Noticed a small bug in Mathematics Stack Exchange, though perhaps it might exist anywhere using MathJax. The winter hat (in this case, &quot;Gimme Space&quot;) shows up inside the rendered expressions. I do not know why it is in that expression specifically. The post shown below is .
My &quot;hat&quot; is shown twice in my answer, once in my sig, and again above. MacOS 10.14.6; Safari Version 14.0.2 (14610.3.7.1.10)
How do I remove the Me-User-Indicator from the Panel? How do I remove the Me-User-Indicator from the Panel?  The indicator contains two items: the written user name and the icon of a human. Set value of textarea in jQuery I am attempting to set a value in a textarea field using jquery with the following code:  $("textarea#ExampleMessage").attr("value", result.exampleMessage);   The issue is, once this code executes, it is not altering the text in the textarea?  However when performing an alert($("textarea#ExampleMessage").attr("value")) the newly set value is returned? Prove that $n! + k$ is a composite number Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3) Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)  SO this is for abstract algebra and I am really struggling with this. Here are some of the definitions and theorems I think would apply.  -an integer a divides and integer b if there is an integer q such that b=aq -Let a and b be integers, with a>0. Then there exist unique integers q and r such that b= aq+r and 0&lt;=r  The question says hint: use 2 cases but I'm not really sure what that means. I'm guessing it wants me to look at it when a is congruent to 0 and when it's not? so i tried to do that but just ended up stuck. below is as far as i got with that attempt  Proof: Case 1 There should be some information about letting variables exist in certain number sets here  assume a is not congruent to 0 mod 3, then 3 does not divide a-0, therefore a-0 does not equal 3q for some q in the integers  I was trying to get this to wind down to thus 3 divides a^2-1 but even tho it seems true (I couldn't find a counter example) I couldn't figure out how to prove it with what I have.  Next i tried reversing it thinking I could contradict it or something.  I think this would be case 2? assume a is congruent to 0 mod 3, then 3 divides a-0 and a-0=3q for some q in the integers, then 3q=a since 0 is the additive identity, therefore 3 divides a,  and again stuck and my brain hurts a little..  Can anyone help me figure out how to work this. Using lasso for feature selection, followed by a non-regularized regression I use Lasso logistic regression in order to identify a smaller subset of important variables. I start with N=51 (28/23) and 32 predictors.   So far it looks pretty promising, because I can identify four important predictors in my optimal model.   Now I would like to take those four predictors and examine them along with some control variables in a standard logistic regression.   My question is, does that analysis strategy make sense? Is there a better way to include controls or other variables that might be interesting?  For a better understanding:   Identify important variables via Lasso logistic regression Do further analysis including identified predictors and other control variables using standard logistic regression (using AIC to check model fit) I understand that division by zero isn't allowed, but we merely just multiplied $f(x) = 1$ by $\frac{x-1}{x-1}$ to get $f(x) = \frac{x-1}{x-1}$ and $a\cdot 1 = 1\cdot a = a$ so they're the same function but with different domain how is this possible?  Or in other words why don't we simplify $f(x) = \frac{x-1}{x-1}$ to $f(x) = 1$ before plotting the points. Is it just defined this way or is there a particular reason ?  Note: my book says the domain of $f(x) = 1$ is $\mathbb{R}$ and the domain of  $f(x) = \frac{x-1}{x-1}$ is $\mathbb{R}$ except $1$. Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = \gcd(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$. Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = hcf(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$.   We know that if $a|c$ and $b|c$ then $a\cdot b\cdot s=c$ (for some positive integer $s$). $(ab|c)$  Then doesn't $ab|dc$ since $ab|c$?  I feel like I'm misunderstanding my givens.    Can we say $\operatorname{lcm}(a,b)=c$,  $\operatorname{hcf}(a,b)=d$, and $\operatorname{lcd}(a,b) \operatorname{hcf}(a*b)=a*b$? Thus, $ab|dc$  as $dc = ab$. Determine Which of the following Sets is a Vector Space Determine which of the following sets is a vector space.  $V$ is the line $y=x$, in the $xy$-plane: $V =\{[x,y]: \; y=x\}$  $W$ is the union of the first and second quadrants in the $xy$-plane: $W=\{[x,y]: \; y\geq0\}$  $U$ is the line $y = x+1$ in the $xy$-plane: $U=\{[x,y]: \; y=x+1\}$    I know that the answer is that "$V$" is the only vector space. However how do I come up with a quick proof to show that $V $is a vector space and that the other $2$ aren't? I'm trying to run a gunicorn_django process in supervisor but it always exits immediately, giving this error:   INFO exited: my_app (exit status 1; not expected) INFO received SIGCLD indicating a child quit INFO gave up: my_app entered FATAL state, too many start retries too quickly   My server script looks like this:  #!/bin/bash set -e LOGFILE=/var/log/gunicorn/my_app.log LOGDIR=$(dirname $LOGFILE) NUM_WORKERS=3 USER=my-www-user GROUP=my-www-user cd /home/my-www-user/my_app source /home/my-www-user/.virtualenvs/my_app/bin/activate test -d $LOGDIR || mkdir -p $LOGDIR gunicorn_django -w $NUM_WORKERS --debug \  --user=$USER --group=$GROUP \  --log-level=debug --log-file=$LOGFILE 2&gt;&gt;$LOGFILE\  --pythonpath=my_app --settings=settings.active \  my_app.wsgi:application   And my supervisor config looks like this:  [program:my_app] directory=/home/my-www-user/my_app/ user=my-www-user command=/home/my-www-user/my_app/server.sh stdout_logfile=/var/log/supervisor/my_app.log stderr_logfile=/var/log/supervisor/my_app-error.log autostart=true autorestart=true   When I su into the my-www-user account the server starts normally. When I do sudo supervisorctl start my_app it simply hangs until I do CTRL-c, then I find the error above in the supervisord.log file.  Does anyone have a clue about what I am missing? Proving the summation formula using induction: $\sum_{k=1}^n \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$ I am trying to prove the summation formula using induction:  $$\sum_{k=1}^n \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$$  So far I have...  Base case:   Let n=1 and test   $\frac{1}{k(k+1)} = 1-\frac{1}{n+1}$  $\frac{1}{1(1+1)} = 1-\frac{1}{1+1}$    $\frac{1}{2} = \frac{1}{2}$   True for n=1   Induction Hypothesis:   Assume the statement is true for the n-th case   $\sum_{k=1}^n  \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$  Inductive Step:   Prove, using the Inductive Hypothesis as a premise, that   $$\sum_{k=1}^{n+1}  \frac{1}{k(k+1)} = \sum_{k=1}^{n}  \frac{1}{k(k+1)} + \frac{1}{(n+1)(n+2)} = 1-\frac1{n+1} + \frac{1}{(n+1)(n+2)} = \frac{(n+1)(n+2)}{(n+1)(n+2)}+\frac{-2-n}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)} = \frac{(n+1)(n+2)-2-n+1)}{(n+1)(n+2)} = \frac{(n+1)(n+2)-n-1}{(n+1)(n+2)} = \frac{n^2+2n+1}{(n+1)(n+2)} = \frac{(n+1)(n+1)}{(n+1)(n+2)} = \frac{n+1}{n+2}$$  To prove $$ 1-\frac{1}{n+2} = \frac{n+1}{n+2} $$ Multiply both sides by $n+2$ to get an equivalent expression. $$ (1-\frac{1}{n+2}) * (n+2) = (\frac{n+1}{n+2}) * (n+2)  $$ $$ n+1=n+2−1 $$  Does this all make sense? How can this be improved upon? What is the best regular expression to check if a string is a valid URL? How can I check if a given string is a valid URL address?  My knowledge of regular expressions is basic and doesn't allow me to choose from the hundreds of regular expressions I've already seen on the web. pgAdmin 4: How to establish a connection through an SSH tunnel/interface in Windows I installed the latest pgAdmin 4 (v2.1) but I found out that (), for the moment, the only way to establish a connection through an SSH tunneling is using another outside process, an  says:  I setup a tunnel as follows: ssh -f dpage(at)hostname(dot)enterprisedb(dot)com -L 5999: hostname.enterprisedb.com:5432 -N  (...) I was running on Mac, but the same procedure should work on Linux. On Windows, there's a tunnelling app in Putty iirc, but I don't know how to use it offhand.  I have found a simple . The connection is established but it seems, when I try to connect through pgAdmin it fails or is refused (using 127.0.0.1:3306 or 127.0.0.1:5432 as the tunnel's destination). Why? How to find the inverse modulo $m$? There are $25$ horses. You can take $5$ horses at a time and race them. Each horse always finishes the race in the same amount of time, and there are no ties. The only information you get from each race is the order that the $5$ horses finished in. What is the smallest number of races you need to find the $3$ fastest horses, in order?   I found this problem on  and cannot figure out the best answer. Please help. Let $X,Y$ be topological and $f:X \to Y$ be open and continuos.  I am studying Baire space and I would like to try the following facts :      $(i)$ The inverse image of dense set is dense.       $(ii)$ The inverse image of comeager set is comeager.   I am studying Baire space and I would like to try the following facts  Can we collaborate to verify the above claims?    Thanks. Ubuntu: Resize the root LVM(2?) partition Linear maps $f: M_n \longrightarrow \mathbb{R}$ such that $f(AB) = f(BA)$ Let $M_n$ be the vector space of $n \times n$ matrices over $\mathbb{R}$. Find (with proof) all linear maps $f: M_n \longrightarrow \mathbb{R}$ such that $f(AB) = f(BA)$ for all matrices $A$ and $B$.   I know that the trace map is one such $f$. The determinant map satisfies $det(AB) = det(BA)$ -- but isn't a linear map, so it doesn't suffice.   Is there a constructive way to find all of the desired linear maps $f$, and show that there are no others ?   Any help would be appreciated. Thanks (= Improve alignment of "Learn more" button in "Newest badge" popup The popup when you click on your newest badge in your user profile has a &quot;Learn more&quot; button, but its alignment is suboptimal; it's too close to the top of its container:  It looks better with a 4px top margin: Determine wether the differential equation is stable or not for the following conditions:  $x'''(0) = x''(0) = x'(0) = x(0) =0$  $\frac{d^3x(t)}{dt^3}+\frac{d^2x(t)}{dt^2}+4\frac{dx(t)}{dt}+4x(t) = e^t(H(t)-H(t-3))$  Now, I got that the solution to this differential equation reduces x(t) by use of Laplace Transforms, however, how do I determine whether or not it is stable?  $x(t) = 0.1e^{-t} + 0.1e^t - 0.1\sin(2t) + 0.1e^3 (\sin(2(t-3)) + e^{-t+3}-e^{t-3})*H(t-3)$  Now I first thought I would look at it graphically and I used Matlab software to get a plot as shown below:    And from the plot above it clearly appears to diverge (though this is not a proof).  Anyone have an idea as to an elegant proof to show the solution is unstable?  Anyone help would be greatly appreciated! How do I check whether a file exists without exceptions?