prove that function mapping is injective iff ker (f) = {e} Heyall, would appreciate some help with abstract algebra because my undergrad brain is fried from doing all the proofs my prof asked me to do. I've hit a bit of a wall with this one; it involves group homomorphisms - super basic but the proof has got to be quite sophisticated cuz my mind is blank.   there are two groups $G$ and $F$ and a mapping $\phi : G \rightarrow F$   the kernel is defined as   $ker (\phi ) := \left \{ x\in  G : \phi (x) = e_{H} \right \}$ whereby $e_H$ is the identity element in $H$  to prove: $\phi$ is injective iff $ker(\phi) = \{e_{G}\}$  whereby $e_G$ is the identity element in $G$  would appreciate any tips too, thanks a bunch xx
Kernel of injective group homomorphism If $\phi$ is a homomorphism from the group $G$ with identity $e$ to the group $G'$ with identity $e'$, then $\phi$ is injective if and only if $\ker\phi=$?  I think the $Ker\phi = \{x\in G:\phi(x)=1\}$  because you are looking at an identity then it is all reals $0$ does not work a because if $x=0, e'x=0$  but then i realize it is injective how does that affect the kernel?
Why can't polynomials have negative exponents or division by a variable Why can't:  $$2x^{-3} - 3x$$  or  $$\frac{1}{2x}$$  be polynomials too?  Why have a definition that excludes these algebraic forms? Where can I find the status of the released versions of Ubuntu? How to enable MySQL client auto re-connect with MySQLdb? I came across PHP way of doing the trick:  my_bool reconnect = 1; mysql_options(&amp;mysql, MYSQL_OPT_RECONNECT, &amp;reconnect);   but no luck with MySQLdb (python-mysql).  Can anybody please give a clue? Thanks. I am currently visiting the USA from the UK (I am a UK citizen) and am intending on staying here for the full 90 days under the Visa Waiver Program.     How long would I have to stay out of the US at the end of this period before returning again?    Would I have to return to the UK, or could I stay with friends in another country (such as Canada) for a while? How to drive multiple LEDs from a Li-ion battery (with varying voltage)? Root of unity for $n+1$ Let $n \ge 1$ and $u_{1},u_{2},...,u_{n}$ are all root of unity for $n+1$ and there are different from $1$.(a) Find polynomial $f \in \mathbb R[x]$ of degree $n$ for which: $f(u_{k})=0$ for $k=1,2,...,n$.(b) Calculate real coefficient and imaginary part for $(2-u_{1})(2-u_{2})\cdot...\cdot(2-u_{n}).$    I have a knowledge about root of unity but I completely don't know how to do it because I think that I need some extra trick to do it but my ideas finished. I know that $\varepsilon_n^{(k)} = \cos\left(\tfrac{2k\pi}{n}\right) + i\sin\left(\tfrac{2k\pi}{n}\right) = e^\frac{2\pi i k}{n}$ and for $n&gt;1$ $\sum _{{k=0}}^{{n-1}}e^{{\frac  {2\pi ik}{n}}}=0$ but I think it is too little knowledge to solve this task.Can you help me? Is it correct to use the word "birthday" for the deceased, or is there a better alternative? How does one refer to the birthdate of someone who is no more, we usually say  Today is my uncle's 80th birth anniversary (Common in Indian English, not sure if it's correct)  or  Today would have been my uncle's 80th birthday   Is it right to say  "Today is my uncle's 80th birthday" for an uncle who is dead? Suppose a package provides whatever.service file so certain events (including whatever.socket) cause a daemon to start with a certain configuration.  I want to disable this completely. In particular, I want the old socket and any other unit files that might refer to this service by name to be unable to start the daemon.  Instead, I want to create my own unit files for this daemon with a different .socket unit and different environment variables. I also want my override to automatically take account of possible upstream updates to the original unit file (which my current solution of copying whatever.service as whatever-modified.service and editing it unfortunately does not provide).  If I create the following symlinks and files: /etc/systemd/system/whatever.service → /dev/null /etc/systemd/system/whatever-modified.service → /usr/lib/systemd/system/whatever.service /etc/systemd/system/whatever-modified.service.d/fix.conf  systemd considers the new service to be masked as well and refuses to start it. How to avoid this? Can I put \fbox along the diagonal so that they touch each other? Not that it's a very good idea to change it, but for fun. According to  post, there are still some problems even after changing entries in /etc/passwd, /etc/shadow, and /etc/sudoers. Any suggestions? Find all files that are NOT of a specific type/extension in folder? I am using Ubuntu 16.04, with Gnome 3.18. Using extensions I have gotten rid of the dash panel and only use the Applications Menu in the upper left-hand corner of the screen. (I don't have Unity.) What I can't figure out is how do you add a new program, move software around to different categories, and create new categories?  Also, any keyboard shortcut to access the Applications Menu? invalid EFI file path on windows 8 Do cell carries have access to FaceTime information? On the newest iPhone, do cell phone carriers have access to FaceTime call data (such as which caller initiated and ended a FaceTime call)? Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory In  gravity, the action is the usual  but now the Torsion tensor is allowed to vary as well (in usual GR, it is just set to zero).  Variation with respect to the metric gives:  $$R_{ab}-\frac{1}{2}R g_{ab}=\kappa P_{ab} \quad (1)$$  where $P_{ab}$ is the canonical stress energy tensor.  Variation with respect to the torsion tensor ${T^{ab}}_c$ gives:  $${T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d = \kappa {\sigma_{ab}}^c \quad (2)$$  where ${\sigma_{ab}}^c$ is the .  By contracting that equation, I can see that if the spin tensor is zero, the Torsion tensor is identically zero:  $${T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d = \kappa {\sigma_{ab}}^c = 0$$ $$ {g^b}_c({T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d) = 0$$ $$ {T_{ab}}^b + {g_a}^b{T_{bd}}^d - {g_b}^b {T_{ad}}^d = 0$$ $$ {T_{ab}}^b + {T_{ad}}^d - 4 {T_{ad}}^d = 0$$ $$ {T_{ad}}^d = 0$$ $${T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d = 0 \quad \Rightarrow \quad {T_{ab}}^c =0$$  My understanding is that:  The spin tensor and the stress energy tensor, are defined entirely in terms of whatever matter Lagrangian we add to the theory.  Therefore, from above, the equations in vacuum are exactly the same as normal GR (so solving for outside matter, only the boundary conditions with the matter could be different).  Assuming my understanding up to here is correct, my line of questioning is:   How is the Spin tensor (and hence Torsion) related to the concept of material with intrinsic spin? Hopefully answered by 1, but if the matter has zero intrinsic spin, yet we have an extended "spinning" body, is the spin tensor still zero (as I'd consider that orbital angular momentum then)? Does this mean Einstein-Cartan predictions are identical to normal Einstein GR if (and only if) the instrinsic spin of any particles and fields in the theory are zero? I have two shapefiles, Point and Polyline. I'm trying to populate my polyline layer with the point data ID (systemID) where the lines start to where they end; eg. create two columns (from and to). The lines already have flow direction assigned to them. I am using ArcHydro tools but Arcpy is also fine.  In ArcHydro I have tried using Attribute Tools > Generate From/To Nodes for lines but it creates values for To and From that are not from my point data. My other alternative was using Feature to Vertices and joining the result to my point data table to get the systemID. But then I would have to update the polyline layer as well. I am sure there is a easier way of doing it.  Note: not all the lines have points at the vertices, blank values are fine for them. Check Constraint with Case Statement I was supposed to find all possible Jordan canonical forms of a $5\times 5$ complex matrix with minimal polynomial $(x-2)^2(x-1)$ on a qualifying exam last semester.  I took the polynomial to mean that there were at least two 2's and one 1 on the main diagonal, and that the largest Jordan block with eigenvalue 2 is $2\times 2$ while the largest Jordan block with eigenvalue 1 is $1\times 1$.  Did I miss any matrices or interrupt the minimal polynomial incorrectly?  \begin{pmatrix}   2 &amp;1  &amp;0  &amp;0  &amp;0\\   0 &amp;2  &amp;0  &amp;0  &amp;0\\   0 &amp;0  &amp;2  &amp;1  &amp;0\\   0 &amp;0  &amp;0  &amp;2  &amp;0\\   0 &amp;0  &amp;0  &amp;0  &amp;1 \end{pmatrix}  \begin{pmatrix}   2 &amp;1  &amp;0  &amp;0  &amp;0\\   0 &amp;2  &amp;0  &amp;0  &amp;0\\   0 &amp;0  &amp;2  &amp;0  &amp;0\\   0 &amp;0  &amp;0  &amp;2  &amp;0\\   0 &amp;0  &amp;0  &amp;0  &amp;1 \end{pmatrix}  \begin{pmatrix}   2 &amp;1  &amp;0  &amp;0  &amp;0\\   0 &amp;2  &amp;0  &amp;0  &amp;0\\   0 &amp;0  &amp;2  &amp;0  &amp;0\\   0 &amp;0  &amp;0  &amp;1  &amp;0\\   0 &amp;0  &amp;0  &amp;0  &amp;1 \end{pmatrix}  \begin{pmatrix}   2 &amp;1  &amp;0  &amp;0  &amp;0\\   0 &amp;2  &amp;0  &amp;0  &amp;0\\   0 &amp;0  &amp;1  &amp;0  &amp;0\\   0 &amp;0  &amp;0  &amp;1  &amp;0\\   0 &amp;0  &amp;0  &amp;0  &amp;1 \end{pmatrix}  \begin{pmatrix}   2 &amp;0  &amp;0  &amp;0  &amp;0\\   0 &amp;2  &amp;0  &amp;0  &amp;0\\   0 &amp;0  &amp;1  &amp;0  &amp;0\\   0 &amp;0  &amp;0  &amp;1  &amp;0\\   0 &amp;0  &amp;0  &amp;0  &amp;1 \end{pmatrix} Points disappear when zooming past certain scale in ArcMap? My points are disappearing when I zoom in past 300,000 in ArcMap.  I tried to repair geometry, but it doesn't fix anything?    I also checked the scale extents and those are turned off.  I can export the points to a new shapefile/feature class, but I'd like to not have to do that.    I'm using 10.1 on Windows 7 x64. If $q$ is coprime to $a$ then $a\mid nq-1,\,$ so $q$ is invertible mod $a$