Question: Let $V$ be span of $(1,1,1)$ and $(0,1,1)∈\mathbb{R^3}$, let $u_1=(0,0,1)$, $u_2=(1,1,0)$, $u_3=(1,0,1)$ then which of the following are correct?   $1.(\mathbb R^3\backslash V)\cup \{(0,0,0)\}$ is not connected.  $2.(\mathbb R^3\backslash V)\cup \{ tu_1+(1-t)u_2:0\le t\le 1\}$  is connected.  $3.(\mathbb R^3\backslash V)\cup \{ tu_1+(1-t)u_3:0\le t\le 1$} is connected.  $4.(\mathbb R^3\backslash V)\cup \{(t,2t,2t):t\in \mathbb R$} is connected.  My attempt: I know what is definition of connected sets, but usually I had applied the definition to subsets of $\mathbb{R}$ but yet i didn't applied it subsets of bigger space $\mathbb{R^3}$. But to start the problem, first I had to find $V$.   Now, $V= span\{(1,1,1),(0,1,1)\}$  $= \{x(1,1,1)+y(0,1,1): x, y ∈\mathbb{R^3}\}$  $=\{(x,x+y,x+y): x,y ∈\mathbb{R}\}$  Now, what will be, $(\mathbb R^3\backslash V)$? it's complement of $V$ in bigger space $\mathbb {R^3}$, beyond my imagination!!  Got stuck! From hours, Please help me..
Let $V$ be a subspace of $\mathbb R^3$ generated by the vectors $\{(1,1,1),(0,1,1)\}.$ There are $3$ vectors $u_1=(0,0,1),u_2=(1,1,0),u_3=(1,0,1)$.Then which of the following sets are connected?  $1.\left(\mathbb R^3\setminus V\right)\cup\{(0,0,0)\}$  $2.\left(\mathbb R^3\setminus V\right)\cup \{tu_1+(1-t)u_3:t\in [0,1]\}$  $3.\left(\mathbb R^3\setminus V\right)\cup\{tu_1+(1-t)u_2:t\in [0,1]\}.$  $4.\left(\mathbb R^3\setminus V\right)\cup \{(t,2t,2t):t\in \mathbb R\}$  Now $V$ is generated by two linearly independent vectors so is like $\mathbb R^2.$ So $\left(\mathbb R^3\setminus V\right)$ will be disconnected but giving the $(0,0,0)$ with them will make it connected.  No. $4$, I think will be connected also because $\{(t,2t,2t),t\in \mathbb R\}$ is in $V$ and contains the point $(0,0,0)$ Now, any neighbourhood of this point will have points from both $V$ and $\left(\mathbb R^3\setminus V\right)$ and so the union is connected.I'm not sure about no. $2$ and $3.$
Do I need a bus/train/plane ticket out of Thailand in order to get a Visa-on-Arrival? Someone told me I need a ticket out of Thailand in order to enter.  I am traveling to Thailand, on March 18th. I have a Ticket from Bangkok to Sweden booked for June 4th.   I will be staying in Thailand for 2 weeks when I arrive there, then traveling around Cambodia, Laos and Vietnam until June 4th when I leave.   With that ticket out of Thailand, will I be able to get in and get a (30 day) Visa-On-Arrival (or not because the ticket out is not within the first 30 days). Or do I need to show proof that I am leaving within 2 weeks of being there at first? Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$. So basically this proof centers around proving that (S,*) is a group, as it's quite easy to see that it's abelian as both addition and multiplication are commutative. My issue is finding an identity element, other than 0. Because if 0 is the identity element, then this group won't have inverses.  The set explicitly excludes -1, which I found to be its identity element, which makes going about proving that this is a group mighty difficult. I have a query going that gets data for an ID for the last 3 months.  I need to tweak it so I get the highest value for each of the three months.  I've tried a couple of things with the aggregate function MAX, but I'm not getting anywhere.  I'm trying to get the max value for each of the past months ....  Here's the data from the query, currently sorted by date (asc):   ID      Date               Value 12410   01/03/2017 12:17    0.000178 12410   01/10/2017 11:36    0.000186 12410   01/17/2017 11:27    0.000189 12410   01/24/2017 13:09    0.000182 12410   01/31/2017 10:37    0.000169 12410   02/07/2017 11:03    0.000214 12410   02/14/2017 11:52    0.000176 12410   02/21/2017 10:51    0.000200 12410   02/28/2017 12:29    0.000194 12410   03/07/2017 08:39    0.000206   Here's the query:   select AnalysisID as "ID" , AnalysisDateTime as "Date", AnalysisValue as "Value" from AnalysisValueTbl where AnalysisID = 12410 and DatePart(m, AnalysisDateTime) = DatePart(m, DateAdd(m, -3, getdate())) and DatePart(yyyy, AnalysisDateTime) = DatePart(yyyy, DateAdd(m,-3, getdate())) or AnalysisID = 12410 and DatePart(m, AnalysisDateTime) = DatePart(m, DateAdd(m, -2, getdate())) and DatePart(yyyy, AnalysisDateTime) = DatePart(yyyy, DateAdd(m,-2, getdate())) or AnalysisID = 12410 and DatePart(m, AnalysisDateTime) = DatePart(m, DateAdd(m, -1, getdate())) and DatePart(yyyy, AnalysisDateTime) = DatePart(yyyy, DateAdd(m,-1, getdate())) order by AnalysisValue desc Looking for the proof of the lemma asserting that the conical surface (envelope) is a closed space. Thank you. In Star Wars: "The Force Awakens," how could this lightsaber have turned up where it did? In The Force Awakens:      Maz Kanata tells Rey that the lightsaber she just picked up and had visions from belonged to Anakin and Luke and now it's hers.   The lightsaber that Luke got from Obi-Wan Kenobi in A New Hope was lost/destroyed when Luke fought Darth Vader in Star Wars: The Empire Strikes Back.  In Return of the Jedi Darth Vader compliments Luke on building a new lightsaber and tells Luke that his skills are now complete.   How could that possibly be the lightsaber that Anakin used originally, and that was then passed on to Luke, when that lightsaber no longer exists?  Writers notes, director's notes, interviews, cast interviews, etc. are welcome. Thanks. Gapless superconductivity I don't understand what is meant by &quot;gapless superconductors&quot;. As far as I know, the superconducting gap is necessairy for both zero resistance and the Meissner effect, the two hallmarks of superconductivity. So, what does it mean to have gapless superconductivity and how does it work? Thanks in advance, Greetings Sum of the series $\sum _{ r=1 }^{ \infty }{ \frac { { r }^{ 3 } }{ r! } } $ How important is the following in purchasing a laptop that will be used for image editing:   processor speed amount of RAM integrated vs. dedicated graphics card HD speed (SSD; or 5400/7200/10000 rpm) Can you give me an example of a function which is sequentially continuous but not continuous? (I know that in first-countable spaces this is equivalent, but what about in spaces without this condition?)  Thank you :) Dell Inspiron 3551 Unresponsive Touchpad I just got my laptop in the mail and I installed Ubuntu 14.04 LTS. It all went smooth until it booted up and I noticed my touch pad doesn’t work at all. It works perfectly on Windows and when I use a wireless mouse it works as well.  Here's my xinput:  ⎡ Virtual core pointer                          id=2    [master pointer  (3)]   ⎜   ↳ Virtual core XTEST pointer                    id=4    [slave  pointer  (2)]   ⎜   ↳ 2.4GHz RF  KEYBOARD  AND  MOUSE               id=11   [slave  pointer  (2)]   ⎜   ↳ DLL06AB:00 06CB:78F1                          id=13   [slave  pointer  (2)]   ⎜   ↳ SynPS/2 Synaptics TouchPad                    id=15   [slave  pointer  (2)]   ⎣ Virtual core keyboard                         id=3    [master keyboard (2)]       ↳ Virtual core XTEST keyboard                   id=5    [slave  keyboard (3)]       ↳ Power Button                                  id=6    [slave  keyboard (3)]       ↳ Video Bus                                     id=7    [slave  keyboard (3)]       ↳ Power Button                                  id=8    [slave  keyboard (3)]       ↳ Sleep Button                                  id=9    [slave  keyboard (3)]       ↳ 2.4GHz RF  KEYBOARD  AND  MOUSE               id=10   [slave  keyboard (3)]       ↳ Integrated_Webcam_HD                          id=12   [slave  keyboard (3)]       ↳ AT Translated Set 2 keyboard                  id=14   [slave  keyboard (3)]       ↳ Dell WMI hotkeys                              id=16   [slave  keyboard (3)]     I would love to have my track pad working and I've done some googling and browsing of this site and nothing I've tried has worked. Any input would be great. Let me know if there’s any other information that can help remedy this. I'm pretty new to Linux. Thank you very much.  Edit: I've updated the kernel to 3.19 and checked my Additional Drivers which had nothing for me. Neither of which worked. "I feel bad for you" versus "I feel badly for you" I ran into an interesting (and very frustrating) issue with the equals() method today which caused what I thought to be a well tested class to crash and cause a bug that took me a very long time to track down.   Just for completeness, I wasn't using an IDE or debugger - just good old fashioned text editor and System.out's. Time was very limited and it was a school project.  Anyhow -   I was developing a basic shopping cart which could contain an ArrayList of Book objects. In order to implement the addBook(), removeBook(), and hasBook() methods of the Cart, I wanted to check if the Book already existed in the Cart. So off I go -  public boolean equals(Book b) {     ... // More code here - null checks     if (b.getID() == this.getID()) return true;     else return false; }   All works fine in testing. I create 6 objects and fill them with data. Do many adds, removes, has() operations on the Cart and everything works fine. I read that you can either have equals(TYPE var) or equals(Object o) { (CAST) var } but assumed that since it was working, it didn't matter too much.  Then I ran into a problem - I needed to create a Book object with only the ID in it from within the Book class. No other data would be entered into it. Basically the following:  public boolean hasBook(int i) {     Book b = new Book(i);     return hasBook(b); }  public boolean hasBook(Book b) {     // .. more code here     return this.books.contains(b); }   All of a sudden, the equals(Book b) method no longer works. This took a VERY long time to track down without a good debugger and assuming the Cart class was properly tested and correct. After swaapping the equals() method to the following:  public boolean equals(Object o) {     Book b = (Book) o;     ... // The rest goes here    }   Everything began to work again. Is there a reason the method decided not to take the Book parameter even though it clearly  was  a Book object? The only difference seemed to be it was instantiated from within the same class, and only filled with one data member. I'm very very confused. Please, shed some light? Replace \input{fileX} by the content of fileX automatically Suppose I have a document with multiple include or input statements  \input{fileA} \input{fileB}   etc.   Is there an easy way to generate a single .tex file where \input{fileA} is replaced by the actual content of fileA etc. without copying it manually? How do you get the logical xor of two variables in Python? If I perform RSA on my input 5 times, I get the original input back out. Why is it that often, iterating RSA encryption just a few times cycles yields back the original value, when the public modulus N is small? Number of ordered pairs $(a,b)$ such that $ab \le 3600$ Find number of ordered pairs $(a,b)$ such that $ab\le 3600$ and $a,b \in N$  My attempt : Well, all $a,b \leqslant 60$ are solutions. These 3600 solutions. After that I have no idea how to count the number of remaining solutions...   I have a feeling that my approach is $very$ bad. But I had to give as it would be marked as off-topic. How can I install .ipa file to my iPhone simulator? I need to install .ipa file to my iPhone simulator.   I tried to change the extension .ipa to .zip and extracted it.  And I got the Payload folder and moved the application inside to my iPhone simulators applications folder found at: /Library/Application\ Support/iPhone Simulator/6.1/Applications/.   Application appears on iPhone Simulator, But the application did't running. How to solve the problem then how to run the app. Remove files with 0 length inside bash I want to create a bash script inside Linux terminal to go into a file and then delete it if it's empty:  #!/bin/bash echo " Enter name of a directory" read Dir for Directory in $( ls $HOME ) do    test -d $HOME/$dir   if [ $# -eq 0 ]   then      for filename in $( ls $HOME/$Dir )     do        test -d $HOME/$dir/$filename       if [ ! -s "$filename"  ]       then         rm $HOME/$dir/$filename       else          echo " not empty"       fi      done    else      echo " $Dir is not a directory"    fi  done   This is the code I am using but it does not work. It recognizes the file created but doesn't delete it even when empty. It Salah displays there is no such file or directory Should I compile release builds with debug info as "full" or "pdb-only"? For a given inner product space $V$, I would like to prove the inequality $$\|\mathbf{x}+\mathbf{y}\|^2 + \|\mathbf{x}-\mathbf{y}\|^2 = 2(\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2).$$  We see that $$\|\mathbf{x}+\mathbf{y}\|^2 + \|\mathbf{x}-\mathbf{y}\|^2 = \langle\mathbf{x}+\mathbf{y},\mathbf{x}+\mathbf{y}\rangle + \langle\mathbf{x}-\mathbf{y},\mathbf{x}-\mathbf{y}\rangle,$$  which by linearity $$=\langle\mathbf{x},\mathbf{x}+\mathbf{y}\rangle + \langle\mathbf{y},\mathbf{x}+\mathbf{y}\rangle + \langle\mathbf{x},\mathbf{x}-\mathbf{y}\rangle - \langle\mathbf{y},\mathbf{x}-\mathbf{y}\rangle.$$  I've been trying to figure out how to go about this proof using linearity in the second argument of an inner product, but my textbook does not say that linearity necessarily holds in the second component. If it does, how would I go about the rest of this proof? If not, should I be potentially using some aspect of conjugate symmetry to prove this statement?