Suppose that we have a torus $T^2$, and if we remove one point onit then it can deform retract onto the wedge of 2 circles, we can achieve it by first removeing one point on the polygonal region that represent the $T^2$ and then the polygonal region with one pointed removed can be retracted on its edge, so how to generalize this process to the case in which finite points are removed
Question 5.33 from "Topology and its Applications"  by Baesner is to compute the fundamental group of the torus ($T^2$) with $n$ points removed. I can "see" in my mind that if we remove one point we get a bouquet of two circles. Less clear is what happens when we remove two (or more) points. Any hints?
Why is "can I get" replacing "could I have"? I noticed the other day when serving the public that when asking for something, people were saying "Can I get an xyz, please". The previous time I had such a job it was "Could I have an xyz", or "May I have..." if they were being very polite.  Does anyone know when this trend started and what caused it? A function such that $f:\mathbb C \to \mathbb C$ such that $f(z)f(iz)=z^2$ satisfies $f(z)+f(-z)=0$.(MADHAVA-2020) Suppose $f$ is a function such that $f:\mathbb C \to \mathbb C$ such that $f(z)f(iz)=z^2$, then we have to show that $\forall z\in \mathbb C f(z)+f(-z)=0 $.   Actually I have solved this problem and have no doubt about it but actually this question came in MADHAVA-2020. So I posted it on this site.   I am not answering the question and inviting all users to answer it.  It is a nice problem.   I will answer it within a week, but I am letting everyone try. Number theory counting sums of squares modulus This is an extra question from an old examination paper:     VI. Let $n&gt;1$ in $\mathbf{Z}$ and let $r(n) = \#\{(a,b)\in \mathbf{Z}^{2};n=a^{2}+b^{2}\}$ Let also $n=n&#39;n&#39;&#39;n&#39;&#39;&#39;$ where $n&#39;,n&#39;&#39;,n&#39;&#39;&#39; \in \mathbf{N}$ and   $p|n&#39; \Longrightarrow p\equiv 1\mod 4 ; p|n&#39;&#39; \Longrightarrow p=2 ; p|n&#39;&#39;&#39; \Longrightarrow p\equiv 3 \mod 4$   for $p\in \mathbf{N}$ prime.   i) Show that if $n&#39;&#39;&#39;$ is not a square, then $r(n)=0$.  ii) Show that if $n&#39;&#39;&#39;$ is a square and $n&#39; = 1$, then $r(n)=4$  iii) Show that if $n&#39;&#39;&#39;$ is a square and $1&lt;n&#39;=p_{1}^{e_{1}}\ddots p_{k}^{e_{k}}$ where $p_{1},...,p_{k}$ are different primes in $\mathbf{N}$ and $e_{1},...,e_{k} \in \mathbf{N}$, then it holds that $r(n)=r(n&#39;)=4(e_{1}+1)\ddots (e_{k}+1)$    I am completely dumbstruck and can't see how to begin (and neither did an older student who took this exam and whom I asked for hints). Help is greatly appreciated. Expressing the area of the cross section of a cylinder I was able to solve similar questions where a triangle was involved, but this one has had me looping and pondering on it for the past week.  The Condition: Let $r$ be a positive constant. Consider the cylinder $x^2+y^2\leq r^2$ and let C be  part of the cylinder that satisfies $0\leq z\leq y$.  The questions:  (1) Consider the cross section of $C$ by the plane $x=t(-r\leq t\leq r)$ and express its area in terms of $r,t$  (2) Calculate the volume of $C$, and express it in terms of $r$.  (3) Let $a$ be the length of the arc along the base of the circle $C$ from the point $(r,0,0)$ to the point $(r cos\theta,r sin\theta,0)(0\leq\theta\leq\pi)$. Let $b$ be the length of the line segment from the point $(r cos\theta, r sin\theta, 0)$ to the point $(r cos\theta, r sin\theta, r sin\theta)$. Express $a$ and $b$ interms of $r,\theta$  (4)Calculate the are of the side of $C$ with $x^2+y^2=r^2$, and express it in terms of $r$.  Here are the expected answers(which I keep failing to get to):  (1) $\frac{1}{2}(r^2-t^2)$  (2) $\frac{2}{3}r^3$  (3) $r\theta$ and $r sin\theta$  (4) $2r^2$ My first issue was when I tried to apt-get update or apt-get upgrade. On upgrade I get the following error:  You might want to run 'apt-get -f install' to correct these. The following packages have unmet dependencies: linux-image-server : Depends: linux-image-3.2.0-27-generic but it is not installed E: Unmet dependencies. Try using -f.   I tried running apt-get install -f and this was the output(after saying yes on prompt)  (Reading database ... 186183 files and directories currently installed.) Unpacking linux-image-3.2.0-27-generic (from .../linux-image-3.2.0-27-generic_3.2.0-27.43_amd64.deb) ... Done. dpkg: error processing /var/cache/apt/archives/linux-image-3.2.0-27-generic_3.2.0-27.43_amd64.deb (--unpack):  failed in write on buffer copy for backend dpkg-deb during `./boot/System.map-3.2.0-27-generic': No space left on device  No apport report written because the error message indicates a disk full error                                                                           dpkg-deb:    error: subprocess paste was killed by signal (Broken pipe) Examining /etc/kernel/postrm.d . run-parts: executing /etc/kernel/postrm.d/initramfs-tools 3.2.0-27-generic   /boot/vmlinuz-3.2.0-27-generic run-parts: executing /etc/kernel/postrm.d/zz-update-grub 3.2.0-27-generic /boot/vmlinuz-3.2.0-27-generic Errors were encountered while processing: /var/cache/apt/archives/linux-image-3.2.0-27-generic_3.2.0-27.43_amd64.deb E: Sub-process /usr/bin/dpkg returned an error code (1)   I tried running apt-get autoremove and it gives me the same error as apt-get upgrade.  When I run df, I get this for /boot:  /dev/sda1                    233191     230297         0 100% /boot   So, I read elsewhere that I should try to purge old kernels. I checked for what kernels I had with:  $ dpkg -l linux-image-\* | grep ^ii ii  linux-image-2.6.38-13-server  2.6.38-13.52  Linux kernel image for version 2.6.38 on x86_64 ii  linux-image-3.0.0-13-server   3.0.0-13.22   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.0.0-14-server   3.0.0-14.23   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.0.0-15-server   3.0.0-15.26   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.0.0-16-server   3.0.0-16.29   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.0.0-17-server   3.0.0-17.30   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.2.0-24-generic  3.2.0-24.39   Linux kernel image for version 3.2.0  on 64 bit x86 SMP ii  linux-image-3.2.0-25-generic  3.2.0-25.40   Linux kernel image for version 3.2.0  on 64 bit x86 SMP ii  linux-image-3.2.0-26-generic  3.2.0-26.41   Linux kernel image for version 3.2.0  on 64 bit x86 SMP   When I try to remove the oldest with this:  $ sudo apt-get purge linux-image-2.6.38-13-server Reading package lists... Done Building dependency tree Reading state information... Done You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies: linux-image-server : Depends: linux-image-3.2.0-27-generic but it is not going to be     installed E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution).   How can I free up or extend boot without messing up my installation? What do you call the symbols list on a map? Every map has symbols that describe things. In the corner of a map, there is a list that explains what each symbol means.  What is the name of that list? (What title should I put above this list)? What is the difference between an int and an Integer in Java and C#? I was reading  when I came across  saying something about a particular type of programmer knowing the difference between an int and an Integer in Java/C# (Object-Oriented Programming Languages).  So, what is the difference? If you draw two cards, what is the probability that the second card is a queen? We had this question arise in class today and I still don't understand the answer given. We were to assume that drawing cards are independent events. We were asked what the probability that the second card drawn is a queen if we take two from the deck. The answer given was 4/52, which seems counter-intuitive to me. How is the probability still 4/52 if there was a card drawn before it? What if the first card drawn was a queen? Why are one-way plane tickets more expensive than return tickets? How do I go about tracking down IO usage in Linux and other *nix OS'? Occasionally, something on my system starts hammering the disk. I would like to log data that would let me know about I/O usage per process, then I could go back to it later and see what keeps doing this. How should I go about solving this problem? Bonus points for solving this for arbitrary block devices. Single word that means "to look down on others"? If the electric field is the gradient of the potential, then can we say that whenever potential is zero, the electric field is zero? For example, in a dipole, at the center of the two charges making up the dipole, the potential is zero but the electric field is non-zero. But if $E = -\operatorname{grad}V$, then why is $E$ not zero?  Another example: In the case of two positive charges, the electric field is zero at the mid-point of line joining them, but potential is non-zero. What am I misunderstanding here? How to send AT commands to modem once the connection is established? I have a zte 3g modem. I use carrier provided dialer for connection establishment. Once the ppp connection is active, i would like to send some AT commands(for ex. Query signal strength, AT+CSQ). But the dialer i use locks the /dev/ttyUSB0 port, which is the command port to send AT commands for my modem. So is there any way, to send the commands, once the connection is active?    Edit: I also tried the additional port /dev/ttyUSB1. But the port is flowing with random data from the modem. A sample is given below.  T^PREFMODE??                                                                     ^PREFMODE:8                                                                       OK                                                                               TC                                                                               ^DSDORMANT:1                                                                      +CSQ:19, 99                                                                       OK                                                                               T^SYSINFO                                                                        ^SYSINFO:2,3,0,4,255                                                              OK                                                                               TT^SYSINFO?                                                                      ^SYSINFO:2,3,0,4,255     I tried adding my commands, i even got output. But the response is very poor. Most of the times, my AT commands went unnoticed. Is the function strcpy always dangerous? How do I need to configure SMTP Auth? I have dedicated FreeBSD box, The email sending without  is working perfectly but very slow. The receiver receives email after almost 15 mins. I checked the maillog it says delays and timeouts that means the server keeps on trying to send email again and again until it successfully sends one. This is consuming lots of CPU.  I am seeking help for configuring smpt module to use my dedicated box's  smtp server, I need help in following places where I have ???? sign:  #SMTP server settings SMTP server -&gt; smpt.mydomain.com  # Tested by pinging, its working SMTP backup server -&gt; mail.mydomain.com  # Also verified by Ping SMTP port -&gt; 25 USE SSL  #SMTP Authentication Username -&gt; ??????? Password -&gt; ???????   I tried using localhost, also username and password of the dedicated box or leaving both lines blank which gives me:     Unable to send e-mail. Contact the site administrator if the problem   persists.   also checking the debug log doesn't show any log, it says check admin/reports/dblog but when I got to the link it just shows me the contents from admin/reports no any dblog Why is my animation not keeping location and rotation correctly? I am currently 3d modeling a twisty puzzle and trying to animate it. I have the puzzle done but when I try to animate it, it randomly twists and turns the individual objects in different directions before converging back to their original form at the end of the animation.  The screencast of me attempting to animate this is here:   File at the end of the screencast:    This is the original file. Just delete the 2 keyframes that are there:    I did make this with a tutorial. It is by rcpongo and I have very slightly edited his design. All credit goes to him for my design. Currently I had built a file upload functionality in Lightning component which accepts the CSV file, read it as text(Each row in CSV is split using \n and each column in the row is split using ,). Then each columns in each row are validated in Javascript and if everything is fine,the data is saved in the backend. While uploading the data, if the file contains , in its data then it is taken as another column. Eg: X-36936,SS-341487,11-WT,414081,"LENDERS, THE HACK",2-Manage,,,,,,,,,,,,12/1/2013,3150,,0,,Dealing,2018,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,  Here "LENDERS, THE HACK" is a single column but it is taken as 2 columns (LENDERS --> 1st column and THE HACK --> 2nd column).Could anyone please help me how to resolve this issue? If I am introducing someone to my daughter's husband's parents can I say "Hi, I'd like to you meet my ___". In-laws would not work here because they are my child's in-laws not my own. Is there a word for this relationship? What's the best way to generate a tag cloud from an array using h1 through h6 for sizing? (a) $S_k=\{A\subset\mathbb{N}: |A|=k\}$ for $k\in\mathbb{N}$  (b) $S = \bigcup_{k=1}^\infty S_k$   Work: For (a), I am not too sure about what approach I should use. I think finding a bijective function between the $S_k$ and $\mathbb{N}$ is hard to do . I also thought about using the Schroeder-Bernstein theorem and find injective functions $f:\mathbb{N}\rightarrow S_k$and $g:S_k\rightarrow\mathbb{N}$ in order to prove that $S_k$ and $\mathbb{N}$ are numerically equivalent. However, I have a hard time finding and proving these injective functions. I haven't really looked at (b) as I have yet to complete (a).