Is it possible to make a prediction for the eigenvalues of the Lagrangian-/action-operator of a quantum system if I know the ones of the Hamiltonian, the position- and momentum-operator?   Can I use the formula   $$ L = \sum_a \frac{1}{2} ( \dot{q}_a \cdot p_a + p_a\cdot \dot{q}_a )- H $$  to find the eigenvalues since the operators $H, p, \dot{q}$ do not commute with each other (and so they are not diagonizable with the same basis)? So the eigenvalues of $L$ are not $T-V$. Which orthonormal basis can we choose for the calculation instead?  I know that the Hamilton and Heisenberg equations also hold for this operator version (see Quantum kinematics and dynamics by J. Schwinger). So I can use  $$ \dot{q}=\frac{\partial H}{\partial p}$$  or a bit more general in terms of the Heisenberg equations:  $$ \dot{q(t)} = {\frac {i}{\hbar }}[H,q(t)]+\left({\frac {\partial q}{\partial t}}\right)_{H}$$  to reformulate the Lagrangian which then depends only on $q,p$ since $H(q,p)$:  $$ L(q(t),p(t)) = \sum_a \frac{1}{2} [{\frac {i}{\hbar }}[H,q_a(t)] \cdot p_a(t) + p_a(t) \cdot {\frac {i}{\hbar }}[H,q_a(t)]+ ]- H(q(t),p(t)) $$  But I also didn't know (with this Lagrangian) which eigenvectors I have to choose.  Since I'm studying Schwinger's quantum action principle I'm interestend in this eigenvalues because they should be minimized by the condition $\delta \hat{S} = 0$
It is often stated that the Lagrangian formalism and the Hamiltonian formalism are equivalent.   We often hear people talk about eigenvalues of Hamiltonians but I have never ever heard a word about eigenvalues of Lagrangians.   Why is this so? Is it not useful? is it not possible to do it?
Problem with solving systems of linear congruence of two variables The problem I am given is finding a solution to the following pair of equations:  $3x + 4y$ $\equiv$ $5$ $mod$ $13$  $2x + 5y$ $\equiv$ $7$ $mod$ $13$  By reading the methodology here:   Since the modulos in my problem are also the same, I am able to use substitution, but this leaves fractions in my expressions.  I then used matrix multiplication as follows:  $\begin{bmatrix} 3 &amp; 4\\ 2 &amp; 5\end{bmatrix}$ $\begin{bmatrix}\ x\\ y\end{bmatrix}$ = $\begin{bmatrix}\ 5\\ 7\end{bmatrix}$   And solving this also gives me a matrix with fractions, so I am a bit lost on how to solve this problem.  Any help is appreciated. Thank you very much. How can you combine all lines that end with a backslash character? Using a common command line tool like sed or awk, is it possible to join all lines that end with a given character, like a backslash?  For example, given the file:  foo bar \ bash \ baz dude \ happy   I would like to get this output:  foo bar bash baz dude happy I have an entry-level Nikon DSLR, and I cant decide if I should upgrade my camera body or upgrade from a  AF-S DX Nikkor 18-200mm to whatever else might be recommended? Asymptotes in a plot I'm using TikZ to plotting some functions for a document. For example, I want to plot 1/(x+1). So I did something like this:  \begin{tikzpicture}[&gt;=stealth,x=3mm,y=3mm] \draw (0,5) node[above]{\fimg $y=f(x+1)=\dfrac{1}{x+1}$}; \clip (-5,-5) rectangle (5,5); \ejes[1]{-4}{4}{-4}{4} \draw[samples=100] plot(\x,{1/(\x+1)}); \end{tikzpicture}   where \ejes[style]{xmin}{xmax}{ymin}{ymax} is command, that I have defined before to draw the axes (join with \fimg). My expected output for the code below was just the function, but I get this:    So, TikZ automagically put the vertical asymptote at x=-1. That isn't a bad thing, but now I want to have some control over it. For example I want dotted lines for vertical asymptotes. How can I do that? How to add a browser tab icon (favicon) for a website? How to use a special character as a normal one in Unix shells? Many questions like 'How to type the double-quote char (")?' are being asked, and we don't want to clutter our community with the same answer (Type it as \" if not enclosed in 's, " if enclosed in 's.) So, the question is here.   You can't type special chars into a terminal like normal ones, e.g. this command will fail:  echo Updates (11)   So, how to type in these chars in the terminal as if they were normal ones?  !#$^&amp;*?[](){}&lt;&gt;~;'"\|&lt;space&gt;&lt;tab&gt;&lt;newline&gt; Google AJAX API - How do I get more than 4 Results? In general, when can I omit that from a sentence?   Can I omit that (emphasized) in sentence (2) below?        We say that such algorithms handle concept drift and can learn from time-changing data streams.   When the statistical properties of the target variable change, we say that concept drift has occurred.   She told me that some of her classmates failed.      To me, sentences (1) and (3) sound better without that. If I omit that from (2), it just doesn't sound right when I read it out loud. (I should point out I'm not a native speaker.)  How do you decide when to omit it? Is it just a matter of taste? (I know that is sometimes mandatory.)  Can you think of sentences that "sound better" with that? Can you think of sentences that "sound better" without that? Can you think of a few sentences where it is wrong to omit that? Mars needs water and women I read this book in high school too many years ago for me to think about. Martian colonist are raiding earth's oceans for water and because of the conditions on Mars where only male children are conceived, they are also on the look out for young women as potential mates. Our main character while on a raid, bags not only a young woman but her brother as well. Returning to Mars they find that Earth has attacked Mars can't remember much more. How to show that $K\leq S_4$ is a normal subgroup? Let $K=\{(1),(1,2),(3,4),(2,4),(1,4),(2,3)\}$ is a subgrope of $S_4$, we want to show that it is a normal subgroup.  I know that i have to examine the condition of normality in $S_4$ i.e. for all $g\in S_4, g^{-1}Kg=K$. I think this takes times or I am missing something near. Please give me the right hint. Thank you.  Edit: $K = \{(1),(12)(34),(13)(24),(14)(23)\}.$ Do sensors wear out? We all know that shutters wear out and they have a limited number of actuation's.  The question I have is, do sensors wear out too? Do they suffer any kind of damage after each shot?  Should I be concerned about this or is the sensor lifespan way longer than the shutter lifespan? I'm wondering about the period character . used as a shortcut for the local computer name.  You can use it when logging into Windows 7, for example, to specify that the user account you're entering belongs to the local computer rather than to a domain:  .\MyLocalUserAccount   And you can use it within a Windows Management Instrumentation (WMI) query to indicate that the target is the local computer rather than a remote computer.  You can also use it when working with named pipes to for similar purposes.  However, you cannot use it as the UNC network name for the computer, like this:  \\.\MyShare   Does this character have an official name (e.g. "LocalHostToken") and when and where can it be used for this purpose? Exploring $ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$. I was exploring the fact that $$ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe,$$ where $B_n$ is the $n$th Bell number.   I found this result by exploring the series on wolframalpha and looking up the sequence of numbers generated. I have no experience with Bell numbers other than knowing that they represent the number of ways of partitioning a set.   What I tried  Looking at the base case we can verify $$\sum_{n=0}^\infty \frac{n}{n!} = \sum_{n=1}^\infty \frac{1}{(n-1)!} = e. $$  But then I realize that $n^2$ is going to be difficult.  Specifically what I want  I am doing this for fun, so I just want to glean some sort of lesson out of this. Resources that let me learn for myself are just as good!   An accepted answer will have any one of the following:   A proof that $\sum_{n = 0}^{\infty} \frac{n^2}{n!} = 2e$ (this interests me a lot) A description of why this series is related to Bell numbers and the number of partitions of a set A link to some resource that introduces Bell numbers and/or their connection to this series. Determine if two rectangles overlap each other? I am trying to write a C++ program that takes the following inputs from the user to construct rectangles (between 2 and 5): height, width, x-pos, y-pos. All of these rectangles will exist parallel to the x and the y axis, that is all of their edges will have slopes of 0 or infinity.  I've tried to implement what is mentioned in  question but I am not having very much luck.  My current implementation does the following:  // Gets all the vertices for Rectangle 1 and stores them in an array -&gt; arrRect1 // point 1 x: arrRect1[0], point 1 y: arrRect1[1] and so on... // Gets all the vertices for Rectangle 2 and stores them in an array -&gt; arrRect2  // rotated edge of point a, rect 1 int rot_x, rot_y; rot_x = -arrRect1[3]; rot_y = arrRect1[2]; // point on rotated edge int pnt_x, pnt_y; pnt_x = arrRect1[2];  pnt_y = arrRect1[3]; // test point, a from rect 2 int tst_x, tst_y; tst_x = arrRect2[0]; tst_y = arrRect2[1];  int value; value = (rot_x * (tst_x - pnt_x)) + (rot_y * (tst_y - pnt_y)); cout &lt;&lt; "Value: " &lt;&lt; value;     However I'm not quite sure if (a) I've implemented the algorithm I linked to correctly, or if I did exactly how to interpret this?  Any suggestions? Does Stack Exchange use caching and if so, how? Why is it considered a bad practice to omit curly braces? Why does everyone tell me writing code like this is a bad practice?  if (foo)     Bar();  //or  for(int i = 0 i &lt; count; i++)     Bar(i);   My biggest argument for omitting the curly braces is that it can sometimes be twice as many lines with them. For example, here is some code to paint a glow effect for a label in C#.  using (Brush br = new SolidBrush(Color.FromArgb(15, GlowColor))) {     for (int x = 0; x &lt;= GlowAmount; x++)     {         for (int y = 0; y &lt;= GlowAmount; y++)         {             g.DrawString(Text, this.Font, br, new Point(IconOffset + x, y));         }      }  }  //versus using (Brush br = new SolidBrush(Color.FromArgb(15, GlowColor)))     for (int x = 0; x &lt;= GlowAmount; x++)         for (int y = 0; y &lt;= GlowAmount; y++)             g.DrawString(Text, this.Font, br, new Point(IconOffset + x, y));   You can also get the added benefit of chaining usings together without having to indent a million times.  using (Graphics g = Graphics.FromImage(bmp)) {     using (Brush brush = new SolidBrush(backgroundColor))     {         using (Pen pen = new Pen(Color.FromArgb(penColor)))         {             //do lots of work         }     }  } //versus using (Graphics g = Graphics.FromImage(bmp)) using (Brush brush = new SolidBrush(backgroundColor)) using (Pen pen = new Pen(Color.FromArgb(penColor))) {     //do lots of work }     The most common argument for curly braces revolves around maintance programming, and the problems that would ensue by inserting code between the original if statement and its intended result:  if (foo)     Bar();     Biz();     Questions:   Is it wrong to want to use the more compact syntax which the language offers? The people that design these languages are smart, I can't imagine they would put a feature which is always bad to use. Should we or Shouldn't we write code so the lowest common denominator can understand and have no problems working with it?  Is there another argument that I'm missing? Everytime I try to sudo something at the terminal, I get a warning message:  alexandre@XPS-15Z $ sudo mount file.iso /mnt/ -o loop [sudo] password for alexandre:  no talloc stackframe at ../source3/param/loadparm.c:4864, leaking memory   It's just annoying to get that all the time. How do I solve that? I'm on Ubuntu 14.04 (upgraded from 13.10) If two functionals have the same kernel, then one is a multiple of the other I would like some help with this exercise.  Suppose that $f_1,\ f_2 \in V^*$ and that $\text{Ker} f_1 = \text{Ker} f_2$. Show that $f_1 = k f_2$ for some scalar $k$.  I expect your suggestions.  Thanks! This question is about finding reasons for compile errors when using tikz.  Consider the files compiles.tex, compiles-not1.tex and compiles-not2.tex. Then it is clear that compiles-not1.tex does not compile, because the name C has not been assigned. The file compiles-not2.tex does not compile, because \draw (A) -- (A -| B ); contains a space after the name B. However, this is difficult to see from the error message.   Is it possible to tweak the error messages generated by tikz to perhaps mark tikz-code and names in some way? What I mean is that in the example, an error message ! Package pgf Error: No shape named 'B ' is known. might be more helpful, as the space in the name B\␣ (B with a trailing space) is more clearly delimited from the message.   File compiles.tex compiles without errors.  \documentclass{article} \usepackage{tikz} \begin{document} \begin{tikzpicture}     \node (A) {A};     \node[above right of=A] (B) {B};     \draw (A) -- (A -| B); \end{tikzpicture} \end{document}   On compilation compiles-not1.tex generates ! Package pgf Error: No shape named C is known.  \documentclass{article} \usepackage{tikz} \begin{document} \begin{tikzpicture}     \node (A) {A};     \node[above right of=A] (B) {B};     \draw (A) -- (A -| C); \end{tikzpicture} \end{document}   compiles-not2.tex generates ! Package pgf Error: No shape named B  is known.  \documentclass{article} \usepackage{tikz} \begin{document} \begin{tikzpicture}     \node (A) {A};     \node[above right of=A] (B) {B};     \draw (A) -- (A -| B ); \end{tikzpicture} \end{document} 3 input XNOR gate operation