Can someone help me:  The map $f : {S}^2  \rightarrow \mathbb{R}^6$ given by $f (x; y; z) = (x^2, y^2, z^2, \sqrt{2}yz, \sqrt{2}xz, \sqrt{2}xy)$ induces an embedding $C^{\infty}$ of $\mathbb{R}P^2$ in $\mathbb{R}^6$?  How can we get from $f$ a embedding $C^{\infty}$ of $\mathbb{R}P^2$ in $\mathbb{R}^4$?
We consider $g : (t,u,v)\in \mathbb{R}^3 \mapsto (t^2,u^2,v^2,\sqrt{2}uv, \sqrt{2}tv,\sqrt{2}tu)\in\mathbb{R}^6$. I have to prove that $g(\mathbb{S}^2)$ is a submanifold of $\mathbb{R}^6$.  $dg_{(t,u,v)}=\begin{pmatrix}2t&amp;0&amp;0\\  0&amp;2u&amp;0\\  0&amp;0&amp;2v\\  0&amp;\sqrt{2}v&amp;\sqrt{2}u\\ \sqrt{2}v&amp;0&amp;\sqrt{2}t\\ \sqrt{2}u&amp;\sqrt{2}t&amp;0\\ \end{pmatrix} $  So it proves that $g$ is an immersion (the rank is $3$ clearly) for $(\mathbb{R}^3-\{(0,0,0)\})$.  Now how to prove that $\mathbb{S}^2$ is homeomorphic to $g(\mathbb{S}^2)$ ? I must prove that $g$ is injective  ?  Thanks in advance !
Tracking disk I/O by application in Windows? Definition of the Pauli group and the Clifford group There seem to be two definitions of the Pauli group. In Nielsen and Chuang, the Pauli group on 1 qubit is defined as \begin{align*} \mathcal{P}_1 = \{\pm I, \pm iI, \pm X, \pm iX, \pm Y, \pm iY, \pm Z, \pm iZ\} = \langle X, Y, Z\rangle. \end{align*} But in other references, such as ,  \begin{align*} \mathcal{P}_1 = \{\pm I, \pm X, \pm iY, \pm Z\} = \langle X, Z\rangle. \end{align*} The second definition makes more sense in the stabilizer formalism, since we usually only consider the action of Pauli strings of $X$ and $Z$. What bothers me is that by the second definition, the phase gate $S$, which is an element of the Clifford group, results in \begin{align*} SXS^\dagger = Y \notin \mathcal{P}_1. \end{align*} I know that the prefactor $\pm i$ somehow doesn't matter, but can someone justify it from a more mathematical/group-theoretical perspective? Word for sentences that make sense when read backwards? Is there a word for sentences that make sense when read normally or backwards (not necessarily by reversing the words)? Example:     "Are you as bored as I am?" How would I find a minimum weight spanning tree for W? What does it mean if a Python object is "subscriptable" or not? How to use hook_node_presave to change field's value Dynamic Table partitioning on a daily basis I have a SQL Server database which contains two tables -- Acks and Logs.  These two tables are related in logic but not in a relational database way. Basically, every message that comes in gets saved in the Log table and, if our server acknowledges it, then that ack gets stored in the Ack table.  We are storing around 5 million Acks and 3 million Logs a day. I am trying to partition these two tables on a daily boundary so that we can easily remove older partitions from the table, and also improve query performance.  I haven't done table partitioning before, so I have been reading some on-line tutorials, however I am stuck on one thing. All the tutorials I have followed seem to manually add filegroups and manually add boundaries.  I want SQL Server somehow to do this daily, and this is what my question is about. I need it to create the new filegroups for the next day, every day at, say, 22:00. Then at 24:00 the inserts should start filling up the new day's partition.  Can anyone point me in the right direction on how to achieve this? Either a comprehensive tutorial or some good old advice will do as well.  My second question: can I somehow apply the same partition function to the two different tables?  They both have a datetime(2) column on which I want to partition, and the same rules  will apply.  How does that then fit in with my filegroups? Do I need a single filegroup for the day? Will each table have a file in that filegroup, or will both tables save to the same file in the filegroup?  Do I have to make a .mdf and .ldf for each filegroup? Or is there still one log file for the entire database? Is multiplication not just repeated addition? If multiplication is not repeated addition, what is it? How do you define multiplication? All of us had to memorize the multiplication table in elementary school, but how did they come up with it if so many people claim that it is wrong to think of it as repeated addition? Voodoo magic? A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$ Union of the conjugates of a proper subgroup Let G be a finite group and H be a proper subgroup. Prove that the union of the conjugates of H is not the whole of G.  Thanks for any help What is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? As the question suggests, what is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? Many thanks in advance. Cannot translate vertice/edge will duplicate "A mice problem" vs. "a mouse problem" Suppose $\displaystyle a_n=\frac{1}{(\text{log}2)^p+(\text{log3})^p+...+(\text{log}n)^p}$ for $n\geq 2$ and $p&gt;0$. For which values of $p$ does $\sum_{n=2}^\infty a_n$ converge?  So, when $p\leq 1$, $\displaystyle a_n\geq \frac{1}{(n-1)(\text{log}n)^p}$ for $n\geq 2$. Let $\displaystyle b_n=\frac{1}{(n-1)(\text{log}n)^p}$. Then $\sum b_n$ diverges since $p\leq 1$. Therefore by comparison test $\sum a_n$ diverges when $p\leq 1$.  When $p&gt;1$, $\displaystyle a_n\leq \frac{1}{(\text{log}n)^p}$. Let $\displaystyle c_n= \frac{1}{(\text{log}n)^p}$. If $\sum c_n$ converges then I will win. But now I am stuck. Is $\sum c_n$ converges? Then how can I prove it? If not how can I conclude the state of convergence of $\sum a_n$ for $p&gt;1$?   Can anybody please help? Java Generics: List, List, List Which SUSY models are affected by the recent LHCb result? The LHCb has recently  the observation of $B_s \rightarrow \mu^+ \mu^-$ with a branching ratio that agrees with the Standard Model (SM). There are many blog posts about it (See: ).  The result initiated a debate over its implication on Supersymmetry in general, which I don't want to discuss here. But it certainly affects some Supersymmetric models that may have a different prediction than that of the SM.   My questions are,   Which models in particular are affected (or variants of models)? What is the MSSM status? The NMSSM? Does the result affect some of the parameter space of a particular model, or does it rule out an entire model/or variant of a model? What antivirus program can I use on Ubuntu? Are there any antivirus applications that work on Ubuntu 16.04? Thanks! Limit with inverse trignometry We have to find the value of $$\lim_{n\to \infty}\sum_{k=2}^n\arccos\left(\frac{1+\sqrt{(k-1)k(k+1)(k+2)}}{k(k+1)}\right)$$  I tried it a lot but could not find any start.   Please help me in this . Adding/updating field to polygon feature class that lists bordering (neighbor) polygons? I have a dataset of zipcodes.  I want to add a new field "NEIGHBORS" that includes bordering polygons in a comma delimited manner such as attached image: I tried to find an answer to this a few years ago, but it seemed it wasn't possible then (I suspect it isn't now either! - but what the hell!)   I have a three monitor system, that I mostly use for remote desktop. However occasionally it would be useful to have remote desktop on two monitors and use the other monitor for the local machine.   I know it is possible to have a single monitor full screen, or to stretch an RDP session across monitors, but this loses things like snapping etc.   P.S. this is a duplicate of  thread, but that was a couple of years ago, I'm hoping things have changed.