Why is the sum of outer products equal to the matrix product of a matrix and its transpose , so $A^TA = \sum_{i=1}^n a_i a_i^T$?
Matrix $A^T A$ as sum of outer products
Find the value of $a^{2m}+a^m+{1\over a^m}+{1\over a^{2m}}$ Let $a$ be a complex number such that $a^2+a+{1\over a}+{1\over a^2}+1=0$  Let $m$ be a positive integer, find the value of $a^{2m}+a^m+{1\over a^m}+{1\over a^{2m}}$  My approach: I factorized equation 1 which yielded $(a+{1\over a})={-1^+_-(5^{1\over2})\over2}$  I don't know if it is of any help. Map preserving intervals but discontinuous Can't see a glass object through another glass object in Eevee Why shouldn't the denominator of the covariance estimator be n-2 rather than n-1? Is "89.9%" singular or plural? I’m writing a dissertation on network security. I’m unsure which of the following to use:   ...89.9%, which is 57 million users.  ...89.9%, which are 57 million users.   Technically it should be are because it refers to a plural amount of people, but it doesn’t sound right to me considering that 57 million is in a sense one group. When closing a question as a duplicate there is a text box which says: "search, or enter a question link or numeric id". You can copy and paste a question link into here, but this fails if the URL starts with https:// instead of http://, with the message "Your search returned no matches; please try a different search". Often when I mistype a command such as ls (e.g. I hit ENTER before I type 's') there is a long (~2s) delay after the terminal displays:  bash: l: command not found...   I can understand the reasons for a similar delay after an incorrect password is entered, per . But why delay after an unrecognized command? Does FAIL_DELAY in /etc/login.defs affect this also? Is it possible to install MySQL 5.5 or 5.6 on Ubuntu 16.04? I recently upgraded to Ubuntu 16.04. Unfortunately I need to work MySQL 5.5 or 5.6. I tried many solutions to have MySQL 5.5 or 5.6 run on Ubuntu 16.04 like this  . But none of them is working. Ubuntu detect the unmet dependencies but says "but it is not going to be installed" for 5.5 or says Package 'mysql-server-5.6' has no installation candidate...  Is it not possible to force an earlier version of MySQL on Ubuntu? Why is that? I was using Windows XP, I like Ubuntu, but I'm wondering,because as  I'm a professional photographer &amp; photo editor.  I do my work with Photoshop, so will photoshop work in ubuntu? Construction of $\mathbb{CP}^2$ from $S^2\times S^2$ Show that $\mathbb{CP}^2$ is homeomorphic to the orbit space of $(S^2\times S^2)/\mathbb{Z}_2$. The action is given by $(x,y) \mapsto (y,x)$. My attempt: I can see that $S^2\times S^2$ is a subset of $S^5$. I am unsuccessfully trying to restrict the action of $S^1$ on $S^5$ to $S^2\times S^2$ to get the required map. Unable to log out of chat. I attempted to log out with the following actions:    This accomplished absolutely nothing since I remained logged in after clicking "log out" on the confirmation page. I then tried logging out of Stack Exchange from one of the network sites and repeated the steps above. The results were the same. I remained logged in no matter which combination of steps I performed.  In fact, the only way I could log out at all was to clear all of the cookies that were set. There is a  being thrown out of a flask, literally hovering out of the container  My question is how detailed a physical explanation of this phenomenon we have?  Question: More precisely, if we used the best available mechanical physics simulator software, and we tried to simulate this behavior, would the software be able to reproduce the effect or not? My guess is not but I would like to know why not.  Or do we have to content ourselves only with handwaved explanations? I am from the school of thought that you don't understand physics until you have a reasonable certainty that you can reproduce the physics in a computer. DOS of Van Hove singularity in 2D square lattice tight binding model For the simplest example, 2D square lattice tight binding model gives the energy band as $$\varepsilon_k=-2t(\cos k_x+\cos k_y) \, .$$ We know that $\mathbf{k}=(0,\pi)$ and related momentum points are saddle points which give $|\nabla_k \varepsilon_k|=0$ and thus some kind of singularity in density of state (DOS) since $$\rho(\varepsilon)\propto \int_{\varepsilon=\text{const}} \frac{d S}{|\nabla_k \varepsilon_k|} \, .$$ How can I get the $\ln$ divergence for DOS near $\varepsilon=0$? Should I only care about those singularity points and omit the integral from normal parts and do Taylor expansion near those saddle points?   Moreover, why are points like $(\pi/2,-\pi/2)$ or $(2\pi/3,-\pi/3)$ not called Van Hove points when those points are also lie in the $\varepsilon=0$ line and give $|\nabla_k \varepsilon_k|=0$? System.LimitException: Too many query rows: 50001 What I am trying to do is to set a date in my date field and also update the object. I am using the class below to do so which works just fine in my sanbox but in production I am getting this error:      System.LimitException: Too many query rows: 50001   public with sharing class updateObject{     public  List&lt;NRProducts__c&gt;  values;     public PageReference updateObject() {         values=[SELECT Refreshed_Date__c FROM NRProducts__c ];         for (NRProducts__c s :values){            s.Refreshed_Date__c =Date.today();          }          update values;         return null;                       }     }   VF Page:  &lt;apex:page showheader="false" controller="updateObject" sidebar="false"&gt; &lt;apex:form&gt;  &lt;apex:commandButton action="{!updateObject}" value="Update"/&gt; &lt;/apex:form&gt; &lt;/apex:page&gt; What happens if light/particles exceeded the speed of light for a particular medium? So I had 20+ Pokemon on my account and today when I tapped on the app, it brought me back to the original screen when you first start playing. My friend who signed up tested using my username when she first started and my username was unavailable, meaning the account still exists. I don't have the password for the account, but I do have the email. So I wanted to know if it was possible to log back into an account? Prove that the power set of an $n$-element set contains $2^n$ elements Theorem. Let $X$ denote an arbitrary set such that $|X|=n$. Then $|\mathcal P(X)|=2^n$.      Proof. The proof is by induction on the numbers of elements of $X$.      For the base case, suppose $|X|=0$. Clearly, $X=\emptyset$. But the empty set is the only subset of itself, so $|\mathcal P(X)|=1=2^0$.      Now the induction step. Suppose $|X|=n$; by the induction hypotesis, we know that $|\mathcal P(X)|=2^n$. Let $Y$ be a set with $n+1$ elements, namely $Y=X\cup\{a\}$. There are two kinds of subsets of $Y$: those that include $a$ and those that don't. The first are exactly the subsets of $X$, and there are $2^n$ of them. The latter are sets of the form $Z\cup\{a\}$, where $Z\in\mathcal P(X)$; since there are $2^n$ possible choices for $Z$, there must be exactly $2^n$ subsets of $Y$ of which $a$ is an element. Therefore $|\mathcal P(Y)|=2^n+2^n=2^{n+1}$. $\square$      From the above explanation, I don't understand why the set that contains $\{a\}$ will contain $2^{|n|}$ elements when it should clearly be $2^{|1|}$.  The construction of a new set $S$ is the union of the old set with cardinality $n$ and an new element $\{a\}$, therefore the set that do not contain $\{a\}$ still has cardinality $n$ and the set that contains $\{a\}$ is just $\{a\}$, one element...  Can someone please elucidate? Back transforming regression results when modeling log(y) I'm fitting a regression on the $\log(y)$.  Is it valid to back transform point estimates (and confidence/prediction intervals) by exponentiation?  I don't believe so, since $E[f(X)] \ne f(E[X])$ but wanted other's opinions.  My example below shows conflicts with back transforming (.239 vs .219).  set.seed(123)  a=-5 b=2  x=runif(100,0,1) y=exp(a*x+b+rnorm(100,0,.2)) # plot(x,y)  ### NLS Fit f &lt;- function(x,a,b) {exp(a*x+b)}  fit &lt;- nls(y ~ exp(a*x+b),  start = c(a=-10, b=15))  co=coef(fit) # curve(f(x=x, a=co[1], b=co[2]), add = TRUE,col=2,lwd=1.2)  predict(fit,newdata=data.frame(x=.7)) [1] 0.2393773  ### LM Fit # plot(x,log(y)) # abline(lm(log(y)~x),col=2) fit=lm(log(y)~x) temp=predict(fit,newdata=data.frame(x=.7),interval='prediction') exp(temp)         fit       lwr       upr 1 0.2199471 0.1492762 0.3240752 Code blocks are not prettified on mobile site On Chrome on Android, code blocks on the mobile site have no syntax highlighting, are not a monospaced font, and words are wrapped.  Mobile site on mobile browser:    Desktop site on mobile browser:    Inline code has the same issue, it isn't a monospaced font.  It has been this way for as long as I can remember. Wouldn't be surprised if this is a dupe. It is always told as a fact without explaining the reason. Why do two objects get charged by rubbing? Why one object get negative charge and other get positive ?