What do the numbers to the right of my name (point score) mean?
What are the numbers / symbols to the right of my reputation counter?
Polygamma reflection formula Converting a DC motor to a generator Can ANY DC motor become a generator if torque is applied to the shaft? example:   And by any, I mean the really cheap ones too.  Or is there a specific type of motor (of which I do not know the name) that can work in both directions (battery to power the shaft OR torque-on-shaft to create a voltage difference between the terminals). Cylindrical vs azimuthal symmetry What is the difference between cylindrical and azimuthal symmetry, are they the same? I have not been able to find a clear explanation of these terms wrt. a cylinder. I need to know which length I should integrate over to solve the problem below.  A cylinder of radius $R$ and length $L$ has a cylindrically symmetric volume charge density of $\rho(x)$. Calculate the total charge on the cylinder. Let $a,b$ be positive integers such that $a\mid b^2 , b^2\mid a^3 , a^3\mid b^4 \ldots$ so on , then $a=b$? Let $a,b$ be positive integers such that $a\mid b^2 , b^2\mid a^3 , a^3\mid b^4 \ldots$ that is $a^{2n-1}\mid b^{2n} ; b^{2n}\mid a^{2n+1} , \forall n \in \mathbb Z^+$ , then is it true that $a=b$ ? Rings, ideals and units-proof Let $R$ be a ring and let $J$ be an ideal of $R$. Assume $J$ contains   a unit of $R$. Prove $J=R$, rigorously.   Again I feel here as if the statement is intuitive so I am not able to make it rigorous. If you could show all steps in the proof that would be nice. I want to prove that $\lim_{(x,y)\to (0,0)} \frac{\sin(x+y)}{x+y} = 1$  Is it sufficient to say that if $u = x+y$ then as $(x,y) \rightarrow (0,0)$ then $u \rightarrow 0$ and then evaluate $\lim_{u\to 0} \frac{\sin u}{u}$ which is 1 by L'Hôpital's rule?  I feel that this is somehow "cheating" and I would really like to prove it in a more rigorous way, i.e., by using an $\epsilon$ - $\delta$ argument.  I have tried for a long time to use the one variable version and prove that if $0 \lt \sqrt{x^2 + y^2} \lt \delta$ then somehow $0 \lt |x+y| \lt \delta$ and it would follow from the one variable case that $| \frac{\sin(x+y)}{x+y} - 1| \lt \epsilon$. However, because $|x+y| = 0$ when $y = -x$ I am not getting anywhere with this.  Any input on this would be helpful :) Story about space explorers who communicate but then decide they must destroy each other I'm looking for a sci-fi story or novel from many years ago where a deep space explorer encounters an alien explorer. The explorers learn to communicate; but on parting, each decides to destroy the other in order to protect his own world. Short story in book collection about war machines and armies out of control I read in an anthology book in 1975. It starts with two people watching a “robot” plane glide high and then dive into an opening of a factory causing a massive explosion. I’m not sure but I think the people may have orchestrated the attack on purpose to get the robot armies fighting each other in the hope that it would diminish robot army capabilities and bring them under control. The robot wars did occur, but at the end of the story the people discover a tube sticking out of the ground that is periodically shooting something into the air. Nearby they find a canister that did not get launched out of the “cannon” successfully and it was broken open. After close examination they discover tiny “ants” on the canister that are actually tiny robots which are rebuilding the robot war factories. The robots have adapted and will keep trying to fulfill their mission. This story may  have been in a booked edited by Roger Elwood, but I cannot guarantee it. What do ref, val and out mean on method parameters? I'm looking for a clear, concise and accurate answer.    Ideally as the actual answer, although links to good explanations welcome.  This also applies to VB.Net, but the keywords are different - ByRef and ByVal. Today I came across an OBVIOUSLY good answer in /review. It had 7 upvotes, contained lists, paragraphs of explanation, and well-applied sample code. It was also moderately long. How does a question like this get flagged for review? I don't really know how the algorithm works.   If this is low-quality then I feel bad about all of my answers.  Here's a  if you care to see it in context. It's not really a huge problem (as Bart's comment emphasizes), but this seems like an extreme outlier in every measurable aspect. How do I determine the result of a Saving Throw? I'm a pretty new player, and I'm building a bard in D&amp;D 5e. I understand how saving throws work in general, but I am a little confused about one thing. Say I cast Blindness/Deafness on a humanoid. They must then make a Constitution saving throw. My question is: what value do they have to roll in order to make the save? Does their roll (plus their constitution modifier) have to beat my constitution (plus my modifier)? This is more of a general player question than one specific to the Bard or that spell. Matrices that commute with all matrices Books on Number Theory for anyone who loves Mathematics?(Beginner to Advanced &amp; just for someone who has a basic grasp of math) What compiler warning level do you recommend for different C/C++ compilers?  gcc and g++ will let you get away with a lot on the default level.  I find the best warning level for me is '-Wall'.  And I always try to remove fix the code for the warnings it generates.  (Even the silly ones about using parenthesis for logical precedence rules or to say I really mean 'if (x = y)')  What are your favorite levels for the different compilers, such as Sun CC, aCC (HPUX ?), Visual Studio, intel?  Edit:  I just wanted to point out that I don't use "-Werror" (but I do understand it's utility) on gcc/g++ because, I use:   #warning "this is a note to myself"   in a few places in my code.  Do all the compilers understand the #warning macro? Complex differential equation How do I solve $$x'=e^{it}\overline{x}?$$  This is a complex differential equation, but I don't see how to solve it.  Edit: the original ODE is given by $$(x', y')=\begin{pmatrix}\cos t&amp; \sin t\\ \sin t&amp;-\cos t\end{pmatrix}(x,y)$$ I want to show that solutions of this ODE do not remain bounded for all $t$, and the idea was to solve the complex equation. stab(S) is isomorphic to $ S_k \times S_{n-k} $ Show that the group $S_n$ (right) acts on the set of subsets of $\{1,2,\ldots n\}$ by $\rho(S,\sigma)=S\sigma$. Show that there are $n+1$ orbits, one for each possible value of $|S|$. Show that if $|S|=k$ then $\mathrm{Stab}(S)\cong S_k\times S_{n-k}$.  I have shown that there are n+1 orbits but now I am stuck with the last bit. my initial thought is that if I let $S =\{1,2,3,...,k\}$ then $\mathrm{Stab}(S)$ would involve $(1)(2)(3)...(k)$ i.e. mapping to itself and then $k+1$ to $n$ mapped in any ways ($S_{n-k}$) I feel like $\mathrm{Stab}$ is isomorphic to $S_{n-k}$ what am I misunderstanding here? How would I prove the last part? Thanks a lot How can I chroot sftp-only SSH users into their homes? How to find the shortest path between opposite vertices of a cube, traveling on its surface? I am stuck with the following problem that says:      Let $A,B$ be the ends of the longest diagonal of the unit cube . The length of    the shortest path from $A$ to $B$ along the surface is :     $\sqrt{3}\,\,$  2.$\,\,1+\sqrt{2}\,\,$  3.$\,\,\sqrt{5}\,\,$ 4.$\,\,3$    My Try:   So, the length of the longest diagonal $AB=\sqrt{3}$. If I reach from $A$ to $B$ along the surface line $AC+CD+BD$, then it gives $3$ units. But the answer is given to be option 3.   Can someone explain? Thanks in advance for your time. How to check if Node connected to another node input in Python I am doing a process to some switch links in materials. I need to know if something is ever linked or not before to process. I can't find out how to :/ The script works well if something is linked only.  import bpy for m in bpy.context.object.material_slots:     mat = m.material     nodes = mat.node_tree.nodes      # get node input (socket)     alphasocket = mat.node_tree.nodes.get("Principled BSDF").inputs[18]     # get node link that connects into the socket     lnk = (link.from_socket for link in mat.node_tree.links if link.to_socket == alphasocket)      #add emission node     emi= nodes.new('ShaderNodeEmission')     emi.name= "emibak"     outepute= mat.node_tree.nodes.get("Material Output").inputs[0]      #make new link destination     #here 's where I cant find out how to check     if lnk != None:         print(mat.name + " alphanod found")        # mat.node_tree.links.new(lnk, emi.inputs[0])     else :         print(mat.name + " alphanod missing")      mat.node_tree.links.new(emi.outputs[0], outepute)     here is another way I am searching...  import bpy  for m in bpy.context.object.material_slots:      mat = m.material     nodes = mat.node_tree.nodes          print("**** Material " + mat.name+ " ****")      for mat_node in mat.node_tree.nodes:         if mat_node.type == 'BSDF_PRINCIPLED':             print("From " + mat_node.name )                         alphasocket = mat_node.inputs[18]             #print(alphasocket)             node_in = mat_node.inputs             #print(node_in)                    for n in alphasocket.links:                                print("Alpha connected to " + n.from_node.name)   This show me what is connected to the alpha channel but if nothing is connected I want to connect a new node. Proving that $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c)$ where $\lim_{k\to \infty}x_k=\lim_{k\to \infty}y_k=c$ with $x_k<c<y_k$ Question.  Suppose that a function $f$ is defined on an interval $I$, $c$ is a point of $I$, and {$x_k$} and {$y_k$} are any two sequences in $I$ such that $\lim_{k\to \infty}x_k=\lim_{k\to \infty}y_k=c$ with $x_k&lt;c&lt;y_k$ for all $k$. Prove that, if $f'(c)$ exists, then $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c).$   My Attempt.  Let $\epsilon_k=\frac{1}{k}$ be given.  For $\epsilon_1=1,$ there exists $\delta_1&gt;0$ such that $x_1,y_1\in I, x_1,y_1\in N'(c;\delta_1)$ implies $\vert \frac{f(x_1)-f(y_1)}{x_1-y_1} -f'(c) \vert&lt;\epsilon_1.$  For $\epsilon_2=\frac{1}{2},$ there exists $0&lt;\delta_2&lt;$min$\{\vert x_1-c \vert,\vert y_1-c \vert,\epsilon_2\}$ such that $x_2,y_2\in I, x_2,y_2\in N'(c;\delta_2)$ implies $\vert \frac{f(x_2)-f(y_2)}{x_2-y_2} -f'(c) \vert&lt;\epsilon_2.$  Similarly, for $\epsilon_{k}=\frac{1}{k},$ there exists $0&lt;\delta_k&lt;$min$\{\vert x_{k-1}-c \vert,\vert y_{k-1}-c \vert,\epsilon_k\}$ such that $x_k,y_k\in I, x_k,y_k\in N'(c;\delta_k)$ implies $\vert \frac{f(x_k)-f(y_k)}{x_k-y_k} -f'(c) \vert&lt;\epsilon_k.$  Then by taking the limit to infinity, we obtain $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c).$   It really took me a lot of time to finish the proof and I need this to be checked by an expert:) Is the proof okay? Any response would be appreciated!