I understand I must find some element that map to identity but what condition  $\phi(1)=4$ is related ? and how to compute $(\phi )(25)$? I must find mapping from 25 to $Z_{7}$ isn't it ? is that 4 ?
Give $\ker(\phi)$  for $\phi: \mathbb{Z} \to \mathbb{Z_7}$ such that $\phi(1) = 4$  So the generator of $\mathbb{Z}$ maps to $4$ and $\langle 4 \rangle = \{0,1,2,3,4,5,6\}$ so it has order seven.  so $\ker(\phi) = 7\mathbb{Z}$ because every seventh element will map to the identity under the given map. Is that the correct reasoning?
Why does demand for stock rise when a company appears to have high future value? How do I change the default file manager back to Nautilus? When in a group chat, if many messages are exchanged between other people in the group, when I check my Whatsapp, the messages are grouped according to sender and not chronologically. I know that my phone isn't the problem as I have recently changed my phone, however, I'm still experiencing the same problem.  I have tried uninstalling and re-installing WhatsApp on my phone, and the same problem still occurs. Two different expansions of $\frac{z}{1-z}$ This is exercise 21 of Chapter 1 from Stein and Shakarchi's Complex Analysis.  Show that for $|z|&lt;1$ one has $$\frac{z}{1-z^2}+\frac{z^2}{1-z^4}+\cdots +\frac{z^{2^n}}{1-z^{2^{n+1}}}+\cdots =\frac{z}{1-z}$$and   $$\frac{z}{1+z}+\frac{2z^2}{1+z^2}+\cdots \frac{2^k z^{2^k}}{1+z^{2^k}}+\cdots =\frac{z}{1-z}.$$  Justify any change in the order of summation.   [Hint: Use the dyadic expansion of an integer and the fact that $2^{k+1}-1=1+2+2^2+\cdots +2^k$.]  I don't really know how to work this through. I know that $\frac{z}{1-z}=\sum_{n=1}^\infty z^n$ and each $n$ can be represented as a dyadic expansion, but I don't know how to progress from here. Any hints solutions or suggestions would be appreciated. Circuit tester trips new GFCI I replaced a non-protected receptacle with a GFCI type and now my circuit tester trips the GFCI as soon as it is plugged in.  The receptacle tested fine before replacement, ground, neutral, hot, etc.  I've double checked the hot/neutral/ground connections.  I've also tried it with the downstream load disconnected.   The GFCI trips the moment I plug in my circuit tester (before pushing the test button).  Otherwise things check out OK.  I can plug in a power tool without it tripping.  Also, the integrated test/reset button work correctly.   I've read that the integrated test button is the only one that truly matters. Is this correct or should I be concerned? I'm sure a home inspector will red flag this if their tester has the same result.  Update  My tester is by "Commercial Electric".  The tester is one of the palm sized three prong type with three leds to indicate the condition of the circuit and a button on top to test GFCI. It's new from the local box store. Setup for a wine bottle with multiple labels I have modeled a wine bottle and created various wine label designs for a client. The labels were created in Illustrator and so I have PNGs. I want to present a render of the same bottle with each label the client. The labels are not simple rectangles.   Should I:    "model" each label and wrap it on the bottle, or  try to map each label image to a portion of the bottle's surface?    I've seen lots of tutorials showing option #2, but something is telling me that the glass of the bottle should be separate from the "paper" label.  Also, with #1, couldn't I use the same bottle blend file and use a different layer for the each label, then turn on a different layer for each new render?  Maybe there's an even better strategy?  thanks, Conditional expectation of $\mathbb{E}(X - Y | (X, Y)\in\mathcal{A})$ Given two independent random variables $X \sim \mathcal{U}[-1,5]$ and $Y \sim \mathcal{U}[-5,5]$, what is $$\mathbb{E}\{Y - X | X \le 1, Y &gt; X, Y \in [-1,1] \}\,?$$  I managed to compute the respective probability   $$ \mathbb{P}\{ X \le 1, Y &gt; X, Y \in [-1,1] \} = \mathbb{P}(X \le 1) \mathbb{P}(Y &gt; X) \mathbb{P}(Y \in [-1,1]) = 1/3 * 1/2 * 1/5 = 1/30$$  How do I continue from there? I can't retag or otherwise edit a question while waiting for an edit I reviewed from queue to be approved or rejected the structure "like for somebody to do something" Can I shrink the transaction log file on a mirror database? This is a followup question to  about why I couldn't shrink the log file on the principal database.  To make a long story short, I setup database mirroring but forgot to make sure the job that backed up the transaction log was running again, and the transaction log grew to almost 60GB.   Since mirroring was set up, this size increase was duplicated on the mirrored server, and eventually took up all the disk space and made the mirror database unusable.  Per  about transaction log maintenance on a mirror database, you can't back up the log on the mirror, but when asked specifically in  about how to shrink an overgrown log file on a mirror database, a  was left that      One way to do this would be to failover to the mirrored database and   do the shrink there. Thoroughly test this out in a non-production   environment to ensure it has the behavior you want/expect.   This seems to suggest that there might be other ways to shrink the log file on a mirror, and this method might not necessarily be safe to do on a production server.  Is there a way to safely shrink the transaction log file on a database mirror? How to describe a guy who is popular with girls? Perhaps I should make it clear:  - He naturally attracts girls.  - He doesn't chase girls and have no intention for any relationship.  - You just see him often together with girls. We roll a six-sided die ten times. What is the probability that the total of all ten rolls is divisible by 6? So the question is really hard I think. I tried using a simple way by calculating the probability of each combination that makes a sum divisible by six, but it would take forever. Does anyone have any ideas?  Suppose that we roll a six-sided die ten times. What is the probability that the total of all ten rolls is divisible by six? How can I cast int to enum? How can an int be cast to an enum in C#? Solution check: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$ How do I get rid of "U3 System" on my USB drive? I have a USB drive (SanDisk SDCZ40-016G) that mounts in a way I've never seen before. When inserted, two items appear in Places -> Computer:    "SanDisk Ultra Backup: 16GB" contains my data, and "CD Drive: U3 System" contains a Windows executable. The part with the Windows executable doesn't look useful to me so I'd like to remove it. I used GParted to delete the only partition listed on the device and then I created and formatted a new partition:    Strangely, the "CD Drive" containing the Windows executable was perfectly intact after this operation.  What's going on? I have been developing in the windows space with Visual Studio for a while now with work, but I have also been using Ubuntu for a while and am keen to get into some software development for linux.  I should also note. I am not looking for .NET and I am aware of mono. I am also familiar with c++ development and some python, so the language isn't so much relevant as the "all in one" aspect.  I was interested to know if there is a useful all in one code/debug/design(gui) IDE similar to something like Visual Studio but for linux? What are the limitations on the Shape Water cantrip? I have been having a discussion regarding the Shape Water cantrip from the Elemental Evil Player's Companion. A friend and I are trying to come to an accurate view of how this spell actually functions.   Some example questions:   Must the targeted water be within a 5 foot cube area, or does it need to be of a volume not exceeding 125 cubic feet? If you shape water that is within a 5 foot cube, can the final shape you animate it to extend outside the cube? Can you cause the shaping to affect ice and snow, or only liquid water? Would you be able to carry a globe of water alongside you over distances?   Really, there are two parts:    What can be targeted by the spell? What are the limitations on the 'shape and animate' portion of the spell?   The answer I am looking for, if possible, would use the language of the spell, possibly with precedent from other game text, to clearly define the limitations of this cantrip. Find the limit: $\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$ Find the limit: $$\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$$   I have no idea what to do, but I thought that this is the limit of the derivative of $f(x)=\left(1+x\right)^{1/x}$, as $x$ tends to 0. Any help? spectral radius in normed space is subadditive. I am reading the book "characterizations of c*-algebras" by Doran and Belfi. On the page 299, the book reads "It can also be shown that if $xy=yx$, then $\nu(x+y)\le\nu(x)+\nu(y)$." The definition of $\nu$ is as follows: in normed algebra,  $$\nu(x)=\limsup_{n\to\infty}\|x^n\|^{\frac1n}$$  Note that the algebra is not assumed to be complete and is also not assumed to be complex scalar(we can have real scalars)  This is my trial: I just tried to mimic the proof of the Minkowski inequality, but failed.  Thank you Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ I've read through the elementary proof of why there exists a solution $x$ to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ for $p$ an odd prime. Is there a group theory generalization for this fact as well?