Consider the ring of integers $\mathbb Z$. Prove that if $p$ is a prime number, then $\langle p \rangle$ is a maximal ideal.  I know I need to show that there is no ideal $J$ such that $\langle p \rangle$ is a subset of $J$. But I do not know how to do it.
I am trying to prove $(a)$ is a maximal ideal in $\mathbb{Z}$,  if and only if $a$ is prime number.  Now I wrote: assume $a \in\mathbb{Z}$, while it's not prime number we can write as $a=xy$, for some integers $x$ and $y$. then $(a)\subset (x)$ and $(a)\subset (y)$  while if $(a)$ is maximal ideal, it's not exist $(k)$ such that $(a)\subset (k)\subset\mathbb{Z}$. so $(a)$ cannot be maximal ideal. we can say by contradictory if $a$ is prime, $(a)$ is maximal ideal.  is this correct?  and how to prove the other way.
I am looking for an example of a sequence which shows that, $a_{n+1} - a_n \to 0$ as $n \to \infty$ does not imply that sequence $a_n$ converges. I have a feeling that the sequence should be an oscillatory one but I am unable to think of an example. I just replaced a broken light fixture in my closet. (The switch in the light fixture broke when I pulled too hard on the chain.)   When replacing the light fixture I noticed that there is only 30 V AC between the white and black wires. The ground wire is cut back and is not being used. By the way, the wiring in my house is all aluminum.   I know the new light fixture works. I checked it out with my multimeter, and I tried it in a different closet. But the light bulb won't turn on because there is only 30 V. (Even if I disconnect the light fixture completely there is still only 30 V.   It seems to me that it must be a problem with the wiring.  My attic does provide some (difficult) access to the wiring, but I am not sure where to start looking for the problem or what to look for. Can anyone offer some suggestions? Personal Hotspot not working over Wi-Fi on iOS 13 After upgrade to iOS 13 from 12.4, I noticed that personal hotspot isn't working anymore over Wi-Fi (via USB is ok).  Tested on two computers with Win7. The PC can connect to Wi-Fi, but no valid IP (IPv4) is assigned and no connection is working to internet. What is the right way to get the current users profile in an apex class?  When I tried to print the following in visualforce page,  i am able to get the profile name  &lt;apex:outputText value="{!$Profile.Name}"/&gt;    But when i tried to print in the Apex class using System.debug(Profile.Name), I couldn't. Derivatives: enter "evaluated at" vertical bar with multiple values as subscripts Sorry for the long title, but how do I enter evaluated at (for a derivative), with multiple valuation criteria at the bottom of the the big | line?  Thanks! Since it seems to me, that a lot of people still need sidebar workarounds, I would be interested to learn about the use-cases, experienced challenges and (most important) possible strategies to avoid them by using different approaches. This is related to the methods discussed here:   In my experience there are a few extremely important and constantly repeating patterns, which to my knowledge are still very hard to be solved using more official techniques. I will also contribute my own top-N use-cases, which are mostly related to UX and UI concerns. You are very welcome to add your own scenarios as answers and add comments on how you would avoid a workaround. My dream would be to team-elaborate strong and repeatable solution-patterns for the most common use-cases.  Finally I would be interested in the experiences of all of you who participate in the Aura pilot, which hopefully could make this kind of workarounds obsolete. It's the first time I'm using Tikz on Latex. I have to draw this block diagram (this is a low resolution snip from a paper)   In Tikz I have done this:   That is very similar but I need some adjustments, like the u_i arrow and the node sum after the H(s) block.  Is there anybody that can help me with this?  my code:      \tikzstyle{block} = [draw, fill=white, rectangle,  minimum height=2em, minimum width=3em] \tikzstyle{sum} = [draw, fill=white, circle, radius=1mm, node distance=1.5cm] \tikzstyle{input} = [coordinate] \tikzstyle{output} = [coordinate] \tikzstyle{pinstyle} = [pin edge={to-,thin,black}]  \begin{tikzpicture}[auto, node distance=2cm,&gt;=latex'] \node [input, name=input] {}; \node [input, above of=input](input1) {}; \node [sum, right of=input] (sum) {}; \node [block, right of=sum] (controller) {$K(s)$}; \node [sum, right of=controller] (sum1) {}; \node [block, right of=sum1] (filterinv) {$H^{-1}(s)$}; \node [block, right of=filterinv] (system) {$G(s)$}; \node [output, right of=system] (output) {}; \node [output, above of=output] (output1) {}; \node [block, below of=filterinv] (filter) {$H(s)$}; \node [block, above of=controller] (delay) {$D(s)$}; \node [sum, below of=sum1] (sum2) {};  \draw [-&gt;] (filterinv) -- node[name=u] {$u_{i}$} (system); \draw [draw,-&gt;] (input) -- node {$s_{i-1}$} (sum); \draw [-&gt;] (sum) -- node {$e_{i}$} (controller); \draw [-&gt;] (system) -- node [name=q] {$q_{i}$}(output); \draw [-&gt;] (q) |- (filter); \draw [-&gt;] (filter) -- node {} (sum2); \draw [-&gt;] (controller) -- node {} (sum1); \draw [-&gt;] (sum1) -- node[name=xi] {$\xi_{i}$} (filterinv); \node [input, below of=sum1](input2) {}; \draw [draw,-&gt;] (input2) -- node {$r_{i}$} (sum2); \draw [-&gt;] (sum2) -| node[pos=0.99] {$-$}      node [near end] {$\tilde{s}_i$} (sum); \draw [-&gt;] (delay) -| node[] {}      node [near end] {} (sum1); \draw [draw,-&gt;] (input1) -- node[pos=0.15] {$u_{i-1}$} (delay); \draw [-&gt;] (u) |- node[pos=0.99] {$u_i$}      node [near end] {} (output1);  \end{tikzpicture} You can reach this IP address  (if my computer is online). I want to assign a domain to it so I wonder how is that possible when I have no DNS? I do own my domain which is ".com". My IP is static.  Thanks C# equivalent to wildcard imports in Java Is there a way in C# to import everything inside a namespace like there is in Java with the wildcard character?  import java.awt.*; I just came across this idiom in some open-source Python, and I choked on my drink.  Rather than:  if isUp:     return "Up" else:     return "Down"   or even:  return "Up" if isUp else "Down"   the code read:  return isUp and "Up" or "Down"   I can see this is the same result, but is this a typical idiom in Python? If so, is it some performance hack that runs fast? Or is it just a once-off that needs a code review? How should an educational institute woo good faculty? What steps must an educational institution wanting to recruit good faculty take? There is a cycle in the academia:    Good doctorates from top universities invariably join other top universities as faculty members.  Good students aspire to study in top universities under excellent faculty members.   Suppose a new educational institution or department springs up and desires to set up a top-quality group, how does it break into this cycle? I suppose this could take some years, but is money-power the only thing that could be wielded by the institute to achieve its end? I have modeled a simple pocket watch in Blender. Every time I import it into Unity3D, regardless of whether I export it from Blender as an FBX or 3ds file, it always comes out missing faces! Why is this happening? I have never had problems importing models into Unity3D like this before.  Blender Screenshot    Unity Screenshots     It seems to me that different faces appear when I rotate it differently. How can I prevent this? Thanks! I find that at the end of a long day, I sometimes have a large number of programs running. All which I will need to use tomorrow. Normally, this isn't an issue, I can simply lock the machine and come back tomorrow. My problem arrises when windows update launches in the middle of the night and force restarts my computer. That in turns closes all my open software. I of course save everything regularly so I don't loose anything, but I waste time reopening all of those resources whenever there is a restart.   [EDIT]  I should clarify that I still want to be able to restart my computer when an update comes down. Preventing the restart only delays the problem until later. I should have been more specific in that I want to be able to recover my working environment after a restart for any reason. Things like scheduled maintence, power loss, updates, and software installs.  [EDIT]  I can't simply have them setup to launch at startup becasuse those files change from week to week. So I need something that monitors what I have open, and gives me the option to "recover" those software sessions when I log back in.  Anyone have any suggestions on what I can do? I'd even be willing to purchase software to do this for me if that is the only option.  Thanks I know how to use the subfig package to fit many subfigures (each with its own caption) into a larger figure (with a general caption).  I know how to use the sidecap package, the put a figure's caption on the side of the figure.  What I do not know is how to display a caption inside the figure, as if it were a subfigure.  More precisely, I have three subfigures that I would like to fit on a 2x2 layout, with the remaining quadrant used for the caption.    Is that easy to achieve? How does dark matter halo outside a galaxy help to explain galaxy rotation curve? How does a   outside a galaxy help to explain a ?  Suppose for simplicity we use a model of a star rotating about a more massive star in a fixed circular orbit. For another star at larger radius, the velocity should be smaller but observationally for spiral galaxies, the opposite happens (moves at even more speed or at least at the same speed as the closer star).  People hypothesize that dark matter halo may be present “surrounding” the system (i.e., halo exists outside the system) to account for such observation. I feel that exactly the opposite should be the case that is, dark matter halo should be inside the galaxy rather than outside. Reason being, if dark matter is outside the galaxy then the effective gravitational pull on the rotating star towards the center will be reduced due to outward attraction on the star exerted by dark matter existing outside. Thus, its speed of rotation should be reduced; but we observe the opposite. So how does dark matter halo "surrounding" a galaxy help to explain galaxy rotation curve? Where am I making mistake in argument? Stupidly, I had been using a condition like this as part of a script:  if [ $(ls FOO* 2&gt; /dev/null) ] # if files named "FOO*" were downloaded  then     echo "Files found"      # ... process and email results else     echo "Not found"     # ... email warning that no files were found (against expectations) fi   That works for zero and one files named FOO*, but fails if there are more than one. From logs I found several different error messages stemming from this:  [: FOO_20131107_082920: unary operator expected [: FOO_20131108_070203: binary operator expected [: too many arguments   My question is: what is the correct way to check, in a Bash if condition, whether one or more files whose name begins with FOO exist?  GNU bash, version 4.2.25(1)-release (x86_64-pc-linux-gnu) If $Ax = Bx$ for all $x \in C^{n}$, then $A = B$. Today I was thinking about composition of functions.  It has nice properties, its always associative, there is an identity, and if we restrict to bijective functions then we have an inverse.  But then I thought about commutativity.  My first intuition was that bijective self maps of a space should commute but then I saw some counter-examples. The symmetric group is only abelian if $n \le 2$ so clearly there need to be more restrictions on functions than bijectivity for them to commute.  The only examples I could think of were boring things like multiplying by a constant or maximal tori of groups like $O(n)$ (maybe less boring).  My question: In a euclidean space, what are (edit) some nice characterizations of sets of functions that commute? What about in a more general space?   Bonus: Is this notion of commutativity important anywhere in analysis? let $\gamma$ be a closed continuosly differentiable path in the upper half plane  not passing through $i$. What are the possible values of the integral $$\frac{1}{2\pi i}\int_{\gamma}\frac{2i}{z^2+1}dz$$ well the integral can be broken like  $$\frac{1}{2\pi i}\int_{\gamma}\frac{2i}{(z+i)(z-i)}dz=$$  $$\frac{1}{2\pi i}\int_{\gamma}\frac{dz}{z-i}dz-\frac{1}{2\pi i}\int_{\gamma}\frac{dz}{z+i}dz=$$ by Cauchy Integral Formuale $$f(i)-f(-i)$$,so the second integral is $0$ as it is analytic in the upper half plane, but the first integral iss just $n(\gamma,i)$, which iss the windding number of $\gamma$ around $i$ what more I can say? Thank you. A conceptual doubt regarding Forced Oscillations and Resonance While studying about the Resonance and Forced Oscillations, I came across a graph in my textbook that is given below:-  Now, the author writes      As the amount of damping increases, the peak shifts to lower frequencies.   Why does this happen? And does this imply that at higher damping levels one cannot achieve a higher amplitude by setting the period of the forced oscillation to be equal to the natural frequency? This seems strange because, the highest amplitude is achieved when the natural frequency is equal to the driving frequency. But, I guess that the rules are somehow different for the damping case.