What is the meaning of //N in mathematica?
Question on operator: // N
In an electromagnetic theory class, my professor introduced the concept of "action at a distance in physics". He said that:     If two charges are at some very large distance, and if any one of the charge moves, then the force associated with the charges changes instantaneously.   But according to Einstein, no information can travel faster than the speed of light.   So photons (the information carriers in electromagnetic force) cannot instantaneously deliver information.   So that we associate a field with the two charges and if any charge moves, there is a deformation in that field and this deformation travels with the speed of light and conveys the information.   If the field deformation information cannot travel more than the speed of light, how does the force instantaneously change at very large distances? Evaluate $\int \cos(3x) \sin(2x) \, dx$. Evaluate the indefinite integral    \begin{align} \int \cos(3x) \sin(2x) \, dx \end{align}   Please see my work attempt as my answer below. I'm looking for a quick start or template, or something, to get a web site's basic layout done quickly and looking fairly professional, and then let me do the coding.  I don't want a CMS, as this is for a highly customised reporting application, but I need something like frame header and two columns and frame footer.  I can design the pretty reports for the wide right hand column myself, as the left hand column menu. Is every metric space a subspace of some connected metric space? Using awk to sum the values of a column, based on the values of another column Show that  $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$  This is an exercise I was trying to do ( not homework ) and I got stuck as follows:  Working from $A \setminus ( B \setminus C )$:  $x \in A \wedge x \notin ( B \setminus C )$ $x \in A \wedge ((x \in B \wedge x \in C) \vee (x \notin B \wedge x \in C) \vee (x \notin B \wedge x \notin C) )$  At this point I'm not sure how to proceed.  Working from the other side $ ( A \setminus B) \cup ( A \cap C ) $:  $ x \in ( A \setminus B) \cup ( A \cap C ) $  $ x \in ( A \setminus B) \vee x \in ( A \cap C ) $  $ ( x \in A \wedge x \notin B ) \vee ( x \in A \wedge x \notin C ) $  $ x \in A \wedge (x \notin B \vee x \notin C ) $  $ x \in A \wedge x \notin ( B \cap C ) $  $ A \setminus ( B \cap C ) $  I could also use some formating tips like how display logical not and how to get the lines to display properly without blank lines inbetween. :-) I can follow the proofs in which the asymptotic normal-distribution of a maximum likelihood estimator $\tilde{\theta}_n$ is derived.   however, does this already imply that the maximum likelihood estimates are asymptotically unbiased, i.e. do we have $$E(\tilde{\theta}_n) \to \theta \text{ as } n \to \infty?$$  Since I am aware that in general it is not true that convergence in distribution implies convergence in moments, an explanation would be nice.  The result is often stated e.g.  as "...it means that the bias of the maximum likelihood estimator is equal to zero up to the order $n^{-1/2}$")  1.) Are  some kind of regularity conditions from the mle theory used to establish this result?  2.) Or is a $\sqrt{n}$-convergence of an estimator (to a normal distribution) in general already enough to establish convergence of its moments?  Note: the Wikipedia article mentions Cox, David R.; Snell, E. Joyce (1968).  , as a source where the order of the bias is derived (formula (12) or (20)).   However in this paper I can't follow the arguments 100%, since their Taylor approximation of $L'(\widehat{\beta})$ is lacking the remainder term. What is the argument used here to discard it completely? "viewed 1 times" When a post is first created, it says "viewed 1 times" in the sidebar. This should say "viewed 1 time". How to make pipes work with Runtime.exec()? Progress bar with dynamic text & text color update What program can be used to package software into one executable? I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ? How to prove that minimum of two exponential random variables is another exponential random variable? How can I prove that the minimum of two exponential random variables is another exponential random variable, i.e. Z = min(X,Y) Can I configure it to allow:   select right-click to copy right-click or ctrl-v to paste? Why is the application of probability in QM fundamentally different from application of probability in other areas? The given differential equation $$\lvert{dy\over{dx}}\rvert\,+\mid{y}\mid+1=0 $$  I don't know how to deal with modulus sign in order to prove the statement  Reference-Different Equations by Shipley L Ross Ch1 Exercise 7(a) Assume you have some objects which have several fields they can be compared by:  public class Person {      private String firstName;     private String lastName;     private String age;      /* Constructors */      /* Methods */  }   So in this example, when you ask if:  a.compareTo(b) &gt; 0   you might be asking if a's last name comes before b's, or if a is older than b, etc...  What is the cleanest way to enable multiple comparison between these kinds of objects without adding unnecessary clutter or overhead?   java.lang.Comparable interface allows comparison by one field only Adding numerous compare methods (i.e. compareByFirstName(), compareByAge(), etc...) is cluttered in my opinion.   So what is the best way to go about this? Proving $M_n(I)\lhd M_n(R)$ and proofs about ideal rings. I'm trying to do a few proofs about rings: $(1)$ I want to show that for a ring $R$ and the ring of matrices $M_n(R)$, if $I\lhd R\implies M_n(I)\lhd M_n(R)$ and that all ideals of $M_n(R)$ are of the form $M_n(I)$. $(2)$ Prove that if $R$ is a ring with division then $M_n(R)$ is a simple ring.  The relation $I\lhd R$ implies that with $r\in R$ and $x\in I$ we have $rx\in I, xr\in I$. But how can I prove the statement for matrices?, here's what I would do: Consider $\psi \in M_n(R)$, if is given any matrix $\phi\in M_n(I)$ it must be $\psi \phi \in M_n(I),\phi \psi \in M_n(I)$, if we denote the element $(i,j)$ of each matrix by $\phi_{ij}$ and $\psi_{ij}$, then $(\psi \phi)_{ij}=\sum_{k=1}^n \psi_{ik}\phi_{kj}$ and $(\phi \psi)_{ij}=\sum_{k=1}^n \phi_{ik}\psi_{kj}$.  Since $I\lhd R$ is $\psi_{ik}\phi_{kj}\in I$ and $\phi_{ik}\psi_{kj}\in I$, which means that $I\lhd R\implies M_n(I)\lhd M_n(R)$.  To show that every ideal of $M_n(R)$ are of the form $M_n(I)$ with $I$ ideal: Suppose $I$ is not an ideal of $R$, then exists $r\in R$ such that $rx\notin I$ or $xr\notin I$, then with taking a matrix $\phi\in M_n(I)$ such that $x$ is an entry of $\phi$ and a matrix $\psi$ such that $r$ is an entry of $\psi$ we get that the matrices $\phi$ and $\psi$ have an entry $xr$ or $rx$, in either case this is not an element of $I$ then the matrix is not in $M_n(I)$. This concludes my proof (is it?) for $(1)$.  For $(2)$ I'd like to proceed by absurdum . Let's say that $M_n(R)$ is not a simple ring, then it has at least one ideal. Denote by $\varphi$ this ideal in $I$, then for any matrix $\psi$ we have $\varphi\psi \in M_n(I)$ and $\psi \varphi \in M_n(I)$.  Now, can $R$ be a ring with division?.I'm not sure how to prove that it can't. I believe I could use this theorem: "If $I\lhd R$ and $1\in I$ then $I=R$. The case $I\neq R$ implies $1\neq R$ and since $\varphi \psi$ and $\psi \varphi$ are in $M_n(I)$ then there are at entries in the product matrix which belong to $I$ (using $I \lhd R$) but are not $1$ (1\notin I); then there are elements in $I$ (and therefore in $R$ by $I\subset R$) that are not invertible, follows that $R$ is not a ring with division. But if $I=R$?, here I can't see a why $R$ can't be a ring with division. How must I deal with anti-virus software on Windows 7 I know this question is looking like an easy question or not a question at all but it's not really easy question. I'm not related to security and anti-viruses and just don't trust every provided anti-virus.  So first question is a real question - Is it OK (safe) to not use anti-virus at all. I know how to use firewall, usually I use only trusted connections and surfing trusted sites. My chances to get virus is really low. Even I've got a bit outdated PC and don't want to pay the performance for it. I can do monthly scans without having one installed - is it OK to be safe ?  If not can you recommend a good free antivirus solution? What are the general rules regarding when can you refreeze food that was frozen and then thawed in the fridge? and what are the reasons behind them? (Hygiene? Food quality?). Let's assume for the sake of the argument that the items were in a non-frozen state only a few hours.  To be more specific, I'm interested in the application of these rules for  - Ginger garlic paste - Dough - Indian Curries - Green Chilly paste - Frozen green peas