Open subgroups of $\mathbb{R}$
what are all the open subgroups of $(\mathbb{R},+)$
When we define the S-matrix, what are "in" and "out" states? I have seen the scattering matrix defined using initial ("in") and final ("out") eigenstates of the free hamiltonian, with  $$\left| \vec{p}_1 \cdots \vec{p}_n \; \text{out} \right\rangle = S^{-1} \left| \vec{p}_1 \cdots \vec{p}_n \; \text{in} \right\rangle$$  so that  $$\left\langle \vec{p}_1 \cdots \vec{p}_n \; \text{out} \mid \vec{q}_1 \cdots \vec{q}_m \; \text{in} \right\rangle = \left\langle \vec{p}_1 \cdots \vec{p}_n \; \text{in} \mid S \mid \vec{q}_1 \cdots \vec{q}_m \; \text{in} \right\rangle.$$  1) What are "in" and "out" states?   2) Are they Fock states?   3) In Schrödinger or Heisenberg or interaction representation?   4) How are they related? (I believe that I see what they handwavily represent physically, but not formally.)  My main issue is that, if "in" and "out" states are one-particle eigenstates of the free hamiltonian, i.e. if $\left| \vec{p}_1 \text{out} \right\rangle$ describes a free particle with momentum $\vec{p}_1$, and $\left| \vec{p}_1 \text{in} \right\rangle$ also describes a free particle with momentum $\vec{p}_1$, then $\left| \vec{p}_1 \text{out} \right\rangle = \left| \vec{p}_1 \text{in} \right\rangle$ ... which is false. Still, books (some at least) describe these "in" and "out" states like that.  Moreover, I have seen (e.g. in Wikipedia, but also ) that the scattering matrix is a map between two different Fock states, and I don't understand that.   5) Do states of the interacting system live in the same Fock space that asymptotic free states?   6) And if not, where do they live?  Understandable references would be appreciated. I tried for a while to classifiy the singularities of $\frac{1}{z}-\frac{1}{\sin z}$ at the origin, but I am stucked. Is there someone who is able to help me at this point? I am trying to make folder that everyone can view and can't copy,modify.    For all answers.I thinking about specially ms office files. I tried following methods  Software  - copy protect    - anti copy    This method can't protect save as and upload  MS office macro    Users can enable,disable macros.  MS office restriction    This is best choice but can't protect from file uploading.   Almost I found the solution. Way is combination of MS office restriction and restrict access from browsers using firewall or antivirus. But I don't exact how to do this. How to return only the Date from a SQL Server DateTime datatype Pass extra parameters to an event handler? I'm trying to figure out how to use "not only... but also" properly. Basically, my goal is to combine two clauses by using "not only".  For negations, I've figured out two styles that both sound correct:   He knew that if he fractures his finger, not only would he not be able to compete in the water polo tournament, he would not be able to take the SAT on Monday, either. He knew that if he fractures his finger, neither would he be able to compete in the water polo tournament, nor would he be able to take the SAT on Monday.   However, I don't know how to form a sentence that does not include a negation while using "not only". Here are a few possibilities that I've considered:   Thanks to his remarkable performance on the SAT, not only was he accepted into Harvard, but he was also given a full scholarship. Thanks to his remarkable performance on the SAT, not only was he accepted into Harvard, but he was given a full scholarship. Thanks to his remarkable performance on the SAT, not only was he accepted into Harvard, he was given a full scholarship. Thanks to his remarkable performance on the SAT, not only was he accepted into Harvard, he was also given a full scholarship.   In those four possibilities, I'm varying the usage of "but" and "also" in the last clause.  What is the proper usage of "not only"? How do you detect where two line segments intersect? How do I determine whether or not two lines intersect, and if they do, at what x,y point? I am wondering whether there is any difference between a gerund acting as subject and an infinitive acting as a subject. My computer does not store the history of the entries entered in RUN (WIN+R).   Thus, how to enable Windows to store the history? How can I view the time of a text message in iOS 7? Prove by mathematical induction that: $\forall n \in \mathbb{N}: 3^{n} > n^{3}$ Prove by mathematical  induction that:   $$\forall n \in \mathbb{N}: 3^{n} &gt; n^{3}$$  Step 1: Show that the statement is true for $n = 1$:  $$3^{1} &gt; 1^{3} \Rightarrow 3 &gt; 1$$  Step 2: Show that if the statement is true for $n = p$, it is true for $n = p + 1$  The general idea I had was to start with $(p+1)^{3}$ and during the process substitute in $3^{p}$ for $p^{3}$ as an inequality.  $$(p+1)^{3} = p^{3} + 3p^{2} + 3p + 1 &lt; 3^{p} + 3p^{2} + 3p + 1$$  Now, if it can be shown that:  $$\forall p \in \mathbb{N}: 3p^{2} + 3p + 1 \leq 2 \cdot 3^{p}$$  ...the proof is complete. This is because $3^{p+1} = 3 \cdot 3^{p}$ and one of those three have already been used.   We do this by mathematical induction. First, the base case of $n = 1$:  $$3\cdot 1^{2} + 3 \cdot 1 + 1 \not \leq 2 \cdot 3^{1}$$  ..which turns out to be false.  What are some more productive approaches to this step? counting number of specific string in each row Why was Padmé made to clean an astromech? Reading  (the Amidala stand-in) I remembered that I always wondered why she ordered the actual queen to scrub R2-D2. There were enough actual servants who could have done it, so why Padmé? If that wasn't enough, after giving the actual queen (Padmé) the order to go away, she tells Panaka to continue with the discussion. How does that make sense?  The significant piece of dialogue:     AMIDALA : Thank you, Artoo Detoo. You have proven to be very loyal...Padme!    ...    AMIDALA : (Cont'd) Clean this droid up the best you can. It deserves our gratitude...(to Panaka) Continue, Captain.   Note that this question is two-fold:   Why was Padmé chosen in the first place? Why would an important meeting with the ship's captain be continued without her? A topological space $X$ is simply connected if it is pathwise connected and each closed path $u : I = [0,1] \to X$ is path homotopic to the constant path at $x_0 = u(0) = u(1)$. Recall that  A closed path $u$ is one with $u(0) = u(1)$. A closed path will also be called a loop and we write $\mathcal L'(X)$ for the set of all loops in $X$. If we want to be more precise, we say $u$ is a loop based at $x_0 \in X$ provided $x_0 = u(0) = u(1)$. By  $\mathcal L'(X,x_0)$ we denote the set of all loops in $X$ which are based at $x_0$. Also note that each constant path is a loop.  Two paths  $u, v : I \to X$ with the same initial point $x_0 = u(0) = v(0)$ and the same terminal point $x_1 = u(1) = v(1)$ are path homotopic, $u \simeq_p v$, if there exists a  homotopy $H : I \times I \to X$ such that $H(t,0) = u(t)$ and $H(t,1) = v(t)$ for all $t$ and $H(0,s) = x_0, H(1,s) = x_1$ for all $s$ (such a homotopy $H$ is stationary on the boundary $\partial I = \{0,1\}$; it is called a path homotopy). It is well-known that $\simeq_p$  is an equivalence relation on the set $\mathcal P(X)$ of all paths in $X$. Clearly $\simeq_p$ restricts to an equivalence relation on $\mathcal L'(X)$ and to an equivalence relation on each $\mathcal L'(X,x_0)$.  The set of equivalence classes $\mathcal L'(X,x_0)/\simeq_p$ is nothing else than the fundamental group $\pi_1(X,x_0)$ with base point $x_0$.   Question: Find a list, as complete as possible, of properties characterizing topological spaces which are simply connected. Each of these properties can then be taken as an alternative definition of simply connected. Note that many questions in this forum deal with aspects of simple connectedness. The answer to the present question will hopefully cover most of them. Compare double in VBA precision problem I have trouble comparing 2 double in Excel VBA  suppose that I have the following code  Dim a as double Dim b as double a = 0.15 b = 0.01   After a few manipulations on b, b is now equal to 0.6  however the imprecision related to the double data type gives me headache because  if a = b then  //this will never trigger end if   Do you know how I can remove the trailing imprecision on the double type? What are the most important things I need to do to encourage Google Sitelinks? I read following in a web site ()     Lenses with maximum apertures of f2.8 allow the camera to use all high   precision sensors. In low light or other situations that are hostile   to autofocus, that’s a big deal. Lenses with a maximum aperture of   f4.0 use only the center focus sensor in its “high precision” mode,   and use the other sensors in their “horizontal line” only mode. Lenses   with a maximum aperture of f5.6 use all sensors in their   “horizontal-only” mode, and lenses with a maximum aperture of f8 use   only the center sensor point, and that with horizontal sensitivity   only.   I want to understand if this is completely true? I am raising this because recently I have purchased a Nikon 70-200 f/4 lens and would like to understand if I have made a wise investment, or, should I collect some more money and buy a 70-200 f/2.8 VR-II lens. The determinant of block triangular matrix as product of determinants of diagonal blocks I am given the following partitioned, upper-triangular matrix:  $$ \begin{bmatrix}   A_1 &amp;*  &amp;*  &amp;*  &amp;*  &amp;*  \\    0&amp; A_2 &amp;*  &amp;*  &amp;*  &amp;*  \\    .&amp;  0&amp; A_3 &amp;*  &amp;*  &amp;*  \\    .&amp;  0&amp;  0 &amp;...  &amp;*  &amp;. \\    .&amp;  0&amp;  0&amp;  0&amp; ... &amp;.  \\    0&amp;  .&amp;  ...&amp;  0&amp;0 &amp; A_m \end{bmatrix} $$  where each of the $A_i$ are block-matrices.  I have to prove that the determinant of this general matrix is: $$ \prod_{n=1}^m \det A_i. $$  We already proved in class that it is true for $ m=2 $ and we can use it in our proof.  Thanks for your answers! Why is the electric field inside a conductor zero in equilibrium? My textbook says the field inside a conductor must be zero in order for the system to be equilibrium and therefore there must be no excess charge inside.  Their proof:  1) Place a gaussian surface inside the conductor. Since the system is at equilibrium, all points on the surface must have an electric field of zero.  2) Therefore the net flux is zero, implying the charge inside is zero.  3) If there is no charge inside, all excess charge must lie on the surface.  My problem:  "Since the system is at equilibrium, all points on the surface must have an electric field of zero."  Why?  Consider an conductor with no excess charge. To simplify things, say the conductor consists of a 100 free electrons and a 100 ions of charge 1$e^+$. The conductor is neutral and at equilibrium.  Now consider the field on one specific electron. There a total of 99 other electrons and 100  ions. So there would be a net field acting on that specific electron. So even though the system is at equilibrium, the net field on the individual particles is non-zero.  So a system can be at equilibrium, even if the net field inside is non-zero. What am I misunderstanding?  Another example: Consider a system consisting of one electron and one proton. If nothing is holding them back they will attract each other until they touch. After they touch, the force acting on them is still non-zero, yet they wouldn't accelerate. Get Last Friday of Month in Java