How do I parse XML with PHP?
How do you parse and process HTML/XML in PHP?
System.Windows.Threading.Dispatcher and WinForms? How can I get the DateTime for the start of the week? How do I run two isolated instances of firefox? For testing purposes, I want to install two different instances of firefox on ubuntu - so that they don't share cookies. (I want to simultaneously login with both instances as different users to our website).  How do I do that? I do it with Firefox &amp; Opera, but I prefer Firefox (because of firebug). I am fairly sure that I understand the how Monte Carlo integration works but I am not understanding the formulation of how it is used to estimate Pi.  I am going by the procedure outlined in the 5th slide of this presentation   I understand the preliminary steps.  Pi is equal to 4 times the area of a quarter of the unit circle.  And the area of the top-right quarter of the unit circle centered at (0,0) is equivalent to the integral of the curve that is the top-right quarter of the unit circle in $0&lt;x&lt;1$ and $0&lt;y&lt;1$.    What I don't understand is how this integral is  $\iint I((x^2+y^2)&lt;1)P(x,y)dxdy$  where $P(x,y)$ is uniformly distributed in the unit square around the quarter circle (i.e. it is always equal to 1 if $0&lt;x&lt;1$ and $0&lt;y&lt;1$ and 0 otherwise).  So this would mean that  $I((x^2+y^2)&lt;1)P(x,y)$ is the function that is the top-right quadrant of the unit circle at $0&lt;x&lt;1$ and $0&lt;y&lt;1$ but I do not understand how this is true since the indicator function can only be 1 or 0.  I understand that it is probably written in this way to make Monte Carlo sampling easy (i.e. it is an expectation so just sample from $P(x,y)$ and get the average of the samples applied to $I((x^2+y^2)&lt;1)$) but it just does not make intuitive sense to me why that integral represents the area under that curve.  Could someone provide an intuitive explanation of this.  Maybe show how that integral was derived in a step-by-step way?  EDIT:  I was able to gain a better understanding by relating the expectation to an area.  I will explain it here in case it helps anyone.  First start with relating Pi to the area of the top-right quadrant of the unit circle  $\pi=4\times A_{tr}$  Then we place the top-right quadrant into the unit square.  And under a uniform distribution over the unit square, the area of the circle quadrant is proportional to the probability of obtaining a sample from it.  It follows that the following equality holds  $P(x^2+y^2&lt;1)=\frac{A_{tr}}{A_{square}}$  and $A_{square}=1$ so  $P(x^2+y^2&lt;1)=A_{tr}$  And substituting into the original equation  $\pi=4\times P(x^2+y^2&lt;1)$  and it is also true that $P(x^2+y^2&lt;1)=E[I(x^2+y^2&lt;1)]$ which is equal to the original double integral.  So I understood it by relating the area to a probability then relating that probability to an expectation that is equivalent to the integral.  Let me know if I have made any mistakes. As homework, I have to prove that     $\forall n \in \mathbb{N}: n^3-n$ is divisible by 6   I used induction  1) basis: $A(0): 0^3-0 = 6x$ , $x \in \mathbb{N}_0$ // the 6x states that the result is a multiple of 6, right?  2) requirement: $A(n):n^3-n=6x$  3) statement: $A(n+1): (n+1)^3-(n+1)=6x$  4) step: $n^3-n+(n+1)^3-n=6x+(n+1)^3-n$  So when I resolve that I do get the equation: $n^3-n=6x$ so the statement is true for $\forall n \in \mathbb{N}$  Did I do something wrong or is it that simple? Should 'using' directives be inside or outside the namespace? I have been running  over some C# code, and it keeps reporting that my using directives should be inside the namespace.  Is there a technical reason for putting the using directives inside instead of outside the namespace? How do I implement a draggable tab using Java Swing? Instead of the static JTabbedPane I would like to drag-and-drop a tab to different position to rearrange the tabs.  EDIT: . MySQL Conditional Insert Meaning of $\int \phi^\dagger \hat A \psi \:\mathrm dx$ While analysing a problem in quantum Mechanics, I realized that I don't fully understand the physical meanings of certain integrals. I have been interpreting:    $\int \phi^\dagger \hat A \psi \:\mathrm dx$  as "(square root of) probability that a particle with state $|\psi\rangle$ will collapse to a state $|\phi\rangle$ when one tries to observe the observable corresponding to $\hat A$" $\int \phi^\dagger  \psi \:\mathrm dx$ as "(square root of) probability that a particle with state $|\psi\rangle$ will collapse to a state $|\phi\rangle$".   All integrals are over all space here, and all $\phi$s and $\psi$s are normalized.  Now, I realize that  interpretation of the first integral doesn't really make sense when put side by side with the second one.   For example, when $|\phi\rangle$ is an eigenstate of $\hat A$, I get two different expressions for "the square root of probability that one will get the corresponding eigenvalue of $\psi$ when one tries to observe the observable corresponding to $\hat A$". As far as I can tell, $\int \phi^\dagger \hat A \psi \:\mathrm dx \neq\int \phi^\dagger  \psi \:\mathrm dx $, even if $|\phi\rangle$ is an eigenstate of $\hat A$. I am inclined to believe that the second integral is the correct answer here (it comes naturally when you split $|\psi\rangle$ into a linear combination of basis vectors). But I am at a loss as to the interpretation of the first integral.  So, my question is, what are the physical interpretations of $\int \phi^\dagger \hat A \psi \:\mathrm dx$ and $\int \phi^\dagger  \psi \:\mathrm dx$?  (While I am familiar with the bra-ket notation, I have not used it in this question as I don't want to confuse myself further. Feel free to use it in your answer, though) "Oriented" vs. "orientated" Tangent and Normal lines that pass through the origin Let $\alpha$ be a regular curve in $\mathbb{R}^2$ and let all of its tangent lines pass through the origin. Also, let $\beta$ be a regular curve in $\mathbb{R}^2$ and let all of its normal lines pass through the origin.   How can I show that $\alpha$ is contained in a straight line through the origin and that $\beta$ is contained in a circle around the origin?  I know that the tangent line $\alpha(t)$ is the line that points in the direction of the tangent vector $T(t)$ and that the normal line of $\beta(t)$ is the line that points in the direction of $N(t)$, but how can I put this all together to complete the proof? Getting the ID of the element that fired an event This question attempts to collect a community-maintained list of quality books on the  programming language, targeted at various skill levels.  C is a complex programming language that is difficult to pick up on-the-go by reading online tutorials. A comprehensive book is often the best way to learn the language, and finding a good book is the first step. It is important to avoid badly-written books, and even more importantly, books that contain serious technical errors.  Please suggest edits to the accepted answer to add quality books, with an approximate skill level and a short blurb/description about each book. (Note that the question is locked, so no new answers will be accepted. A single answer is being maintained with the list)  Feel free to debate book choices, quality, headings, summaries, skill levels, and anything else you see that is wrong. Books that are deemed satisfactory by the C community here will stick around on the list; the rest will be regularly removed.  For books that have reviews by the Association of C and C++ Users (ACCU), a link to those reviews should be added along with the book.  See also:   Other C-related resources in the   A similar list for :      This question was discussed on  as part of the .The consensus was to keep it undeleted and actively maintained. A subring of the field of fractions of a PID is a PID as well. Let $A$ be a PID and $R$ a ring such that $A\subset R \subset \operatorname{Frac}(A)$, where $\operatorname{Frac}(A)$ denotes the field of fractions of $A$. How to show $R$ is also a PID?   Any hints? Separate long math text under sum symbol into different lines? Which way is up? (electric outlet) How can I have a different input method for each app window on windows 8. When I change input language to ENG all app window also uses this method and I don't want that. Like windows 7 that has a different input method for this purpose. thanks How to get the ASCII value of a character Where can I get the original Unix? By default  indents with a hard tab character. How do I change it to spaces?