Whether range of matrix $A^TA$ is the same as that of $A^T$
Row space and column space of $A^T A$ and $A A^T$
How do I fix this "E: Type '*' is not known on line * in source list ..." update error? 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) Background  Recently  was implemented. With this feature the amount of questions with new activity is shown in parenthesis at the beginning of the page title, e.g.     (2) Meta Stack Overflow   means that MSO has two questions with new activity since the user last viewed it.   this feature is unrelated to inbox updates. Here is an example to illustrate this:    Even though there are two inbox messages in this screenshot this is not reflected by the page title. The page title is currently only sensitive to questions with new activity.  Suggestion  I suggest that the page title also reflects inbox updates. For the screenshot above this would mean that the page title shows that there are two inbox updates and two questions with new activity.  I am not sure what format is best for showing this. For chat * in the page title is used to show that there are messages directed at you which is similar in meaning to inbox updates. However, * if is placed after the number in parenthesis to denote inbox updates there might be confusion as there is no connection between the number in parenthesis and * like in chat. Thus, one might prefer another format such as one of the following (all are meant to reflect the status of the screenshot above):      * (2) Meta Stack Overflow     2* (2) Meta Stack Overflow     [2] (2) Meta Stack Overflow     2i 2q  Meta Stack Overflow     (2/2)  Meta Stack Overflow     (2|2)  Meta Stack Overflow    For the alternatives without an explicit hint of which number refers to what it might be best to place the number for questions with new activity first as it will in most cases be greater. This will to a greater degree avoid combinations such as "(0/2)" which may be confused zero of two.  An advantage of the suggested feature is that if the user has a browser tab with a Stack Exchange site then they will be instantly notified of inbox updates. They do not need to focus a tab with a Stack Exchange site to know if there are inbox updates, they merely have to check the relevant page titles among their tabs.  A disadvantage of the suggested feature is that it might be experienced as verbose. I suggest that it is implemented such that it is as unobtrusive as possible while still being informative. Think of a situation where a user has   one Meta Stack Overflow tab with two inbox updates relating to Meta Stack Overflow and two questions with new activity, and one Super User tab with one inbox updates relating to Super User and three questions with new activity.   Then the sum of the inbox notification in the page titles should be local to sites and not global. The page titles for the tabs above should be something like the following pair:      2i 2q  Meta Stack Overflow     1i 3q  Super User Why don't the White Walkers take the sea route to Westeros After watching 6 seasons of Game of Thrones, we know that the only roadblock for White Walkers is "The Wall" to stop them from attacking Westeros. However, we see that King Stannis attacked wildlings through a ship route without crossing the Wall.  What is stopping the White Walkers from doing the same? They can ask wights to make Ships or rowboats. They have a lot of forests to get wood for making ships. They can simply land at the coast of the North and attack Winterfell or whatever they want. Double Factorial: Number of possibilities to partition a set of $2n$ items into $n$ pairs I know that the partition of $2n$ items into $n$ pairs has something to do with double factorial, but I am not sure how many possibilities we exactly have.   We can choose such a partition into pairs by first choosing $n$ elements to be in one half and the remaining $n$ in the other half, which gives ${{2n} \choose n}$ possibilities. Then we can take a permutation of both sets and match them accordingly.   This gives $${{2n} \choose n} n! n!=\frac{(2n)! n! n!}{n! n!}=(2n)!$$ possibilities. But with this we overcount, since we do not want the pairs to be ordered. Each of the $n$ pairs is counted twice, wherefore we have to divide by 2 for every pair, hence $\frac{(2n)!}{2^n}$?   Well, another possiblity to do that is just to iteratively pick $2$ elements among the remaining ones, which gives ${{2n} \choose 2} {{2n-2} \choose 2} \dotsb {4 \choose 2}$  I am not sure, but I do not think that this is the same and also, I do not see the connection to double factorial. Would be very glad if you could help me with these 3 possiblities.   Thank you. Flight cancelled but doesn't show up on status I booked a SpiceJet flight from Dubai to Ahmedabad on Dec 16 (SG-16) scheduled to depart at 7:25 PM. At noon I received an SMS from SpiceJet to advise me that my flight has been delayed by 4h 50m. Sure enough, when I checked flight status on SpiceJet's website, that was the case. But on other websites like flightstats.com and dubaiairports.ae, the status was that flight was "delayed by 2h 25m".    At 5 PM, I got an SMS that flight had been cancelled. I double checked on Spicejet's website and that too showed the flight to be cancelled. But both flightstats.com and dubaiairports.ae still showed the flight as delayed by 2h 25m.    I went to the airport and met SpiceJet officials. Sure enough, the flight was cancelled. But somehow, the flight cancellation status never made it to the system. I have a Tripit Pro account, and even that never sent me an alert for that. I contacted the travel agent through which I booked my flight (makemytrip) for a refund, and they said that their system doesn't show that flight was cancelled. Now, even the Spicejet website has removed the cancellation status from that flight.    Apart from the SMS, I now have no proof that my flight was cancelled.    Can someone tell me, how can I proceed with a refund claim? Select translation: where? Is $\{\frac{m}{10^n}\mid m,n\in\mathbb Z,\ n\geq 0\}$ dense in $\mathbb R$? The set $S$ of real numbers of the form $m/10^n$, $m,n$ integers and $n$ greater than or equal to $0$, is a dense subset of $\mathbb R$ or not??   I know dense means closure of $S$ in $\mathbb R$ is $\mathbb R$ then it will be dense. How to prove or disprove I have no idea. The literature defines multinomial coefficients (or extended bnomial coefficients) as   $$ \binom{n}{r_1,r_2,\cdots,r_l} = \frac{n!}{r_1!r_2!\cdots r_l!}$$ where $$ r_1+r_2+\cdots+r_l = n$$  Which is the coefficient of $x_1^{r_1}x_2^{r_2}\cdots x_l^{r_l}$ in expansion of $(x_1+x_2+\cdots+x_l)^n$.  Question: How do we call the coefficients of $x^r$ in expansion of $(x+x^2+\cdots+x^l)^n$. Is it bad to name them multinomial coefficients?  In case of binomial coefficients the two are same in expansion of $(x_1+x_2)^n$ and the expansion of $(x+x^2)^n$. Obviously the two are different for multinomial expansion.   Example: As $(x_1+x_2+x_3)^2= x_1^2+2 x_2 x_1+2 x_3 x_1+x_2^2+x_3^2+2 x_2 x_3$. The coefficients are $\{ 1,2,2,1,1,2 \}$. Are these MULTI-NOMIAL?  $ As (x+x^2+x^3)^2=x^2 + 2 x^3 + 3 x^4 + 2 x^5 + x^6$. The coefficients are $\{1,2,3,2,1\}$. Are these too the MULTI-NOMIAL?  I prefer different terminologies to avoid confusion. Therefore want to know how these are distinguished in existing literature. Is it "a SSD" or "an SSD"? Possible Duplicates:             Since SSD (solid-state drive) is pronounced es-es-dee, I'm wondering whether one should write "an SSD" or "a SSD". Saying "a SSD" out loud feels a bit off... Fastest method to determine my PC's IP address (Windows) This may seem trivial but I'm looking for the quickest method to determine my PC's IP address within my network.   This changes regularly as I connect from one network to the next via DHCP The connection changes from wired to wireless Windows 7 PC (although ideally the perfect solution would work on all versions of Windows) Various adapters installed (e.g. VMWare, Tunnels, etc.)   I need to know this as I often run a local web server where I need to access files over the local network... and since it changes regularly I want the fastest method to retrieve the address.  My current flow is:   WindowsKey + R to open the Run dialog "cmd" + Enter to open a command prompt "ipconfig" + Enter to get the diagnostics info Scroll to or resize the window so that I can see the line in my Ethernet adapter Local Area Connection for my ipv4 Address Remember it to type elsewhere... or  Right click > select Mark > highlight the address > press Enter to copy it to the clipboard   This wasn't so bad with Windows XP when I had no additional adapters, tunnels, wireless connections etc. but the amount of data returned with this command makes it hard to pluck out.  Surely there must be a better, faster way! (bonus points if adding it to the clipboard is easily accomplished) Is there a way to calculate the improper integral $\int_0^\infty \big(\frac{\ln x}{x - 1}\big)^2 dx$?  What have I tried:  $$\int_0^\infty \Big(\frac{\ln x}{x - 1}\Big)^2 dx = \int_0^1 \Big(\frac{\ln x}{x - 1}\Big)^2 dx + \int_1^\infty \Big(\frac{\ln x}{x - 1}\Big)^2 dx = \int_1^\infty \Big(\frac{\ln x}{x - 1}\Big)^2dx + \int_1^\infty\Big(\frac{\ln t}{\big(\frac{1}{t} - 1\big)t}\Big)^2dt = 2\int_1^\infty \Big(\frac{\ln x}{x - 1}\Big)^2dx$$  From the above statement we can conclude, that the integral does indeed converge, as $0 \leq (\frac{\ln x}{x - 1})^2 \leq \frac{C}{(x - 1)^{\frac{3}{2}}}$, for some constant $C$. However, this new form of the integral is not much helpful in finding its exact value.  I also tried substituting $z = \ln x$ , then $dz = \frac{dx}{x}$. Then the integral becomes reduced to $2\int_{0}^{\infty} \frac{z^2}{e^z(e^z - 1)^2}dz$, which, unfortunately, also does not seem to be any easier. Add "Delete with reason" button/dialog to existing moderator tools for flagged answers While this is an old problem, it's one that will only increase with time.  As Stack Exchange sites gain in popularity, more and more users are coming to the sites in search of answers to their specific problems.  Unfortunately, not all of them understand the basic operating principals behind Stack Exchange, and ask questions, or start discussions as separate answers.  Deleting these answers is fine, but more often (I would argue almost always) than not, a new user doesn't understand why.  Granted, comments are a way of doing this, but there are drawbacks to comments:   They require extra, replicated effort on the part of the moderator for something that they do repetitively The message will be inconsistent; messages like this would be consistent, cite the relevant resources that guide the user in the right direction The larger size would have a higher chance of grabbing the user's attention, as well as put more weight behind the action (as it's officially attached to the answer in a consistent manner). The banner will serve as an example of what not to do to other users who can see the deleted post.   Additionally, there are tools that can help with auto-commenting on posts, but these have their drawbacks as well:   Not all moderators use these tools The messages that are delivered by the tools are ultimately generated initially by the moderators, so the message will not be consistent These tools are not always available on other devices (e.g., mobile platforms)   It would be helpful to the user if they had a reason indicating why the answer was deleted.  Something along the lines of:    In order to accomplish this, I propose an additional (not a replacement) button in the moderation tools labeled "delete with reason".  Mods would still be able to delete answers outright, they just have the option now of stating why in an easy manner which will allows for more time to moderate harder.  Placement is not important (screen real estate is valuable, yes), but I envision it looking something like this in the moderator tools:    When clicking "delete with reason", a dialog, much like the "close" dialog will appear with options:    Initial options for the reasons could be:   Asking a new question Starting discussion   A mechanism like this has the potential for other benefits as well:   Users with an answer deleted with a reason might actually be directed to perform the right action now, and in the future (creating better site denizens) Users who wish to contest the deletion of an answer on meta can cite that specific reason, which mods can specifically address (which actually might reduce noise on meta about "why was my answer deleted?") Can reduce flags on answers asking "why was this deleted?" (admitted, I've not personally experienced this to be a pervading problem on Stack Overflow, but this might be the case for other mods on other Stack Exchange sites) Other Stack Exchange sites which don't have that many users will give a better experience to their users when their first action isn't ideal for a Stack Exchange site; this will enhance the chance that the site will retain the user and get good contributions from them (as per Farseeker's comments: , ).   Additionally, if this works well for the moderator tools, I imagine that it would not be such a leap to eventually have this graduated to users with the rep to cast delete votes, thus allowing non-mods to deliver a more clear message to new users about how to best participate on a Stack Exchange site. How can I create a progress bar with bash?  This is my script :  #!/bin/bash  pass='number1 number12 number13 number14 number15 number16'  chk='number14'  for i in $pass ; do    if [ "$i" == "$chk" ]; then      echo ' Found ^_^'    else      echo 'loading 50%'    fi  done   I want to replace echo 'loading 50%' with anything to create an progress bar. What's the word for the when you suck snot back up your nose? My mom and I say 'soup' like: "Why are you souping the snot back up your nose?" But I realise that this is not accurate. So what's the word? What documents to show if I'm a US citizen with no US passport to re-enter the US? Suppose I am a naturalized citizen of the US, but also a citizen of another country, "XYZ". I then boarded a plane and left the US to another country (maybe not necessarily to XYZ), without first getting a US passport, but using my old, but valid XYZ passport.   In this case what would be the best documents to have for me to gain entry back into the US as a US Citizen without a passport?  Would I need to contact the US Embassy for help in order to be able return?  If the airline does let me aboard to return to the US, even without the US passport, what are the best documents could I show at the border to prove I am a US citizen but that I just don't have a passport, for whatever reason? Post apocalyptic journey in search of knowledge from fallen society (ours) I'm racking my memory for a post-apocalyptic novel that I borrowed from a library maybe 15 years ago, but I'm sure it was published a few years before that though (maybe early 90s?).  I remember a party of people travelling across the flooded USA after some kind of apocalypse (climate change/flooding?). Towards the end of the book the group makes their way to New York(?) and finds a partially flooded library, discovering a set of books that gives hope for the return/rebuilding of society. Geometric series and big theta Consider the following function:  $$S(n)=1+ c + c^2 + ··· + c^n,$$  where c is a positive real number.  (A) This function is the sum of a geometric series. Give a precise closed-form formula for $S(n)$, interms of $c$ and $n$, in the case where $c \neq 1$.  (B) Show that $S(n)$  is:  $\theta(1)$ if $c&lt;1$  $ \theta(n)$ if $c=1$  $\theta(c^n)$ if $c&gt;1$    can someone please explain how i can go about finding the solutions for these problems, i have never encountered these and am stuck, i know that big theta means that the function is bond on both sides but how do i got about finding the answer?  (edit) im still having trouble understanding part B. can someone please explain? (can someone walk me through the second one) so far i have $n \leq c^{n+1} \leq ???$  (edit #2) i tried doing the first one can someone verify if i did it right  $S(n)$ is $\Theta(1)$ if c&lt;1  (1)$S(n)$ is $O(1)$ if c&lt;1  (2)$S(n)$ is $\Omega(1)$ if c&lt;1  (1) $$\frac{C^{n+1}-1}{C-1} \lt (C_1)(1)$$ $$\frac{C^{n+1}-1}{C-1} \lt (C_1)$$ $$C^{n+1} \lt C_1(C-1)$$ $$C^nC-1 \lt C_2C-1$$ $$C^n \lt C_2$$ $$C^n \lt 1$$ since c&lt;1 this means that any number that will be $C^n$ will be less then 1 for all positive n  i did the omega part pretty much the same way and got $$C^n \gt C_2$$ $$C^n \gt 0$$ since c&lt;1, $C^n$ can potentially become very very small so the only thing bounding it is 0 for all positive n Since arrays and hashes can only contain scalars in Perl, why do you have to use the $ to tell the interpreter that the value is a scalar when accessing array or hash elements? In other words, assuming you have an array @myarray and a hash %myhash, why do you need to do:  $x = $myarray[1]; $y = $myhash{'foo'};   instead of just doing :  $x = myarray[1]; $y = myhash{'foo'};   Why are the above ambiguous?   Wouldn't it be illegal Perl code if it was anything but a $ in that place? For example, aren't all of the following illegal in Perl?  @var[0]; @var{'key'}; %var[0]; %var{'key'}; How to label using just numbers from start of text field?