500 distribution is definitely OK, and a 950000/50000 distribution definitely is a no-no, so where does the switch-over happen? Connected graph with colored edges We have connected undirected graph with colored edges in two way (green or blue). And also each vertex have the same number of green and blue edges. How to prove that there are alternate colored (green-blue) path between any two nodes? If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian. I need to prove the equivalence of the following two statements, if $G$ is a group:     $G$ is abelian $\Leftrightarrow$  for each $a,b$ in $G$: $(ab)^n = a^n b^n$ for each integer $n$   I have done the $\Rightarrow$ implication, but I have no idea how to approach $\Leftarrow$, any hints are very welcome. One word/phrase to describe the reaction when you eat a very spicy-hot food I've read the discussion here about .  I've also read this one: .  But I did not find any word or phrase that describes the reaction someone has when eating a too spicy-hot food. You know, they tend to sweat, catch their breath, and even sometimes shed tears; due to the spiciness of the food.  So my questions:        What do you say to describe the reaction someone has when being excitedly "suffered" by a very spicy-hot food? In Indonesian, we would call it kepedasan.   What do you say when you get that thing [the answer to question 1] with you, other than the list I enclose below:                Oh, spicy!     My God, it's spicy-hot!                      Is that it? Idea behind proof that $\frac{21n+4}{14n+3}$ is irreducible for all $n$ What kind of lubrication should I use for a lock? I recently made the Yale lock on my front door much easier to open by spraying some WD-40 in there.  However, I just saw  on WD-40 not being a “true” lubricant, and wondered whether it was appropriate to use on Yale locks? What did Auggie's father give to his wife in the movie Wonder? There is a scene roughly in the last third of Wonder where Auggie's parents Nate and Isabel (Owen Wilson and Julia Roberts) spend an evening together while the kids are out. Nate has a surprise gift for his wife. She opens the box just enough to see it, laughs, and closes it again. So I asked myself the question:  What's in the box?